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Surface Operator, Topological B-model, and Topological Recursion

Masahide Manabe

Nagoya University

December, 2010 @ Taiwan

Collaboration with H.Awata, H.Fuji, H. Kanno and Y.Yamada, arXiv:1008.0574 [hep-th]

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 1 / 37

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.

.

Contents

.

. . 1 1. Introduction and overview

.

. .

2 2. Irregular conformal block

.

. .

3 3. Free energy on Seiberg-Witten curve

.

. .

4 4. Free energy on mirror curve

.

. .

5 5. Conclusion and outlook

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 2 / 37

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1. Introduction and overview

.

Contents

.

. . 1 1. Introduction and overview

.

. .

2 2. Irregular conformal block

.

. .

3 3. Free energy on Seiberg-Witten curve

.

. .

4 4. Free energy on mirror curve

.

. .

5 5. Conclusion and outlook

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 3 / 37

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1. Introduction and overview

.

.

1. Introduction and overview

References:

1. [AGT]:L.F.Alday, D.Gaiotto and Y.Tachikawa, “Liouville Correlation

Functions from Four-dimensional Gauge Theories,” Lett. Math. Phys. 91, 167 (2010) [arXiv:0906.3219 [hep-th]].

2. [AGGTV]:L.F.Alday, D.Gaiotto, S.Gukov, Y.Tachikawa and H.Verlinde,

“Loop and surface operators in N=2 gauge theory and Liouville modular geometry,” JHEP 1001, 113 (2010) [arXiv:0909.0945 [hep-th]].

3. [DV]:R.Dijkgraaf and C.Vafa, “Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems,” arXiv:0909.2453 [hep-th].

4. [KPW]:C.Kozcaz, S.Pasquetti and N.Wyllard, “A & B model approaches to surface operators and Toda theories,” JHEP 1008, 042 (2010)

[arXiv:1004.2025 [hep-th]].

5. [DGH]:T.Dimofte, S.Gukov and L.Hollands, “Vortex Counting and Lagrangian 3-manifolds,” arXiv:1006.0977 [hep-th].

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 4 / 37

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M 0 1 2 3 4 5 6 7 8 9 10 twoM5 Cg,n Cg,n Cg,n Cg,n

M2

Cg,n: genus g Riemann surfice with n punctures g: genus of (Gaiotto) quiver

n: number of SU(2)flavor group 4d viewpoint:

TheM2-brane is considered as asurface operatorwithN = (2,2) SUSY on(0,1)-plane.

2d viewpoint:

This theory is described by the Liouville CFT onCg,n[AGT]. The M2-brane is considered as a degenerate primary operator Φ1,2(z) =exp(−φ(z)/b)in the Liouville CFT.

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 5 / 37

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1. Introduction and overview

The Liouville CFT onCg,ndescribes theUV regionof the gauge theory, and the Seiberg-Witten (Gaiotto) curveCSWis obtained as aramified double coverofCg,n.

.

.

.. .

.

.

We introduce mass scale~as1=b~,2=b1~, and rescaled a,mi as a→a/~,mi →mi/~, and then the SW curveCSWis obtained from the “semiclassical limit”1,2a,mi of thestress tensor T(z):

hT(z)Vm2(q)Vm1(1)i{a}∼ − 1

~2φSW(z)hVm2(q)Vm1(1)i{a}, where Vm(z) =exp(2mφ(z))is the primary operator with conformal weight∆m=m(Q−m), (Q=b+b1).

Above discussion can be generalized toasymptotically free theory. In the case, on the CFT side“irregular” state (Gaiotto state) “|Gi”is introduced. −→Section 2

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 6 / 37

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1. Introduction and overview

Irregular conformal block withΦ1,2insertion has a form hG0|Φ1,2(z)Vm1(z1)· · ·Vmn(zn)|Gi{ai}=exp

(−F(ai)

12 +W(ai,z)

1 +· · ·) ,

and then by combining the above discussion, we obtain W(ai,z) =±

z

φSW(z0)dz0, CSW : x2= φSW(z).

