Morfismos, Vol 11, No 2, 2007 by Morfismos, Department of Mathematics, Cinvestav

VOLUMEN 11 NÚMERO 2 JULIO A DICIEMBRE DE 2007 ISSN: 1870-6525

Morfismos Comunicaciones Estudiantiles Departamento de Matem´aticas Cinvestav

Editores Responsables • Isidoro Gitler • Jes´ us Gonz´alez

Consejo Editorial • Luis Carrera • Samuel Gitler • Onésimo Hernández-Lerma • Hector Jasso Fuentes • Miguel Maldonado • Ra´ ul Quiroga Barranco • Enrique Ram´ırez de Arellano • Enrique Reyes • Armando Sánchez • Mart´ın Solis • Leticia Zárate

Editores Asociados • Ricardo Berlanga • Emilio Lluis Puebla • Isa´ıas L´opez • Guillermo Pastor • V´ıctor P´erez Abreu • Carlos Prieto • Carlos Renter´ıa • Luis Verde

Secretarias Técnicas • Roxana Mart´ınez • Laura Valencia ISSN: 1870 - 6525 Morfismos puede ser consultada electrónicamente en “Revista Morfismos” en la dirección http://www.math.cinvestav.mx. Para mayores informes dirigirse al teléfono 57 47 38 71. Toda correspondencia debe ir dirigida a la Sra. Laura Valencia, Departamento de Matemáticas del Cinvestav, Apartado Postal 14-740, México, D.F. 07000 o por correo electrónico: laura@math.cinvestav.mx.

VOLUMEN 11 NÚMERO 2 JULIO A DICIEMBRE DE 2007 ISSN: 1870-6525

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Lineamientos Editoriales “Morfismos” es la revista semestral de los estudiantes del Departamento de Matem´ aticas del CINVESTAV, que tiene entre sus principales objetivos el que los estudiantes adquieran experiencia en la escritura de resultados matem´ aticos. La publicaci´ on de trabajos no estar´ a restringida a estudiantes del CINVESTAV; deseamos fomentar también la participaci´ on de estudiantes en México y en el extranjero, as´ı como la contribuci´ on por invitaci´ on de investigadores. Los reportes de investigaci´ on matem´ atica o res´ umenes de tesis de licenciatura, maestr´ıa o doctorado pueden ser publicados en Morfismos. Los art´ıculos que aparecer´ an ser´ an originales, ya sea en los resultados o en los métodos. Para juzgar ésto, el Consejo Editorial designar´ a revisores de reconocido prestigio y con experiencia en la comunicaci´ on clara de ideas y conceptos matem´ aticos. Aunque Morfismos es una revista con arbitraje, los trabajos se considerar´ an como versiones preliminares que luego podr´ an aparecer publicados en otras revistas especializadas. Si tienes alguna sugerencia sobre la revista hazlo saber a los editores y con gusto estudiaremos la posibilidad de implementarla. Esperamos que esta publicaci´ on propicie, como una primera experiencia, el desarrollo de un estilo correcto de escribir matem´ aticas.

Morfismos

Editorial Guidelines “Morfismos” is the journal of the students of the Mathematics Department of CINVESTAV. One of its main objectives is for students to acquire experience in writing mathematics. Morfismos appears twice a year. Publication of papers is not restricted to students of CINVESTAV; we want to encourage students in Mexico and abroad to submit papers. Mathematics research reports or summaries of bachelor, master and Ph.D. theses will be considered for publication, as well as invited contributed papers by researchers. Papers submitted should be original, either in the results or in the methods. The Editors will assign as referees well–established mathematicians. Even though Morfismos is a refereed journal, the papers will be considered as preliminary versions which could later appear in other mathematical journals. If you have any suggestions about the journal, let the Editors know and we will gladly study the possibility of implementing them. We expect this journal to foster, as a preliminary experience, the development of a correct style of writing mathematics.

Morfismos

´ ˜ de existencia de Con este numero festejamos los primeros diez anos Morfismos. Con el apoyo decidido del personal académico, estudiantil, técnico y administrativo del Departamento de Matemáticas del CINVES´ de los autores y revisores que activa TAV, as´ı como con la participacion y entusiastamente han colaborado durante este tiempo, Morfismos se ha ´ del conociubicado como un medio de la más alta calidad para la difusion miento académico. Gracias a todos.

Feliz D´ecimo Aniversario, Morfismos!

We celebrate with this issue the tenth anniversary of Morfismos. Thanks to the support of staff and students at the Mathematics Department of CINVESTAV, as well as to the very valuable collaboration of authors and referees, nowadays Morfismos has been recognized as a high quality journal for the communication of results in Mathematics. Thanks to all of them.

Happy Tenth Anniversary, Morfismos!

Contenido Hodge structures in non-commutative geometry Maxim Kontsevich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

A stringy product on twisted orbifold K-theory Alejandro Adem, Yongbin Ruan, and Bin Zhang . . . . . . . . . . . . . . . . . . . . . . . . . 33

Morfismos, Vol. 11, No. 2, 2007, pp. 1–32

XI Solomon Lefschetz Memorial Lecture Series: Hodge structures in non-commutative geometry ∗ (Notes by Ernesto Lupercio)

Maxim Kontsevich

Abstract Traditionally, Hodge structures are associated with complex projective varieties. In my expository lectures I discussed a noncommutative generalization of Hodge structures in deformation quantization and in derived algebraic geometry.

2000 Mathematics Subject Classification: 14A22, 14G10. Keywords and phrases: Mathematical Physics, K-Theory and Homology.

Lecture 1. September 8th, 2005

1.1 This talk deals with some relations between algebraic geometry and non-commutative geometry, in particular we explore the generalization of Hodge structures to the non-commutative realm.

1.3

Hodge Structures.

Given a smooth projective variety X over C we have a naturally defined pure Hodge structure (HS) on its cohomology, namely: ∗

Invited article. Lecturer of the XI Solomon Lefschetz Memorial Lecture Series, Departamento ´ de Matem´ aticas del CINVESTAV, MEXICO, 2005. 1

Maxim Kontsevich

! H n (X, C) = p+q=n, p,q≥0 H p,q (X) by considering H p,q to be the cohomology represented by forms that locally can be written as "

zj1 ∧ d¯ zj2 ∧ . . . ∧ d¯ zjq ai1 ,...,ip ;j1 ,...,jq dzi1 ∧ dzi2 ∧ . . . ∧ dzip ∧ d¯

H n (X, C) is the complexification H n (X, Z)⊗C of a lattice of finite rank. H p,q = H q,p .

1.4 To have this Hodge structure (of weight n) is the same as having the decreasing filtration F p H n :=

′

H p ,q

′

p′ ≥p′ , p′ +q ′ =n

for we have H p,q = F p H n ∩ F q H n .

1.5 What makes this Hodge structure nice is that whenever we have a family Xt of varieties algebraically dependent on a parameter t we obtain a bundle of cohomologies with a flat (Gauss-Manin) connection Htn = H n (Xt , C) over the space of parameters, and Ftp is a holomorphic subbundle (even though Htp,q is not). Deligne developed a great theory of mixed Hodge structures that generalizes this for any variety perhaps singular or noncompact.

1.6

Non-commutative geometry.

Non-commutative geometry (NCG) has been developed by Alain Connes [3] with applications regarding foliations, fractals and quantum spaces in mind, but not algebraic geometry. In fact it remains unknown what a good notion of non-commutative complex manifold is.

XI Solomon Lefschetz Memorial Lecture Series

1.7 There is a calculus associated to NC spaces. Suppose k is a field and for the first talk the field will always be C, only in the second talk finite fields become relevant. Let A be a unital, associative algebra over k. An idea of Connes is to mimic topology, namely differential forms, and the de Rham differential in this framework. We define the Hochschild complex C (A, A) of A as a negatively graded complex (for we want to have all differentials of degree +1), ∂

∂

−→ A ⊗ A ⊗ A ⊗ A −→ A ⊗ A ⊗ A −→ A ⊗ A −→ A, where A⊗k lives on degree −k + 1. The diﬀerential ∂ is given by ∂(a0 ⊗ · · · ⊗ an ) = a0 a1 ⊗ a2 ⊗ · · · ⊗ an − a0 ⊗ a1 a2 ⊗ · · · ⊗ an + . . . + (−1)n−1 a0 ⊗ a1 ⊗ · · · ⊗ an−1 an + (−1)n an a0 ⊗ a1 ⊗ · · · ⊗ an−1 . This formula is more natural when we write the terms cyclically: a0

(1)

an ⊗ .. .

⊗

a1 ⊗ .. .

for a0 ⊗ · · · ⊗ an . It is very easy to verify that ∂ 2 = 0.

1.8 The homology of the Hochschild complex has an abstract meaning op −mod

Ker ∂/Im ∂ = Tor A⊗k A

(A, A).

1.9 An idea in NC geometry is that as A replaces a commutative space the Hochschild homology of A replaces in turn the complex of diﬀerential forms.

Maxim Kontsevich

Theorem 1.9.1 (Hochschild-Konstant-Rosenberg, [6]). Let X be a smooth aﬃne algebraic variety, then if A = O(X) we have HHi (X) := H −i (C (A, A); ∂) ∼ = Ωi (X) where Ωi (X) is the space of i-forms on X. ∆

Proof. The proof is very easy: consider the diagonal embedding X −→ X × X and by remembering that the normal bundle of ∆ is the tangent bundle of X we have Quasi−coherent(X×X)

HH (X) = Tor

(O∆ , O∆ )

this together with a local calculation gives the result.!

1.10 The Hochschild-Konstant-Rosenberg theorem motivates us to think of HHi (A) as a space of diﬀerential forms of degree i on a non-commutative space. Note that if A is non-commutative we have H 0 (C (A, A); ∂) = A/[A, A]. Also, for commutative A = O(X), given an element a0 ⊗ · · · ⊗ an in 1 a0 da1 ∧ . . . ∧ dan . C (A, A) the corresponding form is given by n!

1.11 red There is a reduced version of the complex C (A,A) with the same cohomology obtained by reducing modulo constants all but the first factor

−→ A ⊗ A/(k · 1) ⊗ A/(k · 1) −→ A ⊗ A/(k · 1) −→ A.

1.12 Connes’ main observation is that we can write a formula for an additional differential B on C (A, A) of degree −1, inducing a differential on HH (A) that generalizes the de Rham differential: B(a0 ⊗ a1 ⊗ · · · ⊗ an ) =

! (−1)σ 1 ⊗ aσ(0) ⊗ · · · ⊗ aσ(n) σ

XI Solomon Lefschetz Memorial Lecture Series

where σ ∈ Z/(n + 1)Z runs over all cyclic permutations. It is easy to verify that B 2 = 0, B∂ + ∂B = 0, ∂ 2 = 0, which we depict pictorially as ··· "

B ∂

! A ⊗ A/1 ⊗ A/1

# A ⊗ A/1

∂

%A ∂

and taking cohomology gives us a complex (Ker ∂/Im ∂; B). A naive definition on the de Rham cohomology in this context is the homology of this complex Ker B/Im B.

1.13 We can do better by defining the negative cyclic complex C − (A), which is formally a projective limit (here u is a formal variable, deg(u) = +2): (C red (A, A)[u]/uN ; ∂ + uB). C − := (C red (A, A)[[u]]; ∂ + uB) = lim ←− N

1.14 We define the periodic complex as an inductive limit (u−i C red (A, A)[[u]]; ∂ + uB). C per := (C red (A, A)((u)); ∂ + uB) = lim −→ i

This turns out to be a k((u))-module and this implies that the multiplication by u induces a sort of Bott periodicity. The resulting cohomology groups called (even, odd) periodic cyclic homology and are written (respectively) HPeven (A), HPodd (A). This is the desired replacement for de Rham cohomology.

1.15 Let us consider some examples. When A = C ∞ (X) is considered as a nuclear Fr´echet algebra, and if we interpret the symbol ⊗ as the topological tensor product then we have the canonical isomorphisms: HPeven (A) ∼ = H 0 (X, C) ⊕ H 2 (X, C) ⊕ · · · HPodd (A) ∼ = H 1 (X, C) ⊕ H 3 (X, C) ⊕ · · ·

Maxim Kontsevich

Theorem 1.15.1 (Feigin-Tsygan, [4]). If X is a aﬃne algebraic variety (possibly singular) and Xtop its underlying topological space then HPeven (A) ∼ = H even (Xtop , C) and

HPodd (A) ∼ = H odd (Xtop , C)

(these spaces are finite-dimensional). There is a natural lattice H (X, Z) but we will see later that the “correct” lattice should be slightly diﬀerent.

1.16 Everything we said before can be defined for a diﬀerential graded algebra (dga) rather than only for an algebra A. Recall that a dga (A, d) consists of ! A = n∈Z An a graded algebra with a graded product An1 ⊗ An2 −→ An1 +n2 .

dA : An −→ An+1 a diﬀerential satisfying the graded Leibniz rule.

For example given a manifold X on has the de Rham dga (Ω (X); d).

1.17 The definition of the degree for C (A, A) is given by " deg(a1 ⊗ · · · ⊗ an ) := 1 − n + deg(ai ). i

It is not hard to see that

rank(HP (A)) ! rank(HH (A)), and therefore, if the rank of the Hochschild homology is finite so is the rank of the periodic cyclic homology.

1.18

Hodge filtration on HP (A).

We define F n HPeven (A) as# the classes represented by sequences γi ∈ γi ui/2 ) such that i ≥ 2n. Similarly we Ci (A, A), i ∈ 2Z (namely n+1/2 define F HPodd (A).

XI Solomon Lefschetz Memorial Lecture Series

1.19 We have an interesting instance of this situation in ordinary topology A = C ∞ (X). Here we have: HPeven (A) = H 0 (X) ⊕ H 2 (X) ⊕ H 4 (X) ⊕ H 6 (X) ⊕ · · · and F 0 = HPeven (A), F 1 = H 2 (X) ⊕ H 4 (X) ⊕ H 6 (X) ⊕ · · · and so on. In non-commutative geometry this filtration is the best you can do, for there is no individual cohomologies.

1.20 Let X be an algebraic variety (not necessarily aﬃne). Weibel [15] gave a sheaf-theoretic definition of HH (X) and HP (X). Namely, if X is covered by aﬃne open charts ! Ui , X= 1!i!r

we obtain not an algebra, but a cosimplicial algebra Ak := ⊕(i0 ,...,ik ) O(Ui0 ∩ . . . ∩ Uik ), whose total complex Tot(C (A )) = C (X) still has two diﬀerentials B and ∂ as before. In fact Quasi−coherent(X×X)

H (C (X), ∂) = Tor

(O∆ , O∆ )

is graded in both positive and negative degrees. Weibel observed that one can recover a tilted version of the Hodge diamond in this manner. For a smooth projective X one has HP (X) := HP (A ) = H (X) and the filtration we defined becomes " F i (HP ) = F p H n (X), p=i+n/2

reshuﬄing thus the usual Hodge filtration. Observe that in this example we have: H p,q ⊂ F

p−q 2

1 i ∈ Z, 2

Maxim Kontsevich

1.21 In general one can directly replace an algebraic variety by a dga using a theorem by Bondal and Van den Bergh. For example, let E be a suﬃciently “large” bundle over a smooth projective variety X. You may take for instance E = O(0) + O(1) + · · · + O(dim X). Take the algebra A to be ¯ A := (Γ(X, End(E) ⊗ Ω0,1 ); ∂).

