Dualities and Topological Strings

Transcription

1 Dualities and Topological Strings Strings 2006, Beijing - RD, C. Vafa, E.Verlinde, hep-th/ work in progress w/ C. Vafa & C. Beasley, L. Hollands Robbert Dijkgraaf University of Amsterdam

2 Topological Strings Toy model (cf topology versus geometry) Exact BPS sector of superstrings Mathematical experiments to test and develop physical intuition

3 Exact Effective Actions CY 4 CY d x d θ F () t W 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

4 A-model: Gromov-Witten Invariants Exact instanton sum d H 2 ( X, ) F t = GW e g () gd, d dt genus g # maps

5 M-theory duality

6 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

7 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 )

8 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

9 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

10 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 )

11 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

12 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)

13 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 )

14 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

15 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

16 Topological String Triality Peturbative IIA strings Gromov-Witten strong-weak 9-11 flip M2-branes D2-branes Gopakumar-Vafa Donaldson-Thomas Taub-NUT

17 Universal Topological Wave Function?

18 D-brane charge lattice (B-model) H 3 (X, Z) symplectic vector space H 3 ( X, ) a b X

19 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

20 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)

21 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

22 Topology Change Finite energy transitions X X 0 Universal wave function, components on all geometries Ψ H

23 Disconnected spaces Baby Universes Second quantization X X 1 + X 2 H Sym H

24 Hawking-Hartle Wave Function Sum over bounding geometries Ψ = X B Bi X = B Include singularities (branes, black holes)

25 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

26 AdS/CFT duality string theory on the near horizon geometry of the black hole supersymmetric gauge theory on the brane AdS S CY superconformal quantum mechanics

27 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

28 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)

29 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)

30 Space of All Calabi-Yau s?

31 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

32 2. Combinatorical approach closed strings: operator product expansion O 1 O 1 O 2 O 2 open strings: matrix models

33 1. Factorization

34 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

35 Decomposition X = X 0 #Σ g b 3 =0 b 2 =0 = Core X X 0 Σ g

36 Non-Kähler CY are unique Σ g =# g S 3 S 3 Σ g Moduli space of complex structures dim M g = g 1

37 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

38 2. Combinatorics

39 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

40 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.

41 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 +

42 Monodromy in SL(2,Z) u 1,u 2 S 1 shrinks u u 1 0 u 2 2 u 1 1 u

43 Two Vertices topological vertex dual Mirror Symmetry local Riemann surface

44 The Quintic [Wei-Dong Ruan] S 3 4 = B 4 4 B Δ 4-simplex

45 The Quintic glue

46 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?

47 2 2 1 K 3 T S S... S 2 24 Wilson loops Z top = 1 η(t) 24

48 Universal Moduli Space of CYs?

49 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...