.. .

.

.

Therefore the subleading termW(ai,z)is obtained as the Abel-Jacobi map on the SW curveCSW.

In sectiron 3, higher order terms are discussed by theEynard-Orantin topological recursion relationon SW curve.

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 7 / 37

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1. Introduction and overview

Geometric engineering:(Dijkgraaf-Hollands-Sulkowski-Vafa, arXiv:0709.4446[hep-th])

M 0 1 2 3 4 5 6 7 8 9 10 (S1)

twoM5 C0,4 C0,4 C0,4 C0,4

M2

radius of S10

IIA 0 1 2 3 4 5 6 7 8 9

twoNS5 C0,4 C0,4 C0,4 C0,4

D2

T-duality[Ooguri-Vafa]& mirror symmetry IIA : R4×toric CY3(with alocal A1singularity)

..

DGH

inserted atoric brane

..

Brane D4-brane

IIA 0 1 2 3 4 5 6 7 8 9

D4 L L L

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 8 / 37

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1. Introduction and overview

By lifting to M theory we obtain Claim 2

M : R4×S1(β)×toric CY3(with alocal A1singularity) inserted atoric braneM5-brane Therefore one can expect that

. .

.. .

.

.

topological open string amplitudeon toric CY3inserted atoric brane givesinstanton partition functionwith asurface operatorinR4×S1. Using local mirror symmetry, one obtains a mirror curve which

describes the moduli of the toric brane:

Cmirror={x,y ∈ C |H(x,y) =0} ⊂ C× C.

In section 4, theEynard-Orantin topological recursionon mirror curve is discussed.

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 9 / 37

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1. Introduction and overview

Summary:

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KPF

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)CWIGVJGQT[ %(6

59EWTXG)CKQVVQHQTO GPGTI[OQOGPVWO

VGPUQT

%QWNQOD DTCPEJ

C59

COKTTQT

6QRQNQIKECNUVTKPIUQP

FWCNKV[

/

VQTKEDTCKP OKTTQTU[O

QRGPUVTKPI OQFWNK /KTTQTEWTXG

C

0,4+1

We will see that somefree energies onCSWagree with somefree energies onCmirrordefined by theEynard-Orantin topological recursion.

..

Recursion

..

Section3

..

Section4

..

Conclusion

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 10 / 37

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2. Irregular conformal block

.

Contents

.

. . 1 1. Introduction and overview

.

. .

2 2. Irregular conformal block

.

. .

3 3. Free energy on Seiberg-Witten curve

.

. .

4 4. Free energy on mirror curve

.

. .

5 5. Conclusion and outlook

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 11 / 37

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2. Irregular conformal block

.

.

2. Irregular conformal block

In this talk, for simplicity we only concentrate on thepure SU(2)case.

.

One point irregular conformal block withΦ1,2:

.

.

.

.. .

.

.

The Gaiotto state|∆a, Λiin the Verma module with the conformal dimension∆a= ∆(a) = (b+b1)2/4−a2is defined by the condition:

L1|∆a, Λi = Λ2|∆a, Λi, L2|∆a, Λi =0.

Let us consider an one point block:

Ψ(z,a, Λ) =h∆, Λ|Φ1,2(z)|∆+, Λi, ∆±= ∆(a± 1 4b), whereΦ1,2is the degenerate primary operator with the momentum

1/2b.

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 12 / 37

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2. Irregular conformal block

.

.. .

.

.

Φ1,2satisfies thenull state condition:

(b2L21+L21,2(z) =0,

and thusΨ(z,a, Λ) =z−∆+h2,1Y(z,a, Λ)satisfies:[Awata-Yamada]

[(

bz

∂z )2

+2abz

∂z + Λ2(z+z1) + Λ 4

∂Λ ]

Y(z,a, Λ) =0.

