**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

** / **

^{京都大学}

^{ 基礎物理学研究所}**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**

**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 × R^{4} ** gauge theory on**

^{N = 1, 2} R

^{4}

**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

^{i}

### , X

^{¯}

^{i}

### i, ¯i = 1, · · · , d

**Let us consider a complex manifold**

### d dim

_{C}

### (= 2d dim

_{R}

### )

### dω = 0

^{for}### ω = i G

_{i¯}

_{j}

### dX

^{i}

### ∧ dX

^{¯}

^{j}

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

### c

_{1}

### (M) = 0

### R

_{ij}

### = 0

**- Calabi-Yau 1-fold**

### T

^{2}

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

### X

^{1}

### = z X

^{¯}

^{1}

### = ¯ z

### z ∼ z + 1

**- Calabi-Yau 1-fold**

### T

^{2}

### α

### β α β

### [Σ] = pα + qβ ∈ H

^{1}

### (T

^{2}

### , 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

^{2}

### + y

^{2}

### + u

^{2}

### + v

^{2}

### = 0 C

^{4}

### S

^{3}

### S

^{2}

### 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

_{g}

### → Σ ⊂ M N

_{g,Σ}

**“number” of holomorphic maps**

### F _{g} = �

### Σ∈H

^{2}

### (M, Z)

### N _{g,Σ} e ^{−t}

^{Σ}

### t _{Σ}

**Kahler parameter (area) of 2-cycle**

### Σ

**- Mathematical “definition” of topological strings**

### X :

X : Σ_{g} → Σ ⊂ M

N_{g,Σ} _{maps}

### F _{g} = �

### Σ∈H

^{2}

### (M, Z)

### N _{g,Σ} e ^{−t}

^{Σ}

### t

_{Σ}

Σ **area**
**- Mathematical “definition” of topological strings**

### Z(t) = exp �

g=0

### (g

_{s}

### )

^{2g−2}

### F

_{g}

### (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

**[Witten, ’88]**

### S

^{A}

### =

### �

Σg

### d

^{2}

### z √

### gG

_{i¯}

_{j}

### g

^{µν}

### ∂

_{µ}

### X

^{i}

### ∂

_{ν}

### X

^{¯}

^{j}

### + i�

^{µν}

### ∂

_{µ}

### X

^{i}

### ∂

_{ν}

### X

^{¯}

^{j}

### + · · ·

**we can model the mathematical definition (curve counting).**

### X

^{i}

### (z), X

^{¯}

^{i}

### (z)

### ρ

^{i}

_{µ}

_{¯}

### (z), ρ

^{¯}

^{i}

_{µ}

### (z)

**scalars on**

**Grassmannian one-form on**

### χ

^{i}

### (z), χ

^{¯}

^{i}

### (z)

**Grassmannian scalar on**

### 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

^{A}

### = {Q

^{BRST}

### , V (X, ρ, χ) } +

### �

Σ_{g}

### X

^{∗}

### (ω)

### V = 1 4

### �

Σ_{g}

### d

^{2}

### z √

### gg

^{µν}

### G

_{i¯}

_{j}

### �

### ρ

^{i}

_{µ}

### ∂

_{ν}

### X

^{¯}

^{j}

### + ρ

^{¯}

^{j}

_{µ}

### ∂

_{ν}

### X

^{i}

### �

### [Q, X

^{i}

### ] = 0 {Q, χ

^{i}

### } = 0

### {Q, ρ

^{i}

_{z}

_{¯}

### } = 2i∂

^{z}

^{¯}

### X

^{i}

### + Γ

^{i}

_{jk}

### ρ

^{j}

_{z}

_{¯}

### χ

^{k}

### �

Σ∈H^{2}(M,Z)

