Dualities and Topological Strings Strings 2006, Beijing - RD, C. Vafa, E.Verlinde, hep-th/0602087 - work in progress w/ C. Vafa & C. Beasley, L. Hollands Robbert Dijkgraaf University of Amsterdam
Topological Strings Toy model (cf topology versus geometry) Exact BPS sector of superstrings Mathematical experiments to test and develop physical intuition
Exact Effective Actions CY 4 CY d x d θ F () t W 4 4 2 g g grav Z top = 2g 2 exp λ Fg ( t) g 0 F-terms for Weyl muliplet in 4 dim supergravity action top string partition function λ = gf s grav
A-model: Gromov-Witten Invariants Exact instanton sum d H 2 ( X, ) F t = GW e g () gd, d dt genus g # maps
M-theory duality
Gopakumar-Vafa invariants At strong coupling g s one can integrate out (light) electric charges D0-D2 to obtain the effective action F( λ, t) : logdetδq charges Q M-theory limit g s virtual loops of M2 branes CY 3 S 1 time
5d Black Holes in M-theory CY 4 M2-branes with charge Transversal rotations time Q H 2 (X, Z) M2 M2 SO(4) SU(2) L SU(2) R Internal spin quantum numbers (m L,m R )
BPS degeneracies Index of susy ground states (GV-inv) N m R Q = X m L ( 1) m L N m L,m R Q 4d Quantum Hall system: wave functions lowest Landau level Ψ(z 1,z 2 )= X a n1,n 2 z n1 n 1,n 2 Orbital angular momentum (n 1,n 2 ) 1 zn 2 2 M2 space CY 4 2 self-dual flux rotation
GV Partition function Gas of 5d charged & spinning black holes Z(λ,t)= 5d entropy Y n 1,n 2 Q,m ³ 1 e λ(n 1+n 2 +m)+tq N m Q N m Q p Q 3 m 2
6+1 dim SUSY Gauge Theory Witten index counts D-brane bound states Z =Tr ( 1) F e βh Induced charges: non-trivial gauge bundle (P, Q) ch (E) Reduction to moduli space of vacua Z Euler(M E )
Donaldson-Thomas Invariants Single D6: U(1) gauge theory + singularities k = D0 =ch 3 Tr F 3 q = D2 =ch 2 Tr F 2 instanton strings Z(λ,t)= X k,q DT(k, q)e kλ+qt
Lift to M-theory [Gaiotto, Strominger,Yin] D6 Taub-NUT geometry 1 ds 2 TN = R 2 V (dχ + A ~ d~x) 2 + Vd~x 2 Kaluza-Klein momentum angular momentum 3 1 S U(1) SO(3) 4 SO(4)
Bound states with D0-D2 R Gauge theory quantum numbers spinning M2-branes q = X i k = X i Q i (n i + m i )
4 dim limit: R 0 3 Bound state of D6-D2 Donaldson-Thomas Invariants Z(λ,t)= X k,q DT(k, q)e kλ+qt
5 dim limit: R 4 Free gas of M2-branes Gopakumar-Vafa Invariants Z(λ,t)= Y ³ 1 e λ(n 1+n 2 +m)+tq N m Q n 1,n 2 Q,m
Topological String Triality Peturbative IIA strings Gromov-Witten strong-weak 9-11 flip M2-branes D2-branes Gopakumar-Vafa Donaldson-Thomas Taub-NUT
Universal Topological Wave Function?