After scaling a→a/~, Λ → Λ/~, and introducing

1=b~, 2= ~ b,

a series solution Y(z,a, Λ) =1+∑

n=1Λ2nYn(z,a)of the above differential equation is obtained:

Yn(z,a) =

k=−∞

An,kzk, A0,k = δ0,k, An,k = An1,k1+An1,k+1

1(

2ak + 1k2+12n2).

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 13 / 37

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2. Irregular conformal block

.

.

.. .

.

.

The lower order terms are Y1(z,a) =− 1

1(−2a+ 1+ 22)z z

1(2a+ 1+22),

Y2(z,a) = 1

221(−2a+ 1+22)(−2a+21+22)z2

+ 21+ 2

212(−2a+ 1+22)(2a+ 1+22) + z2

221(2a+ 1+22)(2a+21+22),

Y3(z,a) =− 1

631(−2a+ 1+22)(−2a+21+22)(−2a+31+22)z3

1621+1412+322−16a1−4a2

4312(−2a+ 1+22)(−2a+21+22)(−2a+ 1+ 322)(2a+ 1+22)z +· · · .

These solutions agree with the instanton partition function with a surface operator insertion (“simple type”).

..

Summary

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 14 / 37

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3. Free energy on Seiberg-Witten curve

.

Contents

.

. . 1 1. Introduction and overview

.

. .

2 2. Irregular conformal block

.

. .

3 3. Free energy on Seiberg-Witten curve

.

. .

4 4. Free energy on mirror curve

.

. .

5 5. Conclusion and outlook

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 15 / 37

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3. Free energy on Seiberg-Witten curve

.

.

3. Free energy on Seiberg-Witten curve

SW curve via the stress tensor:

h∆a, Λ|T(z)|∆a, Λi =

n=±1,0

zn2h∆a, Λ|Ln|∆a, Λi

= (Λ2

z + Λ2 z3

)h∆a, Λ|∆a, Λi + 1

z2h∆a, Λ|L0|∆a, Λi

= [Λ2

z + Λ2 z3 + 1

2z2 (1

∂Λ+2∆a

)]h∆a, Λ|∆a, Λi,

where in the final equality we used 1

∂Λ|∆a, Λi = (L0− ∆a)|∆a, Λi.

By scaling a→a/~, Λ → Λ/~, and taking classical limit~ →0, we get h∆a, Λ|T(z)|∆a, Λi −→ −1

~2φSW(z)h∆a, Λ|∆a, Λi.

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 16 / 37

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3. Free energy on Seiberg-Witten curve

SW curve (Gaiotto form) for pure SU(2): pure SU(2) Mirror curve

.

.

.. .

.

.

φSW(z) =M(z)2σ(z), σ(z) :=−z(

z2 u

Λ2z+1)

, M(z) := Λ z2

Note that in the above derivation, we replacedclassical modulus a withquantum modulus u.

As was discussed in the introduction, the leading term of h∆, Λ|Φ1,2(z)|∆+, Λi

h∆a, Λ|∆a, Λi =exp(W(a,z)

1 +· · ·) is given by

..

Free.Egy

..

Claim 1

..

Disk on Mirror

W(a,z) =

z

λSW(z0), λSW(z) =M(z)

σ(z)dz.

In the following we define free energies on SW curve by theEO recursion, and discuss a relation to CFT correlators.

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 17 / 37

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3. Free energy on Seiberg-Witten curve

Eynard-Orantin topological recursionis derived as the loop equation of the hermitian one-matrix model:

ZM = 1

Vol(U(N))

dM exp ( 1

gs

TrV(M) )

, V(M) =

n=1

gn

nMn,

where M is N×N hermitian matrix. By diagonalizing the matrix M as M = diag(λ1, . . . , λN), this is rewritten as

ZM = 1 N!(2π)N

dλ∆2(λ)exp ( 1

gs

V(λ) )

, dλ :=

N i=1

i, ∆2(λ) =

i<j

i− λj)2.

