Introduction to Topological Strings on Local Geometry Masato Taki/

Introduction to Topological Strings on Local Geometry

Masato Taki/ YITP, Kyoto University

Lecture @ 国家理論科学研究中心 15, October 2010

瀧 雅人

/

Topological Strings

Today we will study topological string theory (on the so-called “local geometry”).

It is a solvable toy model of string theory, and has many interesting mathematical feature. So, topological strings give a theoretical playground for string theorists.

∗

∗

∗

dualities relates many physical & mathematical theories solvable model for string theory

non-perturbative dynamics of gauge theories Moreover this theory is a toy model of varied applications.

topological strings

curve counting

Chern-Simons theory matrix model

Seiberg-Witten-Nekrasov theory Dijkgraaf-Vafa theory twister string

Math Physics

Applications

knot theory

black holes mirror symmetry

Calabi-Yau compactification

. . . . . .

・ Hori et al , “Mirror Symmetry” , Clay Mathematics Monographs

・ M. Marino, “Les Houches lectures on matrix models and topological strings”, hep-th/0410165

・ M. Marino, “Chern-Simons theory and topological strings”, hep-th/0406005

References

・ Vonk, “A mini-course on topological strings”,

hep-th/0504147

・ Worldsheet formulation of topological strings

・ Conifold and geometric transition

Plan of the lecture

・ Geometric engineering of gauge theories

・ Application to AGT relation

Part 1

Part 2

Calabi-Yau compactification of superstring theory is an important topics in string theory.

1. Worldsheet Formulation of Topological Strings

- Why Topological String Theory

superstring theory (heterotic,

Type II, ...) on M × R⁴ gauge theory on

^{N = 1, 2} R

The low-energy effective theory of the Calabi-Yau compactification of Type II A(B) superstring theory ??

Topological sting A(B)-model captures the holomorphic information of the low-energy theory exactly !!

In general it is very hard to compute stringy corrections on such a curved background.

So, you will get exact result about stringy correction if you can compute topological string theory.

In this lecture we study topological strings propagating inside a CY. So we define it at the beggining.

- Calabi-Yau space

We impose the Kahler condition

X

, X

i, ¯i = 1, · · · , d

Let us consider a complex manifold

d dim

(= 2d dim

)

dω = 0

ω = i G

dX

∧ dX

Then the Calabi-Yau condition requires that the Ricci curvature of a Kahler manifold is flat :

c

(M) = 0

R

= 0

- Calabi-Yau 1-fold

T

Torus is a simple example of Calabi-Yau 2-fold.

X

= z X

= ¯ z

z ∼ z + 1

- Calabi-Yau 1-fold

T

α

β α β

[Σ] = pα + qβ ∈ H

(T

, Z)

There are two basic 2-cycles .

Generic cycles are spanned by these two cycles as

- Calabi-Yau 3-fold

The Calabi-Yaus of our interest are 3-fold . Moreover we deal with

only non-compact cases.

d = 3

conifold : hypersurface inside

x

+ y

+ u

+ v

= 0 C

S

S

xy = uv

By changing the coordinates, we have another difiniton eq.

t

Let us recall that string theory is a quantum mechanics of one-dimensional objects propagating the target space M. A string sweeps out the two-dimensional surface which we call “world-sheet”

- String Thoery

string

worldsheet Σ_g

M

t

Let us recall that string theory is a quantum mechanics of one-dimensional objects propagating the target space M. A string sweeps out the two-dimensional surface which we call “world-sheet”

- String Thoery

In order to formulate topological string theory propagating inside a target space M, we consider holomorphic embedding of the worldsheet into a Calabi-Yau M.

X : Σ _g � → M

X :

X : Σ

→ Σ ⊂ M N

F _g = �

Σ∈H

(M, Z)

N _g,Σ e ^−t

t _Σ

Σ

- Mathematical “definition” of topological strings

X :

X : Σ_g → Σ ⊂ M

N_{g,Σ maps}

F _g = �

Σ∈H

(M, Z)

N _g,Σ e ^−t

t

Σ area - Mathematical “definition” of topological strings

Z(t) = exp �

g=0

(g

)

F

(t)

2D nonlinear sigma model whose target space is - Physical realization of topological strings

N = (2, 2) M

topological twist

(redefinition of the Lorentz charges) topological sigma model whose target space is

M

S

=

�

Σg

d

z √

gG

g

∂

X

∂

X

+ i�

∂

X

∂

X

+ · · ·

we can model the mathematical definition (curve counting).

