LOCALIZATION AND VORTICES Kazutoshi Ohta Meijigakuin University

LOCALIZATION AND

VORTICES

Kazutoshi Ohta

Meijigakuin University

The Localization Theorem (Duistermaat-Heckman theorem) essentially works behind the integrability.

Lower dimensional gauge theories

(3-dim Chern-Simons, 2-dim Yang-Mills, etc.) Supersymmetric quantum mechanics

Topological (supersymmetric) gauge theories Reduced matrix model from 3d CS or 2d YM Topological strings

Non-critical strings

There exist various kinds of exact solvable systems:

Introduction

1. Introduction

2. What is the localization?

3. Application to 2d field theory

4. Volume of the moduli space of the Abelian vortex 5. Other applications

6. Wilson loops 7. Conclusion

Plan of my talk

What is the localization?

Originally it appears in the partition function of the statistical model.

The integration formula by Duistermaat and Heckman states that if H determines the symplectic action of U(1) on the phase space the integral localizes to the critical points of H

Z = 1 βⁿ

�

dH=0

� det |ω^ij|

det |∂^ijH|e^−βH Z =

�

M

ωⁿ e^{−βH(�x,�p)}

where is a symplectic 2-form on the 2n-dim phase space M.

ω

Outline of the proof

There exists a vector field X associated with H and it satisfies (d + ι_X)(H + ω) = dH + ι_Xω = 0

Z = 1 βⁿ

�

M

e^−β(H+ω)

If we add exact term to the “action”, the partition function is independent of the additional term

namely

indep. of λ

(ι_Xω = X^µω_µνdx^ν) If we define , it satisfies and forms an equivariant cohomology on M.

(d + ι_X)² = L^X

d_X ≡ d + ι^X

(d_XΩ = 0 & Ω �= d^Xψ & L^XΩ = 0) d_X

∂Z

∂λ ∝ 1 βⁿ

�

M

d_X(ψe^{−β(H+ω)−λdXψ}) = 0 Z = 1

βⁿ

�

M

e^{−β(H+ω)−λdXψ}

indep. of λ

Let us now choose

Bi-Hamiltonian structure (H,ω)↔(K,Ω) naturally appears if X is an isometry of M.

which satisfies

d_X(H + ω) = 0 d_X(K + Ω) = 0

Recall that the partition function is independent of the coupling of the exact term, we can take the large coupling limit

without changing the value of the integral:

d_X

Z = 1 βⁿ

�

M

e−β(H+ω)−λ(g^µνX^νX^ν+··· )

Z ∼ 1 βⁿ

�

M

√detgδ(X)e1 −β(H+ω)−···

λ → ∞

d_Xψ = d_X(ι_Xg) = K + Ω, K = ι_Xι_Xg, Ω = dι_Xg

metric

If we carefully integrate out all of the variables and calculate Jacobians in the large coupling limit, we obtain the DH

localization formula

Thus, the integral of the partition function is localized at X=0, that is the fixed points of the Hamiltonian vector flow (dH=0)

Z = 1 βⁿ

�

dH=0

� det |ω^ij|

det |∂^ijH|e^−βH

Example Harmonic oscillator

The gaussian integral also can be understood as the sum over the fixed points of the Hamiltonian vector flow

H = p�²

2m + 1

2mω²�x²

Z = �

d�xd�p e^{−β(2m�p2 +12mω2�x2)}

= 1 βⁿ

� 2π ω

�n

e^−βH(0,0)

There exists the following correspondence

DH formula Supersymmetric theory

super (BRST) charge Q

isometry gauge symmetry

coordinates and forms bosons and fermions(ghosts) bi-Hamiltonian BRST exact action

d_X = d + ι_X

The localization is very important to the integrability of the supersymmetric systems (owing to a cancellation of the

functional determinants between boson and fermions).

Relation to SUSY systems

[Witten (1992)]

We first consider the BF-type 2d U(N) gauge theory on a Riemann surface with genus g

Application to 2d Yang-Mills theory

Z =

�

DΦDAe^−Tr

R

Σg iΦF

where ._{F = dA + A ∧ A}

Integrating out Φ first, we get Z =

�

DA δ(F )

This means that the partition function of the BF theory calculates the volume of the moduli space (space of solutions) of the flat

connections F=0.