### e

^{−}

^{α�}

^{1}

^{t}

^{Σ}

### �

[X(Σg)]=Σ

### DXDχDρe

^{−}

^{α�}

^{1}

^{{Q,V }}

**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

### ∂

_{z}

_{¯}

### X

^{i}

### = ∂

_{z}

### X

^{¯}

^{j}

### = 0

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

### S

_{boson}

^{A}

### = �

Σ_{g}

### d

^{2}

### z G

_{i¯}

_{j}

### (∂

_{z}

### X

^{i}

### ∂

_{z}

_{¯}

### X

^{¯}

^{j}

### + ∂

_{z}

_{¯}

### X

^{i}

### ∂

_{z}

### X

^{¯}

^{j}

### )

### = 2

### �

Σ_{g}

### d

^{2}

### z G

_{i¯}

_{j}

### ∂

_{z}

_{¯}

### X

^{i}

### ∂

_{z}

### X

^{¯}

^{j}

### +

### �

Σ_{g}

### X

^{∗}

### (ω)

### X : Σ

_{g}

### → Σ ⊂ M

**we count“number” of holomorphic maps**

### F

_{g}

### := �

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

^{2}

### 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

^{2}

### R = Im(z) T

^{2}

### u → e

^{iβ}

### u, v → e

^{−iβ}

### v x → e

^{iα}

### x, y → e

^{−iα}

### y

### z = xy

### {

**These cycles collapse on the locus or **

### z = 0 z = �

### z = uv + �

### z = 0

### z = � α

### β

### {

### z = 0

### z = � S

^{3}

### |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 + �

**at the locus**

### |x|

^{2}

### + |y|

^{2}

### = � **. . . ** S

^{3}

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

### R × T

^{2}

**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

^{2}

### = P

^{1}

### t

**- Geometric Transition**

**we can define closed **
**topological string on it**

**[Witten, ’93]**

### S

^{3}

**we can wrap topological 3-**
**branes on** S^{3}

**gauge theory on the branes is **
**the Chern-Simons theory on** S^{3}

**open string theory**

**we will find open/closed duality duality between them !**

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

### F (g

_{s}

### , t) = �

d≥1

### 1 d

### �

### 2 sin dg

_{s}

### 2

### �

_{−2}

### e

^{−dt}

**By using this formula, we obtain**

� 1

2 sin ^{z}_{2}�2 = − 1

z^{2} + �

g=1

(−1)^{g−1}B_{2g}

2g(2g − 2)! z^{2g−2}
1

e^{z} − 1 = 1

z − 1

2 + �

g=1

(−1)^{g−1}B_{2g}

(2g)! z^{2g−1}

F = − 1
g_{s}^{2}

�

d=1

e^{−dt}

d^{3} − 1

12 log(1 − e^{−t}) + �

g=2

g_{s}^{2g−2} (−1)^{g−1}B_{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

_{gauge}

### = �

g

### g

_{s}

^{2g−2}

### F

_{g}

### (λ) = log Z

_{string}

### = �

g,h

### g

_{s}

^{2g−2}

### λ

^{2g−2+h}

### F

_{g,h}

### λ = N g

_{s}

### Z

^{CS}

### (S

^{3}

### ) = 1

### (k + N)

^{N/2}

N −1

### �

j=1

### 2 sin

^{N −j}

### jπ

### k + N

### Z

^{CS}

### (S

^{3}

### ) = 1

### (k + N)

^{N/2}

N −1

### �

j=1

### 2 sin

^{N −j}

### jπ k + N

### g

_{s}

### = 2π k + N

### t = iλ = 2πiN k + N

### F = log Z

^{CS}

### = − N

### 2 log(k + N) +

### �

N j=1### (N − j) log(2 sin jπ

### k + N )

### = − N

### 2 log(k + N) −

### �

N j=1### (N − j) log( 2πj

### k + N )

### +

### �

N j=1### (N − j)

### �

∞ n=1### log(1 − g

_{s}

^{2}

### j

^{2}

### 4π

^{2}

### n

^{2}

### )

**By using the formula** sin(πz) = πz

�∞ n=1

(1 − z^{2}
n^{2} )

### F

^{non-pert}

### F

^{pert}

### :

### :

### F

^{pert}

### F

^{pert}

### =

N −1

### �

j=1

### (N − j)

### �

∞ n=1### �

∞ k=1### 1 k

### 1

### n

^{2k}

### j

^{2k}

### � g

_{s}

^{2}

### 4π

^{2}

### �

^{k}

### ζ(2k)

=

�∞ k=1

ζ(2k) k

� g_{s}^{2}
4π^{2}

�k N −1�

j=1

(N − j)j^{2k}

�

j=1

j^{k} = N^{k}
2 +

[^{k}_{2}]