X

(z), X

(z)

ρ

(z), ρ

(z)

scalars on

Grassmannian one-form on

χ

(z), χ

(z)

Σ _g

Σ _g Σ _g

The action of topological sigma model whose target space is is given by

M

This action is BRST-exact for a topological symmetry Q

S

= {Q

, V (X, ρ, χ) } +

�

Σ_g

X

(ω)

V = 1 4

�

Σ_g

d

z √

gg

G

�

ρ

∂

X

+ ρ

∂

X

�

[Q, X

] = 0 {Q, χ

} = 0

{Q, ρ

} = 2i∂

X

+ Γ

ρ

χ

�

Σ∈H²(M,Z)

e

�

[X(Σg)]=Σ

DXDχDρe

Then the path-integral is

topological : independent of and worldsheet metric

α

Then the path-integral is dominated by the minimum

≥

�

Σ_g

X

(ω)

Let us evaluate at

α

→ 0

∂

X

= ∂

X

= 0

The path-integral localizes to the integral over the holomorphic maps !!

S

= �

Σ_g

d

z G

(∂

X

∂

X

+ ∂

X

∂

X

)

= 2

�

Σ_g

d

z G

∂

X

∂

X

+

�

Σ_g

X

(ω)

X : Σ

→ Σ ⊂ M

we count“number” of holomorphic maps

F

:= �

hol.maps X

e

R

Σg X^∗(ω)

In this way the path-integral for genus- worldsheet implies the counting of the holomorphic maps from the worldsheets to the Calabi-Yau

g

In general it is very hard to compute such a function mathematically. . . Fortunately, for a wide class of non-compact CY, we can compute these partition function exactly by utilizing string dualities !!

So, in the following, we focus on the dual pictures of the topological strings.

2. Conifold and Geometric Transition

- Deformed Conifold

We can deform the conifold by introducing the (complex structure) deformation

� ∈ R

Let us regard it as a fibration. Base is

R × T

z = xy

xy = uv + �

{ ^|x|

|y|

|u| |v|

|x| = |y| = �

|z|

|u| = |v| = �

|z − �|

z = uv + �

|x|

|y|

|u| ^|v|

Re(z)

Fiber is and phases

R × T

R = Im(z) T

u → e

u, v → e

v x → e

x, y → e

y

z = xy

{

These cycles collapse on the locus or

z = 0 z = �

z = uv + �

z = 0

z = � α

β

{

z = 0

z = � S

|x| = |y| = |z|

|u| = |v| = |z − �|

Im(z) = 0

z = xy

z = uv + �

x = ¯ y

u = ¯ v

1/2

1/2

By substituting it into the definition eq., we get the form of the minimal line

xy = uv + �

|x|

+ |y|

= � . . . S

We choose a point of the non- compact fiber direction :

R × T

fiber

{

- Resolved Conifold

z = 0

z = �

At , the 3-sphere vanishes, and we recover the conifold

� = 0

xy = uv

We can smooth the singular conifold by the resolution

S

= P

t

- Geometric Transition

we can define closed topological string on it

[Witten, ’93]

S

we can wrap topological 3- branes on S³

gauge theory on the branes is the Chern-Simons theory on S³

open string theory

we will find open/closed duality duality between them !

- A-model topological string on resolved conifold [Gopakumar-Vafa, ’98]

F (g

, t) = �

d≥1

1 d

�

2 sin dg

2

�

e

By using this formula, we obtain

� 1

2 sin ^z₂�2 = − 1

z² + �

g=1

(−1)^g−1B_2g

2g(2g − 2)! z^2g−2 1

e^z − 1 = 1

z − 1

2 + �

g=1

(−1)^g−1B_2g

(2g)! z^2g−1

F = − 1 g_s²

�

d=1

e^−dt

d³ − 1

12 log(1 − e^−t) + �

g=2

g_s^2g−2 (−1)^g−1B_2g 2g(2g − 2)!

�

d=1

e^−dt d^3−2g

- large-N duality as gauge/gravity duality

amplitudes of gauge theory

‘tHooft’s idea

closed string genus expansion

!?