This “bosonic” 2d BF theory is nicely embedded in the 2d topological (cohomological) field theory:

BRST transformations is given by

The following BF theory action is Q-closed (but not Q-exact) under the above BRST symmetry

QA = iλ, Qλ = −d^AΦ, QH = i[Φ, χ], Qχ = H,

Q ¯Φ = iη, Qη = [Φ, ¯Φ]

QΦ = 0

O ≡ Tr

�

Σ_g

iΦF + 1

2λ ∧ λ

QO = 0, O �= Q(something)

Hamiltonian

The “supersymmetric action” is Q-exact (of course Q-closed).

The partition function is independent of the coupling constant of the SUSY action and localized at the flat connection F=0. So the observable of the cohomological operator gives the partition

function of BF theory.

F = 0 S = 1

h²Q

� Tr

�

η[Φ, ¯Φ] + 1

2χ(H − F ) + λd^AΦ¯

�

VEV in TFT

PF of bosonic FT

bi-Hamiltonian

�e^{−TrR iΦF +12λ∧λ}�

TFT = �

DΦDADλ · · · e^{−TrR iΦF +12λ∧λ+h21 Q{∗}}

= �

DΦDADλe^{−TrR iΦF +12λ∧λ}

=

�

DΦDAe^{−TrR iΦF}

This structure of the BF theory means that the localization theorem (DH formula) also works:

After fixing the gauge Φ=diag(φ1,φ2,...,φN), we find

�

DΦDAe^{−Tr R iΦF}

= 1 N !

�

�k∈Z^N

� �^N

a=1

dφ_a �

a<b

(φ_a − φ^b)^2−2ge^{−i PNa=1 kaφa}

= 1 N !

�

�

n∈Z^N

�

a<b

(n_a − n^b)^2−2g

where ka is the Chern number of a-th Cartan and we have used the Poisson resummation formula:

�

�k∈Z^N

e^{−i Pa kaφa} = �

�

n∈Z^N

�

a

δ(φ_a − n^a)

We can deform the 2d BF theory to 2d Yang-Mills theory without spoiling the localization properties, since we can add the Q-

closed operator:

where .

Deformation to 2d Yang-Mills theory

S = Tr

�

Σ_g

iΦF + µ

2 Φ²

Integrating out Φ

S = 1

4g_YM²

�

Σ_g

F² g_YM² = µ/2

The partition function has the expression of the discrete sum Z = 1

N !

�

�n∈Z^N

�

a<b

(n_a − n^b)^2−2ge^{−gYM2 A Pa n2a}

area [Migdal]

2d BF and YM theory evaluate essentially the volume of the moduli space of the flat connection F=0. We now consider the Abelian vortex equations:

Volume of moduli space of vortices

The BRST transformations:

[A. Miyake, N. Sakai, KO (work in progress)]

F = ω

2 (1 − |h|²)

∂h = ∂¯¯ h = 0

QA = iλ, Qλ = −dφ, Qh = iψ, Qψ = φh, Q¯h = iψ^†, Qψ^† = −φ¯h, Qφ = 0.

To impose the holomorphic constraint for h (F-term constraints), we introduce extra fields and BRST transformations:

The Q-cohomological (Q-closed but not Q-exact) action:

QY_z = iφχ_z, Qχ_z = Y_z, QY_z¯ = −iφχ^z¯, Qχ_z¯ = Y_z¯.

S₀ = �

Σ_g

�

iφ �

F − ω

2 (1 − |h|²)�

+ 1

2λ ∧ λ + ω

2 ψψ^†

�

We can also add the Q-exact action (SUSY action) without changing the partition function, and integrate out all fields exactly. The result of the partition function is

where k is the Chern number: k = 1 2π

�

Σ_g

F

Z^k = 1

(2π)^2−2g

� _∞

−∞

dφ � 2π iφ

�k+1−g

e^iφ(¹₂^A−2πk)

This integral of the partition function reduces to a simple residue integral. So the partition function is localized at a pole. Finally we obtain

For example, if g=0, we have Z^k = π^k

�g j=0

(4π)^j (A − 4πk)^k−j g!

j!(k − j)!(g − j)! .

Z^k = π^k (A − 4πk)^k

k! .

This volume is smaller than the moduli space of the point like objects on the sphere:

The vortex has a hard core

vs

Z^k < πA^k k!

