String Geometry Postgraduate Conference in Complex Geometry Cambridge, 2015 Johar M. Ashfaque University of Liverpool
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Aim of the Talk
To show how geometry has played a key role To highlight some of the various connections or links
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Outline
Strings Attached The Role of Geometry Calabi Yaus Orbifolds Extra Dimensions Coulomb Branch & Higgs Branch in 3D N = 4 Supersymmetric Gauge Theories
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
The Fundamental Forces of Nature
Weak Nuclear Force Strong Nuclear Force Electromagnetism Gravity
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Bosons & Fermions
Matter: Fermions Interactions: Bosons
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Standard Model of Particle Physics
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Then Why String Theory?
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Then Why String Theory?
To incorporate GRAVITY with the Standard Model gauge group SU(3) × SU(2) × U(1) | {z } | {z } Strong
| |
Electroweak
{z Rank=4 {z SO(10)
Johar M. Ashfaque
} }
Postgraduate Conference in Complex Geometry, Cambridge
Why (Bosonic) String Theory Is Not The Complete Story?
Two major setbacks
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Why (Bosonic) String Theory Is Not The Complete Story?
Two major setbacks The ground state of the spectrum always contains a tachyon. As a consequence, the vacuum is unstable.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Why (Bosonic) String Theory Is Not The Complete Story?
Two major setbacks The ground state of the spectrum always contains a tachyon. As a consequence, the vacuum is unstable. Does not contain fermions. Where is the matter?
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Z2 Graded Lie Algebra Let g be a Lie algebra. Then g decomposes as g = g0 ⊕ g1 where g0 represents even part and g1 represents the odd part. For the linear map [ , ]:g×g→g we have g0 × g0 → g0 g0 × g1 → g1 g1 × g0 → g1 g1 × g1 → g0 where it can be seen that the linear map on g0 acts as a commutator but on g1 acts as an anti-commutator. Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Why Superstrings?
Supersymmetry is the symmetry that interchanges bosons and fermions.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Superstring Theories
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Branes
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Dualities
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
T -Duality
T -duality relates a theory with a small compact dimension to a theory where that same dimension is large.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Modular Invariance
The symmetry group of the torus is SL(2, Z ) which is much bigger than the U(1) symmetry of the circle. A string model is consistent whenever all physical quantities are invariant under these symmetries. This needs to be checked by looking at the simplest quantity: The integrand of the 1-loop vacuum-to-vacuum amplitude known as the partition function.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
The Partition Function
Partition function is used to include all physical states Taking the one-loop partition function transforms the worldsheet into a torus. A torus has two non-contractible loops often referred to as the ”toroidal” and ”poloidal” directions.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
The Role of Geometry
Useful insight Model Building Low Energy Effective Models of String theory
Models with 4 flat space-time dimensions 3 generations of matter N = 1 SUSY
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Calabi-Yaus: Motivation
Superstrings conjectured to exist in 10D: M4 Ă— CY 3 (CY 3 is 3 complex dims or 6 real dims) Compactification of extra dimensions on CY manifolds is popular as it leaves some of the original SUSY unbroken
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Calabi-Yaus: Motivation
Superstrings conjectured to exist in 10D: M4 Ă— CY 3 (CY 3 is 3 complex dims or 6 real dims) Compactification of extra dimensions on CY manifolds is popular as it leaves some of the original SUSY unbroken Several other motivations for studying these: F-theory compactifications on CY 4 allow to find many classical solutions in the string theory landscape
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Calabi-Yaus: Motivation
Superstrings conjectured to exist in 10D: M4 × CY 3 (CY 3 is 3 complex dims or 6 real dims) Compactification of extra dimensions on CY manifolds is popular as it leaves some of the original SUSY unbroken Several other motivations for studying these: F-theory compactifications on CY 4 allow to find many classical solutions in the string theory landscape First attempts at obtaining standard model from string theory used the now standard compactification of E8 × E8 heterotic string theory. In such compactifications gens = 12 |χ| where χ is the Euler characteristic. For 3 generation models, χ = ±6.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Complex Manifolds (1)
Note. Essentially, holomorphic transition functions ⇒ complex manifold.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Lightening Review: Vector Bundles
A section s of a vector bundle is a map s : B → E such that πs(x) = x for all x ∈ B.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
The Various Types
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Complex Manifolds (2)
If (M, J) is an almost complex 2n-fold with N = 0 then J is called a complex structure and M a complex n-fold. This condition of integrability of J being satisfied allows M to be covered by complex coordinates.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Kähler Manifolds
Kähler manifolds are a subclass of complex manifolds and, as such, are naturally oriented. In addition to J, Kähler manifolds have a Hermitian metric g (+ associated connection) and can thus be denoted by the triplet (M, g , J).
