Dualities andTopological Strings

Strings 2006, Beijing

- RD, C. Vafa, E.Verlinde, hep-th/0602087 - work in progress w/ C. Vafa & C. Beasley, L. Hollands

Robbert DijkgraafUniversity 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

4CY × ｡

4 4 2 ( ) gg gravd x d F t Wθ∫F-terms for Weyl muliplet in

4 dim supergravity action

CY

top string partition function

2 2

0exp ( )gtop g

gZ F tλ −

≥

= ∑

s gravg Fλ =

A-model: Gromov-Witten Invariants

,( ) g ddt

gd

F t GW e−= ∑Exact instanton sum 2 ( , )d H X∈ ｢

# maps∈､genus g

M-theory duality

Gopakumar-Vafa invariantsAt strong coupling one can integrate out(light) electric charges D0-D2 to obtain theeffective action

gs →∞

charges

( , ) log det QQ

F tλ Δ∑:

3 1CY S time× × ×｡

M-theory limit

gs

virtual loopsof M2 branes

5d Black Holes in M-theory4CY time× ×｡

Transversal rotationsSO(4) ≡ SU(2)L × SU(2)R

M2-branes with charge

Q ∈ H2(X,Z)M2M2

Internal spin quantum numbers

(mL,mR)

BPS degeneracies

M2

Index of susy ground states (GV-inv)

NmRQ =XmL

(−1)mLNmL,mRQ

4d Quantum Hall system:wave functions lowest Landau level

Ψ(z1, z2) =Xn1,n2

an1,n2zn11 z

n22

CY

4 2≅｡ ｣Orbital angular momentum

(n1, n2)self-dual flux

rotation

space

GV Partition function

Gas of 5d charged & spinning black holes

Z(λ, t) =Yn1,n2Q,m

³1− eλ(n1+n2+m)+tQ

´−NmQ5d entropy

NmQ ∼pQ3 −m2

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(ME)

Donaldson-Thomas InvariantsSingle D6: U(1) gauge theory + singularities

q = D2 = ch2 ∼ TrF 2

instanton strings

k = D0 = ch3 ∼ TrF 3

Z(λ, t) =Xk,q

DT (k, q)ekλ+qt

Lift to M-theoryD6 → Taub-NUT geometry

4｡

SO(4)

angular momentum

ds2TN = R2

·1

V(dχ+ ~A · d~x)2 + V d~x2

¸

3 1S×｡U(1)× SO(3)

Kaluza-Klein momentum

[Gaiotto, Strominger,Yin]

Bound states with D0-D2

spinning M2-branes

R

q =Xi

Qi

k =Xi

(ni +mi)

Gauge theory quantum numbers

4 dim limit:

Bound state of D6-D2

R→ 0

3｡

Z(λ, t) =Xk,q

DT (k, q)ekλ+qt

Donaldson-Thomas Invariants

5 dim limit:

4｡

R→∞

Free gas of M2-branes

Gopakumar-Vafa Invariants

Z(λ, t) =Yn1,n2Q,m

³1− eλ(n1+n2+m)+tQ

´−NmQ

Topological String TrialityPeturbative IIA strings

Gromov-Witten

M2-branesGopakumar-Vafa

D2-branesDonaldson-Thomas

stron

g-weak 9-11 flip

Taub-NUT

Universal TopologicalWave Function?

D-brane charge lattice (B-model)

3( , )

X

H X

a b∧∫｣

symplecticvector space

H3(X,Z)

Period Map & Quantization

3( , )

X

H X

a b∧∫｣

moduli space of CY

Lagrangian cone L=graph (dF0)semi-classical state ψ ~ exp F0

L

symplecticvector space

hol 3-form dz1 dz2 dz3

Ω

Top String Partition Function = Wave Function

2 2exp ( ), gtop gg

Z F tλ λ−= Ψ = =∑ h

Transforms as a wave function underSp(2n,Z) change of canonical basis (A,B)

Wave Function of String TheoryCompactify on a 9-space

X × time

Ψ ∈ HX

Flux/charge/brane sectors

HX =MQ

HQX

Topology Change

Finite energy transitions

X → X 0

Ψ ∈ H

Universal wave function, components on all geometries

Baby UniversesDisconnected spaces

X → X1 +X2Second quantization

H → Sym∗H

Hawking-Hartle Wave FunctionSum over bounding geometries

X = ∂B

Include singularities (branes, black holes)

