Topological G2 Strings

Jan de Boer, Amsterdam

M-theory in the city

Based on:hep-th/0506211, JdB, Asad Naqvi and Assaf Shomerhep-th/0610080, JdB, Paul de Medeiros, Sheer El-Showk and Annamaria Sinkovicswork in progress

Motivation

• M-Theory on G2 manifolds can give rise to realistic N=1 physics in four dimensions

•Attempt to unify topological string theories: topological M-theory?

•Understand terms in the low-energy effective action in three dimensions

•Understand the relation between 3d and 4d physics: c-map

•Unify branes and world-sheet instantons

•Better understanding of S-duality

World-sheet approach

General N=1 supersymmetric σ-model:

Has an N=1 superconformal algebra on the world-sheet with generators G,T

Generically, there is no spacetime supersymmetry

S =Rd2xd2µ[G¹ º (X ) +B¹ º (X )]D+X ¹ D¡ X º

Space-time supersymmetry (no fluxes)

Covariantly constant spinors

Special holonomy

Covariantly constant differential forms

Extra generators in the world-sheet chiral algebra:

A = ! ¹ 1 :::¹ kù 1 : : :ù k ¹@A = 0obeys

! ¹ 1:::¹ k

(plus superpartner)

G2 manifoldsHavea covariantly constant three-formÁ and four-form¤Á

weight

3/2 G Φ

2 T K X

5/2 M

Á ¤Á

These six generators form a non-linear algebra, the G2 algebra

Form an N=1

Subalgebra with

C=7/10:

Tricritical Ising Model

Á» e1 ^e2 ^e3+e1 ^e4 ^e5 +e1 ^e6 ^e7+e2 ^e4 ^e6

¡ e2 ^e5^e7 ¡ e3^e4 ^e7 ¡ e3 ^e5 ^e6

Primaries aredenoted by jhI ;hr i with hI theweightwrt the stress-tensor X of the tricritical Isingmodel,and ¢ = hI +hr the total conformal weight

There is no U(1) current, but there is a BPS bound thatindicates when multiplets are short or long:

hI +hr ¸1+

p1+80hI8

Thereare thereforeonly four types of chiral primaries:j 0;0i j 1

10; 25i j 6

10; 25i j 3

2;0i

T 2

G+ 3/2

G- 3/2

J 1

T+∂J 2

G+ 2

G- 1

J 1

Twisting for Calabi-Yau:

De neQB RST =HG¡ , then Q2

B RST = 0

If J =@Â, then twisting is like adding a backgroundcharge for Â, and

It turns out that eÂ=2 is a vertex operator that generatesa Ramond ground state (related to spectral °ow)

hO1 :: :On i twisted =heÂ=2(1 )O1 : ::OneÂ=2(0)iuntwisted

In theG2 case, there is no U(1) symmetry!

G(z) has weights ( 110;

75) and using the fusion

rules it acts as follows on theHilbert space:

H 110 ;¤

H0;¤

H 610 ;¤

H 32 ;¤

Wecan thereforesplitG(z) =G" (z) +G#(z)

Proposal: Q =HG#

¡ 1=2

De necorrelators as beforewith suitable insertion of VRR

RESULTS

BRST cohomology consists precisely of the chiral primaries

Three-point functions exist and are independent of the insertion points of the operators

Evidence that the path integral localizes on constant maps

Evidence that the theory also exists at higher genus

BRST operator turns out to have a nice geometrical interpretation

Recall that G2 ½SO(7)

G2 rep: 1 7 14 27

0-forms |0,0>

1-forms |1/10,2/5>

2-forms |6/10,2/5> |0,1>

3-forms |3/2,0> |11/10,2/5> |1/10,7/5>

4-forms |2,0> |16/10,2/5> |6/10,7/5>

5-forms |21/10,2/5> |3/2,1>

6-forms |26/10,2/5>

7-forms |7/2,0>

Dolbeault complex for G2 manifolds

For example:

Q(±g¹ ºÃ¹L Ã

ºR ) = 0 , Á¹ º½@[º±g½]¾= 0

is the known equation for metric moduli

Theseare in one-to-one correspondencewith H 3(M ).

Three-point functions give a map

H 3(M ) £ H 3(M ) £ H 3(M ) ¡ ! R.

Geometric interpretation?

Thereexists a prepotential F (ti ) such that

hOiOjOk i / @3

@ti @tj @tkF (t):

where

ti =RA i Á;

@F@ti =

73

RB i

¤Á

exactly as in special geometry. It turns out thatF (t) is exactly given by theHitchin functional

F (t) =RÁ^¤Á

The topological G2 string computes all quantities that appear in the low energy effective action of M-theory compactified on a G2-manifold: the Kähler potential and gauge couplings.

We can compute the genus one partition function in the topological G2 string and compare to a one-loop calculation done using the Hitchin functional. Fails for ordinary Hitchin, may work for generalized Hitchin (work in progress).

Spin(7) does not seem to work at all.

F (CY £ S1) » (FA + ¹FA )1=3(FB + ¹FB )2=3

Kodaira Spencer Theory

Holomorphic Chern-Simons Theory

6d Hitchin functional

Kähler Gravity

Chern-Simons Theory

??

Open Topological G2

??

7d Hitchin functional

FG2

FB

FA

Open string field theory seems to exist!

S =RM ¤Á^CS3(A)

Plus its dimensional reductions to 0,3,4 dimensions

Can incorporate various world-sheet instantons and branes

World-sheet instantons U in CY map to associative cycles U x S1 in CY x S1

Open world-sheet instantons ending on Slag branes map to three-branes ending on four-branes

Or they map to a single smooth associative cycle in CY x S1

D2

D2

F1

OSFT may yield an all-order definition of topological G2 string theory. Is it renormalizable?

Moduli space of associative cycles resums various non-perturbative effects

S-duality exchanges associative and coassociative cycles??

Applications to CY x S1 – G2 string computes hypermultiplet moduli?

Applications to (singular) G2 compactifications

Add fluxes

OUTLOOK