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
de Rham Cohomology (1) A p-form ω is called closed if dω = 0. Denote the set of closed p-forms by Z p (M, R). A p-form ω is called exact if ω = dη for some (p − 1)-form η. Denote the set of exact p-forms by B p (M, R). Since d 2 = 0, exact p-forms are closed. So, the set of exact p-forms is a subset of the set of closed p-forms, that is B p (M, R) ⊂ Z p (M, R), but closed p-forms are not necessarily exact. A closed differential form ω on a manifold M is locally exact when a neighbourhood exists around each point in M in which ω = dη. (Poincaré Lemma) Any closed form on a manifold M is locally exact.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
de Rham Cohomology (2) The de Rham cohomology class of M is defined as p HdR (M, R) =
Z p (M, R) . B p (M, R)
The dimension of the de Rham cohomology is given by the p-th Betti number p bp (M) = dim HdR (M, R). Poincaré duality states H k (M) ' H n−k (M) and thus bk = bn−k for an n-dimensional manifold M. Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Chern Classes
Chern classes encode topological information about bundle.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Calabi-Yaus
CY manifold of real dimension 2m is a compact KaĚˆhler manifold (M; J; g ) with KaĚˆhler Metric has vanishing Ricci curvature First Chern class vanishes a globally defined, nowhere vanishing holomorphic m-form
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Hodge Diamond: Calabi-Yau 3-folds Interested in CY 3 Hodge numbers hp,q run over p, q = 0, ..., 3. These can’t exceed the top form on the manifold - in this case a (3, 3)-form. This gives Hodge Diamond. 1 0
0 h1,1
0 h2,1
1
0 h1,2
h1,1
0 0
1 0
0 1
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Remarks Unfixed Hodge numbers in CY 3 Hodge diamond are h1,1 , h1,2 , h2,1 , h2,2 Not Independent as h1,1 = h2,2 (Hodge Dual) h1,2 = h2,1 (Complex Conjugation)
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
K 3 Surface
K 3 surfaces are examples of Calabi-Yau two-folds. K 3 serves as the simplest non-trivial example of Calabi-Yau compactification. It has also played a crucial role in string dualities since the mid 1990s. The Hodge diamond is given by 1 0 22 0 1
1 0 1
0 20
0
1 0
1
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
The Quintic Hypersurface in CP4 We have that the total Chern class for Q is given by c(Q) = 1 + 10x 2 − 40x 3 The Euler characteristic is given by the integral over M of the top Chern class of M which in the case of the Calabi-Yau 3-fold is Z χ= c3 (M) M
The Euler characteristic for the quintic is χ(Q) = −200
h1,1 = 1 always ⇒ h2,1 = 101
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
The Role of Hodge Diamond
Euler characteristic for CY 3 χ = 2(h1,1 − h2,1 ) Interested in CY 3 for 3 generation models with χ = ±6 we can further restrict to only CY 3 with h1,1 − h2,1 = ±3 It may be tricky to compute h1,1 , h2,1 for certain CY 3. However, there are many ways of computing χ. Often it’s easier to find χ and one of the Hodge numbers. This then fixes the other and all topological info is fixed.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Mirror Symmetry
There is a fascinating symmetry of CY manifolds, called mirror symmetry, that can be seen on Hodge Diamond. Given a CY manifold M, ∃ another CY manifold M 0 of same dimension h(p,q) (M) = h(3−p,q) (M 0 ) This mirror symmetry exchanges h1,1 and h2,1 on Hodge diamond Although two CY manifolds M, M 0 may look very different geometrically, string theory compactification on these manifolds leads to identical effective field theories Means that CY manifolds exist in mirror pairs (M, M 0 )
