topological g2 strings
DESCRIPTION
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. Motivation. - PowerPoint PPT PresentationTRANSCRIPT
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