JSPS/RFBR collaboration
KKR Karasaki
March 2014

Instanton partition function, DDAHAand recursion formula
Yutaka Matsuo (U. Tokyo)
With S. Kanno and H. Zhang
arXiv:????.????, 1306.1523

Mostly a review ofSchiffmann and Vasserot, arXiv:1202.2756

Introduction

AGT conjecture

Alday-Gaiotto-Tachikawa (2009)

Toda

N=2 SYM
Nekrasov partition function=Correlation function

From the viewpoint of M5 brane, it may be understood as the consequenceof two types of compactification,
Riemann surface
with punctures

N=2 SYM

Four sphere

Toda/Liouville field theory“Geometric Langrands program” (Witten)
Here we pursue more direct correspondence through the symmetry behind the scene.

Mathematical foundation of AGT
AGT conjecture
Nekrasov Instanton partition
function

Correlation function of
Liouville/Toda field theory

Natural action
on fixed points oftorus action

equivalence with
Virasoro/W algebra
Spherical Degenerate Double
Affine Hecke algebra (SH)
Reference: Schiffmann and Vasserot
arXiv:1202.2756

Spherical Degenerate Double
Affine Hecke AlgebraDouble Affine Hecke Algebra (DAHA)
Introduced by Cherednik to investigate propertiesof Macdonald polynomial
DAHA for GLnGenerators:

Braid Relation:
Hecke Relation:
Extra generators:

Graphical representation: Burella,Watts, Pasquier,Vala, Annals of Henri Poincarre 2013

Degenerate DAHATake a limit:

A representation:
Dunkl operatorSH (Spherical DDAHA)DDAHA:
SH

generated byrestriction on the symmetric part
generators:: projection to totally symmetric part

reduction toCalogero-Sutherland and Dunkl operator
The algebra SH (spherical degenerate double affine Hecke algebra) is most naturally
understood in the context of Calogero-Sutherland system with the Hamiltonian:The eigenstate of this Hamiltonian is called Jack polynomial which is labeled by Young diagram

Higher conserved charges:

Recursion relation for Jack polynomials
The operators zi and Di represent Dunkl operators of Calogero-Sutherland system.
A natural action is defined for their eigen-function – Jack polynomials.
They can generate every Jack polynomial through recursion relations.
In particular, the action of D1,n and D0,n takes a simple form,

Y

x(s)

y(s)

sW
YSHc : Central extension of SH

Nonlinear part of the algebra:

arbitrary parameters (central charges)

Action of SH on cohomology ofinstanton moduli spaceInstanton moduli space and 2d CFT
An Intuitive Picture (1)Rough illustration of the correspondence between
the cohomology of instanton moduli space and 2D CFT and SH
We start from a simplified version of moduli space where we keep the locationof instantons as a moduli: Hilbert scheme of k points on euclidian 4D space C2.
(C2)k is divided by Sk since these points are not distinguishable.

Sk: permutation group
When the location of some instantons overlaps, we meet orbifold singularities.
We need blow up such singularity in order to have a smooth space.

Instanton moduli space and 2d CFT
An Intuitive Picture (2)When two instantons overlap, we need attach two cocycle to resolve the singularity,
which represents the direction where two instantons collides.
For three instanton case, we have four cycle.

In general, 2(n-1) cycle is needed to resolve the singularity from n-points getting
together. This resembles the Fock space of free boson if we sum overthe moduli space of arbitrary number of instantonsTorus action and equivariant
cohomology
More precise definition of Hilbert scheme
Torus action:

fixed points of torus action: parameterized by a Young table Y

Action of DDAHA on fixed points

which implies a correspondence:

DDAHA is a natural symmetry acting on the cohomology ofinstanton moduli spaceCoproduct of DDAHA and Walgebra

DDAHA is a natural symmetry acting on the cohomology of
instanton moduli space.
It does not, however, describe the instanton moduli space.
For SU(N) case, it is parametrized by,Location of instanton
size of each instantonmutual SU(N) orientation
In order to describe the extra moduli, one needs totake the coproduct of DDAHA and pick up theirreducible representation out of it, just as composition of spins

For SH, the “fundamental representation” is described bya free boson Fock space F since it is parameterized by Young diagam Y
One may assign (in the large N limit)
Representation of SH