Let us study SU(N) Chern-Simons theory as the gauge theory [Witten, `88]:

F = log Z

= �

g

g

F

(λ) = log Z

= �

g,h

g

λ

F

λ = N g

Z

(S

) = 1

(k + N)

N −1

�

j=1

2 sin

jπ

k + N

Z

(S

) = 1

(k + N)

N −1

�

j=1

2 sin

jπ k + N

g

= 2π k + N

t = iλ = 2πiN k + N

F = log Z

= − N

2 log(k + N) +

�

(N − j) log(2 sin jπ

k + N )

= − N

2 log(k + N) −

�

(N − j) log( 2πj

k + N )

+

�

(N − j)

�

log(1 − g

j

4π

n

)

By using the formula sin(πz) = πz

�∞ n=1

(1 − z² n² )

F

F

:

:

F

F

=

N −1

�

j=1

(N − j)

�

�

1 k

1

n

j

� g

4π

�

ζ(2k)

=

�∞ k=1

ζ(2k) k

� g_s² 4π²

�k N −1�

j=1

(N − j)j^2k

�

j=1

j^k = N^k 2 +

[^k₂]

�

g=0

k+1C_2g N^k+1−2g B_2g k + 1

+

�

2k

C

N

B

2g

N^2k+2 B₀

(2k + 1)(2k + 2) power-sum formula

N

F

F

=

N −1

�

j=1

(N − j)

�

�

1 k

1

n

j

� g

4π

�

ζ(2k)

=

�∞ k=1

ζ(2k) k

� g_s² 4π²

�k N −1�

j=1

(N − j)j^2k

+

�

2k

C

N

B

2g

N^2k+2 B₀

(2k + 1)(2k + 2)

λ = N g

F

= g

(· · · ) + g

(· · · )

+ �

g=2

�

k=1

g

λ

ζ(2k) k

B

(2π)

C

`tHooft expansion !!

F

= �

g=2

g

F

(λ)

F

(λ) = B

�

h=0

λ

ζ(2g + 2h − 2) (2π)

2g+2h−2

C

g(2g + 2h − 2)

2g+2h−3

C

g(2g − 2)

�

n=1

1

(2πn)

+

�

g

B

2g(2g − 2)

B

(2g − 2)!

constant maps

F

= B

2g(2g − 2)

�

�

2 λ

C

1

(2πn)

= B

2g(2g − 2)

�

� 1

(2πn + λ)

+ 1

(2πn − λ)

�

= B

2g(2g − 2)

�

1

(2πn + λ)

F

= · · · + �

g≥2

g

B

2g(2g − 2)

1

(λ)

non-pert. part gives the n=0 part

binomial theorem

+

�

g

B

2g(2g − 2)

�

n∈Z

1

(2πn + λ)

F = ( g

, g

-terms ) + �

g=2

g

B

2g(2g − 2)

B

(2g − 2)!

�

g=2

g

B

2g(2g − 2)

(−1)

(2g − 3)!

�

d

e

This is precisely the world sheet instanton corrections for A- model on resolved conifold !!

�

n∈Z

1

n + z = 2πi

�

e

t = iλ

S

geometric transition

closed topological string on resolved conifold

open topological string on deformed conifold

=

In this way, we find out that the Chern-Simons partition function is equal to the topological string amplitude for the resolved conifold.

Worldsheet formulation of topological strings Conifold and geometric transition

Plan of the lecture

・ Geometric engineering of gauge theories

・ Application to AGT relation

Part 1

Part 2

Revision of Part 1

F (g

, t) = �

g≥0

�

β∈H²(X,Z)

�

d≥1

n

1 d

�

2 sin dg

2

�

e

S

geometric transition

closed topological string on resolved conifold

open topological string on deformed conifold

=

3. Geometric Engineering of Gauge Theory

In the following, we focus on string theory on toric CY (generalization of conifold)

toric Calabi-Yau : Local models of Calabi-Yau manifolds

(describe the structure in neighborhood of singurarity) Geometric engineering

AdS/CFT . . .

ADE singularity

ADE gauge symmetry

fibration over

R(|z|) C

S

(θ) z = |z|e

- complex plane

t

CP

- resolved conifold

- Symplectic quotient

toric date s.t.

moment map

: Calabi-Yau condition

� v

∈ Z

Q

∈ Z

µ

(z) =

N +3

�

j=1

Q

|z

|

z

→ e

z

M = C

//G

= ∩

µ

(t

)/G

�

j

Q

= 0

G = U (1)

N +3

�

i=1

Q

� v

= 0

a = 1, · · · , N

- Symplectic quotient

Y K

Y K

Y K

toric date

Calabi-Yau condition

toric diagram

web diagram

dual graph

- resolved conifold

Q

= +1, +1, −1, −1

� v

=



 1 0 0



 �v

=



 1 1 0



 �v

=



 1 0 1



 �v

=



 1 1 1





µ = �

µ_i ∈ Z≥0 | µ¹ ≥ µ² ≥ · · ·� - topological vertex formalism

2. Assign Young diagrams for each edges of these parts

1. Decompose a toric web-diagram into vertices and propagators

3. Glue them to get topological string partition function

How to compute topological string amplitudes for toric Calabi-Yau manifolds ?