Then one can show that one can repeat the previous constructions obtaining the corresponding filtration. Namely, the periodic cyclic homology (and the Hodge filtration) of the dga A coincides with those of X.

1.22 Take Xalg to be a smooth algebraic variety over C and let XC ∞ be its underlying smooth manifold. Consider the natural map XC ∞ −→ Xalg . This map induces an isomorphism HP (XC ∞ ) ←− HP (Xalg ). This isomorphism is compatible with the Hodge filtrations but the filtrations are diﬀerent.

1.23 The next important ingredients are the integer lattices. Notice that H (X, C) has a natural integer lattice H (X, Z), which allows us to speak of periods, for example. There is also another lattice commen (X) := surable with H (X, Z), namely, the topological K-theory Ktop K even (XC ∞ ) ⊕ K odd (XC ∞ ). Let A be an algebra, then we get K0 (A) by considering the projective modules over A. There is a Chern character map K0 (A)

!!! !!! !!! !"

! F 0 (HPeven (A)) ! " "# """ " " "" """

HC0− (A)

! HPeven (A)

XI Solomon Lefschetz Memorial Lecture Series

Here it may be appropriate to recall that HP (A) is a Morita invariant and therefore we can replace A by A ⊗ Matn×n for Matn×n a matrix algebra. If π ∈ A is a projector (namely π 2 = π) we have explicitly ch(π) = π −

4! 2! (π − 1/2) ⊗ π ⊗ π · u + (π − 1/2) ⊗ π ⊗ π ⊗ π ⊗ π · u2 + . . . . 1! 2!

There is a similar story for K1 (A) −→ HPodd (A), and also for higher K-theory.

1.24 0 (X) is up to torsion If A = C ∞ (X) then the image of K0 (A) = Ktop

H 0 (X, Z) ⊕ H 2 (X, Z) · 2πi ⊕ H 4 (X, Z) · (2πi)2 ⊕ · · · . 0 (X)⊗Q = H even (X, Q) but the lattice is diﬀerent We have of course Ktop and so Bott periodicity is broken. In order to √ restore it we must rescale the odd degree part of the lattice by the factor 2πi, and then we obtain √ √ H 1 (X, Z) · 2πi ⊕ H 3 (X, Z) · ( 2πi)3 ⊕ · · ·

We call this new lattice the non-commutative integral cohomology HNC (X, Z) ⊂ HP (C ∞ (X)).

Proposition 1.24.1. For A = C ∞ (X) the image up to torsion of ch : Kn (A) −→ HP(n mod 2) (A) is

(n mod 2)

(2πi)n/2 HNC

(X, Z).

1.25 We are ready to formulate one of the main problems in non-commutative geometry. Let A be a dga over C. The problem is to define a nuclear Fr´echet algebra AC ∞ satisfying Bott periodicity Ki (AC ∞ ) ∼ = Ki+2 (AC ∞ ),

i≥0

together with an algebra homomorphism A → AC ∞ satisfying:

Maxim Kontsevich

The homomorphism A → AC ∞ induces an isomorphism HP (A) ∼ = HP (AC ∞ ), ch : Kn (AC ∞ ) −→ HP (AC ∞ ) is a lattice, i.e. when we tensor with C we obtain an isomorphism ∼ =

Kn (AC ∞ ) ⊗Q C −→ HP(n mod 2) (AC ∞ ).

1.26 Consider for example the case of a commutative algebra A. Every such algebra is an inductive limit of finitely generated algebras A = lim An →

where each An can be thought of as a singular aﬃne variety. In general HP (A) ̸= lim→ HP (An ), but the right-hand side is a better definition for HP (A). In this case we find that the lattice we are looking for is simply (Spec An (C)). lim Ktop →

1.27 We will attempt now to explain some non-commutative examples that are close to the commutative realm, and are obtained by a procedure called deformation quantization. Let us consider first a C ∞ noncommutative space. Let Tθ2 be the non-commutative torus (for θ ∈ R) so that C ∞ (Tθ2 ) is precisely all the expressions of the form ! an,m zˆ1n zˆ2m , an,m ∈ C, n,m∈Z

such that for all k we have am,n = O((1 + |n| + |m|)−k ), and zˆ1 zˆ2 = eiθ zˆ2 zˆ1 . For θ ∈ 2πZ we get the usual commutative torus.

XI Solomon Lefschetz Memorial Lecture Series

1.28 We will also consider some non-commutative algebraic spaces obtained by deformation quantization. Start by taking !2 a smooth aﬃne algebraic variety X and a bi-vector field α ∈ Γ(X, T X). Define, as is usual, the bracket by {f, g} := ⟨α, df ∧ dg⟩.

The field α defines a Poisson structure iﬀ the bracket satisfies the Jacobi identity. We will call α admissible at infinity if there exists a smooth ¯ − X so that α extends ¯ ⊃ X and a divisor X ¯∞ = X projective variety X ¯ to X and the ideal sheaf IX¯ ∞ is a Poisson ideal (closed under brackets).

1.29 n n ¯ The simplest instance of this is "when X = C , X = CP and the admissibility condition for α = i,j αi,j ∂i ∧ ∂j reads deg(αij ) ! 2.

1.30

We have the following [10]: Theorem 1.30.1. If X satisfies ¯ O) = H 2 (X, ¯ O) = 0, H 1 (X,

(e.g. X is a rational variety) then there exists a canonical filtered algebra A! over C[[!]] (actually a free module over C[[!]]) that gives a ∗-deformation quantization, and when we equal the deformation parameter to 0 we get back O(X). While the explicit formulas are very complicated the algebra A! is completely canonical.

1.31 This theorem raises the interesting issue of comparison of parameters. ∂ ∂ ∧ ∂y . Here we can Take for example the case X = C2 and α = xy ∂x guess that $ # ˆ . ˆ Yˆ ⟩/ X ˆ Yˆ = e!Yˆ X A! ∼ = C[[!]]⟨X,

On the other hand the explicit formula for A! involves infinitely many graphs and even for this simple example it is impossible to get the explicit parameter e!. A priori one only knows that just certain universal ˆ Yˆ = q(!)Yˆ X. ˆ series q(!) = 1 + . . . should appear with X

Maxim Kontsevich

1.32 A slightly more elaborate example is furnished by considering (Sklyanin) ¯ = CP 2 and α = p(x, y) ∂ ∧ ∂ with elliptic algebras. Here we take X ∂x ∂y ¯ ∞ ⊂ CP 2 in this case is a cubic curve. We deg(p) = 3. The divisor X ¯ −X ¯ ∞ , which is an aﬃne algebraic surface and since it has take X = X a symplectic structure it also has a Poisson structure α. Its quantum algebra A! depends on an elliptic curve E and a shift x $→ x + x0 on E. The question is then: How to relate E and x0 to the bi-vector field α and the parameter !? Again there is only one reasonable guess. Start with the bi-vector ¯∞. field α and obtain a 2-form α−1 on CP 2 with a first order pole at X ¯ ∞ . Taking residues we obtain a holomorphic Our guess is that E = X −1 1 1-form Res(α ) ∈ Ω (E). The inverse of this 1-form is a vector field (Res(α−1 ))−1 on E. Finally: ! " ! x0 = exp , Res(α−1 ) but to prove this directly seems to be quite challenging.

1.33 It is a remarkable fact that this comparison of parameters problem can be solved by considering the Hodge structures.

1.34 Let us consider X to be either a C ∞ or an affine algebraic variety and A to be C ∞ (X) (respectively O(X)). The theory of deformation quantization implies that all nearby non-commutative algebras and related objects (such as HP , HH , etc.) can be computed semi-classically. In particular nearby algebras are given by Poisson bi-vector fields α. Also C (A!, A!) is quasi-isomorphic to the negatively graded complex (Ω−i (X), Lα ) where the differential is Lα = [ια , d]. If you want to see this over C[[!]] simply consider the differential L!α . We just described what Brylinski calls Poisson homology. The differential B in this case is simply the usual de Rham differential B = d. We would like to consider now HP (A!). This is computed by the complex (

Ωi (X)[i][[!]]((u)), ud + !Lα ) ,

XI Solomon Lefschetz Memorial Lecture Series

which is the sum of infinitely many copies of some finite-dimensional ˆ C[[!]] C((!)) is C((!)) tensored with the complex. Namely HP (A!)⊗ finite-dimensional cohomology of the Z/2-graded complex: ΩN

d Lα

! ΩN −1

d Lα

# ···

# Ω1

Lα

# Ω0 Lα

1.35 We claim that the cohomology of this complex is H (X). The reason for this is really simple, for we have that exp(ια )d exp(−ια ) = d + [ια , d] +

1 [ια , [ια , d]] + . . . = d + Lα . 2!

Here we used the fact that [α, α] = 0 to conclude that only the first two terms survive.

1.36 Let us turn our attention to the lattice. Our definition uses Kn (A) but we may get this lattice by using the Gauss-Manin connection. If we have a family of algebras At depending on some parameter t, Ezra Getzler [5] defined a flat connection on the bundle HPt over the parameter space. This allows us to start with the lattice ⊕k H k (X, (2πi)k/2 · Z) (up to torsion) at t := ! = 0 in our situation. The parallel transport for the Gauss-Manin connection comes from the above identification of periodic complexes given by the conjugation by exp(!ια ).

1.37 To compute the filtration we will assume that X is symplectic, and therefore α is non-degenerate. Again we set dim(X) = N = 2n, and ω = α−1 is a closed 2-form. We also set ! := 1. The following theorem is perhaps well known but in any case is very simple: Theorem 1.37.1. For (X, ω) a symplectic manifold the Hodge filtration is given by HP even :

F n−k/2 = eω (H 0 ⊕ · · · ⊕ H k ), k ∈ 2Z eω H 0 ⊂ eω (H 0 ⊕ H 2 ) ⊂ · · ·

Maxim Kontsevich

HP odd : F n−k/2 = eω (H 1 ⊕ · · · ⊕ H k ), k ∈ 2Z + 1 eω H 1 ⊂ eω (H 1 ⊕ H 3 ) ⊂ · · · Notice that this is not the usual Hodge filtration coming from topology: H 2n ⊂ H 2n ⊕ H 2n−2 ⊂ · · · Proof. Consider the Z/2-graded complex Ω2n

! Ω2n−1

Lα

# ···

Lα

# Ω1

# Ω0

Lα

where Ωk lives in F k/2 . The diﬀerential is not compatible with the filtration, nevertheless after the conjugation by exp(ια ) we can use instead the complex d

Ω2n ←− Ω2n−1 ←− · · · ←− Ω0 and we would like to understand what happens to the filtration. Let ∗ be the Hodge operator with respect to ω (the Fourier transform in odd variables). Under this map the original Z/2-graded complex becomes Ω0

Lα

# Ω1

Lα

$ # ···

Lα d

! Ω2n−1

Lα

! Ω2n

where the filtration has been reversed. The important remark here is that now eια does preserve this filtration, transforming the last complex into the complex Ω0 deg

% Ω1

n − 1/2

% ···

···

% Ω2n

Finally, all that remains to be seen is that eω∧· = eια ∗ e−ια .

XI Solomon Lefschetz Memorial Lecture Series

1.38 This theorem is related to the Lefschetz decomposition formula 2 for K¨ahler manifolds. In fact we have obtained HP (X!α ) = H (X), furthermore we have lim F i HP (X!α ) = F i HP (X)

!→0

if and only if we have a Lefschetz decomposition for the symplectic manifold. If (X, ω) is K¨ahler compact we define the multiplication operator ω∧ : H (X) −→ H +2 (X), which is clearly nilpotent. The Lefschetz decomposition corresponds to the decomposition into Jordan blocks for this operator. In the case at hand the Lefschetz decomposition becomes the Hodge decomposition of a non-commutative space.

1.39 Consider Tθ2 the non-commutative torus. A result of Marc Rieﬀel [12] states that Tθ2 is Morita equivalent to Tθ2′ if and only if ! " aθ + b a b , ∈ SL2 (Z). θ′ = c d cθ + d If you consider in this case HPeven (Tθ2 ) = H 0 ⊕ H 2 ←− K0 (Tθ2 ) you can see that K0 (Tθ2 ) contains the semigroup of bona fide projective modules, producing a half-plane in the lattice bounded by a line of slope θ/(2π), which from our point of view can be identified with the Hodge filtration F 1 . This helps to clarify the meaning of Rieﬀel’s theorem. In this example we get an interesting filtration only for HP even , and nothing for HP odd . 2 Lefschetz influence in mathematics is clearly so large that is would be hard to give a talk in his honor without having the opportunity to mention his name at many points.

Maxim Kontsevich

1.40 Consider an elliptic curve: E = C/ (Z + τ Z) ,

ℑ(τ ) > 0.

Here HPeven (E) = H 0 (T 2 ) ⊕ H 2 (T 2 ), HPodd (E) = H 1 (T 2 , C) ⊃ H 1,0 (E) {terms in the Hodge filtration}. This becomes in terms of generators C ⊗ (Ze0 ⊕ Ze1 ) ⊃ C · (e0 + τ e1 ). While in the corresponding situation for Tθ2 the filtration can be written as θ e˜1 ). C ⊗ (Z˜ e0 + Z˜ e1 ) ⊃ C · (˜ e0 + 2π All this confirms a general belief that non-commutative tori are limits of elliptic curves as τ → R. Also E can be seen as a quotient of a 1-dimensional complex torus C× , while Tθ2 plays the role of a real circle modulo a θ-rotation.

1.41 I shall finish this lecture with one final puzzle. Namely, in the previous example the Hodge filtrations do not fit. In the elliptic curve the interesting Hodge structure detects parameters in odd cohomology while in the non-commutative torus the Hodge filtration detects parameters in even cohomology. A reasonable guess for the solution of this puzzle is that one should tensor by a simple super-algebra (discovered by Kapustin in the LandauGinzburg model) given by A = C[ξ]/(ξ 2 = 1) with ξ odd. Here HP (A) = C0|1 . The question is: How does this super-algebra naturally arise from the limiting process τ → R sending an elliptic curve to a foliation?

XI Solomon Lefschetz Memorial Lecture Series

2 2.1

Lecture 2. September 9th, 2005. Basic Derived Algebraic Geometry.

This field started by A. Bondal and M. Kapranov in Moscow around 1990. Derived algebraic geometry is much simpler than algebraic geometry. While algebraic geometry starts with commutative rings and builds up spectra via the Zariski topology and the theory of sheaves, in derived algebraic geometry there is no room for many of these concepts and the whole theory becomes simpler.

