Topological G2 Strings

1. Topological G2 Strings Jan de Boer, Amsterdam M-theory in the city Based on: hep-th/0506211, JdB, Asad Naqvi and Assaf Shomer hep-th/0610080, JdB, Paul de Medeiros, Sheer El-Showk and Annamaria Sinkovics work in progress

2. 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

3. 2 2 R [ ( ) ( ) ] d d µ S G X B X D X D X ¹ º + x = + ¡ ¹ º ¹ º 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

4. ¹ Ã Ã A @ ¹ ¹ A 0 k 1 ! = = ¹ ¹ : : : k 1 : : : ! ¹ ¹ k 1 : : : Space-time supersymmetry (no fluxes) Covariantly constant spinors Special holonomy Covariantly constant differential forms Extra generators in the world-sheet chiral algebra: obeys (plus superpartner)

5. l h f Á d f f Á H i 1 2 3 1 4 5 1 6 7 2 4 6 t t t t Á ¤ a v e a c o v a r a n y c o n s a n r e e - o r m a n o u r - o r m + + + ^ ^ ^ ^ ^ ^ ^ ^ e e e e e e e e e e e e » 2 5 7 3 4 7 3 5 6 ¡ ¡ ¡ ^ ^ ^ ^ ^ ^ e e e e e e e e e Á ¤ Á G2 manifolds These six generators form a non-linear algebra, the G2 algebra

6. j i ( ) d d b h h h h h h P h b h b d h T i i U i B P S i i i 1 t t t t t t t t t r m a r e s a r e e n o e y w e w e g e r e s n o c u r r e n u e r e s a o u n a h h f l f I f I h l T i i i t t r ; , e r e a r e e r e o r e o n y o u r y p e s o c r a p r m a r e s : h f h l d l X I d h l l h l i i i i i i i t t t t t t t t t t t w r e s r e s s - e n s o r o e r c r c a s n g m o e 1 2 6 2 3 n c a e s w e n m u p e s a r e s o r o r o n g : j i j i j i j i 0 0 0 , ; 1 0 ; 5 1 0 ; 5 2 ; d h h h l f l h ¢ i t t t t + a n e o a c o n o r m a w e g p = I h 1 1 8 0 + + r h h I ¸ + I r 8

7. 2 H ¡ ¯ h D Q G Q 0 t e n e e n = = B R S T B R S T , Twisting for Calabi-Yau:

8. = 2 h h I i  t t t t t t t t t t f @ h l k d d b k d I J u r n s o u a e i i s a i v e r i e x o p e r i a o r a g e n e r a e s t t t  e n w s n g s e a n g a a c g r o u n = = = , 2 2 h i h ( ) ( ) i O O O O   0 ( ) d d l d l ° R t t t t t e 1 e ( ) = a a m o n g r o u n s a e r e a e o s p e c r a o w d d h f d h i h ! i I G U 1 1 i t t t t 1 t t t n w s e n u n w s e : : : : : : c a r g e o r  a n n e c a s e e r e s n o s y m m e r y 2 , ,

9. 1 7 ( ) ( ) h h d h f G i i i t t z a s w e g s a n u s n g e u s o n h f l W ¯ l i b f h b l f D V i i i i t t t t t t e c a n e r e o r e s p e n e c o r r e a o r s a s e o r e w s u a e n s e r o n o 1 0 ; 5 R R # H l P Q G l f l l h l b H i i t t t t " # r o p o s a : H H H H r u e s a c s a s o o w s = o n e e r s p a c e : ( ) ( ) ( ) G G G + = 1 2 ¡ z z z 3 6 1 = 0 ¤ ¤ ¤ ¤ ; ; ; ; 1 2 1 0 0

10. 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

11. ( ) l l h R G S O 7 t t ½ e c a a 2 Dolbeault complex for G2 manifolds

12. l F o r e x a m p e : ¹ ( ) ± Ã Ã Á @ ± Q º º ½ 0 0 g g = , = [ ] R L ¹ º ¹ º ½ ¾ h k f d l i i i i t t t s e n o w n e q u a o n o r m e r c m o u 3 ( ) h d h T H M i i t t e s e a r e n o n e - o - o n e c o r r e s p o n e n c e w . h f T i i i t t r e e - p o n u n c o n s g v e a m a p 3 3 3 ( ) ( ) ( ) H M H M H M R £ £ ¡ ! . ? G i i i t t t t e o m e r c n e r p r e a o n

13. i ( ) h l h h T F i i t t t t t t e r e e x s s a p r e p o e n a s u c a 3 @ h i ( ) O O O F t / k i j k i j @ @ @ : t t t h w e r e @ F 7 i R R Á Á t ¤ = = i i @ A B ; 3 t i l l h I i i t t t t t t t e x a c y a s n s p e c a g e o m e r y u r n s o u a . ( ) l b h h f l F H i i i i i t t t t t s e x a c y g v e n y e c n u n c o n a R ( ) Á Á F t ^ ¤ =

14. = = ¹ ¹ 1 1 3 2 3 ( ) ( ) ( ) F C Y S F F F F £ + + » A A B B • 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.

15. F G F F 2 B A Kodaira Spencer Theory Holomorphic Chern-Simons Theory 6d Hitchin functional Kähler Gravity Chern-Simons Theory ?? ?? Open Topological G2 7d Hitchin functional

16. R ( ) Á S C S A ^ ¤ = 3 M Open string field theory seems to exist! 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

17. D2 F1 D2

18. OUTLOOK • 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