�

g=0

k+1C_{2g} N^{k+1−2g} B_{2g}
k + 1

### +

### �

k g=12k

### C

_{2g−2}

### N

^{2k+2−2g}

### B

_{2g}

### 2g

N^{2k+2} B_{0}

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

N

### F

^{pert}

### F

^{pert}

### =

N −1

### �

j=1

### (N − j)

### �

∞ n=1### �

∞ k=1### 1 k

### 1

### n

^{2k}

### j

^{2k}

### � g

_{s}

^{2}

### 4π

^{2}

### �

^{k}

### ζ(2k)

=

�∞ k=1

ζ(2k) k

� g_{s}^{2}
4π^{2}

�k N −1�

j=1

(N − j)j^{2k}

### +

### �

k g=12k

### C

_{2g−2}

### N

^{2k+2−2g}

### B

_{2g}

### 2g

N^{2k+2} B_{0}

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

### λ = N g

_{s}

### F

^{pert}

### = g

_{s}

^{−2}

### (· · · ) + g

_{s}

^{0}

### (· · · )

### + �

g=2

### �

k=1

### g

_{s}

^{2g−2}

### λ

^{2k+2−2g}

### ζ(2k) k

### B

_{2g}

### (2π)

^{2k 2k}

### C

_{2g−2}

**`tHooft expansion !!**

### F

_{(g≥2)}

### = �

g=2

### g

_{s}

^{2g−2}

### F

_{g}

### (λ)

### F

_{g(≥2)}

### (λ) = B

_{2g}

### �

h=0

### λ

^{2h}

### ζ(2g + 2h − 2) (2π)

^{2g+2h−2}

2g+2h−2

### C

_{2g−2}

### g(2g + 2h − 2)

2g+2h−3

### C

_{2h}

### g(2g − 2)

### �

n=1

### 1

### (2πn)

^{2g+2h−2}

### +

### �

∞ g=2### g

_{s}

^{2g−2}

### B

_{2g}

### 2g(2g − 2)

### B

_{2g−2}

### (2g − 2)!

**constant maps**

### F

_{g (≥2)}

### = B

_{2g}

### 2g(2g − 2)

### �

∞ n=1### �

∞ h=0### 2 λ

^{2h}

_{2g+2h−2}

### C

_{2h}

### 1

### (2πn)

^{2g+2h−2}

### = B

_{2g}

### 2g(2g − 2)

### �

∞ n=1### � 1

### (2πn + λ)

^{2g−2}

### + 1

### (2πn − λ)

^{2g−2}

### �

### = B

_{2g}

### 2g(2g − 2)

### �

∞ n∈Z�=0### 1

### (2πn + λ)

^{2g−2}

### F

^{non-pert}

### = · · · + �

g≥2

### g

_{s}

^{2g−2}

### B

_{2g}

### 2g(2g − 2)

### 1

### (λ)

^{2g−2}

**non-pert. part gives the n=0 part**

**binomial theorem**

### +

### �

∞ g=2### g

_{s}

^{2g−2}

### B

_{2g}

### 2g(2g − 2)

### �

n∈Z

### 1

### (2πn + λ)

^{2g−2}

### F = ( g

_{s}

^{−2}

### , g

_{s}

^{0}

### -terms ) + �

g=2

### g

_{s}

^{2g−2}

### B

_{2g}

### 2g(2g − 2)

### B

_{2g−2}

### (2g − 2)!

### �

g=2

### g

_{s}

^{2g−2}

### B

_{2g}

### 2g(2g − 2)

### (−1)

^{g}

### (2g − 3)!

### �

∞ d=1### d

^{2g−3}

### e

^{−dt}

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

### �

n∈Z

### 1

### n + z = 2πi

### �

∞ d=0### e

^{−2πidz}

### t = iλ

### S

^{3}

**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

_{s}

### , t) = �

g≥0

### �

β∈H^{2}(X,Z)

### �

d≥1

### n

^{g}

_{β}

### 1 d

### �

### 2 sin dg

_{s}

### 2

### �

_{2g−2}

### e

^{−d�β,t�}

### S

^{3}

**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

^{1}

### (θ) z = |z|e

^{iθ}

^{:}**- complex plane**

### t

### CP

^{1}

**- resolved conifold**

**- Symplectic quotient**

**toric date** **s.t.**

**moment map**

**: Calabi-Yau condition**

### � v

_{i}

### ∈ Z

^{3}

### Q

^{a}

_{i}

### ∈ Z

### µ

_{a}

### (z) =

N +3

### �

j=1

### Q

^{a}

_{j}

### |z

^{j}

### |

^{2}

### z

_{j}

### → e

^{i}

^{P}

^{a}

^{Q}

^{j}

^{a}

^{α}

^{a}

### z

_{j}

### M = C

^{N +3}

### //G

### = ∩

^{N}

_{a=1}

### µ

^{−1}

_{a}

### (t

_{a}

### )/G

### �

j

### Q

^{j}

_{a}

### = 0

### G = U (1)

^{N}

N +3

### �

i=1

### Q

^{a}

_{i}

### � v

_{i}

### = 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

_{i}

### = +1, +1, −1, −1

### � v

_{1}

### =

###

### 1 0 0

###

### �v

_{2}

### =

###

### 1 1 0

###

### �v

_{3}

### =

###

### 1 0 1

###

### �v

_{4}

### =

###

### 1 1 1

###

###

µ = �

µ_{i} ∈ Z≥0 | µ^{1} ≥ µ^{2} ≥ · · ·�
**- 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