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
The Diophantine Equation
n+r +1−
r X
να
=0
α=1
By requiring that να ≥ 2, for any fixed value of n there is a finite number of solutions. This can be immediately seen as r X
να = 1 + n + r ≥ 2r ⇒ 1 + n ≥ r
α=1
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Final Remarks: Calabi-Yau 3-Manifolds For r = 1, ν1 = 5 and N = r + n = 4 we have CP4 [5],
χ(CP4 [5]) = −200
For r = 2, ν1 + ν2 = 6 and N = r + n = 5 we find CP5 [2, 4],
χ(CP5 [2, 4]) = −176
CP5 [3, 3],
χ(CP5 [3, 3]) = −144
For r = 3, ν1 + ν2 + ν3 = 7 and N = r + n = 6 CP6 [2, 2, 3],
χ(CP6 [2, 2, 3]) = −144
For r = 4, ν1 + ν2 + ν3 + ν4 = 8 and N = r + n = 7 CP7 [2, 2, 2, 2], Johar M. Ashfaque
χ(CP7 [2, 2, 2, 2]) = −128 Postgraduate Conference in Complex Geometry, Cambridge
Summary
Connect String Theory To Low Energy Effective Field Theory Internal geometry determines 4D theory Examples Heterotic Strings on CY 3 F-theory on elliptic CY 4
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
A Few Words On Orbifolds
Orbifolds are simply quotient spaces of a manifold modulo some discrete group. If G is freely acting, M having no fixed points under G action then the orbifold is smooth. However, if G was to have fixed points then the orbifold has singularities. One-dimensional orbifolds are very simple. There are only two of them, the circle and the interval.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Example: The Real Line
Consider the real line R. It has a Z2 symmetry. This symmetry has one fixed point (a singularity) at x = 0 The orbifold R Z2 is a half line. The real line has another infinite symmetry group namely the translations x → x + 2πλ The resulting orbifold is smooth which is a circle of radius λ.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Strings Once More
Heterotic strings are hybrid strings with either
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Strings Once More
Heterotic strings are hybrid strings with either left-moving sector being supersymmetric and right-moving sector being bosonic or left-moving sector being bosonic and right-moving sector being supersymmetric.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Strings Once More
Heterotic strings are hybrid strings with either left-moving sector being supersymmetric and right-moving sector being bosonic or left-moving sector being bosonic and right-moving sector being supersymmetric. There are two heterotic string theories, one associated to the gauge group E 8 Ă— E8 and the other to SO(32)
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
The Free Fermionic Construction
A general boundary condition basis vector is of the form n o 1,...,5 1,2,3 1,...,8 α = ψ 1,2 , χi , y i , ω i |y i , ω i , ψ ,η ,φ where i = 1, ..., 6 ψ φ
1,...,5
1,...,8
- SO(10) gauge group - SO(16) gauge group
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
The Very Simple Rules
The ABK Rules [Antoniadis, Bachas, Kounnas, 1987]
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
An Example: The NAHE Set