2.2 Let us start by commenting on the algebraization of the notion of space. If we begin with a (topological) space X, first one can produce an algebra A = O(X), its algebra of functions. Next we assign an abelian category to this algebra, the abelian category A − mod of A-modules. At every stage we insist in thinking of the space as the remaining object: The category A − mod is the space. The abelian category A − mod has a nice subfamily, that of vector bundles, namely finitely generated projective modules. Recall that projective modules are images of (n × n)-matrices π : An → An satisfying π 2 = π. The final step consists in producing from the category A − mod a triangulated category D(A − mod) that goes by the name of the derived category. X $→ O(X) = A $→ A − mod $→ D(A − mod).

2.3 While the abelian category A − mod is nice we are still forced to keep track of whether a functor is left-exact or right-exact, etc. This is greatly simplified in the derived category D(A − mod). The derived category D(A − mod) is built upon infinite Z-graded complexes of free A-modules and considering homotopies.

2.4 We take one step further and consider the subcategory CX ⊂ D(A − mod)

Maxim Kontsevich

of perfect complexes. A perfect complex is a finite length complex of finitely generated projective A-modules (vector bundles). All of the above can be generalized to dg algebras.

2.5 We are ready to make important definitions: Definition 2.5.1. Let k be a field. A k-linear space X is a small triangulated category CX that is Karoubi closed (namely all projectors split), enriched by complexes of k-vector spaces. In particular for any two objects E and F we are given a complex Hom CX (E, F) such that HomCX (E, F) = H 0 (Hom CX (E, F)). Definition 2.5.2. A k-linear space X is algebraic if CX has a generator (with respect to taking cones and direct summands).

2.6 The following holds: Proposition 2.6.1. The category CX has a generator if and only if there exists a dga A over k such that CX â&#x2C6;ź = Perfect(A â&#x2C6;&#x2019; mod). This proposition allows us to forget about categories and consider simply dga-s (modulo a reasonable definition of derived Morita equivalence).

2.7 There is a nice relation with the notion of scheme: Theorem 2.7.1 (Bondal, Van den Bergh [2]). Let X be a scheme of finite type over k, then CX has a generator. The moral of the story in derived algebraic geometry is that all spaces are aďŹ&#x192;ne.

XI Solomon Lefschetz Memorial Lecture Series

2.8 The following example is due to Beilinson [1]. Consider X = CP n . Then Db (Coherent(X)) = Perfect(X) = Perfect(A − mod),

where A = End(O(0)⊕· · ·⊕O(n)). A finite complex of finite-dimensional representations of A is the same as a finite complex of vector bundles over X = CP n .

2.9 We make a few more definitions. Definition 2.9.1. An algebraic k-linear space X is compact if for every pair of objects E and F in CX we have that ! rank Hom(E, F[i]) < ∞. i∈Z

In the language of the dga (A, dA ) this is equivalent to: ! rank H i (A, dA ) < ∞. i∈Z

Definition 2.9.2. We say that an algebraic k-linear space X is smooth if A ∈ Perfect(A ⊗ Aop − mod).

Definition 2.9.3. (a version of Bondal-Kapranov’s) X is saturated if it is smooth and compact. This is a good replacement for the notion of smooth projective variety.

2.10 The following concerns the moduli of saturated spaces: Proposition 2.10.1 (Finiteness Property). The moduli space of all saturated k-linear spaces X modulo isomorphisms can be written as a countable disjoint union of schemes of finite type: " Si / ∼ i∈I

modulo an algebraic equivalence relation.

Maxim Kontsevich

2.11

Operations with saturated spaces.

We have several basic operations inherited from the operations on algebras: (i) Given a space X we can produce its opposite space X op by sending the algebra A to its opposite Aop . (ii) Given two spaces X and Y we can define their tensor product X ⊗ Y by multiplication of their corresponding dga’s AX ⊗ AY .

(iii) Given X, Y we define the category Map(X, Y ) := Aop X ⊗AY −mod.

(iv) There is a nice notion of gluing which is absent in algebraic geometry. Given f : X → Y (namely a AY − AX -bimodule Mf ) construct a new algebra AX∪f Y by considering upper triangular matrices of the form ! " ax mf , 0 ay with ax ∈ AX , ay ∈ AY and mf ∈ Mf .

Beilinson’s theorem can be interpreted as stating that CP n is obtained by gluing n + 1 points. This does not sound very geometric at first. An interesting outcome is that we get an unexpected action of the braid group in such decompositions of CP n . Notice that the cohomology of the gluing is the direct sum of the cohomologies of the building blocks.

2.12

Duality theory

The story is again exceedingly simple for saturated spaces. There is a canonical Serre functor SX ∈ Map(X, X)

satisfying

Hom(E, F)∗ = Hom(F, SX (E))∗

In terms of the dga AX we have that

−1 = RHomA⊗Aop (A, A ⊗ Aop ). SX

In the commutative case this reads

where KX = Ωdim X .

SX = KX [dim X] ⊗ · ,

XI Solomon Lefschetz Memorial Lecture Series

2.13 It seems to be the case that there is a basic family of objects from which (almost) everything can be glued up: Calabi-Yau spaces. Definition 2.13.1. A Calabi-Yau saturated space of dimension N is a saturated space X where the Serre functor SX is the shifting functor [N ]. There are reflections of various concepts in commutative algebraic geometry such as positivity and negativity of curvature in the context of separated spaces. Here we should warn the reader that sometimes it is impossible to reconstruct the commutative manifold from its saturated space: several manifolds produce the same saturated space. Think for example about the Fourier-Mukai transform.

2.14 We also have a Z/2-graded version of this theory. We require all complexes and shift functors to be 2-periodic.

2.15

Examples of saturated spaces.

Smooth proper schemes. They form a natural family of saturated spaces. Deligne-Mumford stacks that look locally like a scheme X with a finite group Γ acting X. By considering (locally) the algebra A = O(X) ! k[Γ] we can see immediately that they also furnish examples of saturated spaces. Quantum projective varieties. Suppose we start by considering an ample line bundle L over a smooth projective variety X. Say we have αX a bi-vector field defined over L − 0 the complement of the zero section. We assume that αX is invariant under Gm = GL1 . Deformation quantization implies that we obtain a saturated noncommutative space over k((")). Landau-Ginzburg models. This is a Z/2-graded example. Here we are given a map f : X −→ A1 ,

f ∈ O(X),

f ̸≡ 0,

Maxim Kontsevich

where X is a smooth non-compact variety and A1 is the affine line. The idea comes from the B-model in string theory. The category C(X,f ) consists of matrix factorizations. In the affine case the objects are super-vector bundles E = E even ⊕ E odd over X, together with a differential dE such that d2E = f · Id.

In local coordinates we are looking for a pair of matrices (Aij ) and (Bij ) so that A · B = f · Id.

˜ d ˜)) to be the complex of Z/2-graded We define Hom((E, dE ), (E, E ˜ with diﬀerential spaces HomO(X) (E, E) d(φ) = φ · dE − dE˜ · φ.

It is very easy to verify that d2 = 0.

The generalization of the definition of C(X,f ) to the global case is due to Orlov [11]. Consider Z = f −1 (0) as a (possibly nilpotent) subscheme of X. Then C(X,f ) := Db (Coherent(Z))/Perfect(Z).

We expect C(X,f ) to be saturated if and only if X0 = Critical(f ) ∩ f −1 (0) is compact. This is undoubtedly an important new class of triangulated categories. A beautiful final example is obtained by starting with a C ∞ compact symplectic manifold (X, ω), with very large symplectic form [ω] ≫ 0 (This can be arranged by replacing ω by λω with λ big.) The Fukaya category F(X, ω) is defined by taking as its objects Lagrangian submanifolds and as its arrows holomorphic disks with Lagrangian boundary conditions (but the precise definition is not so simple.) Paul Seidel [13] has proposed an argument showing that in many circumstances F(X, ω) is saturated. This is a manifestation of Mirror Symmetry that says that F(X, ω) is equivalent to C(X,ω)∨ , where (X, ω)∨ is the mirror dual to (X, ω). While originally mirror symmetry was defined only in the Calabi-Yau case, now we expect that the mirror dual to a general symplectic manifold will be dual to a category of Landau-Ginzburg type. In any case in many known examples the category is glued out of Calabi-Yau pieces.

XI Solomon Lefschetz Memorial Lecture Series

2.16 In derived algebraic geometry there are a bit more spaces but much more identifications and symmetries than in ordinary algebraic geometry. For instance, when X is Calabi-Yau then Aut(CX ) is huge and certainly much bigger than Aut(X). Another example: there are two diﬀerent K3 surfaces X, X ′ that have the same CX = CX ′ , but in fact X and X ′ need not be diﬀeomorphic. Again think of the symmetries furnished by the Fourier-Mukai transform.

2.17

Cohomology

Let us return to the subject of cohomology. First, let us make an important remark. If A is a saturated dga then its Hochschild homology H (A, A) := H (C (A, A)) is of finite rank, and the rank of the periodic cyclic homology is bounded by rankHP (A) ! rankH (A, A). In the case in which A is a commutative space we have HP (A) ∼ = HdeRham (X) and H (A, A) ∼ (X). = HHodge

2.18 This motivates the following definition: Definition 2.18.1. For a saturated space X over k the Hodge to de Rham spectral sequence is said to collapse if rank H (A, A) = rank HP (A). This happens if and only if for all N " 1 we have that H (C red (A, A)[u]/uN , ∂ + uB) is a free flat k[u]/uN -module.

2.19

The Degeneration Conjecture:

For any saturated X the Hodge to de Rham spectral sequence collapses.3 This conjecture is true for commutative spaces, for quantum projective schemes and for Landau-Ginzburg models (X, f ). 3

See the very promising work of Kaledin that has appeared since, [7, 8].

Maxim Kontsevich

There are two types of proofs in the commutative case. The first uses K¨ahler geometry and resolution of singularities. This method seems very hard to generalize. The second method of proof uses finite characteristic and Frobenius homomorphisms, this is known as the Deligne-Illusie method and we expect it (after D. Kaledin) to work in general.

2.20 Let us assume this conjecture from now on. We have then a vector bundle H over Spec k[[u]] and we will call Hu the fiber. The space of sections of this bundle is HC − (A). This bundle carries a canonical connection ∇ with a first or second order pole at u = 0. In the Z-graded case we have a Gm -action, λ ∈ k× ,

u #→ λ2 u,

defining the connection. The monodromy of the connection is 1 on HP even (A) and −1 on HP odd (A). In this case the connection has a first order pole at u = 0 and the spectrum of the residue of the connection is 21 Z. The Z/2-graded case is even nicer, for the connection can be written in a universal way with an explicit but complicated formula containing the sum of five terms (see [9]). There is a reason for this connection to exist, and we explain it in two steps. Recall that if you have a family of algebras At over a parameter space you get a flat connection on the bundle HP (At ) whose formula tends to be very complicated. Consider the moduli stack of Z/2-graded spaces. We have an action of Gm : (A, dA ) #→ (A, λdA ). This corresponds in string theory to the renormalization group flow. The fixed points of this action contain Z-graded spaces (but there are also the elements of fractional charge, and the quasihomogeneous singularities.) This corresponds to a scaling u #→ λu and therefore produces the desired connection. In this case we have a second order pole at u = 0.

XI Solomon Lefschetz Memorial Lecture Series

2.21 The basic idea is that the connection ∇ replaces the Hodge filtration. Notice that a vector space together with a Z-filtration is the same as a vector bundle over k[[u]] together with a connection with first order pole at u = 0 and with trivial monodromy. Of course, our connection is more complicated but it is generalizing the notion of filtration.

2.22 We will use now the Chern character ch : K0 (X) −→ {covariantly constant sections of the bundle H}. If in particular we consider the Chern class of idX ∈ CX×X op we have that ch(idX ) is a covariantly constant pairing ∼ =

∗ Hu −→ H−u

that is non-degenerate at u = 0.

2.23 Let us now describe the construction of an algebraic model for a string theory of type IIB. Let X be a saturated algebraic space together with: A Calabi-Yau structure (this exists if the Serre functor is isomorphic to a shift functor)4 To be precise a Calabi-Yau structure is a section Ωu ∈ HC − (A) = Γ(H) such that as Ωu=0 is an element in Hochschild homology of X, that in turn gives a functional on H (A, dA ) making it into a Frobenius algebra, see [9]. A trivialization of H compatible with the pairing Hu ⊗ H−u −→ k and so the pairing becomes constant. 4

In a sense a Calabi-Yau structure is more or less a choice of isomorphism between Serre’s functor and a shift functor.

Maxim Kontsevich

If the main conjecture is true and the Hodge to de Rham degeneration holds then from such X we can construct a 2-dimensional cohomological quantum field theory. The state space of the theory will be H0 = HH (X). As a part of the structure one gets maps H0⊗n −→ H (Mg,n , k). We will not describe the whole (purely algebraic) construction here, but we shall just say that it provides solutions to holomorphic anomaly equations. It seems to be the case that when we apply this procedure to the Fukaya category we recover the usual Gromov-Witten invariants for a symplectic manifold. It is very interesting to point out that the passage to stable curves is dictated by both, the degeneration of the spectral sequence, and the trivialization of the bundle. This fact was my main motivation for the Degeneration Conjecture. In the Z-graded case a Calabi-Yau structure requires a volume element Ω and a splitting of the non-commutative Hodge filtration compatible with the Poincar´e pairing.

2.24 We can make an important definition. Definition 2.24.1. A non-commutative Hodge Structure over C is a holomorphic super vector bundle Han over D = {|u| ! 1, u ∈ C} with connection ∇ outside of u = 0, with a second order pole and a regular singularity5 together with a covariantly constant finitely generated Z/2graded abelian group Kutop for u ̸= 0 such that Kutop ⊗ C = Huan . In the Z-graded case the pole has order one, and the lattice Kutop comes from the topological K-theory.

2.25

The Non-commutative Hodge Conjecture.

Let X be a saturated space. Consider the map Q ⊗ Image(K0 (CX ) −→ Γ(Han (X))) −→ Q ⊗ HomNC−Hodge−structures (1, Han (X)). 5

By a regular singularity we mean that covariantly constant sections grow only polynomially. Therefore under a meromorphic gauge transformation we end up with a first order pole.

XI Solomon Lefschetz Memorial Lecture Series

The conjecture says that this map is an isomorphism. One can introduce a notion of a polarized NC Hodge structure. The existence of a polarization in addition to the Hodge conjecture imply that the image of K0 (CX ) is K0 (CX )/numerical equivalence, where numerical equivalence is the kernel of a pairing ⟨, ⟩ : K0 ⊗ K0 −→ R, given by ⟨E, F⟩ = χ(RHom(E, F )).

This pairing is neither symmetric nor anti-symmetric, so a priori it could have left and right kernels, but the Serre functor ensures us that they coincide.