^{1}

### C

^{3}

### ν

### λ

### µ

### µ _{ν}

**vertex function**

**framing factor & propagator**

### f

_{µ}

### (q)

^{n}

### (−1)

^{|µ|}

### e

^{−t|µ|}

### δµ, ν

^{t}

### C

_{λ,µ,ν}

### (g

_{s}

### )

**Gluing along a leg is done by the following procedure**

**vertex function** **propagator** **framing factro**

### �

ν

### C

_{λµν}

### (−1)

^{|ν|}

### e

^{−t|ν|}

### (f

_{ν}

### )

^{n}

### C

_{λ}

^{�}

_{µ}

^{�}

_{ν}

^{t}

### Z = · · · · · ·

### ν ν

^{t}

### λ

### µ

^{�}

### µ

### λ

^{�}

### t

### C

_{λµν}

### (q) = q

^{κ}

^{µ}

^{/2}

### s

_{ν}

^{t}

### (q

^{−ρ}

### ) �

η

### s

_{λ}

^{t}

_{/η}

### (q

^{−ν−ρ}

### )s

_{µ/η}

### (q

^{−ν}

^{t}

^{−ρ}

### )

### S

_{λ}

### (x

_{1}

### , x

_{2}

### , · · · , x

^{N}

### ) = det(x

_{i}

^{λ}

^{j}

^{+N −j}

### ) det(x

_{i}

^{N −j}

### )

### s

_{µ}

### (x)s

_{ν}

### (x) = �

λ

### c

^{λ}

_{µ,ν}

### s

_{λ}

### (x)

### s

_{λ/µ}

### (x) = �

λ

### c

^{λ}

_{µ,ν}

### s

_{ν}

### (x)

**Schur function**

**skew Schur function**

### q

^{−ρ}

### q

^{−µ−ρ}

### x

_{i}

### = q

^{−µ}

^{i}

^{+i−}

^{1}

^{2}

### x

_{i}

### = q

^{i−}

^{1}

^{2}

i = 1, 2, 3, · · ·

### κ

_{µ}

### = �

i

### µ

_{i}

### (µ

_{i}

### + 1 − 2i)

### Q = e

^{−t}

### q = e

^{−g}

^{s}

**Gluing rule**

### C

_{λµν}

### (q) = q

^{κ}

^{µ}

^{/2}

### s

_{ν}

^{t}

### (q

^{−ρ}

### ) �

η

### s

_{λ}

^{t}

_{/η}

### (q

^{−ν−ρ}

### )s

_{µ/η}

### (q

^{−ν}

^{t}

^{−ρ}

### )

**vertex funstion**

### {

### Z = · · · �

ν

### C

_{λµν}

### (−Q)

^{|ν|}

### (f

_{ν}

### )

^{n}

### C

_{λ}

^{�}

_{µ}

^{�}

_{ν}

^{T}

### · · ·

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

### F = �

d

### 1 d

### Q

^{d}

### (q

^{d/2}

### − q

^{−d/2}

### )

^{2}

### = �

n

### − log(1 − Qq

^{n}

### )

^{n}

^{Z =} ^{�}

n=1

### (1 − Qq

^{n}

### )

^{n}

### F (g

_{s}

### , t) = �

d≥1

### 1 d

### �

### 2 sin dg

_{s}

### 2

### �

_{−2}

### e

^{−dt}

### Q = e

^{−t}

### q = e

^{−g}

^{s}

### −

**- topological vertex for the resolved conifold**

### Z = �

ν

### C

_{φφν}

### (−Q)

^{|ν|}

### C

_{φφν}

^{t}

### = �

ν

### s

_{ν}

### (q

^{−ρ}

### )(−Q)

^{|ν|}

### s

_{ν}

^{t}

### (q

^{−ρ}

### )

### �

µ

### s

_{µ}

^{t}

### (x)s

_{µ}

### (y) = �

i,j

### (1 + x

_{i}

### y

_{j}

### )

### s

_{µ}

### (Qx) = Q

^{|µ|}

### s

_{µ}

### (x)

### =

### �

∞ n=1### (1 − Qq

^{n}

### )

^{n}

### {

### t ν

### C

_{ΦΦν}

### C

_{ΦΦν}

^{T}

**- 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 ... glueball**N = 1

**effective glueball superpotential is given by the open topological **
**string**

### W (S) = N �

h

### h S

^{h−1}

### F

_{g=0,h}

### = N F

_{g=0}

^{�}

### (S)

**[BCOV, ’94]**

S = TrW_{α}W ^{α}

### F

_{0}

### (S) = 1

### 2 S

^{2}

### log S + �

h=2

### B

_{2h−2}

### (2h − 2)(2h)! S

^{2h}

### W (S) = N S log S + · · ·

### = − �

n∈Z

### (S + 2πin) log(S + 2πin)

^{N}

**Veneziano-Yankielowcz superpotential**

**infinitely many domain walls**

**generalization**
**Dijkgraaf-Vafa theory**

**- geometric engineering **

**A singularity CY 3-fold**_{1}

### S

^{2}

**small**

**3-form field**

### C

_{µνλ}

**massless U(1) gauge **
**field**

### A

^{3}

_{µ}

**brane on**

### D2

### D2 ¯

^{brane on}### W

_{µ}

^{+}

### W

_{µ}

^{−}

## }

**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

_{eﬀ}

### =

### �

### d

^{4}

### x ∂

_{t}

_{i}

### ∂

_{t}

_{j}

### F

_{0}

### (t) F

^{i}

### ∧ F

^{j}

### + �

### d

^{4}

### x �

g=1

### F

_{g}

### (t) (T

^{2}

### )

^{g−1}

### R

^{2}

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

### F

_{g}

**is the topological string free energy !**

### �T � ∼ �

### � ∼ g

^{s}

**- Seiberg-Witten theory**

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

### �Φ� = diag(a

^{1}

### , · · · , a

^{N}

### )

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

**are Cartan gauge fields**

### S

_{eﬀ}

### =

### �

### τ

_{ij}

### F

_{µν}

^{i}

### F

^{jµν}

### d

^{4}

### x + · · ·

### τ

_{ij}

### = ∂

^{2}

### ∂a

_{i}

### ∂a

_{j}

### F(a, Λ)

### F

_{µν}

^{i}

_{i = 1,} · · · , N − 1 U (1)