Locally they look like a conifold Geometric transition enable us to

calculate these amplitudes using Chern- Simons theory

Topological vertex [AMKV, ‘03]

Decomposition of toric web-diagram Parts

: trivalent vertex . . . local patch

: edes . . . 1.

2.

CP

C

ν

λ

µ

µ _ν

vertex function

framing factor & propagator

f

(q)

(−1)

e

δµ, ν

C

(g

)

Gluing along a leg is done by the following procedure

vertex function propagator framing factro

�

ν

C

(−1)

e

(f

)

C

Z = · · · · · ·

ν ν

λ

µ

µ

λ

t

C

(q) = q

s

(q

) �

η

s

(q

)s

(q

)

S

(x

, x

, · · · , x

) = det(x

) det(x

)

s

(x)s

(x) = �

λ

c

s

(x)

s

(x) = �

λ

c

s

(x)

Schur function

skew Schur function

q

q

x

= q

x

= q

i = 1, 2, 3, · · ·

κ

= �

i

µ

(µ

+ 1 − 2i)

Q = e

q = e

Gluing rule

C

(q) = q

s

(q

) �

η

s

(q

)s

(q

)

vertex funstion

{

Z = · · · �

ν

C

(−Q)

(f

)

C

· · ·

- A-model topological string on resolved conifold [Gopakumar-Vafa, ’98]

F = �

d

1 d

Q

(q

− q

)

= �

n

− log(1 − Qq

)

^{Z = �}

n=1

(1 − Qq

)

F (g

, t) = �

d≥1

1 d

�

2 sin dg

2

�

e

Q = e

q = e

−

- topological vertex for the resolved conifold

Z = �

ν

C

(−Q)

C

= �

ν

s

(q

)(−Q)

s

(q

)

�

µ

s

(x)s

(y) = �

i,j

(1 + x

y

)

s

(Qx) = Q

s

(x)

=

�

(1 − Qq

)

{

t ν

C

C

- Large-N computation of the glueball superpotential

Type IIA theory with N D6-branes wrappiing on 3-sphere inside the deformed conifold

IR

SU(N) super Yang-Mills theory ... glueballN = 1

effective glueball superpotential is given by the open topological string

W (S) = N �

h

h S

F

= N F

(S)

[BCOV, ’94]

S = TrW_αW ^α

F

(S) = 1

2 S

log S + �

h=2

B

(2h − 2)(2h)! S

W (S) = N S log S + · · ·

= − �

n∈Z

(S + 2πin) log(S + 2πin)

Veneziano-Yankielowcz superpotential

infinitely many domain walls

generalization Dijkgraaf-Vafa theory

- geometric engineering

A singularity CY 3-fold₁

S

small

3-form field

C

massless U(1) gauge field

A

brane on

D2

D2 ¯

W

W

}

Let us consider compactification of Type IIA on the CY.

SU(2) Yang-Mills field !

Type IIA on

ADE singularity

N=2 4D ADE gauge theory

Topological A-model on [BCOV, ’94]

S

=

�

d

x ∂

∂

F

(t) F

∧ F

+ �

d

x �

g=1

F

(t) (T

)

R

U(1) gauge field strength in N=2 SUGRA multiplet LEEA takes the following form

F

�T � ∼ �

� ∼ g

- Seiberg-Witten theory

Low energy effective action of gauge theory is described by Seiberg- Witten prepotential ^{N = 2}

�Φ� = diag(a

, · · · , a

)

Low energy effective theory is Higgsed by vevSU (N ) → U(1)^{N −1}

are Cartan gauge fields

S

=

�

τ

F

F

d

x + · · ·

τ

= ∂

∂a

∂a

F(a, Λ)

F

_{i = 1,} · · · , N − 1 U (1)

- Nekrasov formulae [Nekrasov, ‘02]

Nekrasov gave the generating function of Seiberg-Witten prepotential via instanton caluculus

Z

(a, Λ, �) = exp

�

�

F

(a, Λ)

F

(a, Λ) = F

(a, Λ)

The higher genus terms correspond to the corrections from the N=2 supergravity background fields (graviphotons)

