The Topological G 2 String

1. The Topological G2 String Asad Naqvi (University of Amsterdam) (in progress) with Jan de Boer and Assaf Shomer hep-th/0506nnn

2. 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 Physical Superstrings Topological Strings

3. Schematics of topological strings Twisting Scalar SUSYQ Q Cohomology Topological Observables Chiral Primaries

4. 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 Topological strings on CY 3-folds depends on complex strucutre depends on Kahler strucutre A and B models have been conjectured to be S-dual

5. 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.

6. Outline • G2manifolds • 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

7. G2 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

8. G2 sigma models Lets start with a (1,1) sigma model where This model has (1,1) supersymmetry

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

10. Kahler manifolds-an example 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

11. Extended G2 algebra A G2 holonomy manifold has acovariantly constant 3-form which implies the existence of where and There is also acovariantly constant 4 formwhich leads to a dimension 2 current Xand a dimension 5/2 current M

12. OPEs

13. 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

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

15. 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)

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

17. 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.

18. Consider the three states Matrix of inner products is given by Unitarity Eigenvalues > 0

19. States which saturate the bound will be calledchiral 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

20. 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 Shatashvili+Vafa 1994 is a candidate for an exactly marginal deformation

21. 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/0311253

22. large volume Also The OPE The K0 eigen-value of the this operator should be zero.

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

24. 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.

25. Twisting the G2 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

26. Back to the G2 twist Correlation functions Effectively, the background charge changes from and c changes as BRST and anti-ghost We can show that This splits as

27. Projectors 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

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

29. 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.

30. 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

31. G2 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.

32. 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.

33. Descent Relations We saw earlier that the moduli are related to the operatorsA which has dimension (1/2,1/2) We can then deform the action by Only certain conformal blocks of Aare Q-invariant, so it is not obvious if is Q-invariant. We will now show that this is the case.

34. Define We saw earlier which implies Then

35. Dolbeault Cohomology for G2 and the chiral BRST Cohomology 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

36. BRST Cohomology Geometrically The following table summarizes theL0 and X0eigenvalues of these operators

37. 7+14 1 Projection operator onto the 7 when acting on 2 forms is We can repeat this analysis for the two and three forms

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

39. Differential Complexes

40. Total BRST Cohomology If we combine the left movers with the right movers, we get a more interesting cohomology Full de Rham cohomology

41. Themetric andB-fieldmoduli should be given by operators of the form with

42. 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 G2 metrics.

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

44. G2 Special Geometry Define, and In fact, and

45. Topological G2 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

46. 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

47. Back to topological G2 strings Charge Screened Antighost Charge2 which is exactly the right value to cancel the background charge of

48. Conclusions • 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 ?