^{N −1}

**- Nekrasov formulae [Nekrasov, ‘02]**

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

### Z

^{Nek.}

### (a, Λ, �) = exp

### �

∞ g=0### �

^{2g−2}

### F

^{g}

### (a, Λ)

### F

^{0}

### (a, Λ) = F

^{SW}

### (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 �

^{2}

**instanton**

### F + �F = 0

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

### Z =

### �

### DA · · · e

^{−}

^{g2}

^{1}

^{{Q,V }}

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

### Z = �

Y_{1},··· ,Y^{N}

### q

^{| �}

^{Y |}

### z

_{vector}

### (�a, �

_{1,2}

### ; � Y )z

_{matters}

### (�a, m, �

_{1,2}

### ; � Y )

### z

_{vector}

### (�a, �

_{1,2}

### ; � Y )

### =

### �

N a,b=1### �

(i,j)∈Y^{a}

### (a

_{a}

### − a

^{b}

### − �

^{1}

### (Y

_{bj}

^{T}

### − i + 1) + �

^{2}

### (Y

_{ai}

### − j + 1))

^{−1}

× �

(i,j)∈Y^{b}

(a_{a} − a^{b} + �_{1}(Y_{bi} − j + 1) − �^{2}(Y_{aj}^{T} − i + 1))^{−1}

### �

_{1}

### = −�

^{2}

### = �

**We will focus on the case of**

**Example : local Hirzebruch surface **

### K → P

^{1}

### × P

^{1}

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

### A

_{1}

### P

^{1}

### Y

_{1}

### Y

_{2}

### P

^{1}

_{base}

### P

^{1}

_{fiber}

### Z = �

µ_{1},µ_{2},µ_{3},µ_{4}

### Q

_{F}

^{|µ}

^{1}

^{|+|µ}

^{3}

^{|}

### Q

_{B}

^{|µ}

^{2}

^{|+|µ}

^{4}

^{|}

### q

^{−κ}

^{µ1}

^{/2 +κ}

^{µ2}

^{/2 −κ}

^{µ3}

^{/2 −κ}

^{µ4}

^{/2}

### × C

^{φ µ}1 µ

_{4}

^{t}

### C

φ µ_{2}

^{t}µ

_{1}

^{t}

### C

^{µ}2 φ µ

_{3}

### C

φ µ_{4}µ

_{3}

^{t}

### = �

µ_{2},µ_{4}

### Q

_{B}

^{|µ}

^{2}

^{|+|µ}

^{4}

^{|}

### q

^{+κ}

^{µ2}

^{/2 −κ}

^{µ4}

^{/2}

### K

^{µ}

_{4}

^{µ}

_{2}

### (Q

_{F}

### ) K

^{µ}

_{2}

^{t}

^{µ}

_{4}

^{t}

### (Q

_{F}

### )

**N=2 SU(2) gauge theory**

### K

_{Y}

_{1}

_{,Y}

_{2}

### (Q) = S

_{Y}

^{T}

1

### (q

^{−ρ}

### )S

_{Y}

_{2}

### (q

^{−ρ}

### ) �

W

### Q

^{|W |}

### S

_{W}

### (q

^{−Y}

^{1}

^{−ρ}

### )S

_{W}

### (q

^{−Y}

^{2}

^{T}

^{−ρ}

### )

� �

W

Q^{|W |}S_{W} (q^{−ρ})S_{W} (q^{−ρ})

### = S

_{Y}

^{T}

1

### (q

^{−ρ}

### )S

_{Y}

_{2}

### (q

^{−ρ}

### )

### �

∞ i,j=1### 1 − Qq

^{i+j−1}

### 1 − Qq

^{−Y}

^{1i}

^{−Y}

^{2 j}

^{T}

^{+i+j−1}