1/N expansion of the h-point function of a gauge invariant observable O(p) = Trp1M ∼ ∂φ(p):

W(p1, . . . ,ph) =hO(p1)· · · O(ph)i(c)=

g=0

gs2g2+hW(g,h)(p1, . . . ,ph),

is invariant under the infinitesimal transformation λi → λi+ 

p− λi

, p6= λi, || 1.

This invariance gives us theloop equation.

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 18 / 37

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3. Free energy on Seiberg-Witten curve

Topological recursion: BKMP .

.

.. .

.

.

The multilinear meromorphic differentialsfW(g,h)onCSWare recursively defined by the Eynard-Orantin topological recursion:

..

Summary

..

Higher pure SU(2)

Wf(0,1)(z) :=0, Wf(0,2)(z1,z2) :=B(z1,z2), Wf(g,h+1)(z,z1,· · · ,zh) :=

qi∈CSW Resq=qi

dEq,¯q(z) λSW(q)− λSWq)

{Wf(g1,h+2)(q, ¯q,z1, . . . ,zh)

+

g

`=0

J⊂H

fW(gl,|J|+1)(q,zJ)fW(l,|H|−|J|+1)q,zH\J) }

,

dEq,¯q(z) :=1 2

q¯ q

B(z, ξ), near a branch point qi.

where H={1, . . . ,h},J={i1, . . . ,ij} ⊂H,zJ ={zi1, . . . ,zij}.

1 1 h

1 z z z

g

h

1 z z

z

g z

g 





 q

q q q

k zk

j j

i zi

J

= + Σ

z

z1 h

l

l l

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 19 / 37

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3. Free energy on Seiberg-Witten curve

.

Bergman kernel B(z1,z2):

..

pure SU(2)

.

.

.

.. .

.

.

B(z1,z2)

z1z2

dz1dz2

(z1z2)2 +finite. Holomorpic except z1=z2.

I

Ai

B(z1,z2) =0, i=1, . . . ,g=the genus ofCSW. In the case that SW curve has g=1 with two sheets as

x2=M(z)2σ(z), σ(z) =

4 i=1

(zqi) =z4+S1z3+S2z2+S3z+S4, this is given by theAkemann’s formula:

B(z1,z2) = dz1dz2

2(z1z2)2

(2f(z1,z2) +G(k)(z1z2)2 2

σ(z1)σ(z2) +1 )

,

f(z1,z2) :=z12z22+1

2z1z2(z1+z2)S1+1

6(z12+4z1z2+z22)S2+1

2(z1+z2)S3+S4, G(k):=1

3S2+ (q1q2+q3q4)E(k)

K(k)(q1q3)(q2q4), k2=(q1q2)(q3q4) (q1q3)(q2q4). K(k)(resp. E(k)) is the complete elliptic integral of the first (resp. the second) kind.

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 20 / 37

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3. Free energy on Seiberg-Witten curve

Free energyFSW(g,h): .

.

.. .

.

.

FSW(g,h)(z1, . . . ,zh) :=

z1

· · ·

zh

W(g,h)(z10, . . . ,zh0),

W(0,1)(z) := λSW(z), W(0,2)(z1,z2) :=B(z1,z2) dz1dz2 (z1−z2)2, W(g,h)(z1, . . . ,zh) := fW(g,h)(z1, . . . ,zh), (g,h)6= (0,1), (0,2).

In the matrix model, these free energies are obtained from 1/N expansion ofhφ(x1)· · · φ(xh)i(c)on spectral curve.

The disk free egyFSW(0,1)(z)is nothing but the leading termW(a,z) of the (normalized) conformal block.

..

Disk

In this definition, there exist ambiguities of the constants of integration.

In the following, we only consider theuniversal termswhich do not depend on choices of the constants.