2.26 We can now go on `a la Grothendieck and define a category of noncommutative pure motives. Consider X a saturated space over k. We define now Hom(X, Y ) = Q ⊗ K0 (Map(X, Y ))/numerical equivalence. Ordinarily one takes algebraic cycles of all possible dimensions on the product of two varieties. In our situation we must be careful to add direct summands. This should be equivalent to a category of representations of the projective limit of some reductive algebraic groups over k. This non-commutative motivic Galois group responsible for such representations is much larger than usual because of the Z/2-gradings.

2.27 We can also discuss mixed motives in this context. They are a replacement of Voevodsky’s triangulated category of mixed motives. We start again by considering saturated spaces X but now we want to define a new Hom(X, Y ) space as the K-theory spectrum of Map(X, Y ) (an infinite loop space.) We can canonically form the triangulated envelope. Notice that this construction contains ordinary mixed motives for usual varieties modulo the tensoring by Z(1)[2].

Maxim Kontsevich

2.28

Crystalline cohomology and Euler functors.

Consider an algebra A flat over Zp (the p-adic integers) and saturated over Zp . We expect a canonical Frobenius isomorphism ∼ =

Frp : H (C red (A, A)((u)), ∂ + uB) −→ H (C red (A, A)((u)), ∂ + puB), as Zp ((u))-modules preserving connections. Such isomorphism does exist in the commutative case, given a smooth X over Zp we have H (X, ΩX ; d) ∼ = H (X, ΩX ; pd) for p > dim X. Using the holonomy of the connection ∇ ∂ we can go from u to pu ∂u and get an operator Frp with coeﬃcients in Qp . We can state

2.29

The non-commutative Weil conjecture.

Let λα ∈ SpecF rp then λα ∈ Q ⊂ Qp . For all ℓ ̸= p, then |λα |ℓ = 1. $λα |C = 1.

2.30 For the Landau-Ginzburg model X = A1 and f = x2 we get that the cohomology is 1-dimensional, λ ∈ Qp and λ=

" p−1 ! ( mod p), 2 λ4 = 1.

√ The period for the Hodge structure is 2π. There is a reasonable hope for the existence of the Frobenius isomorphism (cf. the work of D. Kaledin.)

XI Solomon Lefschetz Memorial Lecture Series

2.31 For every associative algebra A over Z/p there is a canonical linear map H0 (A, A) −→ H0 (A, A) given by a #→ ap . Recall that H0 (A, A) = A/[A, A]. There are two things to verify, that this map is well defined and that it is linear. (This is a very pleasant exercise.) Moreover this map lifts to a map H0 (A, A) −→ HC0− (A). There is an explicit formula for this lift. For p ! 3 we have " # ! p−1 n−1 p−1 a #→ ap + (coeﬃcients)ai0 ⊗ · · · ⊗ ain u 2 + ! a⊗p u 2 , 2 n even, P p−3!n!2

iα =p

where the last coeﬃcient is non-zero. The formula for p = 2 reads: a #→ a2 + 1 ⊗ a ⊗ a · u.

2.32 One can calculate various simple examples and this seems to suggest a potential mechanism for the degeneration of the Hodge-to-de Rham spectral sequence in characteristic p > 0. The situation is radically diﬀerent for p = 0. This mechanism works as follows. Let us consider polynomials in u: H (C red (A, A)[u], ∂ + uB). There is no obvious spectral sequence in this case. What we have is a quasi-coherent sheaf over A1 with coordinate u. In characteristic p = 0 we have that this sheaf vanishes when u ̸= 0, namely (C red (A, A)[u, u−1 ], ∂ + uB) is acyclic (this is an early observation by Connes). So we have everything concentrated in an infinitedimensional stalk at u = 0, in the form of an infinite Jordan block, plus some finite Jordan blocks. It certainly looks like nothing resembling a vector bundle, it is very singular. The degeneration we seek would

Maxim Kontsevich

mean that we have no finite Jordan blocks. The situation is unfortunately quite involved. In contrast, in finite characteristic p, it seems to be the case that the cohomology of the complex (C red (A, A)[u], ∂+uB) is actually a coherent sheaf, and it will look like a vector bundle if the desired spectral sequence degeneration occurs. The following conjecture explains why we obtain a vector bundle.

2.33

Conjecture.

Let A be a flat dga over Zp . Let A0 = A ⊗ Z/pZ over Z/pZ. Then (C red (A0 , A0 )[u, u−1 ], ∂ + uB) is canonically quasi-isomorphic to (C red (A0 , A0 )[u, u−1 ], ∂) as Z/p[u, u−1 ]-modules. In this conjecture there is no finiteness condition at all.

2.34 The reason for this is as follows. The complex C red (A, A) admits an obvious increasing filtration Fil!n = A ⊗ (A/1)⊗!n−1 + 1 ⊗ (A/1)⊗n . Let V := A/1[1], on grn (Fil) we can write ∂ + B as: 1+σ+···+σ n−1

⊗n

1−σ

! V ⊗n

where σ is the generator of Z/nZ (this would be acyclic in characteristic 0). On the other hand ∂ is simply: 1−σ

V ⊗n −→ V ⊗n . For any free Z/pZ-module the above complex grn (Fil) with diﬀerential ∂ + B is acyclic if (n, p) = 1. At the same time if n = kp the complex is canonically isomorphic to 1−σ

V ⊗k −→ V ⊗k . The hope is that some finite calculation of this sort could allow us to go deep into the spectral sequence and prove the desired degeneration.

XI Solomon Lefschetz Memorial Lecture Series

2.35 We finish by making some remarks regarding the Weil conjecture. Let A be over Z. It would be reasonable to hope that we can define local L-factors by ! " Frp −1 Lp (s) = det 1 − s , p and we should get a sort of non-commutative L-function. We define for a saturated space X an L-function: # L(X) = Lp (s).

This L(X) should satisfy

The Riemann hypothesis. Namely its zeroes lie on ℜ(s) = 12 . The Beilinson conjectures. They state that the vanishing order and leading coeﬃcients at s ∈ 12 Z, s ! 1/2 are expressed via K1−2s (X) that should in turn be finite dimensional.

2.36 This L-function diﬀers from the traditional L-function defined as a product over all points of a variety over finite fields Fq weighted by 1/q s . We can imagine an L-function defined on a saturated space X as the sum over objects of CX weighted in some way. We do not know the exact form of these weights but we expect them to depend on certain stability condition. Such sums appear in string theory as sums over D-branes (for example in the calculation of the entropy of a black hole [14]). It is reasonable to imagine that there is a big non-commutative L-function one of whose limiting cases is arithmetic while the other is topological string theory. Acknowledgement I am grateful to Giuseppe Dito, Yan Soibelman and Daniel Sternheimer for their help and comments. Maxim Kontsevich ´ IHES, 35 Route de Chartres, F-91440, France. maxim@ihes.fr

Maxim Kontsevich

References [1] Beilinson A. A., Coherent sheaves on Pn and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68–69. [2] Bondal A.; van den Bergh M., Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1–36, 258. [3] Connes A., Noncommutative Geometry, Academic Press Inc., San Diego, CA, 1994. [4] Feigin B. L.; Tsygan B. L., Additive K-theory and crystalline cohomology, Funktsional. Anal. i Prilozhen. 19 (1985), no. 2, 52–62, 96. [5] Getzler E., Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology, Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992), Israel Math. Conf. Proc., vol. 7, Bar-Ilan Univ., Ramat Gan, 1993, pp. 65–78. [6] Hochschild G.; Kostant B.; Rosenberg A., Differential forms on regular affine algebras, Trans. Amer. Math. Soc. 102 (1962), 383–408. [7] Kaledin D., Non-commutative Cartier operator and Hodge-to-de Rham degeneration, arXiv:math.AG/0511665v1. [8] Kaledin D., Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie, arXiv:math.KT/0611623v2. [9] Kontsevich M.; Soibelman Y., Notes on A∞ -algebras, A∞ -categories and noncommutative geometry. I, arXiv:math.RA/0606241. [10] Kontsevich M., Deformation quantization of algebraic manifolds, Lett. Math. Phys. 56 (2001), no. 3, 271–294. [11] Orlov D. O., Triangulated categories of singularities and D-branes in LandauGinzburg models, Tr. Mat. Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 240–262. [12] Rieffel M. A., C ∗ -algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415–429. [13] Seidel P., Fukaya categories and deformations, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) (Beijing), Higher Ed. Press, 2002, pp. 351–360. [14] Strominger A., Black hole entropy from near-horizon microstates, J. High Energy Phys. (1998), no. 2, Paper 9, 11 pp. (electronic). [15] Weibel C., The Hodge filtration and cyclic homology, K-Theory 12 (1997), no. 2, 145–164.

Morfismos, Vol. 11, No. 2, 2007, pp. 33–64

A stringy product on twisted orbifold K-theory Alejandro Adem

Yongbin Ruan

∗

Bin Zhang

Abstract In this paper we define an associative stringy product for the twisted orbifold K–theory of a compact, almost complex orbifold X. This product is defined on the twisted K–theory τ Korb (∧X) of the inertia orbifold ∧X, where the twisting gerbe τ is assumed to be in the image of the inverse transgression H 4 (BX, Z) → H 3 (B ∧ X, Z).

2000 Mathematics Subject Classification: 55N15, 55N45, 19L47. Keywords and phrases: orbifolds, twisted K-theory.

Introduction

Over the last twenty years, there has been a general trend towards the infusion of physical ideas into mathematics. One of the successful examples in the last few years is the subject of twisted K-theory. Interest in it originates from two diﬀerent sources in physics, the consideration of a D-brane charge on a smooth manifold by Witten [30] and the notion of discrete torsion on an orbifold by Vafa [29]. In mathematics, there have been important developments connected to this. On the one hand, it inspired a new subject often referred to as stringy orbifold theory. On the other hand, it revitalized and re-established connections to many classical topics such as equivariant K-theory, groupoids, stacks and gerbes. ∗

Invited article. Partially supported by the NSF and NSERC. 2 Partially supported by the NSF. 1

Alejandro Adem, Yongbin Ruan, and Bin Zhang

For smooth manifolds, the mathematical foundation of twisted K-theory has been worked out and for any cohomology class α ∈ H 3 (X, Z), one can associate a twisted K-theory α K(X) (see [6], [7], [17], [21], [4]). One interesting phenomenon is the diﬀerence between a torsion class and a non-torsion one: for torsion α, we have a natural notion of twisted vector bundle or twisted sheaf; for a non-torsion α, there is no geometric notion of vector bundle and one has to use infinite–dimensional analysis. The case of an orbifold or even a more general singular space is much more interesting. This naturally relates to equivariant theories if we specialize to the case of X = [M/G] where M is a smooth manifold and G is a compact Lie group acting almost freely on M . One can consider a cohomology class α ∈ H 3 (BG, Z) and its corresponding twisted Ktheory, where BG is the classical classifying space for G. This was the set-up of [3] for orbifold twisted K-theory α Korb (X) using discrete torsion. Twisted K-theory has been generalized to K-theory twisted by gerbes (see [19]), and also using the framework of groupoids (see [18]). In the general case, one can think that the twisting is a cohomology class3 α ∈ H 3 (BX, Z) where BX is now the classifying space of the orbifold X. One advantage of working with orbifolds is the nontrivial cohomolog∗ (X, C). ical counterpart called Chen-Ruan cohomology of orbifolds, HCR One can use discrete torsion [27] (more generally torsion gerbes [25]) ∗ (X, L ). Moreover, to obtain a twisted Chen-Ruan cohomology HCR α the twisted Chen-Ruan cohomology has an important internal product ∗ (X, L ) a ring. On the other hand, the tensor product making HCR α produces a map α

Korb (X) ⊗

Korb (X) →

α+β

Korb (X).

Note that it shifts the twisting to α + β; one natural question is if there is an internal “stringy” product for α Korb (X)? Freed, Hopkins and Teleman [14] have proved the beautiful result that the twisted equivdim G (G) for the adjoint action is isomorphic to the ariant K-theory α KG Verlinde algebra of representations of the central extension of the loop 3

To be totally precise, we will actually be twisting with cocycles.

A stringy product on twisted orbifold K-theory

algebra LG for a semi-simple Lie group G. This algebra carries a very important ring structure via the Verlinde product, whose structure constants encode the information for so–called conformal blocks. Using the group structure of G, one can also construct a ring structure (via dim G (G); these rings turn out to be the Pontryagin product) for α KG isomorphic. Due to the importance of the Verlinde product in representation theory, the existence of a stringy product on the twisted K-theory for more general spaces becomes an important question. This is the problem we will address in this article and its sequel. Our main observation is that there is indeed a stringy product for the twisted K-theory of orbifolds. Moreover, the key information determining such a stringy product does not lie in H 3 (BX, Z) as one conventionally believes; instead, it lies in H 4 (BX, Z). Given a class φ ∈ H 4 (BX, Z), it induces a class θ(φ) ∈ H 3 (B ∧ X, Z) where ∧X is the inertia stack of X and thus we can define a twisted K-theory θ(φ) K(∧X). The inertia stack ∧X can be viewed as the moduli space of constant loops on X. Furthermore, there is a key multiplicative formula for θ(φ) characterized by the eﬀect of φ on the moduli space M of constant morphisms from a Riemann surface. This map, which can be thought of as the inverse of the classical transgression map, appears in [12] for finite group cohomology. Based on this we derive a simple extension for orbifold groupoids and explicitly prove its multiplicative property (a more geometric version of this formula appears in [20]). Our second ingredient is more subtle: experience from Chen-Ruan cohomology tells us that a naive definition does not give an associative product. The reason lies in the fact that the fixed–point sets Xg , Xh for g ̸= h in general do not intersect each other transversely. It is known that in Chen-Ruan cohomology theory one can correct the naive definition by introducing a certain obstruction bundle. Combining these two ingredients, we obtain an associative product which can be viewed as a K-theoretic counterpart of the Chen-Ruan product for orbifold cohomology. Theorem 1.1 Let X denote a compact, almost complex orbifold, and

Alejandro Adem, Yongbin Ruan, and Bin Zhang

let τ be a U (1)–valued 2–cocycle for the inertia orbifold ∧X which is in the image of the inverse transgression. Then there is an associative product on τ Korb (∧X) which generalizes both the Pontryagin and the orbifold cohomology product. Our construction is in fact motivated by the so–called Pontryagin product on KG (G), for G a finite group, which is what our construction amounts to, for X = ∧[∗/G] in the untwisted case. As an application, we use our construction to clarify the twisted Pontryagin product; it may not always exist, and when it exists, it may not be unique either. We provide an explicit calculation of the inverse transgression map for the cohomology of finite groups, showing that in fact it can be computed using the natural multiplication map Z × ZG (h) → G, where ZG (h) denotes the centralizer of h ∈ G. Using this we exhibit a group, G = (Z/2)3 and an integral cohomology class φ ∈ H 4 (G, Z) such that under the inverse transgression it maps non–trivially for every properly twisted sector, yielding an interesting product structure on θ(φ) KG (G). One of the original motivations for the introduction of the twisted theory in orbifolds was the hope of describing the cohomology of desingularizations of an orbifold. Joyce constructed five classes of topologically diﬀerent desingularization of T 6 /Z4 [16], arising from a representation Z/4 ⊂ SU (3). It is known that Joyce’s desingularizations are not captured by discrete torsion. For a while, there was the expectation that they may be captured by 1-gerbes. The computation in [2] shows that the high hopes for 1-gerbes is probably misplaced; however, we notice that H 4 (B(T 6 /Z4 ), Z) seems to contain precisely the information related to desingularization. We hope to return to this question later. We would like to make a comment about notation: throughout this paper we will be using the language of orbifold groupoids, hence given an orbifold X we will be thinking of it in terms of a Morita equivalence class of orbifold groupoids, represented by G; in this context τ Korb (X) is interpreted as τ K(G), using the notion of twisted K–theory of groupoids, which we will summarize in Section 3. The results in this article were first announced by the second author at the Florida Winter School on Mathematics and Physics in December,

A stringy product on twisted orbifold K-theory

2004. Here, we present our construction for the orbifold case. The construction for general stacks will appear elsewhere. During the course of this work, we received an article by Jarvis-Kaufmann-Kimura [15] which also deals with a stringy product in K-theory; indeed the restriction of our twisted K-theory τ K(∧X) to the non-twisted sector gives their small orbifold K-theory K(| ∧ X|). The authors would like to thank MSRI and PIMS for their hospitality during the preparation of this manuscript, and the third author would like to thank the MPI– Bonn for its generous support.