The NAHE set is the set of basis vectors B = {1, S, b1 , b2 , b3 } where 1 = {ψµ1,2 , χ1,...,6 , y 1,...,6 , ω 1,...,6 |ȳ 1,...,6 , ω̄ 1,...,6 , ψ̄ 1,...,5 , η̄ 1,2,3 , φ̄1,...,8 }, S = {ψµ1,2 , χ1,...,6 }, b1 = {ψµ1,2 , χ1,2 , y 3,...,6 |ȳ 3,...,6 , ψ̄ 1,...,5 , η̄ 1 }, b2 = {ψµ1,2 , χ3,4 , y 1,2 , ω 5,6 |ȳ 1,2 , ω̄ 5,6 , ψ̄ 1,...,5 , η̄ 2 }, b3 = {ψµ1,2 , χ5,6 , ω 1,...,4 |ω̄ 1,...,4 , ψ̄ 1,...,5 , η̄ 3 }.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
The NAHE: The Gauge Group
SO(44)
SO(10) × E8 × SO(6)3 with N=4
N=2
N=1
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
The Various SO(10) Breakings
α+β
SO(10)
/ SU(5) × U(1)
α
SO(6) × SO(4)
β
SU(3)C × U(1)C × SU(2)L × U(1)L SO(10)
α+β+γ
SU(3)C × U(1)C × SU(2)L × SU(2)R
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Extra Dimensions
In models with extra dimensions the usual (3 + 1)-dimensional space-time x Âľ = (x 0 , x 1 , x 2 , x 3 ) is extended to include additional spatial dimensions parametrized by coordinates x 4 , x 5 , ..., x 3+N where N is the number of extra dimensions. String theory arguments would suggest that N can be as large as 6 or 7. Depending on the type of metric in the bulk, ED models fall into one of the following two categories: Flat, also known as universal ED models (UED). Warped ED models.
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
UED Models The metric on the extra dimensions is chosen to be flat. However, to implement the chiral fermions of the SM in UED models one must use an orbifold S 1 /Z2 . The size of the extra dimension is simply parametrized by the radius of the circle R.
In the case of N = 2, one of the many is known as the chiral square corresponding to T 2 /Z4 . The two extra dimensions have equal size and the boundary conditions are such that adjacent sides of the chiral square are identified. Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Supersymmetric Gauge Theories Supersymmetric Gauge Theories in 3D N = 4 are subject to a strange duality: Mirror Symmetry 3D mirror symmetry exchanges Coulomb branch and Higgs branch of two dual theories.
Mirror Symmetry
←−−−−−−−−→ Coulomb branch: moduli space parametrised by scalars in the V-plet Higgs branch: moduli space parametrised by scalars in the H-plet Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
The Idea Study moduli space of instantons ⇒ calculate Hilbert Series for Higgs Branch. What is the Hilbert series? It is the partition function that counts chiral gauge invariant operators Why Bother? The chiral gauge invariant operators parametrize the moduli space Hilbert Series encodes all the information: dimension of the moduli space, generators and relations
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
Example: Hilbert Series for C2 /Z2
C2 with action of Z2 : (z1 , z2 ) ↔ (−z1 , −z2 ) Holomorphic functions invariant under this action: z12 , z22 , z1 z2 , z14 , ... All polynomials constructed from 3 generators subject to 1 relation X = z12 , Y = z22 , Z = z1 z2 XY
Johar M. Ashfaque
= Z2
Postgraduate Conference in Complex Geometry, Cambridge
Example Contd.: Hilbert Series for C2 /Z2 Isometry group of C2 = U(2) Cartan Subalgebra: U(1)2 Choose counters t1 is the U(1) charge of z1 t2 is the U(1) charge of z2 2
HS(t1 , t2 , ; C ) =
1+t12 +t22 +t1 t2 +...
=
∞ X
t1i t2j =
i,j=0
Y i
1 1 − ti
Unrefine HS(t) =
∞ X
t i+j = 1 + 3t 2 + 5t 4 + ... =
i,j,...
X
(2k + 1)t 2k
k=0
Dimension of the moduli space = pole of the unrefined Hilbert series Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge
THANK YOU!!!
Johar M. Ashfaque
Postgraduate Conference in Complex Geometry, Cambridge