Ψ =XB

|Bi

“Entropic Principle”Natural probability density on moduli space of string compactifications

eS = |Ψ|2

Depends on massless & massive d.o.f.

peaked aroundmoduli space

string theory on the near horizon geometry of the

black hole

AdS/CFT duality

22

3AdS S CY× ×

supersymmetricgauge theory on

the brane

superconformalquantum mechanics

Hawking-Hartle Wave Function[OSV, Ooguri,E.Verlinde,Vafa]

2

BH topZ ψ=

Euclideantime

22

3AdS S CY× ×

topψ topψ

M2

CY

Entropic principleM-theory on CY + membrane wrapped around

Entropy

F (t) =

ZX

1

6t3S(Q) = −F (t) +Q · t

If b2(X)=1

Prefers small d (d=5 for Quintic)

F (t) =d

6t3 S ∼ Q

3/2

d1/2

Q ∈ H2(X,Z)

Supersymmetry breaking

Non-susy boundary conditions

Z(β) = Tr e−βH β

Positivity of Hβ < β0 ⇒ Z(β) > Z(β0)

Ground states

Prefers symmetric CY’s (accidental zero-modes)

Z(∞) = dimH0 = #harmonic forms≥ Euler

Space of AllCalabi-Yau’s?

cf Space of Riemann Surface/CFTs1. Deligne-Mumford compactification Mg

g g1 g2

• boundary contains lower genus surfaces

• factorization: local operators in CFT

O1 O2

2. Combinatorical approach

• closed strings: operator product expansion

O1

O2O1 · O2

• open strings: matrix models

1. Factorization

Topology of Calabi-Yau spaces

Diffeomorphism type of X is completely fixed (in case of zero torsion) by b3(X) and b2(X) plus invariantsZ

X

1

6x3, x ∈ H2(X,Z)Z

X

x ∧ c2 X

Decomposition

=

X0X Σg

X = X0#Σg

b3 = 0 b2 = 0

Core

Non-Kähler CY are unique

Σg

Σg = #g S3 × S3

Moduli space of complex structures

dimMg = g − 1

Miles Reid’s Fantasy:“There is only one CY space”

Mg

b2 = 0

All CY connected through conifoldtransitions S3 → S2

b2 = 1Kähler CYs

2. Combinatorics

CY

3S

3T

SYZ: fibrations of CY by special Lagr T3

network ofsingularities

S1 shrinks

6d Gauge Theory

3d Gauge Theory

CY 3M

Potential Φ(u) satisfies Monge-Ampère equation

ds2 = gij(u)duiduj , gij(u) = ∂i∂jΦ(u),

det ∂i∂jΦ = 1.

Limit Vol(T3) → 0, integral affine manifold

R3 o SL(3,Z)

Large complex structure

3｡

Stringy cosmic stringu1, u2

ds2 =1

τ2|du2 − τdu1|2 + (du3)2

τ(z) ∼ 12πlog z + · · ·

u1 = Re(z), u2 = Re(

Z zτ(w)dw),

u3

Monodromy in SL(2,Z)u1, u2

u3

1 1

2 2

1 0

1 1u uu u⎛ ⎞ ⎛ ⎞⎛ ⎞

→⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠

S1 shrinks

3｡

Two Vertices

+ -

dualMirror

Symmetrytopological vertex local Riemann surface

3｡

The Quintic [Wei-Dong Ruan]

3 4S B= ∂ 4 4B ≅ Δ4-simplex

The Quintic

glue+ -

--- - -

- - -- - - - -

-

++

++

+ + + + ++

++

++

OSV: Large N 3d Susy Gauge Theory

• in IR dominated by CS term (after deformation)• Wilson lines carry adjoint fields• 3d top field theory realization?

3S

24 Wilson loops

2 2 13 K T S S× → ×

... 2S

Ztop =1

η(t)24

Universal Moduli Space of CYs?

Topological Strings

• Compute BPS black hole degeneracies(gauge-gravity dualities)

• Interesting probability distribution onthe moduli space of vacua

• Universal Calabi-Yau wave function?• Combinatorical models?• Many more surprises...