Preliminaries on Orbifolds and Groupoids

In this section, we summarize some basic facts about orbifolds, using the point of view of groupoids. Our main reference is the book [1], but [24] is also a useful introduction. Recall that an orbifold structure can be viewed as an orbifold Morita equivalence class of orbifold groupoids; we shall present all of our constructions in this framework. Suppose that G = {s, t : G1 → G0 } is an orbifold groupoid, namely, a proper, ´etale Lie groupoid, we will use |G| to denote its orbit space, i.e., the quotient space of G0 under the equivalence relation: x ∼ y iﬀ there is an arrow g : x $→ y. Conversely, we call G an orbifold presentation of |G|. Recall that a groupoid homomorphism φ : H → G between (Lie) groupoids H and G consists of two (smooth) maps, φ0 : H0 → G0 and φ1 : H1 → G1 , that together commute with all the structure maps for the two groupoids H and G. Obviously, a groupoid homomorphism φ : H → G induces a continuous map |φ| : |H| → |G|. Definition 2.1 Let φ, ψ : H → G be two homomorphisms. A natural transformation α from φ to ψ is a smooth map α : H0 → G1 , giving for each x ∈ H0 an arrow α(x) : φ(x) → ψ(x) in G1 , natural in x in the sense that for any h : x → x′ in H1 , the identity ψ(h)α(x) = α(x′ )φ(h) holds. Definition 2.2 Let φ : H → G and ψ : K → G be homomorphisms of Lie groupoids. The groupoid fibered product H ×G K is the Lie

Alejandro Adem, Yongbin Ruan, and Bin Zhang

groupoid whose objects are triples (y; g; z) where y ∈ H0 , z ∈ K0 and g : φ(y) → ψ(z) in G1 . Arrows (y; g; z) → (y ′ ; g ′ ; z ′ ) in H ×G K are pairs (h; k) of arrows, h : y → y ′ in H1 and k : z → z ′ in K1 with the property that g ′ φ(h) = ψ(k)g. Composition in H ×G K is defined in the natural way. Next we recall the notion of equivalence of groupoids. Definition 2.3 A homomorphism φ : H → G between Lie groupoids is called an equivalence if the map tπ1 : G1 s ×φ H0 → G0 is a surjective submersion and the square H1 (s, t) ↓

H0 × H0

→ φ×φ

→

G1 ↓ (s, t)

G0 × G0

is a fibered product of manifolds. Definition 2.4 Two orbifold groupoids G and G′ are said to be orbifold Morita equivalent if there is a third orbifold groupoid H and two equivalences φ : H → G and φ′ : H → G′ . An orbifold homomorphism from H to G is a triple (K, ϵ, φ), where K is another orbifold groupoid, ϵ : K → H is an equivalence and φ : K → G is a groupoid homomorphism. The equivalence relation for orbifold homomorphisms is generated by natural transformations of φ and equivalences of K. Definition 2.5 The category of orbifolds is the category whose objects are the orbifold Morita equivalence classes of orbifold groupoids and the morphisms are equivalence classes of orbifold homomorphisms. Remark In this paper, we will use the term homomorphism for a groupoid homomorphism, an orbifold homomorphism will be clearly identified when it arises. There are several important constructions which play a fundamental role in stringy orbifold theory. Given r > 0 an integer, we can consider the r–tuples of composable arrows in G, i.e. Gr = {(g1 , . . . , gr ) ∈ Gr1 | t(gi ) = s(gi+1 ), i = 1, . . . , r}.

A stringy product on twisted orbifold K-theory

These fit together to form a simplicial space, whose geometric realization is the classifying space BG of the groupoid G. In our discussion of homological invariants of groupoids, we will be considering cochains arising from this complex. Recall that the inertia groupoid ∧G is a groupoid canonically associated with G which is defined as follows: Definition 2.6 For any groupoid G, we can associate an inertia groupoid ∧G as (∧G)0 = {g ∈ G1 | s(g) = t(g)}, (∧G)1 = {(a, v) ∈ G2 | a ∈ (∧G)0 } where s(a, v) = a, t(a, v) = v −1 av. More generally, we can define the groupoid of k-sectors Gk as (Gk )0 = {(a1 , a2 , · · · , ak ) ∈ Gk1 | s(a1 ) = t(a1 ) = · · · = s(ak ) = t(ak )} | s(a1 ) = t(a1 ) = · · · (Gk )1 = {(a1 , a2 , · · · , ak , u) ∈ Gk+1 1 = s(ak ) = t(ak ) = s(u)} with s(a1 , · · · , ak , u) = (a1 , · · · , ak ), t(a1 , · · · , ak , u) = (u−1 a1 u, · · · , u−1 ak u). The construction of the inertia groupoid and Gk in general is completely functorial. Namely, a homomorphism of groupoids induces a homomorphism between k-sectors and an equivalence of orbifold groupoids induces an equivalence betweem them. In case of orbifold groupoids, the inertia groupoid can be identified as the space of constant loops on G; more generally, Gk−1 can be identified as the space of constant morphisms from an orbifold sphere with k-orbifold points to G. We will come back to these descriptions later. Another important notion is that of quasi–suborbifold; before defining it we first point out that for a groupoid G, and an open subset V ⊂ G0 , {s, t : s−1 (V ) ∩ t−1 (V ) → V } is a groupoid, which we will denote by G|V .

Alejandro Adem, Yongbin Ruan, and Bin Zhang

Definition 2.7 A homomorphism of orbifold groupoids φ : G → H is a quasi-embedding if φ : G0 → H0 is an immersion. For any y ∈ im(φ) ⊂ H0 , with isotropy group Hy , φ−1 (y) is in an orbit of G, and for any x ∈ φ−1 (y), φ : Gx → Hy is injective. For any y ∈ im(φ), and any x ∈ φ−1 (y), there are neighborhoods Uy of y and Vx of x such that H|Uy = Uy ! Hy , G|Vx = Vx ! Gx and G|φ−1 (Uy ) ∼ = (Hy ×φ(Gx ) Vx ) ! Hy . |φ| : |G| → |H| is proper. Definition 2.8 G together with φ is called a quasi–suborbifold of H. The following are important examples of quasi–suborbifolds. Example 2.9 Suppose that G = X ! G is a global quotient groupoid (i.e. a quotient by a finite group). We often use the stacky notation [X/G] to denote the groupoid. An important object is the inertia groupoid ∧G = ('g Xg ) ! G where Xg is the fixed point set of g and G acts on 'g Xg as h : Xg → Xhgh−1 by h(x) = hx. By our definition, φ : ∧G → G induced by the inclusion map Xg → X is a quasi– embedding. Example 2.10 Let G be the global quotient groupoid defined as in the previous example. We would like to define an appropriate notion of the diagonal ∆ for G × G. We define it as ∆ = ('g ∆g ) ! (G × G) where ∆g = {(x, gx), x ∈ X}. Our definition of quasi–suborbifold includes this example. More generally, we define the diagonal ∆(G) as the groupoid fibered product G ×G G. One can check that ∆(G) = G ×G G is locally of the desired form and hence a quasi–suborbifold of G × G. Notice that the map x → (x, 1x , x) allows us to identify G as a component of ∆(G).

A stringy product on twisted orbifold K-theory

Example 2.11 For l ≤ k, there are natural evaluation morphisms ei1 ,··· ,il : Gk → Gl given by ei1 ,··· ,il (a1 , · · · , ak ) = (ai1 , · · · , ail ). Furthermore, we have e : Gk → G given by e(a1 , · · · , ak ) = s(a1 ) = t(a1 ) = · · · = s(ak ) = t(ak ). The latter one corresponds to taking the image of constant morphism. We leave as an exercise for the reader to check that e and the ei1 ,··· ,il are quasi-embeddings and that Gk is a quasi–suborbifold of Gl . One of the main tools is the notion of a normal bundle. If i : G → H is a quasi-embedding, i∗ T H is a groupoid vector bundle over G such that T G is a subbundle. Then we can define the normal bundle NG|H = i∗ T H/T G. NG|H behaves as the normal bundle does for smooth manifolds. Next we introduce the notion of intersection for quasi–suborbifolds. Definition 2.12 Let f : G1 → H and g : G2 → H denote quasi– suborbifolds. We define their intersection G1 ∩ G2 as the restriction of the pullback G1 ×H G2 to the component H in H ×H H. Note that under this definition it makes sense to intersect a quasi– suborbifold with itself. Under certain conditions these intersections can have nice properties, analogous to the situation for manifolds. We will be interested in the notion of a clean intersection. Definition 2.13 Suppose that f : G1 → H, g : G2 → H are smooth quasi–suborbifolds, we say that G1 intersects G2 cleanly if the intersection orbifold G1 ∩ G2 is a smooth quasi–suborbifold of H (where as before H is viewed as a component of ∆(H)) such that for every x ∈ (G1 )0 ∩ (G2 )0 , T(x,1x ,x) (G1 ∩ G2 ) = Tx G1 ∩ Tx G2 . Example 2.14 As we mentioned before, the evaluation map e : ∧G → G is a quasi–suborbifold. Then, e with itself forms a clean intersection.

Alejandro Adem, Yongbin Ruan, and Bin Zhang

Indeed the question is local, and locally it corresponds to the intersection of fixed point sets V g ∩ V h . This is clearly a clean intersection. More generally, ei1 ,··· ,il : Gk → Gl is an quasi-embedding. Then, two diﬀerent quasi-embeddings ei1 ,··· ,il , ej1 ,··· ,jl : Gk → Gl form a clean intersection. We leave it as an exercise for our readers. As in manifold theory, there is also the notion of transversality for quasi–suborbifolds. Definition 2.15 Suppose that f : G1 → H, g : G2 → H are smooth homomorphisms. We say that f × g is transverse to the diagonal ∆ if locally f × g is transverse to every component of the diagonal ∆ on the object level. We say that f, g are transverse to each other if f × g is transverse to the diagonal ∆. Example 2.16 Suppose that f : G1 → H, g : G2 → H are quasi– embeddings which are transverse to each other. Then the intersection G1 ∩ G2 is a quasi–suborbifold of H. Note that a clean intersection need not be transverse, this failure of transversality plays a role in the definition of orbifold cohomology and K–theory.

Gerbes and Twisted K–Theory

We now consider the cohomology and K–theory of orbifold groupoids. Definition 3.1 Let G denote a Lie groupoid, then we define the continuous U (1)-valued k–cochains on G as C k (G, U (1)) = {φ : Gk → U (1) | φ is continuous}. The diﬀerential on this abelian group (using additive notation) is defined via δφ(g1 , · · · , gk+1 ) = φ(g2 , · · · gk+1 ) +

k ! (−1)i φ(g1 , · · · , gi gi+1 , · · · , gk+1 ) + (−1)k+1 φ(g1 , · · · , gk ). i=1

A stringy product on twisted orbifold K-theory

By a result due to Moerdijk [23], if G is an ´etale groupoid then the ˇ cohomology of this chain complex is the Cech cohomology of BG with coeﬃcients in the sheaf C(U (1)) of U (1)-valued continuous functions over the classifying space BG. By the exact sequence 0 → Z → C(R) → C(U (1)) → 0, we obtain a long exact sequence in cohomology, H k (BG, C(R)) → H k (BG, C(U (1))) → H k+1 (BG, Z) → H k+1 (BG, C(R)). Since C(R) is a fine sheaf, the connecting homomorphism is an isomorphism, and so for k > 0, H k (BG, C(U (1))) ∼ = H k+1 (BG, Z). We recall Definition 3.2 An n-gerbe on G is a pair (H, θ) consisting of ϵ

a refinement G ← H (i.e., ϵ is an equivalence) an (n + 1)–cocycle φ : Hn+1 → U (1). Next we define equivalence of gerbes. Definition 3.3 Given two n–gerbes (H, θ) and (H′ , θ′ ) on G we have the following: (H, θ) is equivalent to (H′ , θ′ ) if there is a common refinement H′ ← H′′ → H such that the induced (n + 1)–cocycles on H′′ are the same. (H, θ) is isomorphic to (H′ , θ′ ) if there is a common refinement H′′ such that the induced (n + 1)–cocycles on H′′ diﬀer by a coboundary. ϵ

Let K → G be an equivalence. If (H, θ) is a n-gerbe over K, by definition, it is a n-gerbe over G. For an n-gerbe (H, θ) over G, H′ = H ×G K is an orbifold which is equivalent to both H and K by projections p1 : H′ → H and p2 : H′ → K. So we have the pull-back n-gerbe (H′ , p∗1 θ) over K. It is easy to see that under these operations, equivalent (isomorphic) gerbes go to equivalent (isomorphic) ones, therefore gerbes behave well under orbifold Morita equivalence. Definition 3.4 An n-gerbe on an orbifold is an equivalence class of pairs (G, θ), where G is a presentation of the orbifold, θ is a n + 1-cocycle on G, and the equivalence relation is gerbe isomorphism.

