PCCG- JM ASHFAQUE

String GeometryPostgraduate 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


Outline

Strings Attached

The Role of Geometry

Calabi Yaus

Orbifolds

Extra Dimensions

Coulomb Branch & Higgs Branch in 3D N = 4Supersymmetric Gauge Theories


The Fundamental Forces of Nature

Weak Nuclear Force

Strong Nuclear Force

Electromagnetism

Gravity


Bosons & Fermions

Matter: Fermions

Interactions: Bosons


Standard Model of Particle Physics


Then Why String Theory?


Then Why String Theory?

To incorporate GRAVITY with the Standard Model gaugegroup SU(3)︸︷︷︸

Strong

×SU(2)× U(1)︸︷︷︸Electroweak︸︷︷︸

Rank=4︸︷︷︸SO(10)


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?



Two major setbacks





Two major setbacks




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 acommutator but on g1 acts as an anti-commutator.


Why Superstrings?

Supersymmetry is the symmetry that interchanges bosons andfermions.


Superstring Theories


Branes


Dualities


T -Duality

T -duality relates a theory with a small compact dimension to atheory where that same dimension is large.


Modular Invariance

The symmetry group of the torus is SL(2,Z ) which is much biggerthan the U(1) symmetry of the circle.

A string model is consistent whenever all physical quantities areinvariant under these symmetries.

This needs to be checked by looking at the simplest quantity: Theintegrand of the 1-loop vacuum-to-vacuum amplitude known asthe partition function.


The Partition Function

Partition function is used to include all physical states

Taking the one-loop partition function transforms theworldsheet into a torus.

A torus has two non-contractible loops often referred toas the ”toroidal” and ”poloidal” directions.


The Role of Geometry

Useful insight

Model Building

Low Energy Effective Models of String theory

Models with

4 flat space-time dimensions3 generations of matterN = 1 SUSY


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 ispopular as it leaves some of the original SUSY unbroken

Several other motivations for studying these: F-theorycompactifications on CY 4 allow to find many classicalsolutions in the string theory landscape

First attempts at obtaining standard model from string theoryused the now standard compactification of E8 × E8 heteroticstring theory. In such compactifications gens = 1

2 |χ| where χis the Euler characteristic. For 3 generation models, χ = ±6.
















Complex Manifolds (1)

Note. Essentially, holomorphic transition functions ⇒ complexmanifold.


Lightening Review: Vector Bundles

A section s of a vector bundle is a map s : B → E such thatπs(x) = x for all x ∈ B.


The Various Types


Complex Manifolds (2)

If (M, J) is an almost complex 2n-fold with N = 0 then J iscalled a complex structure and M a complex n-fold.

This condition of integrability of J being satisfied allows M tobe covered by complex coordinates.


Kahler Manifolds

Kahler manifolds are a subclass of complex manifolds and, as such,are naturally oriented.In addition to J, Kahler manifolds have a Hermitian metric g (+associated connection) and can thus be denoted by the triplet(M, g , J).


de Rham Cohomology (1)

A p-form ω is called closed if dω = 0.

Denote the set of closed p-forms by Zp(M,R). A p-form ω iscalled exact if ω = dη for some (p − 1)-form η.

Denote the set of exact p-forms by Bp(M,R).

Since d2 = 0, exact p-forms are closed. So, the set of exactp-forms is a subset of the set of closed p-forms, that isBp(M,R) ⊂ Zp(M,R), but closed p-forms are not necessarilyexact.

A closed differential form ω on a manifold M is locally exact whena neighbourhood exists around each point in M in which ω = dη.

(Poincare Lemma) Any closed form on a manifoldM is locallyexact.


de Rham Cohomology (2)

The de Rham cohomology class of M is defined as

HpdR(M,R) =

Zp(M,R)

Bp(M,R).

The dimension of the de Rham cohomology is given by the p-thBetti number

bp(M) = dimHpdR(M,R).

Poincare duality states

Hk(M) ' Hn−k(M)

and thusbk = bn−k

for an n-dimensional manifold M.


Chern Classes

Chern classes encode topological information about bundle.


Calabi-Yaus

CY manifold of real dimension 2m is a compact Kahler manifold(M; J; g) with

Kahler Metric has vanishing Ricci curvature

First Chern class vanishes

a globally defined, nowhere vanishing holomorphic m-form


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

0 h1,1 0

1 h2,1 h1,2 1

0 h1,1 0

0 0

1


Remarks

Unfixed Hodge numbers in CY 3 Hodge diamond areh1,1, h1,2, h2,1, h2,2

Not Independent as h1,1 = h2,2 (Hodge Dual)h1,2 = h2,1 (Complex Conjugation)


K3 Surface

K3 surfaces are examples of Calabi-Yau two-folds.K3 serves as the simplest non-trivial example of Calabi-Yaucompactification. It has also played a crucial role in string dualitiessince the mid 1990s.The Hodge diamond is given by

1 10 0 0

22 1 20 10 0 01 1


The Quintic Hypersurface in CP4

We have that the total Chern class for Q is given by

c(Q) = 1 + 10x2 − 40x3

The Euler characteristic is given by the integral over M of the topChern class of M which in the case of the Calabi-Yau 3-fold is

χ =

∫M

c3(M)

The Euler characteristic for the quintic is

χ(Q) = −200

h1,1 = 1 always ⇒ h2,1 = 101


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 wecan 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’seasier to find χ and one of the Hodge numbers. This thenfixes the other and all topological info is fixed.


Mirror Symmetry

There is a fascinating symmetry of CY manifolds, calledmirror symmetry, that can be seen on Hodge Diamond.

