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The Topological G 2 String Asad Naqvi (University of Amsterdam) (in progress) with Jan de Boer and Assaf Shomer hep-th/0506nnn

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Introduction 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 Topological Strings Physical Superstrings

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Schematics of topological strings Twisting Scalar SUSY Q Q Cohomology Topological Observables Chiral Primaries

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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

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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 G 2 holonomy This may have applications to M-theory compactifications on G 2 manifolds It can improve our understanding of the relation between supersymmetric gauge theories in three and four dimensions.

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Outline G 2 manifolds G 2 sigma models (1,1) SUSYExtended symmetry algebra Tricritical Ising model algebra is contained in this extended algebra Topological twist of the G 2 sigma model Relation to Geometry Topological G 2 strings Shatashvili and Vafa

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G 2 manifolds 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

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G 2 sigma models Lets start with a (1,1) sigma model where This model has (1,1) supersymmetry

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G-structures and Extended Chiral Algebra Covariantly constant forms Extra holomorphic currents Given a covariantly constant p-form satisfying the current satisfies dim and dimcurrents

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On a Kahler manifold, a Kahler form implies the existence of a dimension 1 current and a dimensioncurrent 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 example

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Extended G 2 algebra A G 2 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 whereand

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OPEs

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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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Tricritical Ising Model Kac table: Spectrum of conformal primaries Some fusion rules:-

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Coulomb 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)

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Conformal blocks and screened vertex operators The fusion rules imply that = =

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A 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.

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Consider the three states Matrix of inner products is given by UnitarityEigenvalues > 0

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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

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Ramond 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

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Moduli 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/

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large volume Also The OPE The K 0 eigen-value of the this operator should be zero.

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Topological Twist Review of the Calabi-Yau twisting Sigma model action:- A-twist scalar 1-form scalar1-form with Effectively, we are adding background gauge field for the U(1) 1-form scalar B-twist

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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 G 2 sigma model We apply this to the G 2 sigma model The role of will be played by sits purely within the For the G 2 sigma model the role of the U(1) part is played by the tri-critical Ising model

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Back to the G 2 twist Effectively, the background charge changes from and c changes as Correlation functions BRST and anti-ghost We can show that This splits as

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Projectors As we saw before, a generic state in the theory can be labeled by two qunatum numbers:- h I is the weight of the state under the tri-critical Ising part. For primary fields Define P k to be the projector which projects onto the k th conformal family

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The BRST operator that can be written as This squares to zero:- BRST and its Cohomology State Cohomology From the tri-critical fusion rules, we know that Then, by definition

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We can solve for c 1 and c 2 upto an irrelevant phase and c 2 =0 implies This is precisely the unitarity bound that we found earlier.

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Operator 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

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G 2 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.

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An sl(2|1) 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.

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Descent 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

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Define We saw earlier which implies Then

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Dolbeault Cohomology for G 2 and the chiral BRST Cohomology For a G 2 manifold, forms at each degree can be decomposed in irreducible representations of G 2. Cohomology groups decompose as and depend on the G 2 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 Cohomology Geometrically The following table summarizes the L 0 and X 0 eigenvalues of these operators

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Projection 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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Chiral BRST Cohomology with This is exactly the cohomology of the operator Almost trivial since

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Differential Complexes

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Total BRST Cohomology If 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 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 G 2 metrics.

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

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G 2 Special Geometry Define, and In fact, and

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Topological G 2 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

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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

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Back to topological G 2 strings Screened Antighost Charge Charge 2 which is exactly the right value to cancel the background charge of

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Conclusions We have constructed a new topological theory in 7 dimensions which captures the geometry of G 2 manifolds Relation to topological M-theory ? D-branes ? Spin 7 ?

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