Alejandro Adem, Yongbin Ruan, and Bin Zhang

ˇ From the definition, it is clear that an n-gerbe defines a Cech (n+1)-cocycle for the sheaf C(U (1)) of continuous U (1)–valued functions on the classifying space BG (BH and BG are weakly homotopy equivalent). Hence it will define a cohomology class in H n+1 (BG, C(U (1))) ∼ = H n+2 (BG, Z). The image of θ under n+2 (BG, Z) is called its characteristic class the connecting homomorphism in H or Dixmier-Douady class. We can associate twisted K-theory to a 1-gerbe. For simplicity we assume that the 2–cocycle θ is defined on G, i.e., we are dealing with (G, θ), and the twisted K-theory θ K(G) will be defined. We follow the treatment of [18] to describe this. Let H be a separable Hilbert space; it is well-known that the characteristic class of a principal P U (H)–bundle over G also lies in H 3 (BG, Z). Hence, given a 1-gerbe, we should be able to associate a P U (H) bundle with the same characteristic class; in fact we can associate a canonical principal P U (H)–bundle. We outline its construction. For the orbifold groupoid G = {s, t : G1 → G0 }, let R = G1 × U (1) be the topologically trivial central extension, and (g1 , r1 )(g2 , r2 ) = (g1 g2 , θ(g1 , g2 )r1 r2 ), which makes {˜ s, t˜ : R → G0 } a Lie groupoid, where s˜(g, r) = s(g), t˜(g, r) = t(g). Now let Gx = t−1 (x); there is a system of measure (Haar system) λ = (λx )x∈G0 , where λx is a measure with support Gx such that for all f ∈ Cc (G1 ), ! x → g∈Rx f (g)λx (dg) is continuous. By L2x , we denote the space L2 (Gx ) consisting of functions defined on Gx which are L2 with respect to the Haar measure. Let Ex = L2x ⊗ H, E = &x Ex . Then E is a countably generated continuous field of infinite dimensional Hilbert spaces over the finite dimensional space G0 , and therefore is a locally trivial Hilbert bundle according to the Dixmier-Douady theorem [13]. The Lie groupoid R acts naturally on E: U (1) acts on H by complex multiplication. Therefore, E is naturally a Hilbert bundle over {˜ s, t˜ : R → G0 }. Notice E is not a Hilbert bundle over G. However, P (E) is a projective bundle over G with precisely the same characteristic class of θ. Let B be the principal bundle of orthonormal frames of E; it is a U (H)-principal bundle. By our previous argument, P B is a principal P U (H)-bundle over G. Let θ be a 1-gerbe and Pθ be the associated P U (H)-bundle constructed above. Let F red0 (H) be the space of Fredholm operators endowed with the ∗-strong topology and F red1 (H) be the space of self-adjoint elements in F red0 (H). Let K(H) be the space of compact operators endowed with the

A stringy product on twisted orbifold K-theory

norm-topology. Now consider the associated bundles F rediθ (H) := Pθ ×P U (H) F redi (H) → G0 , Kθ (H) := Pθ ×P U (H) K(H) → G0 . By Fiθ , we denote the space of norm-bounded, G1 -invariant, continuous sections x → Tx of the bundle F rediθ (H) → G0 such that there exists a norm-bounded, G1 -invariant, continuous section x → Sx of Kθ (H) → G0 with the property that 1 − Tx Sx and 1 − Sx Tx are continuous sections of Kθ (H) vanishing at infinity of |G|. Definition 3.5 For any section T of Fiθ . We define the support supp(T ) as the set of point x ∈ |G| such that Tx′ is not invertible for any x′ in the preimage of x. Then we have Definition 3.6 Let G be an orbifold groupoid and (G, θ) be a 1-gerbe. We define its θ–twisted K-theory as θ

K i (G) = {[T ] | T ∈ Fiθ },

where [T ] denotes the homotopy class of T where T is compactly supported. Note that since the space of invertible operators is contractible, any T with compact support is homotopic to a section which is the identity outside a compact subset. Suppose that i : U → G0 is an open subset, using the property above, we have a natural extension i∗ :

i∗ θ

K i (G|U ) →

K i (G).

Remark Suppose that we have a cocycle α = β +δρ. Then there is a canonical isomorphism between central extensions of groupoids ψρ : Rα → Rβ given by ψρ (g, r) = (g, ρ(g)r). Hence it induces an isomorphism Pα (E) → Pβ (E) and also a canonical isomorphism ψρ :

K i (G) →

K i (G).

Alejandro Adem, Yongbin Ruan, and Bin Zhang

Suppose that in fact ρ is a cocycle, i.e., δρ = 0. Then α = α + δρ and hence we have an automorphism ψρ : α K i (G) → α K i (G). Furthermore, if ρ = δγ is a coboundary, then ψρ is the identity. Hence H 1 (BG, U (1)) acts as automorphisms of twisted K-theory. It is easy to check in many examples that they are nontrivial automorphisms. In the literature, twisted K-theory is ˇ often referred to as being twisted by a Cech cohomology class or characteristic class of a 1-gerbe. This is a rather ambiguous statement, as cohomologous 1-gerbes induce isomorphic twisted K-theory, but this is not canonical. This observation is particularly important when we define a product structure on twisted K-theory. To summarize: for an orbifold groupoid G, and a 1-gerbe (G, θ), twisted K-theory θ K(G) is well–defined up to isomorphism (see [18]). Definition 3.7 For an orbifold and a 1-gerbe on it, taking a presentation (G, θ) of the gerbe, the twisted K-theory of the orbifold is defined to be θ K(G). There is a natural addition operator for α K ∗ (G) induced by the Hilbert space addition H ∼ = H ⊕ H. On the other hand, the multiplication operation induced by Hilbert space tensor product H ∼ = H ⊗ H shifts the twisting α

K ∗ (G) ⊗

K ∗ (G) →

α+β

K ∗ (G).

Remark Strictly speaking, there is an issue about canonicity because of the way we identify H ⊕ H and H ⊗ H with H . Since U (H) is contractible, any identification will give the same homotopy classes, therefore the same K-theory element. We also want to point out that the product of an element in twisted Ktheory with a vector bundle is to be understood in the following sense. Let P be a family of Fredholm operators on H parameterized by a space M and E a complex vector bundle over M of finite rank. Then P · E is a family of Fredholm operators on H ⊗ E parameterized by M , which at every point x ∈ M has P · E(x) = P (x) ⊗ IdE . Hence it is easy to see that if P ∈ θ K i (G) and E is a G-bundle, then P · E ∈ θ K i (G) (just like the above remark, the way to identify H ⊗ E with H does not change the element). So even though E may not be an element of K 0 (G), the product with E makes sense. To define our stringy product, we will need a version of the push-forward map in the context of twisted K-theory. For smooth manifolds, such a pushforward map has already been worked out by Carey-Wang [9]. In the case of orbifold groupoids, we follow their treatment, the extra eﬀort will be needed to deal with the groupoid structure.

A stringy product on twisted orbifold K-theory

Let G, H be almost complex groupoids, f : G → H be a homomorphism which preserves the almost complex structures, and α be a 1-gerbe on H. We ∗ will define the push-forward map f∗ : f α K ∗ (G) → α K ∗ (H). Let G = {s, t : G1 → G0 } be a groupoid and E a rank n complex vector bundle over G, i.e., π : E → G0 is a complex vector bundle with compatible G∗ action. We first establish the Thom homomorphism Φ : α K ∗ (G) → π α K ∗ (G! E), where G ! E is the transformation groupoid with object set E and arrow set G1 ×G0 E. Fix any invariant hermitian metric on E [28], then for any g ∈ G0 , e ∈ Eg = π −1 (g), we use e∗ to denote the dual of e with respect to the fixed hermitian metric. The complex G-bundle π : E → G0 defines a complex of G-bundles over E, λE = (Λeven π ∗ E, Λodd π ∗ E, φ), where φ(g,e) = e ∧ −e∗ !. Notice that in ordinary K-theory, this is the Thom element. For any element x ∈ α K ∗ (G), it is represented by a G-equivariant section x : G0 → F red∗α (H) with supp(x) compact. Proposition 3.8 For any 1-gerbe α, there is a K 0 (G)-module homomorphism Φ:

K ∗ (G) →

π∗ α

K ∗ (G ! E)

which is the standard Thom isomorphism in the case of equivariant K-theory. Proof. For any element x ∈ α K i (G), it is given by a G-equivariant section x : G0 → F rediα (H). Therefore, we have a section π ∗ (x) : E → π ∗ F rediα (H). Now supp(π ∗ (x)) = π −1 (supp(x)). For any point (g, e) ∈ E, let us consider the operator: D : H ⊗ Λeven Eg ⊕ H ⊗ Λodd Eg → H ⊗ Λeven Eg ⊕ H ⊗ Λodd Eg defined by D(g,e) =

(π ∗ x)(g) ⊗ 1 1 ⊗ φ(g,e)

−1 ⊗ φ∗(g,e) (π ∗ x)(g)∗ ⊗ 1

where ∗ means the adjoint operator. This is the so-called “graded tensor product” of Fredholm operators. It is easy to check that D(g,e) is Fredholm and if e ̸= 0, then it is invertible. Globalizing this construction, we have a family D of Fredholm operators parameterized by E. In fact, it is a fiberwise Fredholm operator on Hilbert

Alejandro Adem, Yongbin Ruan, and Bin Zhang

bundles Λ π ∗ E ⊗ H over E, as we remarked, the identification of fiber with H does not matter. Notice that supp(D) = π −1 supp(x) ∩ i(G0 ), where i : G0 → E is the zero section. D is a section of π ∗ F rediα (H) ∼ = F rediπ∗ α (H). D is G ! E-equivariant. Therefore D represents an element of Now we define Φ:

K ∗ (G) →

π∗ α

K i (G ! E).

π∗ α

K ∗ (G ! E)

x %→ D Up to homotopy, as in ordinary K-theory, this definition does not depend on any choice, so it is well-defined. Because it is the graded tensor product with λE , Φ is a K(G)-module homomorphism. Furthermore, by the definition, we see that it is a generalization of the Thom isomorphism in equivariant Ktheory. ✷ We need a slightly more general version of this. Let U be an open neighborhood of the zero section, from the definition, Φ(x) is supported on the zero section, so by restriction, we have the following Thom homomorphism. Proposition 3.9 For any 1-gerbe α, there is a K(G)-module homomorphism Φ:

K ∗ (G) →

π∗ α

K ∗ (G ! E|U ).

To handle the general situation, let us first recall a lemma for Lie groupoids [22]. Lemma 3.10 Let p : F → G0 be a smooth surjective submersion, then the groupoid F ×p G is equivalent to G, where (F ×p G)1 = (F × F ) p×p ×s×t G1 and (F ×p G)0 = F , and the new source map is s : (F × F ) p×p ×s×t G1 → F, ((x, y), g) %→ x, the new target map is t : (F × F ) p×p ×s×t G1 → F, ((x, y), g) %→ y. For a Lie groupoid homomorphism f : G → H, if we apply the lemma above to the space F = G0 × H0 , and take K to be F ×p H, we can prove the next lemma, where all the maps are the natural ones.

A stringy product on twisted orbifold K-theory

Lemma 3.11 Let f : G → H be a homomorphism of Lie groupoids, then there exists a Lie groupoid K and homomorphisms g : G → K, h : K → H, where g0 : G0 → K0 is an embedding and h is an equivalence, such that f is the composition of g and h. In other words, any homomorphism is an embedding on the object level up to Morita equivalence. Now we can prove the existence of a push-forward map in twisted K-theory. Theorem 3.12 If f : G → H is a homomorphism between almost complex groupoids which preserves the almost complex structures, such that |f | : |G| → |H| is proper, and f1 (G.x) = H.f0 (x) for any x ∈ G0 , then there is a pushforward map ∗ f∗ : f α K ∗ (G) → α K ∗ (H). Proof. Given our last lemma, we may assume that f0 : G0 → H0 is a proper embedding. By our assumption, the normal bundle E of G0 in H0 is a complex G-bundle, and we can identify an open neighborhood U of the zero section in the normal bundle with a neighborhood of f0 (G0 ) in H0 , i.e. we have an embedding j : U → H0 as an open subset. It defines a homomorphism: ∗ j∗ : j α K ∗ (G ! E|U ) → α K ∗ (H|j(U ) ), because the action of G on the normal bundle is induced from the H action, in this case any G-equivariant section is H-equivariant. It is clear that f π is homotopic to j. Therefore, π ∗ f ∗ α = j ∗ α + δρ for some ρ. The choice of ρ is not unique; for example, we can add a 1-cocycle. This corresponds exactly to the non-canonicity of the dependence of twisted K-theory on the cohomology class of a 1-gerbe. However, f π = j on the zero section; therefore, we can choose ρ such that ρ = 0 on the zero section. Since U deformation retracts to the zero section, it fixes ρ uniquely. Now we have homomorphisms: f ∗α

Φ π∗ f ∗ α

K ∗ (G) →

pρ

j∗

K ∗ (G ! E|U ) → α K ∗ (G ! E|U ) → α K ∗ (H|j(U ) ) → α K ∗ (H)

where the last homomorphism is extension for an open saturated subgroupoid. ✷ The composition is the push-forward map f∗ . Given our explicit definition, it is easy to check the following properties of the push-forward map. Proposition 3.13 Let f : G → H as before, then there exists an element ∗ c = c(G, H) such that for any a ∈ f α K ∗ (G) and b ∈ β K ∗ (H), we have f ∗ f∗ (a) = a · c f∗ (a · f ∗ (b)) = f∗ (a) · b.

Alejandro Adem, Yongbin Ruan, and Bin Zhang

In particular for quasi–suborbifolds, we have following result. Corollary 3.14 If i : G1 → G is a quasi–suborbifold, then there is a pushforward map ∗ i∗ : i α K ∗ (G1 ) → α K ∗ (G) satisfying the above properties. For later purposes we would like to introduce Definition 3.15 If E → G is a complex orbifold bundle, then its K–theoretic Euler class eK (E) is defined as i∗ λE , the complex of G–vector bundles obtained by pulling back the Thom element λE using the zero–section i : G → E. Note that we can define the product x · eK (E) as x · eK (E) = x · (∧even E) + (x · ∧odd E)∗ .

The Inverse Transgression for Groupoids

In order to define the stringy product in twisted K–theory, we will need a cohomological formula to match up the levels which appear in the twistings. The basic construction is the inverse transgression, which was defined in [12]. We provide a formulation for groupoids inspired by the case of finite groups. We will also provide some explicit calculations. See [19] for a more geometric view on this. Recall that (∧G)0 = {a ∈ G1 | s(a) = t(a)}, (∧G)1 = {(a, u1 ) ∈ G1 × G1 | s(a) = t(a) = s(u1 )}. It is easy to check that the k–tuples of composable arrows in ∧G are (∧G)k = {(a, u1 , · · · , uk ) ∈ Gk+1 | s(a) = t(a) = s(u1 ), t(ui ) = s(ui+1 )}. Definition 4.1 Define θ : C k+1 (G, U (1)) → C k (∧G, U (1)) by θ(φ)(a, u1 , · · · , uk ) = (−1)k φ(a, u1 , · · · , uk ) +

k ! i=1

where ai = (u1 · · · ui )−1 au1 · · · ui .