Awata, Odake, M, Shiraishi, 1995
in the context of Calogero-Sutherland

It is possible to generate all other generators fromby taking commutation relationsRepresentation by n bosons
We have derived a representation of SH on free boson Fock space
Representation on the direct productis nontrivial since the algebra is nonlinear, i.e. the coproduct is nontrivial.
In particular, the generator Drs is not sum of operators acting on each F

This time the algebra SH has nontrivial central extensionRepresentation by orthonormal basis
We introduce an orthonormal basis for the representation of DDAHA

which gives a representation of DDAHA with central charges

From the expression of coproduct, one may derive thestress energy tensor:

whereThis is the standard representation of WN -algebra in termsof N free bosons with central charge

Further confirmation of correspondence
Structure of Hilbert space is determined by the singular vector.
For W-algebra it is determined by the nonvanishingaction of the screening operator.
For SH, one may find it by the condition.

which is implemented by

It gives a consistent result.

Recursive proof of AGT

Based on algebra SH, one may give a direct proof of AGT conjecture.
Proof for pure super Yang-Mills
Schiffmann and Vassero(’12), Maulik and Okounkov (‘13)
Proof for N=2*
Fateev-Litvinov (‘09)
Proof for linear quiver gauge theories
Alba-Fateev-Litvinov-Tarnopolskiy (‘10)
Belavin-Belavin (‘11)
Fateev-Litvinov (‘11)
Morosov-Mironov-Shakirov (‘11)
Zhang-M (‘11), Kanno-M-Zhang (‘12, ‘13)
Morosov-Smirnov (’13)

Basic strategy
Typical form of Nekrasov partition function

q: moduli space parameterSum over fixed point of localization parameterized by Young diagram Y
f(Y): contribution of each fixed point
SH acts on the fixed points, thus gives recursion relations among
coefficients which typically are the products of factors of the form:

Recursion relation among factors
For the factor from bifundamental matter:

One may derive the recursion relation for other factors from them. Other relation which
we use is the following algebraic identities.

Proof for pure SYM
AGT conjecture:

One may map such conditions to similar ones in SH. One may directly check the following
state satisfies them. They have the desired norm.

Proof for linear quiver
Nekrasov Partition function for conformal inv. SU(N)x…xSU(N) (L factors) quiver
NNN

N

N

is identical to conformal block function of 2D CFT described bySU(N) Toda field theory with L+3 vertex operator insertions
VVwith identification of parameters

VN

Partition function takes the form of matrix
multiplication
NN

N NN

N

Partition function looks like matrix multiplication

N

This time, what we need to prove iswith a vertex operator insertion.

NRecursion for quiver from SH

The coefficients δ-1,n coincide with D-1,n action on the bra and ket.
This recursion formula uniquely fix the partition function thus proves AGT.
On the other hand the coefficient appearing in δ1,n is slightly shifted by ξ.
This small mismatch will be explained by an exotic choice of vertex operator.

Anomaly in vertex operator
The vertex operator is written as a product of U(1) (Heisenberg)
part and WN part:The standard definition in terms of free boson is written as,

We need modify the U(1) part of the vertex operator
Carlsson-Okounkov

Anomaly in the commutator with Virasoro cancels
those in recursion relation.

Summary
1. We reviewed Spherical DDAHA (SH) which describes thestructure of instanton moduli space.
2. SH is equivalent to WN-algebra + U(1) factor when it isrealized by irreducible “symmetric part” in the tensor
product of N “fundamental” representation. In principle, it ismuch larger symmetry such as 𝑊𝑊(1 + ∞)3. SH is much simpler to be treated in the instanton partition
function. Indeed, the recursive proof of AGT is given invariety of cases.

Future prospects
 The symmetry SH is quite interesting mathematical object

and should be pursued further.
 Connection between coproduct of SH and integrable models(Maulick-Okounkov): R-matrix and Yangian Generalization of SH for other gauge groups Treatment of nonlocal objects in SYM.
 Application of SH to higher spin gauge theory and/or
fractional Hall effects
 Nekrasov-Shatashvilli limit: action on noncommutative
Riemann surface, Toda system, Calogero system