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 21 / 37

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3. Free energy on Seiberg-Witten curve

Let us considermulti-point irregular conformal block:

h∆a0, Λ|Φ1,2(z1)· · · Φ1,2(zh)|∆a00, Λi h∆a, Λ|∆a, Λi =exp

{ ∑

k=1

~kFCFT(k)(z1, . . . ,zh) }

.

In the following we only treat theself-dual case1=−2=i~. a0 and a00are determined from thefusion rule, and further we choose these asFCFT(k) becomescompletely symmetric in zi. This can be computed by the same method explained in section 2.

..

MultiPt

This corresponds tomultiple surface operatorson the gauge theory side.

.

Claim 1: generalization of [KPW] discussed in superconformal case

.

.

.

.. .

.

.

FCFT(h2)(z1, . . . ,zh) =FSW(0,h)(z1, . . . ,zh) The case of h=1 is nothing but the claim of

..

Disk .

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 22 / 37

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3. Free energy on Seiberg-Witten curve

.

Computation for pure SU (2)

SW curve

At first we have to compute the period:

da(u) du =

I

A

∂λSW(z)

∂u = 1

I

A

dz

σ(z) = 1

πΛ q3q1

K(k), k2= q1q2

q1q3

, where q1=0, q2= (u

u24)/2Λ2and q3= (u+

u24)/2Λ2, and then we obtain

u(a) =a2+ Λ4 2a2 + 8

32a6+ 12

64a10+1469Λ16

8192a14 + 4471Λ20

16384a18 +O(Λ24).

By the formula

..

Annulus , the annulus free egyFSW(0,2)(z1,z2)is obtained:

..

Annulus on Mirror

FSW(0,2)(z1,z2) = z12z22+1 16a4z1z2

Λ4+(z1+z2)(z13z23+1) 32a6z12z22 Λ6

+10(z12+z22)(z14z24+1) +9z1z2(z14z24+1) +32z12z22(z12z22+1)4z12z22(z12+z22)

512a8z13z23 Λ8

+O(Λ10).

We checked our claim up toΛ8.

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 23 / 37

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3. Free energy on Seiberg-Witten curve

By the recursion

..

Recursion , higher topology free energies are iteratively determined.

These free energies can be expanded by thekernel differentials:

χ(n)i (z):=Res

q=qi

( 2dEq,¯q(z) λSW(q)− λSWq)

dq (qqi)n

) .

For example, W(0,3)(z1,z2,z3)is

W(0,3)(z1,z2,z3) =

qi

Res

q=qi

2dEq,¯q(z)

λSW(q)− λSWq)B(z2,q)B(z3, ¯q)

=1 2

qi

M(qi)2σ0(qi(1)i (z1(1)i (z2(1)i (z3),

χ(1)i (z) = dz 2M(qi0(qi)

σ(z) (

G(k) + 2f(z,qi) (zqi)2 )

, and then we obtain (, and checked our claim up toΛ6):

..

Summary

..

Three-holed on Mirror

FSW(0,3)(z1,z2,z3) = z21z22z321 16a7z1z2z3Λ6+3(

z31z23z33(z1+z2+z3)− (z1z2+z2z3+z3z1)) 64a9z21z22z32 Λ8

+

{z12+z22+z32− (z12z22+z22z32+z32z12) 128a11z1z2z3 +5(

z41z24z34(z12+z22+z32)− (z12z22+z22z23+z32z12)) 128a11z13z23z33

+9(

z13z32z33(z1z2+z2z3+z3z1)− (z1+z2+z3))

256a11z12z22z32 +9(z12z22z321) 64a11z1z2z3

}

Λ10+O(Λ12).

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 24 / 37

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4. Free energy on mirror curve

.

Contents

.

. . 1 1. Introduction and overview

.

. .

2 2. Irregular conformal block

.

. .

3 3. Free energy on Seiberg-Witten curve

.

. .

4 4. Free energy on mirror curve

.

. .

5 5. Conclusion and outlook

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 25 / 37

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4. Free energy on mirror curve

.

