the topological g 2 string
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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 ?
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