S_{Y}_{1}(q^{−ρ}) = q^{�Y} ^{T}^{�} �

t∈Y

(1 − q^{h}^{Y} ^{(t)})^{−1}
**where**

### Q

_{B}

### = (βΛ)

^{4}

F F

F

F

F

### Q

_{F}

### = e

^{−4βa}

### q = e

^{−2β�}

### R

^{4}

### × S

^{1}

^{β}

**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

^{3}

**-patch**

**melting crystal corner**

**grand-canonical ensemble melting **

**crystals** **topological vertex !**

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

### Z

_{λ,µ,ν}

### = �

crystals

### e

^{−g}

^{s}

^{#(boxes)}

^{C}

^{λ,µ,ν}

^{(g}

^{s}

^{)}

### 1

### k

_{B}

### T = g

_{s}

**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**

### �

### �

4 i=1### e

^{α}

^{i}

^{φ(z}

^{i}

^{)}

### � = �

i<j

### (z

_{i}

### − z

^{j}

### )

^{−α}

^{i}

^{α}

^{j}

### �φ(z

^{1}

### )φ(z

_{2}

### )� = − ln(z

^{i}

### − z

^{j}

### )

### z

_{1,··· ,4}

### = 0, q, 1, ∞

### B

free CFT### = (1 − q)

^{−α}

^{2}

^{α}

^{3}

### m

_{1}

### m

_{2}

**We choose**

### (1 − q)

^{−m}

^{1}

^{m}

^{2}

### = �

Y

### q

^{|Y |}

### �

(i,j)∈Y

### �

f =1,2

### φ(m

_{f}

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

^{2}

**Some algebra leads to the following expansion**

### φ(m, (i, j)) = m + i − j

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

_{j}

^{T}

### − i + 1) + (Y

^{i}

### − j)

### �

Y

### χ

_{Y}

### (x)χ

_{Y}

### (y) = exp �

n

### nx

_{n}

### y

_{n}

**where we use**

### ( )

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

### B

^{free CFT}

### = Z _{U (1),N}

_{f}

_{=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

^{2}

### S

^{3}

### S

^{3}

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

**open B-model on**
**B-branes on**

### S

^{2}

### m

_{1}

### m

_{2}

### m

_{3}

### m

_{4}

**mirror symmetry **

**&**

**geometric transition**

### N

_{1}

N_{2}

### N

_{3}

### N

_{4}

### m

_{i}

### = g

_{s}

### N

_{i}

**open B-model = matrix model**

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

### V

_{N}

### = det(Φ − Λ)

^{N}

### �

### �

4 i=1### V

_{N}

_{i}

### � =

### �

N ×N

### dΦ

### �

4 i=1### det(Φ − Λ

^{i}

### )

^{N}

^{i}

=

�

d^{N}z �

I<J

(z_{I} − z^{J})^{2} �

i,I

det(z_{I} − Λ^{i})^{N}^{i}

**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**