Extended Conformal Symmetry and Recursion Formulae for Nekrasov

Partition Function

Yutaka Matsuo (U. Tokyo) With S. Kanno and H. Zhang

arXiv:1306.1523, 1207.5658, 1110.5255

Workshop at Hongo October 2013

INTRODUCTION

Brief History of 4D Instanton physics and 2D integrable models

Seiberg-Witten (1994) Solving Prepotential of N=2 gauge theory by Riemann surface

Donagi-Witten, Gorsky et. al. Itoyama-Morozov, Nakatsu-Takasaki, D’Hoker-Phong (1995~) Connection with integrable model/hierarchy

Nekrasov, Nekrasov-Okounkov, Nakajima-Yoshioka (2003-4) Derivation of instanton partition function Contribution from fixed point labeled by Young diagrams Structure of 2D CFT in the cohomology of moduli space

Alday-Gaiotto-Tachikawa (2009) Direct connection between instanton partition function and conformal block function of 2D CFT

Nekrasov-Shatashvili (2009) Discovery of a limit (NS limit) where integrablity shows up

Intuitive picture of 2D CFT structure in instanton moduli space

Configuration space of k instantons

Sk: permutation group We need blow-up to solve singularities

Singular points: overlap of some instantons

2(n-1)-cycle (pn) appears when we blow up the singularity with n instantons

H. Nakajima Short review: Morozov and Smirnov, arXiv:1307.2576

Ex) Cohomology of five instanton moduli space

This resembles the Fock space of free boson if we sum over the moduli space of n instantons

N=2 quiver gauge theory

4D conformal field theory with N=2 SUSY, SU(N)x.. xSU(N) gauge group

N U(N) gauge group, i=1 ,.. ,L

bifundamental hypermultiplet

N N N N N N

N (N) fundamental hypermultiplet

AGT conjecture Nekrasov Partition function for SU(N)x…xSU(N) quiver

is identical to conformal block function of 2D CFT described by SU(N) Toda field theory with L+2 vertex operator insertions

V

V

V

with identification of parameters

N N N N N N

It is described by WN algebra with extra U(1) factor

Alday-Gaiotto-Tachikawa arXiv:0906.3219 Wyllard arXiv:0907.2189, Mironov-Morozov 0908.2569

Partition function takes the form of matrix multiplication

N N N N N Partition function looks like matrix multiplication

Set of Young diagrams which labels the fixed points of localizations for SU(N)

N N

N

The explicit form of Z will be given shortly.

Comparison with CFT V

V

V

V

V

V V Pants decomposition

Conformal block function can be written by decomposing Riemann sphere in the pants

μ

At the hole, we insert decomposition of “1”

by an orthonormal basis

μ

Alba et. al. arXiv:1012.1312

Main claim of the talk 1. We derive an infinite set of recursion relations

where LHS is the variations by adding/subtracting a box in Y or W with some structure constant

2. The variation δ can be closely related to a quantum symmetry DDAHA (Degenerate Double Affine Hecke Algebra) which is equivalent to WN algebra

3. They can be identified with the extended conformal Ward identities when we have the identification

which gives a recursive proof of AGT for SU(2) gauge group

Some comments Related studies to prove AGT

O. Schiffmann and E. Vasserot, arXiv:1202.2756 D. Maulik and A. Okounkov, arXiv:1211.1287

Proof for pure super Yang-Mills where only gauge group is relevant appearance of symmetry DDAHA (SV) = Yangian (MO)

Alba, Fateev, Litvinov, Tarnopolskiy, arXiv:1012.1312 Fateev, Litvinov arXiv:1109.4042

(Implicit) construction of basis

Morozov and Smirnov, arXiv:1307.2576

Dotsenko-Fateev integral to evaluate inner product Interpretation of basis as generalized Jack polynomial

A proof of the inner product formula

Basic structure behind AGT

2D Toda field theory (W-algebra)

Cohomology Instanton moduli

space

Calogero-Sutherland

Jack polynomial

One and the same objects!

W-algebra =DDAHA AGT conjecture

Maulik-Okounkov Schiffmann-Vassero

Plan of the talk

1. Introduction 2. Recursion formula for Nekrasov partition

function 3. Structure of instanton moduli space and

DDAHA 4. DDAHA and W algebras 5. Derivation of Ward identities 6. Conclusion

RECURSION FORMULA FOR NEKRASOV PARTITION FUNCTION

Nekrasov partition function From the experience of previous works, we suspect if there exists a recursion formula for Nekrasov partition function.

with

Product of factors for each box in Young diagrams

Variation formulae We evaluate the ratio of Nekrasov formula when we add/remove a box in Y and W.

For example:

which looks very complicated..

with

We combine them in the following form,

with

This factor is necessary to absorb extra terms from variations of vector multiplets.

