The Topological GThe Topological G22 String String

Asad Naqvi

(University of Amsterdam)

(in progress) with Jan de Boer and Assaf Shomer

hep-th/0506nnn

Introduction and MotivationIntroduction and Motivation

Topological strings have provided a useful insights into various physical and mathematical questions

• They are useful toy models of string theories which are still complicated enough to exhibit interesting phyiscal phenomena in a more controlled setting

• The describe a sector of superstrings and provide exact answers to certain questions concerning BPS quantities

TopologicalStrings

PhysicalSuperstrings

Schematics of topological strings

Twisting

Scalar SUSY Q

QCohomology

Topological Observables

Chiral Primaries

Topological strings on CY 3-folds

Closed strings:-• A-model only depends on the Kahler structure• B-model only depends on the Complex structure

However the A and B models mix when we couple the closed strings to D branes

• A-brane action

• B-branes action

depends on complex strucutre

depends on Kahler strucutre

A and B models have been conjectured to be S-dual

Several authors have found a seven dimensional theory which unifies and extends features of the A and B models.

This was one of our motivations to define a twist of string theory on a manifold of G2 holonomy

This may have applications to M-theory compactifications on G2 manifolds

It can improve our understanding of the relation between supersymmetric gauge theories in three and four dimensions.

OutlineOutline

• G2 manifolds

• G2 sigma models

(1,1) SUSY Extended symmetry algebra

• Tricritical Ising model algebra is contained in this extended algebra

• Topological twist of the G2 sigma model

• Relation to Geometry

• Topological G2 strings

Shatashvili and Vafa 9407025

GG2 2 manifoldsmanifolds

Special holonomy

Under this embedding

i.e. there is a covariantly constant spinor

is a covariantly constant p-form

This is non-zero for p=0,3,4 and 7

GG22 sigma models sigma models

Lets start with a (1,1) sigma model

where

This model has (1,1) supersymmetry

G-structures and Extended Chiral AlgebraG-structures and Extended Chiral AlgebraCovariantly constant

formsExtra holomorphic currents

Given a covariantly constant p-form satisfying

the current

satisfies

dim and dim currents

On a Kahler manifold, a Kahler form

implies the existence of a dimension 1 current

and a dimension current

which extend the (1,1) algebra to a (2,2) algebra

On Calabi-Yau manifolds, there is a holomorphic 3-form which extends this algebra even more and generates spectral flow

Kahler manifolds-an exampleKahler manifolds-an example

Extended GExtended G22 algebra algebra

A G2 holonomy manifold has a covariantly constant 3-form

There is also a covariantly constant 4 form which leads to a dimension 2 current X and a dimension 5/2 current M

which implies the existence of

where and

OPEsOPEs

An important fact is that

which means that states of the CFT can be labeled by its tri-critical Ising model weight and its weight in the remainder

Tricritical Ising ModelTricritical Ising Model

Kac table: Spectrum of conformal primaries

Some fusion rules:-

Coulomb gas representation of tri-critical IsingCoulomb gas representation of tri-critical Ising

This is a CFT of a scalar field coupled to a background charge

Screeners:

Screened vertex operators (Felder ‘88)

Conformal blocks and screened vertex operatorsConformal blocks and screened vertex operators

The fusion rules

imply that

= =

A BPS boundA BPS bound

Highest weight states are annihilated by the positive modes of all the generators.

Zero modes of the three dimension two bosonic operators commute when action on highest weight states

Highest weight state:-

We want to derive some bounds on that come from unitarity.

Consider the three states

Matrix of inner products is given by

Unitarity Eigenvalues > 0

States which saturate the bound will be called chiral primary

Notice the definition of chiral primaries involve a non-linear inequality.

We will see later that the topological theory keeps only the chiral primary states

Ramond SectorRamond Sector

Ramond sector ground states: dim =

These states imply the existence of some NS sector states

has dimension

So preserves and is dim 1

is a candidate for an exactly marginal deformation

Shatashvili+Vafa 1994

ModuliModuli

Geometrically, the metric moduli are deformations of the metric which preserve the Ricci flatness condition

can be written as the square of a first order operator if the manifold supports a covariantly constant spinor

We can construct a spinor-valued 1-form

It can be shown that

math.dg/0311253

large volume

Also

The OPE

The K0 eigen-value of the this operator should be zero.