(−1)i+k φ(u1 , · · · , ui , ai , ui+1 , · · · , uk ),

A stringy product on twisted orbifold K-theory

A routine but slightly tedious computation shows that this is in fact a cochain map, i.e., δθ = θδ. We should note that θ is a natural map defined for all groupoids. For orbifold groupoids it induces a homomorphism θ∗ : H k (BG, U (1)) → H k−1 (B ∧ G, U (1)), and hence a homomorphism θ∗ : H k+1 (BG, Z) → H k (B ∧ G, Z). The cochain map θ and the induced map in cohomology will be called the inverse transgression. Recall that the moduli space of constant morphisms M3 (G) from an orbifold sphere with three orbifold points can be identified with the 2-sector orbifold G2 , where (G2 )0 = {(a, b) ∈ G2 | s(a) = t(a) = s(b) = t(b)}, (G2 )k = {(a, b, u1 , · · · , uk ) ∈ Gk+2 |

s(a) = t(a) = s(b) = t(b) = s(u1 ), t(ui ) = s(ui+1 )}

with s(a, b, u1 , · · · , uk ) = (a, b), t(a, b, u1 , · · · , uk ) = (ak , bk ) where ai = (u1 · · · ui )−1 au1 · · · ui , bi = (u1 · · · ui )−1 bu1 · · · ui . There are three natural evaluation morphisms e1 : G2 → ∧G by e1 (a, b) = a, e2 : G2 → ∧G by e2 (a, b) = b, e12 : G2 → ∧G by e12 (a, b)) = ab. Furthermore, e1 , e2 , e12 are all quasi-embeddings. Definition 4.2 Define µ : C k+2 (G, U (1)) → C k (G2 , U (1)) by µ(φ)(a, b, u1 , · · · , uk ) = φ(a, b, u1 , · · · , uk ) ! + (−1)i+j φ(u1 , · · · , ui , ai , ui+1 , · · · , uj , bj , uj+1 , · · · , uk )

where the summation is taken over the set of pairs (i, j) satisying 0 ≤ i ≤ j ≤ k and (i, j) ̸= (0, 0).

Alejandro Adem, Yongbin Ruan, and Bin Zhang

A second key multiplicative formula is given by the equation µδ + δµ = e∗1 θ + e∗2 θ − e∗12 θ.

Note that the function µ defines a chain homotopy between e∗1 θ + e∗2 θ and e∗12 θ. If φ is a cocycle, then θ(φ) is a cocycle and the formula above implies that e∗1 θ(φ) + e∗2 θ(φ) = e∗12 θ(φ) + δµ(φ). In particular we see that the diﬀerence between the cocycles is given by a canonical coboundary, expressed explicitly as a function of φ. This will be very important when we make our identifications in twisted K–theory. We will verify and apply this formula in low degree, which is our main interest here. Proposition 4.3 Let φ be an element in C 3 (G, U (1)). Then δµ(φ) + µδ(φ) = e∗1 θ(φ) + e∗2 θ(φ) − e∗12 θ(φ). Proof.

This can be proved by an explicit calculation.

δµ(φ)(a, b, u1 , u2 ) = − φ(a1 , b1 , u2 ) + φ(a, b, u1 u2 )

− φ(a, b, u1 ) + φ(a1 , u2 , b2 ) − φ(a, u1 u2 , b2 ) + φ(a, u1 , b1 )

− φ(u2 , a2 , b2 ) + φ(u1 u2 , a2 , b2 ) − φ(u1 , a1 , b1 ), µδ(φ)(a, b, u1 , u2 ) = φ(b, u1 , u2 ) − φ(u1 , b1 , u2 ) + φ(u1 , u2 , b2 )

+ φ(a1 , b1 , u2 ) − φ(a1 , u2 , b2 ) + φ(u2 , a2 , b2 ) − φ(ab, u1 , u2 )

+ φ(au1 , b1 , u2 ) − φ(au1 , u2 , b2 ) − φ(u1 a1 , b1 , u2 ) + φ(u1 a1 , u2 , b2 ) − φ(u1 u2 , a2 , b2 ) + φ(a, bu1 , u2 ) − φ(a, u1 b1 , u2 ) + φ(a, u1 u2 , b2 )

+ φ(u1 , a1 b1 , u2 ) − φ(u1 , a1 u2 , b2 ) + φ(u1 , u2 a2 , b2 ) − φ(a, b, u1 u2 )

+ φ(a, u1 , b1 u2 ) − φ(a, u1 , u2 b2 ) − φ(u1 , a1 , b1 u2 ) + φ(u1 , a1 , u2 b2 ) − φ(u1 , u2 , a2 b2 ) + φ(a, b, u1 ) − φ(a, u1 , b1 ) + φ(a, u1 , u2 )

+ φ(u1 , a1 , b1 ) − φ(u1 , a1 , u2 ) + φ(u1 , u2 , a2 ).

We now add these two expressions. Using the identities au1 = u1 a1 , bu1 = u1 b1 , a1 u2 = u2 a2 , and b1 u2 = u2 b2 , cancelling and collecting terms, yields the expression [µδ + δµ](φ)(a, b, u1 , u2 ) = φ(a, u1 , u2 ) − φ(u1 , a1 , u2 ) + φ(u1 , u2 , a2 ) + φ(b, u1 , u2 ) − φ(u1 , b1 , u2 ) + φ(u1 , u2 , b2 )

− φ(ab, u1 , u2 ) + φ(u1 , a1 b1 , u2 ) − φ(u1 , u2 , a2 b2 ).

A stringy product on twisted orbifold K-theory

This expression is exactly e∗1 θ + e∗2 θ − e∗12 θ applied to φ, hence the proof is complete. ✷ The inverse transgression formula implies that a 2-gerbe φ on an orbifold groupoid G induces a 1-gerbe θ(φ) on the associated inertia groupoid ∧G. Furthermore, two equivalent (isomorphic) 2-gerbes induce equivalent (isomorphic) 1-gerbes on the inertia groupoid. Recall that there is an embedding e : G → ∧G by e(x) = 1x where 1x is the identity arrow. The image e(G) is often referred as non-twisted sector and other components of ∧G are called twisted sectors. Corollary 4.4 If φ is a cocycle, then e∗ θ(φ) is a coboundary. Proof. e∗ θ(φ)(u, v) = θ(1, u, v). Using the embedding λ : G → G2 given by x → (ix , ix ), we can pull back e∗1 θ(φ) + e∗2 θ(φ) − e∗12 θ(φ) in cohomology. Note that λ∗ e∗1 θ(φ) = λ∗ e∗2 θ(φ) = λ∗ e∗12 θ(φ) = e∗ θ(φ). Therefore, e∗ θ(φ) = δe∗ µ(φ) is a coboundary. This implies that restricted to the untwisted sector, our cocycle θ(φ) gives rise to a trivial cohomology class. ✷

The Inverse Transgression in the Case of a Finite Group

In the case when the original orbifold is [∗/G] where G is a finite group, the inverse transgression has a classical interpretation in terms of shuﬄe products. Recall that ∧[∗/G] can be thought of in terms of G with the conjugation action; this breaks up into a disjoint union of orbits of the form G/ZG (g), indexed by conjugacy classes. Each of these is in turn equivalent to [∗/ZG (g)]; so we have a Morita equivalence ∧[∗/G] ∼ = &(g) [∗/ZG (g)]. Hence we can restrict our attention to these components; in particular we would like to describe each θg : C k (G, U (1)) → C k−1 (ZG (g), U (1)). Now for a finite group G, the cochain complex C ∗ (G, U (1)) is in fact equal to HomG (B∗ (G), U (1)), where B∗ (G) is the bar resolution for G (see [8], page 19). There is natural homomorphism ρg : ZG (g) × Z → G given by ρg (x, ti ) = xg i , where t is a generator for Z; the fact that ZG (g) centralizes g is crucial here. This homomorphism induces a map in integral homology H∗ (ZG (g), Z) ⊗ H∗ (Z, Z) ∼ = H∗ (ZG (g) × Z, Z) → H∗ (G, Z).

Alejandro Adem, Yongbin Ruan, and Bin Zhang

Classically it is known that multiplication is induced by the shuﬄe product on the chain groups (see [8], page 117–118); i.e. there is a chain map B∗ (ZG (g)) ⊗ B∗ (Z) → B∗ (G) which will induce ρg∗ in homology. Let t denote a generator of the cyclic group Z. The shuﬄe product we are interested in is Bk (ZG (g)) ⊗ B1 (Z) → Bk+1 (G) given by ! [g1 |g2 | . . . |gk ] ⋆ [ti ] = σ[g1 |g2 | . . . |gk |gk+1 ] σ

where gk+1 = g i , σ ranges over all (k, 1)–shuffles and σ[g1 |g2 | . . . |gk+1 ] = (−1)sign(σ) [gσ(1) |gσ(2) | . . . |gσ(k+1) ]. A (k, 1)–shuffle is an element σ ∈ Sk+1 such that σ(i) < σ(j) for 1 ≤ i < j ≤ k. These are precisely the cycles: 1, (k k + 1), (k − 1 k k + 1), (k − 2 k − 1 k k + 1), . . . , (1 2 3 . . . k k + 1). Note that there are k+1 of them. This can be dualized, using U (1) coefficients, but for cohomology purposes it’s easier to use integral coefficients. Given a cocycle φ ∈ C k+1 (G, Z), we see that θg (φ) ∈ C k (ZG (g), Z) can be defined as θg (φ)([g1 |g2 | . . . |gk ]) = φ([g1 |g2 | . . . |gk ] ⋆ [g]) where g1 , g2 , . . . , gk ∈ ZG (g). As a consequence of this we see that θg∗ : H k+1 (G, Z) → H k (ZG (g), Z) is induced by the multiplication map ρ∗g : H k+1 (G, Z) → H k (ZG (g), Z) ⊗ H 1 (Z, Z). To be precise, if ν is the natural generator for H 1 (Z, Z), then ∗ ρ∗g (u) = resG ZG (g) (u) ⊗ 1 + θg (u) ⊗ ν.

This discussion clarifies the geometric arguments in [12], and will also allow us to do some computations in cohomology. Example 5.1 Finite group cohomology is diﬃcult to compute, especially over the integers. The simple examples such as cyclic and quaternion groups are not so interesting in this context, as their odd dimensional cohomology (with trivial Z coeﬃcients) is zero. The first interesting example is G = (Z/2)2 . In this case H ∗ (G, F2 ) is a polynomial algebra on two degree one generators x, y. In degree four there is a natural basis given by x4 , y 4 , x3 y, x2 y 2 , xy 3 . For an elementary abelian 2–group, the mod 2 reduction map for k > 0 is a

A stringy product on twisted orbifold K-theory

monomorphism H k (G, Z) → H k (G, F2 ), and so we can understand it as the kernel of the Steenrod operation Sq 1 : H k (G, F2 ) → H k+1 (G, F2 ). Hence we see that H 4 (G, Z) can be identified with the subspace generated by x4 , y 4 and x2 y 2 . These are all squares, hence when we apply θg∗ : H 4 (G, Z) → H 3 (G, Z) for any g ∈ G, the result will always be zero. Next we consider G = (Z/2)3 ; in this case H ∗ (G, F2 ) is a polynomial algebra on three degree one generators x, y, z. In this case we have an element α = Sq 1 (xyz) = x2 yz + xy 2 z + xyz 2 which represents a non-square element in H 4 (G, Z). By analyzing the multiplication map in cohomology we obtain the following. Lemma 5.2 Let g = xa y b z c be an element in G = (Z/2)3 , where we are writing it in terms of the standard basis (identified with its dual by abuse of notation). Then θg∗ (α) = a(y 2 z + z 2 y) + b(x2 z + xz 2 ) + c(x2 y + xy 2 ) and so it is non–zero on every component except the one corresponding to the trivial element in G. Now for an abelian group, the multiplicative formula implies that for all ∗ in cohomology, or up to coboundaries. In particular g, h ∈ G, θg∗ + θh∗ = θgh this implies that the correspondence g #→ θg (α) defines a homomorphism G → H 3 (G, Z) of elementary abelian groups, in this case an isomorphism.

The Twisted Pontryagin Product for Finite Groups

Let G denote a finite group, and consider the orbifold defined by its action on a point. Then the inertia groupoid ∧G can be identified with the groupoid determined by the conjugation action of G on itself. In this case the untwisted orbifold K–theory is simply KG (G), which is additively isomorphic to ! (g) R(ZG (g)), where as before ZG (g)) denotes the centralizer of g in G, and the sum is taken over conjugacy classes. This group can be endowed with a certain product, known as the Pontryagin product, defined as follows. An equivariant vector bundle over G (with the conjugation action) can be thought of as a collection of finite dimensional vector spaces Vg with a G-module structure on ⊕g∈G Vg such that gVh = Vghg−1 . The product of two of these bundles is now defined as: " Vg1 ⊗ Wg2 . (V ⋆ W )g = {g1 ,g2 ∈G, g1 g2 =g}

Alejandro Adem, Yongbin Ruan, and Bin Zhang

This formula has been referred to as the holomorphic orbifold model in the physics literature [11]. This product admits an alternate description, which will admit a geometric generalization. In this case we can identify G2 with the orbifold defined by considering G × G with the conjugation action on both coordinates. Our maps e1 , e2 and e12 correspond to (g, h) "→ g, (g, h) "→ h, (g, h) "→ gh respectively, which are G–equivariant with respect to the conjugation action. Then, if α, β are elements in KG (G), the Pontryagin product can also be defined as α ⋆ β = e12∗ (e∗1 (α) · e∗2 (β)). We propose to extend this definition to twisted K–theory, with certain conditions on the twisting cocycle. Note that given a 2–cocycle τ = θ(φ) in the image of the inverse transgression, then by our multiplicative formula we have e∗1 τ + e∗2 τ = e∗12 τ + δµ(φ). Definition 6.1 Let τ be a U (1) valued 2–cocycle for the orbifold defined by the conjugation action of a finite group G on itself which is in the image of the inverse transgression. The Pontryagin product on τ KG (G) is defined by the following formula: if α, β ∈ τ KG (G), then α ⋆ β = e12∗ (e∗1 α · e∗2 β). Note that if τ = θ(φ) then by our multiplicative formula we have e∗1 τ + e∗2 τ = e∗12 τ + δµ(φ) and so the product e∗1 (α) · e∗2 (β) lies in ∗ e∗ 1 τ +e2 τ

KG (G) =

e∗ 12 τ +δµ(φ)

KG (G) ∼ =

e∗ 12 τ

KG (G).

Now applying e12∗ , this is mapped to τ KG (G); and so we have a product on our twisted K–theory; it is elementary to verify that this defines an associative product. Our approach will define a twisted Pontryagin product for any cocycle in the image of the inverse transgression. This cocycle could very well be a coboundary; but that does not necessarily imply that the corresponding product on KG (G) is the standard Pontryagin product. It is also clear that we may choose twistings which give rise to a twisted K–theory without any product.4 4

For much more on Pontryagin products please consult [14] and its sequels.

A stringy product on twisted orbifold K-theory

Example 6.2 If G is an abelian group, then what we are doing is using the identification θ(φ)g + θ(φ)h = θ(φ)gh to define a product on the abelian group X(G) =

θ(φ)g

R(G)

g∈G

via the pairing θ(φ)g

R(G) ⊗

θ(φ)h

R(G) →

θ(φ)gh

R(G).