.

4. Free energy on mirror curve

We consider top. A-model on a toric CY3:

MA:=X/U(1)n, U(1)n:Xi eiααliαXi, X :=

{

(X1, . . . ,Xn+3)∈ Cn+3

n+3 i=1

liα|Xi|2=rα∈ MA,

n+3 i=1

liα=0, α =1, . . . ,n, liα∈ Z} ,

whereMAis the n dimensional Kähler parameter space.

Examples

01

01 01

10

10 10

10

01

01

11

11 1

X

X

X

1

2

2

3

=0 X1=0

X1=0

=0 X2=0

X2=0

=0

X3=0

X3=0

X4=0 X4=0

X4=0

(-1,1)

(-1,-1)

O1 O1 O0 O2 P O

P

P

P1 4 1

4 P

2

12

21

11 11

10 10 01

11

3

l=(1,1,-1,-1) l=(1,1,0,-2) l=(-3,1,1,1)

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 26 / 37

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4. Free energy on mirror curve

We insert atoric braneon a localC3patch: Duality

X

X X

k k

k k

i

i i

i i

j

j j j

=0

=0 =0

c

L ^X^^X^=2 2

c

QRGPUVTKPIOQFWNK

^X^^X^=02 2

4GXXX S0 +OXXX =0 C

S1

L has a topologyC ×S1.

Here aD4-braneis wrapping aLagrangian submanifold L.

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 27 / 37

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4. Free energy on mirror curve

TheHori-Vafa mirror manifoldMB of MA:

MB: = {

+, ω,x1, . . . ,xn+3)∈ C2×(C)n+3 ω+ω=

n+3

i=1

xi,

n+3

i=1

xiliα=zα,

n+3 i=1

liα=0 }

= {

+, ω,x,y)∈ C2×(C)2 ω+ω=H(x,y) }

By taking an appropriate local coordinate patch whose parameterx represents the open string modulix =exp(−c+i

S1A), we obtain a mirror curve which describes the moduli of thetoric brane:

Cmirror={x,y ∈ C |H(x,y) =0} ⊂ C× C.

By the mirror maps:

Tα(z) = I

Aα

λmirror(x) = I

Aα

log y(x)dx

x , Qα=eTα(z), FA,α(0,0)= ∂FA(0,0)

∂Tα = I

Bα

λmirror(x), the B-model is related to the A-model.

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 28 / 37

(29)

4. Free energy on mirror curve

BKMP (Remodeling the B-model) conjecture:

.

.

.. .

.

.

In the previous discussion Recursion , by replacingλSW(z)with λmirror(x) :=log y(x)dx

x ,

we define thefree energiesFmirror(g,h). And thenBKMP conjecturestates:

Fmirror(g,h)(x1, . . . ,xh) =FA(g,h)(X1, . . . ,Xh).

In this equality the (open & closed) mirror maps are used, and the A-model free energiesFA(g,h)are computed by thetopological vertex.

.

Claim 2:

..

Engineer

.

.

.

.. .

.

.

Fmirror(0,h)(x1, . . . ,xh) β−→ F0 SW(0,h)(z1, . . . ,zh)

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 29 / 37

(30)

4. Free energy on mirror curve

.

.

Computation for pure SU (2)

Qb

Qf

X X1 2

1 x

y

x

y

1

1

VQTKEDTCPG

l= (-2,1,0,1,0) l= (-2,0,1,0,1)

b

f

4dN =2 pure SU(2)gauge theory is engineered by the Hirzebruch surface F0=P1f ×P1b.

We insert a toric brane on an inner leg of local F0. Relation of the parameters is

Qf =e2βa, Qb =X1X2= β4Λ4, X1:= β2Λ2w1, X2:= β2Λ2w whereβ is the radius of the fifth dimension S1in the gauge theory.

Masahide Manabe (Nagoya University) B-model and Recursion December, 2010 @ Taiwan 30 / 37

參考文獻

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