Topological BRST symmetry Q reduces the “path integral” to usual integral.

the local minima of the action dominate the integral

�F �

instanton

F + �F = 0

In the presence of the gravitational background, the N=2 theory becomes a topological field theory

Z =

�

DA · · · e

The localization formula reduces these Nekrasov partition functions to combinatrical expression (without integration)

Z = �

Y₁,··· ,Y^N

q

z

(�a, �

; � Y )z

(�a, m, �

; � Y )

z

(�a, �

; � Y )

=

�

�

(i,j)∈Y^a

(a

− a

− �

(Y

− i + 1) + �

(Y

− j + 1))

× �

(i,j)∈Y^b

(a_a − a^b + �₁(Y_bi − j + 1) − �²(Y_aj^T − i + 1))⁻¹

�

= −�

= �

We will focus on the case of

Example : local Hirzebruch surface

K → P

× P

This geometry is a fibration of singularity over base . The SU(2) gauge symmetry emerges from the string theory on the singularity.

A

P

Y

Y

P

P

Z = �

µ₁,µ₂,µ₃,µ₄

Q

Q

q

× C

C

C

C

= �

µ₂,µ₄

Q

q

K

(Q

) K

(Q

)

N=2 SU(2) gauge theory

K

(Q) = S

1

(q

)S

(q

) �

W

Q

S

(q

)S

(q

)

� �

W

Q^{|W |}S_W (q^−ρ)S_W (q^−ρ)

= S

1

(q

)S

(q

)

�

1 − Qq

1 − Qq

S_Y1(q^−ρ) = q^{�Y T�} �

t∈Y

(1 − q^{hY (t)})⁻¹ where

Q

= (βΛ)

F F

F

F

F

Q

= e

q = e

R

× S

This is precisely the Nekrasov partition function of the N=2 SU(2) pure SYM on

We can recover the 4-dimensional result by taking

β → 0

λ λ µ

µ

ν ν

C

grand-canonical ensemble melting

crystals topological vertex !

- Duality to Crystal melting [Okounkov-Leshetikin-Vafa, ‘04]

Z

= �

crystals

e

^C

^(g

⁾

1

k

T = g

4. AGT conjecture

B ^Liouville (q) = Z _Nekrasov (q)

[Alday-Gaiotto-Tachikawa, ‘09]

“AGT relation” : The equivalence between a Nekrasov instanton partition function of 4D gauge theory and the conformal block of 2D Liouville CFT on the associated Rieman surface

SU(2) quiver gauge theory

pants decomposition

conformal block on the surface

Free-field CFT vs U(1)-theory

Let us consider the free CFT with the background charge Q

�

�

e

� = �

i<j

(z

− z

)

�φ(z

)φ(z

)� = − ln(z

− z

)

z

= 0, q, 1, ∞

B

= (1 − q)

m

m

We choose

(1 − q)

= �

Y

q

�

(i,j)∈Y

�

f =1,2

φ(m

, (i, j)) E(0, Y, (i, j))

Some algebra leads to the following expansion

φ(m, (i, j)) = m + i − j

E(a, Y, (i, j)) = a + (Y

− i + 1) + (Y

− j)

�

Y

χ

(x)χ

(y) = exp �

n

nx

y

where we use

( )

This is the Nekrasov partition function for the “U(1) gauge theory with 2-flavors”!

B

= Z _{U (1),N}

₌₂

- “Derivation” of AGT conjecture [Dijkgraaf-Vafa, `09]

Recall the geometric transition of A-model topological string

N A-branes

Mirror B-model topological string of the setup is

N B-branes

S

S

S

closed A-model geometry corresponding to SU(2) gauge theory with 4-flavors is

open B-model on B-branes on

S

m

m

m

m

mirror symmetry

&

geometric transition

N

N₂

N

N

m

= g

N

open B-model = matrix model

N B-branes insertion of [Dijkgraaf-Vafa, `02]

V

= det(Φ − Λ)

�

�

V

� =

�

N ×N

dΦ

�

det(Φ − Λ

)

=

�

d^Nz �

I<J

(z_I − z^J)² �

i,I

det(z_I − Λⁱ)^Ni

Coulomb-gas expression of the conformal block !!

: Dotsenko-Fateev integral

The geometric transition provides a “proof” of the mysterious AGT relation.

・ formulation of topological strings

・ geometric transition

Summary

・ Applications of geometric transition

AGT relation

glueball superpotential

Nekrasov partition function

Fin