topological g2 strings - abdus salam centre for...
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Topological Strings and Special Topological Strings and Special HolonomyHolonomy ManifoldsManifolds
Asad Naqvi(University of Wales, Swansea)
with Jan de Boer and Assaf Shomer
hep-th/0506211
12th Regional Conference, NCP, Islamabad
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String TheoryString Theory
We introduce an action, which is simply the area of the world sheet.
To find the probability of transition, we follow Feynman’s prescription of integrating over all possible paths between two points.
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Constructing Topological StringsConstructing Topological Strings
Twist
Has SUSY Q which is a scalar on the world sheet
Has SUSY Q which is a spinor on the world sheet
Now consider only those operators which are in the cohomology of Q
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If M is a Calabi-Yau manifold, then there are 2 ways we can twist, the A and the B twist
A twist: Holomorphic mapsB twist: Constant maps
Also, Hence these correlation functions are position independent. They are functions of the moduli of the Calabi-Yau.
•A-model only depends on the Kahler structure moduli•B-model only depends on the Complex structure moduli
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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 phyiscalphenomena in a more controlled setting
• The describe a sector of superstrings and provide exact answers to certain questions concerning BPS quantities
TopologicalStrings
PhysicalSuperstrings
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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
Shatashvili and Vafa 9407025
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GG2 2 manifoldsmanifoldsSpecial 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
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GG22 sigma modelssigma models
Lets start with a (1,1) sigma model
where
This model has (1,1) supersymmetry
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GG--structures and Extended structures and Extended ChiralChiral AlgebraAlgebraCovariantly constant
formsExtra holomorphic
currents
Given a covariantly constant p-form satisfying
the current
satisfies
dim and dim currents
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KahlerKahler manifoldsmanifolds--an examplean exampleOn 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
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Extended GExtended G22 algebraalgebraA G2 holonomy manifold has a covariantly constant 3-form
which implies the existence of
where and
There is also a covariantly constant 4 form which leads to adimension 2 current X and a dimension 5/2 current M
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OPEsOPEs
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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
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TricriticalTricritical IsingIsing ModelModel
Kac table:Spectrum ofconformal primaries
Some fusion rules:-
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Conformal blocksConformal blocks
The fusion rules imply
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A A UnitarityUnitarity BoundBound
States which saturate the bound will be called chiral primary
Notice the definition of chiral primaries involve a non-linearinequality.
We will see later that the topological theory keepsonly the chiral primary states
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has dimension
So preserves and is dim 1
is a candidate for an exactly marginal deformation
Shatashvili+Vafa1994
RamondRamond SectorSector
Ramond sector ground states: dim =
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Topological TwistTopological Twist
Review of the Calabi-Yau twisting
Sigma model action:-
scalar 1-formscalar1-form 1-form scalarA-twistB-twist
Effectively, we are adding background gauge field for the U(1)
with
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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.
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Twisting the GTwisting the G22 sigma modelsigma 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 playedby the tri-critical Ising model
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Back to the GBack to the G22 twisttwistCorrelation functions
BRST Operator (Scalar Q)We can show that
This splits as
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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
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BRST and its BRST and its CohomologyCohomologyThe BRST operator that can be written as
This squares to zero:-
State Cohomology
From the tri-critical fusion rules, we know that
Then, by definition
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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.
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DolbeaultDolbeault CohomologyCohomology for Gfor G22 and the and the chiralchiral BRST BRST CohomologyCohomology
For 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
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BRST BRST CohomologyCohomology GeometricallyGeometrically
The following table summarizes the L0 and X0 eigenvaluesof these operators
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Differential Complexes
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7 + 14 1Projection operator onto the 7 when acting on 2 forms is
We can repeat this analysis for the two and three forms
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ChiralChiral BRST BRST CohomologyCohomology
with
This is exactly the cohomology of the operator
Almost trivial since
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Total BRST Total BRST CohomologyCohomologyIf we combine the left movers with the right movers, we get a more interesting cohomology
Full de Rham cohomology
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The metric and B-field moduli should be given by operators of the form
with
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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.
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In fact, the generating function of all our correlation functionsis given by
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GG22 Special GeometrySpecial GeometryDefine,
and
In fact,
and
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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 ?