This result is previously calculated by Manton and Nasir (1998) in the different way. They need:

1. Details of the solution of the vortex equations 2. Structure of the vortex moduli space metric

3. Integrals over the moduli space with the metric However the localization method does not use the concrete solutions of vortices!

We agin consider the thermodynamical partition function of the vortices (do not confuse with the field theory partition function!):

Thermodynamics of the vortices

For free particles (solitons) , then we have Z = 1

�^2k

�

d^kpd^kx e^−βH(p,x)

H = ¹₂g_ijpⁱp^j

We can calculate the free energy F=-T ln Z and equation of state

Z = � 2π²T

�²

�^k �

M^k

d^kx�

det g_ij

= � 2π²T

�²

�^k

Vol(M^k)

P = kT

A − 4πk

pressure

for g=0 (S²)

Volume of the moduli space of the non-Abelian vortex system (Hitchin system) [Moore, Nekrasov and Shatashvili (1997)]

Other applications 1

F_{z ¯z} + [W_z, W_z¯] = 0, D^z¯W_z = D^zW_z¯ = 0,

Localization

Z = 1 N !

�

φ�∈R^N

µ(φ)^−1+g �

a<b

(φ_a − φ^b)^2−2g �

(φ_a − φ^b)² + �²�1−g

where is solutions of the following Bethe ansatz equations_RN e^2πiφi �

i�=j

φ_j − φⁱ + i�

φ_j − φⁱ − i� = 1 and ε is a regulator of the flat direction in W

Volume of the moduli space of the instantons (ADHM equations) [Nekrasov (2002)]

Other applications 2

Localization

Nekrasov’s formula

[B₁, B₁^†] + [B₂, B₂^†] + II^† − J^†J [B₁, B₂] + IJ = 0

Prepotential of 4d N=2 SUSY YM theory

Reduced matrix model from 3d Chern-Simons theory on S³ or 2d YM theory on S² [Ishiki, KO, Shimasaki, Tsuchiya (2008)]

Other applications 3

dimensional reduction

S = Tr

�

S²

iΦF + µ

2 Φ² S = Tr

�

S³

AdA + A³

dimensional reduction

S_MM = 1 g²Tr

�

M_i � 1

2M_i + i

6�_ijk[M_j, M_k]

��

i = 1, 2, 3

The partition function of the reduced matrix model is exactly calculated by using the localization:

This partition function is much more complicated rather than the 2d YM theory one. But in the large N limit, we recover the

partition function of U(r) 2d YM theory (or 2d CS theory)

Z_2dYM = C �

�n∈Z^r

� �^r

l=1

dy_l �

i<j

((y_l − y^m)² − (n^l − n^m)²)e^{−g2YMA1 Prl=1(yl2+n2l)}

Minahan-Polychronakos form

(Poisson resummation from Migdal’s form)

Z_r = �

d�∈Z^r

�

Rd�

�r l=1

dy_l �

l<m

(yl − y^m)² − (d^l − d^m)²

(y_l − y^m)² − (d^l + d_m)²e^−2g21

P_r

l=1(¹4d_ly_l²+₁₂¹ d³_l−₁₂¹ d_l)

Wilson loop

A perfect circle Wilson loop

or some class of Wilson loops is Q-closed (but not Q-exact) in SUSY theory.

This fact is investigated in 4d N =4 SYM theory ^[Erickson-

Semenoff-Zarembo, Drukker-Gross]. It is also proven by using the localization theorem [Pestun(2007)].

The Q-closed Wilson loop does not change the

localization properties. So we can evaluate the vev of the Wilson loop exactly.

W_R(C) = Tr_R

�

P exp

�� (A_ϕ + Φ cos θ)dϕ

��

It is also important that the Q-closed Wilson loop in SUSY reduces to solvable one in equivalent “bosonic” theory

Conclusion

The localization reduces the infinite dimensional problem to the finite one.

The localization plays important role in supersymmetric (exact solvable) field theory.

A special Wilson loop is also solvable by the localization.

I also would like to apply this method to the calculations of non-perturbative dynamics in supersymmetric theories (string theory, field theory, lattice field theory, matrix models and

more!)

Bosonic theory

Supersymmetric field theory

Supergravity/

string theory Holography

Localization