In the case described in 5.1 for G = (Z/2)3 and a cocycle φ representing the cohomology class xy 2 z + xyz 2 + x2 yz, θ(α) establishes a group homomorphism G → H 3 (G, Z), with image the subgroup generated by xy 2 + x2 y, xz 2 + x2 z and yz 2 + y 2 z. In this case we see that for g ̸= 1, θ(φ)g R(G) has rank equal to two, and so X(G) is of rank equal to twenty-two, with the twisted Pontryagin product described above. This can be made explicit. The case of the Pontryagin product should be considered as motivation for the case of orbifold groupoids. As long as we twist with a cocycle in the image of the inverse transgression, the levels will match up as required. Hence the main diﬃculty is geometric–as we shall see in the next section, there is an obstruction bundle which plays an important role.

Twisted K–theory of Orbifolds

During the course of our investigation of possible stringy products on the twisted K-theory of orbifolds, we came to realize that one needs to use the very same information required to construct the Chen-Ruan cohomology of orbifolds ([10]). We first briefly recall the situation for orbifold cohomology, and then proceed to develop the tools necessary to deal with twisted K–theory. For a very interesting but diﬀerent approach we refer the reader to [15]. ∗ (G, C) is additively the same as H ∗ (∧G, C); what First we recall that HCR is interesting is the ring structure. Recall that there are three evaluation maps e1 , e2 , e12 : G2 → ∧G. A naive definition of the stringy product for orbifold cohomology (which would generalize the Pontryagin product) would be α ⋆ β = (e12 )∗ (e∗1 α ∪ e∗2 β). However, one soon discovers that this product is not associative due to the fact that e1 , e12 are not transverse in general. In fact, the correction term is precisely the obstruction bundle to the transversality of e1 and e12 . A natural idea was to modify the definition of the product via α ⋆ β = (e12 )∗ (e∗1 α ∪ e∗2 β ∪ e(EG2 ))

Alejandro Adem, Yongbin Ruan, and Bin Zhang

where we need to construct a bundle EG2 in a fashion that is consistent with the obstruction to transversality of e1 , e12 , and e(EG2 ) denotes its Euler class. A key observation is that the obstruction bundle in the construction of the Chen-Ruan product provides such a choice. We will adapt this same idea to K–theory. Throughout this section, we assume that G is a compact, almost complex orbifold groupoid. Then Gk naturally inherits an almost complex structure such that the evaluation map ei1 ,··· ,il : Gk → Gl is an almost complex quasiembedding. G2 can be identified with the space of constant morphisms from an orbifold sphere with three marked point to G; we now make this identification precise. Consider an orbifold Riemann sphere with three orbifold points (S2 , (x1 , x2 , x3 ), (m1 , m2 , m3 )); in this context we simply denote it by S2 . Suppose that f is a constant morphism from S2 to G. Here, the term constant means that the induced map |f | : S2 → |G| is constant. Let y = im(|f|) and Uy /Gy be an orbifold chart at y. By the results in [1], f is classified by the conjugacy class of a homomorphism πf : π1orb (S2 ) → Gy . Recall that π1orb (S2 ) = {λ1 , λ2 , λ3 ; λki i = 1, λ1 λ2 λ3 = 1}, where λi is represented by a loop around the marked point xi . πf is uniquely determined by a pair of elements (g1 , g2 ) with gi ∈ Gy where gi = πf (λi ); on the other hand, (g1 , g2 ) ∈ G20 . It is clear that the same method can be used to identify the moduli space of constant morphisms from an orbifold sphere with k + 1 marked points to G with the groupoid of k–multisectors Gk . For any f ∈ G2 viewed as a constant morphism, we can form an elliptic complex ∂¯f : Ω0 (f ∗ T G) → Ω0,1 (f ∗ T G),

where f ∗ T G is a complex vector bundle by our assumption. This defines an orbibundle E with EG2 |f = Coker ∂¯f . We now examine EG2 in more detail. Let g1 , g2 ∈ G20 ; by the definition, g1 , g2 ∈ Gx for x = s(gi ) = t(gi ). Let N be the subgroup of Gx generated by g1 , g2 . By Lemma 4.5 in [1], N depends only on the component of G2 . Let e : G2 → G be an evaluation map. Clearly N acts on e∗ T G while fixing T G2 . There is an obvious surjective homomorphism π : π1orb (S2 ) → N and Ker π ˜ is the orbifold universal is therefore a subgroup of finite index. Suppose that Σ 2 ˜ is smooth. Let Σ = Σ/Ker ˜ π; then Σ is cover of S . By [1] (see Chapter II), Σ 2 compact and we have a quotient map Σ → S = Σ/N . Since N contains the relation gimi = 1, Σ is smooth. It is clear that f lifts to an ordinary constant map f˜ : Σ → Uy ; hence ∗ ˜ f T G = Ty G is a trivial bundle over Σ. Then we can lift the elliptic complex

A stringy product on twisted orbifold K-theory

to Σ ∂¯Σ : Ω0 (f˜∗ T G) → Ω0,1 (f˜∗ T G). The original elliptic complex is just the N -invariant part of the current one. However, Ker(∂¯Σ ) = Ty G and Coker (∂¯Σ ) = H0,1 (Σ) ⊗ Ty G. Now we vary y in a component G2(γ) to obtain e∗(γ) T G for the evaluation map e(γ) : G2(γ) → G and H 0,1 (Σ) ⊗ e∗(γ) T G. N acts on both. It is clear that (e∗(γ) T G)N = T G(γ) as we claim. The obstruction bundle E(γ) we want is the invariant part of H 0,1 (Σ)⊗C e∗(γ) T G, i.e., E(γ) = (H 0,1 (Σ)⊗C e∗(γ) T G)N . We remark that E(γ) can have different dimensions at the different components of G2 . We can obviously use the same method to construct a bundle EGk over Gk whose fiber is the cokernel of ∂¯f for f ∈ Gk . Consider the evaluation maps e1 , e2 , e12 : G2 → ∧G. Suppose that the local chart of G is U/G. Then, the local chart of ∧G is (%g∈G U g )/G. The local chart of G2 is (%g1 ,g2 ∈G U g1 ∩U g2 )/G. The e1 , e2 , e12 are quasi-embeddings and hence G2 can be thought of as a quasi–suborbifold of ∧G via three different quasiembeddings, denoted by e1∗ G2 , e2∗ G2 , e12∗ G2 . It is clear that G3 = G2 ×e12 ,e1 G2 and so G3 can be viewed as the intersection of the quasi–suborbifolds e12∗ G2 and e1∗ G2 : π G3 →2 G2 π1 ↓ ↓ e1 e12 2 G → ∧G The problem is that e12∗ G2 , e1∗ G2 are not intersecting transversely in general. Let ν = (e12 π1 )∗ T ∧ G/π1∗ T G2 + π2∗ T G2 where πi : G3 → G2 are the natural projection maps. This is the so-called excess bundle of the intersection. A crucial ingredient in the proof of associativity for the Chen-Ruan product is the bundle formula: Theorem 7.1 E G3 ∼ = π1∗ EG2 ⊕ π2∗ EG2 ⊕ ν. The proof is given in [10] by gluing arguments. We shall call this the obstruction bundle equation. These bundles will play a key role in our definition of the product. It ties in to the transversality question mentioned before via the following lemma, which is an orbifold analogue of the clean intersection formula first described by Quillen [26]:

Alejandro Adem, Yongbin Ruan, and Bin Zhang

Lemma 7.2 Suppose that i1 : H1 → G and i2 : H2 → G are quasi–suborbifolds of G forming a clean intersection H3 . Then, if u ∈ α K(H1 ) we have i∗2 i1∗ u = π2∗ [π1∗ u · eK (ν)] where π1 : H3 → H1 and π2 : H3 → H2 are the natural projections, ν is the excess bundle of the intersection, and eK (ν) ∈ K(H3 ) denotes its Euler class. Note that this lemma will allow us to connect facts about the geometry of the quasi–suborbifolds G2 and G3 in ∧G with products, and the obstruction bundle E plays a key role here. In fact the only information we need about E is the obstruction bundle equation mentioned above. Definition 7.3 Suppose that G is an almost complex orbifold. Let φ denote a U (1)–valued 3–cocycle for G, and θ(φ) its inverse transgression, i.e. a U (1)– valued 2–cocycle for ∧G. For α, β ∈ θ(φ) K(∧G), we define α ⋆ β = e12∗ (e∗1 α · e∗2 β · eK (EG2 )). Our goal is to show that this defines an associative product. However, we want to do this using very general properties of our construction, so that it will be a natural extension of the Chen–Ruan product. We will abbreviate E2 = EG2 , E3 = EG3 .5 As with the twisted Pontryagin product, we must explain in what sense this defines a product. Given that e1 , e2 , e12 : G2 → ∧G, then e∗i (α) ∈ ei ∗(θ(φ)) K(G2 ) for i = 1, 2. Hence e∗1 (α) · e∗2 (β) · eK (EG2 ) ∈

∗ e∗ 1 θ(φ)+e2 θ(φ)

K(G2 ).

Now here we must be careful. We use the fact that the twisting cocycle is in fact equal to e∗12 θ(φ) + δµ(φ). We have a canonical isomorphism e∗ 12 θ(φ)+δµ(φ)

K(G2 ) ∼ =

e∗ 12 θ(φ)

K(G2 ).

Next we apply the push forward e12∗ to our expression, which now lands in θ(φ) K(∧G). Note that the product is clearly commutative; it is associativity that requires a proof. 5

We have chosen to work only with even dimensional K–theory, but this can be readily extended to odd dimensions; the signs work out appropriately after a tedious computation.

A stringy product on twisted orbifold K-theory

Theorem 7.4 This product is associative: (α ⋆ β) ⋆ γ = α ⋆ (β ⋆ γ). Proof. We pull back the quasi-embeddings e1 , e2 and e12 to G3 as follows: let e˜1 = e1 π1 ; e˜2 = e2 π1 ; e˜3 = e2 π2 and e˜123 = e12 π2 . We will also make use of an operator defined for triples, namely let I3 : G3 → G3 be defined by (g1 , g2 , g3 ) "→ (g2 , g3 , g3−1 g2−1 g1 g2 g3 ). Note that e˜i I3 is equal to e˜i+1 up to conjugation (modulo three); hence they induce the same map in K-theory. Note that eK (E3 ) is invariant under I3∗ by construction. We are now ready to start computing: (α ⋆ β) ⋆ γ = e12∗ (e∗1 (α ⋆ β) · e∗2 γ · eK (E2 ))

= e12∗ (e∗1 (e12∗ (e∗1 α · e∗2 β · eK (E2 ))) · e∗2 γ · eK (E2 )).

We can apply the clean intersection formula to this expression, whence we obtain (α ⋆ β) ⋆ γ = e12∗ (π2∗ (π1∗ (e∗1 α · e∗2 β · eK (E2 )) · e(ν)) · e∗2 γ · eK (E2 )) = e12∗ (π2∗ (e˜1 ∗ α · e˜∗2 β · π1∗ eK (E2 ) · eK (ν)) · e∗2 γ · eK (E2 )) e∗1 α · e˜∗2 β · π1∗ eK (E2 ) · eK (ν) · π2∗ e∗2 γ · π2∗ eK (E2 )). = e12∗ π2∗ (˜ Note here that we are using the following property of the pushforward: π2∗ (x · π2∗ y) = π2∗ (x) · y. Thus we have (α ⋆ β) ⋆ γ = e˜123∗ (˜ e∗1 α · e˜∗2 β · e˜∗3 γ · π1∗ eK (E2 ) · π2∗ e(E2 ) · eK (ν)) = e˜123∗ (˜ e∗1 α · e˜∗2 β · e˜∗3 γ · eK (E3 )). Now we consider α ⋆ (β ⋆ γ) = (β ⋆ γ) ⋆ α = e˜123∗ (˜ e∗1 β · e˜∗2 γ · e˜∗3 α · eK (E3 )) e∗1 β · e˜∗2 γ · e˜∗3 α · eK (E3 )) = e˜123∗ I3∗ I3∗ (˜ = e˜123∗ (I3∗ e˜∗1 β · I3∗ e˜∗2 γ · I3∗ e˜∗3 α · I3∗ eK (E3 )) e∗2 β · e˜∗3 γ · e˜∗1 α · eK (E3 )) = e˜123∗ (˜ e∗1 α · e˜∗2 β · e˜∗3 γ · eK (E3 )) = e˜123∗ (˜ = (α ⋆ β) ⋆ γ. Hence our proof of associativity is complete.

✷

Alejandro Adem, Yongbin Ruan, and Bin Zhang

Alejandro Adem Department of Mathematics University of British Columbia Vancouver, B. C. Canada adem@math.ubc.ca

Yongbin Ruan Department of Mathematics University of Michigan Ann Arbor, MI 14627 USA ruan@umich.edu

Bin Zhang Department of Mathematics Sichuan University bzhang@math.sunysb.edu

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A stringy product on twisted orbifold K-theory

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Holonomy for Gerbes over Orbifolds,

[21] Mathai V.; Stevenson D., Chern character in twisted K-theory: equivariant and holomorphic cases, Comm. Math. Phys. 236 (2003), no. 1, 161–186 [22] Moerdijk I., Classifying toposes and foliations, Ann. Inst. Fourier (Grenoble) 41 (1991), no. 1, 189–209. [23] Moerdijk I., Proof of a Conjecture of A. Haefliger, Topology Vol.37, No.4 (1998), pp. 735–741. [24] Moerdijk I., Orbifolds as groupoids: an introduction, in Cont. Math. 310, AMS (2002), pp. 205–222. [25] Pan J.; Ruan Y.; Yin X., Gerbes and twisted orbifold quantum cohomology, preprint [26] Quillen D., Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7 1971 29–56 (1971). [27] Ruan Y., Discrete torsion and twisted orbifold cohomology, J. Symplectic Geom. 2 (2003), no. 1, 1–24.

Alejandro Adem, Yongbin Ruan, and Bin Zhang

[28] Renault J., A Groupoid Approach to C â&#x2C6;&#x2014; -Algebras, Lecture Notes in Mathematics, 793. Springer, Berlin, 1980 [29] Vafa C., Modular invariance and discrete torsion on orbifolds, Nuclear Phys. B 273 (1986), no. 3-4, 592â&#x20AC;&#x201C;606. [30] Witten E., D-Branes and K-theory, hepth/9810188.

Morfismos, Comunicaciones Estudiantiles del Departamento de Matem´ aticas del CINVESTAV, se termin´ o de imprimir en el mes de julio de 2008 en el taller de reproducci´ on del mismo departamento localizado en Av. IPN 2508, Col. San Pedro Zacatenco, M´exico, D.F. 07300. El tiraje en papel opalina importada de 36 kilogramos de 34 × 25.5 cm consta de 500 ejemplares con pasta tintoreto color verde.

Apoyo t´ecnico: Omar Hern´ andez Orozco.

Contenido Hodge structures in non-commutative geometry Maxim Kontsevich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

A stringy product on twisted orbifold K-theory Alejandro Adem, Yongbin Ruan, and Bin Zhang . . . . . . . . . . . . . . . . . . . . . . . . . 33