D-brane charge lattice (B-model) H 3 (X, Z) symplectic vector space H 3 ( X, ) a b X
hol 3-form dz 1 dz 2 dz 3 Period Map & Quantization Ω symplectic vector space H 3 ( X, ) L X a b moduli space of CY Lagrangian cone L=graph (df 0 ) semi-classical state ψ ~ exp F 0
Top String Partition Function = Wave Function top =Ψ= exp 2g 2 ( ), λ = g g Z λ F t h Transforms as a wave function under Sp(2n,Z) change of canonical basis (A,B)
Wave Function of String Theory Compactify on a 9-space Ψ H X X time Flux/charge/brane sectors H X = M Q H Q X
Topology Change Finite energy transitions X X 0 Universal wave function, components on all geometries Ψ H
Disconnected spaces Baby Universes Second quantization X X 1 + X 2 H Sym H
Hawking-Hartle Wave Function Sum over bounding geometries Ψ = X B Bi X = B Include singularities (branes, black holes)
Entropic Principle Natural probability density on moduli space of string compactifications e S = Ψ 2 Depends on massless & massive d.o.f. peaked around moduli space
AdS/CFT duality string theory on the near horizon geometry of the black hole supersymmetric gauge theory on the brane AdS S CY 2 3 2 superconformal quantum mechanics
Hawking-Hartle Wave Function [OSV, Ooguri,E.Verlinde,Vafa] Z BH = ψ top 2 ψ top Euclidean time ψ top AdS S CY 2 3 2
Entropic principle M-theory on CY + membrane wrapped around Entropy S(Q) = F (t)+q t If b 2 (X)=1 Q H 2 (X, Z) F (t) = d 6 t3 S Q3/2 d 1/2 F (t) = M2 Z X 1 6 t3 CY Prefers small d (d=5 for Quintic)
Supersymmetry breaking Non-susy boundary conditions Z(β) =Tr e βh β Positivity of H β < β 0 Z(β) >Z(β 0 ) Ground states Z( ) =dimh 0 =#harmonic forms Euler Prefers symmetric CY s (accidental zero-modes)
Space of All Calabi-Yau s?
cf Space of Riemann Surface/CFTs 1. Deligne-Mumford compactification M g boundary contains lower genus surfaces g g 1 g 2 factorization: local operators in CFT O 1 O 2
2. Combinatorical approach closed strings: operator product expansion O 1 O 1 O 2 O 2 open strings: matrix models
1. Factorization
Topology of Calabi-Yau spaces Diffeomorphism type of X is completely fixed (in case of zero torsion) by b 3 (X) and b 2 (X) plus invariants Z 1 X 6 x3, x H 2 (X, Z) Z x c 2 X X
Decomposition X = X 0 #Σ g b 3 =0 b 2 =0 = Core X X 0 Σ g
Non-Kähler CY are unique Σ g =# g S 3 S 3 Σ g Moduli space of complex structures dim M g = g 1
Miles Reid s Fantasy: There is only one CY space b 2 =1 Kähler CYs b 2 =0 M g All CY connected through conifold transitions S 3 S 2
2. Combinatorics
SYZ: fibrations of CY by special Lagr T 3 6d Gauge Theory T 3 CY network of singularities S 1 shrinks 3d Gauge Theory S 3
Large complex structure Limit Vol(T 3 ) 0, integral affine manifold CY 3 M R 3 o SL(3, Z) ds 2 = g ij (u)du i du j, g ij (u) = i j Φ(u), Potential Φ(u) satisfies Monge-Ampère equation det i j Φ =1.
Stringy cosmic string u 1,u 2 3 u 3 ds 2 = 1 τ 2 du 2 τdu 1 2 +(du 3 ) 2 u 1 =Re(z), u 2 =Re(Z z τ(w)dw), τ(z) 1 2π log z +
Monodromy in SL(2,Z) u 1,u 2 S 1 shrinks u 3 1 1 u 1 0 u 2 2 u 1 1 u
Two Vertices + - 3 3 topological vertex dual Mirror Symmetry local Riemann surface
The Quintic [Wei-Dong Ruan] S 3 4 = B 4 4 B Δ 4-simplex
The Quintic glue + -- + + - + + - - - + + - - - - + + - - - - -+ + + + + +
OSV: Large N 3d Susy Gauge Theory S 3 in IR dominated by CS term (after deformation) Wilson lines carry adjoint fields 3d top field theory realization?
2 2 1 K 3 T S S... S 2 24 Wilson loops Z top = 1 η(t) 24
Universal Moduli Space of CYs?
Topological Strings Compute BPS black hole degeneracies (gauge-gravity dualities) Interesting probability distribution on the moduli space of vacua Universal Calabi-Yau wave function? Combinatorical models? Many more surprises...