Topological TwistTopological Twist

Review of the Calabi-Yau twisting

Sigma model action:-

A-twist scalar 1-formscalar1-form

with

Effectively, we are adding background gauge field for the U(1)

1-form scalarB-twist

So on a sphere

Since

On higher genus surfaces, we need 2-2g insertions

This effectively adds a background charge for the U(1) part thereby changing its central charge.

Twisting the GTwisting the G22 sigma model sigma model

We apply this to the G2 sigma model

The role of will be played by

sits purely within the

For the G2 sigma model the role of the U(1) part is played by the tri-critical Ising model

Back to the GBack to the G22 twist twist

Effectively, the background charge changes from

and c changes as

Correlation functions

BRST and anti-ghostWe can show that

This splits as

ProjectorsProjectors

As we saw before, a generic state in the theory can be labeled by two qunatum numbers:-

hI is the weight of the state under the tri-critical Ising part.

For primary fields

Define Pk to be the projector which projects onto the kth conformal family

The BRST operator that can be written as

This squares to zero:-

BRST and its CohomologyBRST and its Cohomology

State Cohomology

From the tri-critical fusion rules, we know that

Then, by definition

We can solve for c1 and c2 upto an irrelevant phase and c2=0 implies

This is precisely the unitarity bound that we found earlier.

Operator CohomologyOperator Cohomology A local operator corresponding to the chiral primary states will in general not commute with Q.

In fact, only particular conformal blocks of operators will be Q-closed.

We can show that the conformal blocks

satisfy

GG22 Chiral Ring Chiral Ring

The unitarity bound

implies that there are no singular terms in the OPE, and the leading regular term saturates the bound and so is a chiral primary operator itself.

So we have a ring of chiral operators.

An sl(An sl(22||11) Subalgebra) Subalgebra There exists a subalgebra which is the same as that obeyed by the lowest modes of the N=2 algebra.

Define

Then,

form a closed algebra.

A particularly useful relation is

which means that correlation functions of Q invariant operators are position independent.

Descent RelationsDescent Relations

We saw earlier that the moduli are related to the operators A which has dimension (1/2,1/2)

Only certain conformal blocks of A are Q-invariant, so it is not obvious if is Q-invariant. We will now show that this is the case.

We can then deform the action by

Define

We saw earlier

which implies

Then

Dolbeault Cohomology for GDolbeault Cohomology for G22 and the and the

chiral BRST Cohomologychiral BRST CohomologyFor a G2 manifold, forms at each degree can be decomposed in irreducible representations of G2.

Cohomology groups decompose as and depend on the G2 irrep R only and not on p

We can define a sub-complex of the de Rham complex as follows

We will next see that this operator maps to our BRST operator Q

BRST Cohomology GeometricallyBRST Cohomology Geometrically

The following table summarizes the L0 and X0 eigenvaluesof these operators

17 + 14

Projection operator onto the 7 when acting on 2 forms is

We can repeat this analysis for the two and three forms

Chiral BRST CohomologyChiral BRST Cohomology

with

This is exactly the cohomology of the operator

Almost trivial since

Differential Complexes

Total BRST CohomologyTotal BRST Cohomology

If we combine the left movers with the right movers, we get a more interesting cohomology

Full de Rham cohomology

The metric and B-field moduli should be given by operators of the form

with

Correlation FunctionsCorrelation Functions

Consider three point function of operators

On general grounds, we expect this is the third derivative of a prepotential if suitable flat coordinates are used for the moduli space of G2 metrics.

In fact, the generating function of all our correlation functionsis given by

GG22 Special Geometry Special GeometryDefine,

and

In fact,

and

Topological GTopological G22 Strings Strings

Review of topological strings on Calabi-Yau manifolds:

At genus g, we need to insert 2g-2 operators

Chiral operators have negative charge

So for CY sigma models, there are no interesting correlators at higher genus

We need to go to topological strings to get interesting higher genus amplitudes, which means we need to integrate over the moduli space of Riemann surfaces, which is 3g-3 dimensional

The measure on the moduli space of Riemann surfaces is defined by

has charge +1

So topological strings on a CY are only interesting in d=3

Back to topological GBack to topological G22 strings strings

Screened

Antighost

Charge

Charge 2

which is exactly the right value to cancel the background charge of

ConclusionsConclusions

• We have constructed a new topological theory in 7 dimensions which captures the geometry of G2 manifolds

• Relation to topological M-theory ?

• D-branes ?

• Spin 7 ?