Given a CY manifold M, ∃ another CY manifold M ′ of samedimension h(p,q)(M) = h(3−p,q)(M ′)

This mirror symmetry exchanges h1,1 and h2,1 on Hodgediamond

Although two CY manifolds M, M ′ may look very differentgeometrically, string theory compactification on thesemanifolds leads to identical effective field theories

Means that CY manifolds exist in mirror pairs (M,M ′)


The Diophantine Equation

(n + r + 1−

r∑α=1

να

)= 0

By requiring that να ≥ 2, for any fixed value of n there is a finitenumber of solutions. This can be immediately seen as

r∑α=1

να = 1 + n + r ≥ 2r ⇒ 1 + n ≥ r


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], χ(CP7[2, 2, 2, 2]) = −128


Summary

Connect String Theory To Low Energy Effective Field Theory

Internal geometry determines 4D theory

Examples

Heterotic Strings on CY 3F-theory on elliptic CY 4


A Few Words On Orbifolds

Orbifolds are simply quotient spaces of a manifold modulo somediscrete group.

If G is freely acting, M having no fixed points under G action thenthe orbifold is smooth.

However, if G was to have fixed points then the orbifold hassingularities.

One-dimensional orbifolds are very simple. There are only two ofthem, the circle and the interval.


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 orbifoldRZ2

is a half line.

The real line has another infinite symmetry group namely thetranslations

x → x + 2πλ

The resulting orbifold is smooth which is a circle of radius λ.


Strings Once More

Heterotic strings are hybrid strings with either

left-moving sector being supersymmetric and right-movingsector being bosonic or

left-moving sector being bosonic and right-moving sectorbeing supersymmetric.

There are two heterotic string theories, one associated to thegauge group

E8 × E8

and the other toSO(32)


Strings Once More





E8 × E8



Strings Once More





E8 × E8



The Free Fermionic Construction

A general boundary condition basis vector is of the form

α ={ψ1,2, χi , y i , ωi |y i , ωi , ψ

1,...,5, η1,2,3, φ

1,...,8}

where i = 1, ..., 6

ψ1,...,5

- SO(10) gauge group

φ1,...,8

- SO(16) gauge group


The Very Simple Rules

The ABK Rules[Antoniadis,Bachas,Kounnas, 1987]


An Example: The NAHE Set

The NAHE set is the set of basis vectors

B = {1,S,b1,b2,b3}

where

1 = {ψ1,2µ , χ1,...,6, y1,...,6, ω1,...,6|y1,...,6, ω1,...,6, ψ1,...,5, η1,2,3, φ1,...,8},

S = {ψ1,2µ , χ1,...,6},

b1 = {ψ1,2µ , χ1,2, y3,...,6|y3,...,6, ψ1,...,5, η1},

b2 = {ψ1,2µ , χ3,4, y1,2, ω5,6|y1,2, ω5,6, ψ1,...,5, η2},

b3 = {ψ1,2µ , χ5,6, ω1,...,4|ω1,...,4, ψ1,...,5, η3}.


The NAHE: The Gauge Group

SO(44)

��SO(10)× E8 × SO(6)3

withN = 4

��N = 2

��N = 1


The Various SO(10) Breakings

SO(10)

α

��

α+β // SU(5)× U(1)

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


Extra Dimensions

In models with extra dimensions the usual (3 + 1)-dimensionalspace-time xµ = (x0, x1, x2, x3) is extended to include additionalspatial dimensions parametrized by coordinates x4, x5, ..., x3+N

where N is the number of extra dimensions. String theoryarguments would suggest that N can be as large as 6 or 7.

Depending on the type of metric in the bulk, ED models fall intoone of the following two categories:

Flat, also known as universal ED models (UED).

Warped ED models.


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 onemust use an orbifold S1/Z2. The size of the extra dimension issimply parametrized by the radius of the circle R.

In the case of N = 2, one of the many is known as the chiralsquare corresponding to T 2/Z4. The two extra dimensions haveequal size and the boundary conditions are such that adjacent sidesof the chiral square are identified.


Supersymmetric Gauge Theories

Supersymmetric Gauge Theories in 3D N = 4 are subject to astrange duality: Mirror Symmetry

3D mirror symmetry exchanges Coulomb branch and Higgsbranch of two dual theories.

Mirror Symmetry←−−−−−−−−→

Coulomb branch: moduli space parametrised by scalars in theV-plet

Higgs branch: moduli space parametrised by scalars in theH-plet


The Idea

Study moduli space of instantons ⇒ calculate Hilbert Series forHiggs Branch.

What is the Hilbert series?

It is the partition function that counts chiral gauge invariantoperators

Why Bother?

The chiral gauge invariant operators parametrize the modulispace

Hilbert Series encodes all the information: dimension of themoduli space, generators and relations


Example: Hilbert Series for C2/Z2

C2 with action of Z2: (z1, z2)↔ (−z1,−z2)

Holomorphic functions invariant under this action:z21 , z

22 , z1z2, z

41 , ...

All polynomials constructed from 3 generators subject to 1relation

X = z21 ,Y = z22 ,Z = z1z2

XY = Z 2


Example Contd.: Hilbert Series for C2/Z2

Isometry group of C2 = U(2)Cartan Subalgebra: U(1)2

Choose counterst1 is the U(1) charge of z1t2 is the U(1) charge of z2

HS(t1, t2, ;C2) = 1+t21 +t22 +t1t2+... =∞∑

i ,j=0

t i1tj2 =

∏i

1

1− ti

Unrefine

HS(t) =∞∑

i ,j ,...

t i+j = 1 + 3t2 + 5t4 + ... =∑k=0

(2k + 1)t2k

Dimension of the moduli space

=

pole of the unrefined Hilbert series


THANK YOU!!!


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