The right hand sides take the following simple expression:

with the following replacements:

This expression can be rewritten in the form:

where qn is written by a generating functional:

We arrive at

as claimed.

Due to a combinatorial identity:

Comments • We have seen such recursive formula for special cases

– β=1 Special Young diagram: Zhang-M 1110.5255 – β=1 General case: Kanno-M-Zhang 1207.5658

• Originally we used it to give convincing argument for our

conjecture for the Selberg average

• In the second work, it is embedded into much large symmetry W(1+∞)

• It turns out, for general β, symmetry becomes much more involved (DDAHA)

STRUCTURE OF INSTANTON MODULI SPACE AND DDAHA

Algebraic structure of instanton moduli space

In the following, we would like to identify the symmetry behind the recursion relation. Most of the material, as it turns out, was covered by the following reference where the algebraic structure of Instanton moduli space is clarified (arXiv:1202.2756). (95pages, very unfriendly for the first reading..)

Double Affine Hecke Algebra (DAHA)

DAHA for GLn Generators:

Two parameters

Braid-like Relations

Introduced by Cherednik to investigate properties of Macdonald polynomial

published in 2004

Degenerate DAHA (DDAHA) We take a limit in DAHA:

Generators are replaced by: while keeping

The relations among generators are replaced by

DAHA DDAHA

Macdonald polynomial Jack polynomial

q-Virasoro, q-W Virasoro, W-algebra

5D SYM 4D SYM

“Dunkl operator” for Calogero-Sutherland

Spherical DDAHA We focus on the action of X, y on symmetric polynomial of X The operator is symmetrized as

symmetrization operator

The action of such symmetrized product is called Spherical DDAHA, In the following, we omit “spherical” for simplicity. Such algebra is generated by

In general,

is obtained from their commutation relation.

Action on Jack polynomials The operators X, y represent Dunkl operators of Calogero-Sutherland system. A natural action is defined for their eigenfunction – Jack polynomials. In particular, the action of D1,0 and D0,2 takes a simple form,

In the first line, W is obtained from Y by adding one more box, f(W, Y) is a coefficient. Such formula is called Pieri relation. This will be generalized to the multiplication of pn, and differential of pn which gives the action of Dn,0 The second line implies D0,2 is the Hamiltonian of Calogero-Sutherland system.

Relevance of DDAHA in SYM

Action of product of c1 in the Hilbert space of cohomology

By introducing the equivariant cohomology with deformation parameter ε1,2 the action of c1 is generalized to,

Interestingly, this is the Hamiltonian of Calogero-Sutherland model (D0,2) after replacing,

Awata-Matsuo-Odake-Shiraishi, 1995

As explained in the introduction, there exists a free boson representation of the cohomology of the instanton moduli space.

Cohomology of instanton moduli space The cohomology of SU(N) instanton moduli space has more involved structure than the Hilbert scheme.

We need to introduce extra parameters ai to describe the localization with respect to torus action which are to be identified with VEV for vector multiplet in N=2 SYM. The Jack polynomial should be upgraded to generalized Jack polynomial which is characterized by N array of Young diagram.

where the coefficients C(Y,W) is obtained by solving “integrable lattice model” where the degree of freedom on each lattice site is given by Jack polynomial. (Maulik-Okounkov)

DDAHA AND W ALGEBRA

The algebra DDAHA generators:

looks like affine SU(2)

Nonlinear algebra

infinite number of commuting charges

cl: central charges l=0,1,2,...

The generators E are defined nonlinearly by D0,l with infinite number of central charges

Representation by orthonormal basis We introduce an orthonormal basis for the representation of DDAHA

which gives a representation of DDAHA with central charges

Comparison with the coefficient in recursion formula

The coefficients δ-1,n coincide with D-1,n action on the bra and ket. 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.

The variations resembles the standard representation of SU(2)

which determines the representation of SU(2). (Null state, Unitary representation and so on.)

The action of D±1,n on |a,Y> plays a similar role. Later we will show that DDAHA is closely related to W algebra. In W-algebra, the analysis of null state is very complicated and it is state of art to find it. In DDAHA, it is obvious as SU(2) since it is represented by the orthogonal basis. Identification of null state structure is immediate in this sense.

Embedding of U(1) current and Virasoro generators

r

s Definition of algebra

1 -1

1 2

Definition of Heisenberg (U(1)) and Virasoro

plays a special role “Hamiltonian” for Calogero-Sutherland

From D0,2 and D1,0, D-1,0 one may reconstruct the action of other generators

Heisenberg

Virasoro

Relation with W1+∞ An algebra with similar set of generators: W1+∞ algebra

It is obtained as β-> 1 limit of DDAHA with the correspondence

W1+∞ algebra gives a representation of WN algebra + U(1) current when it is described by N fermions (C=N)

The relevance of such algebra was first conjectured by Kanno-M-Shiba (arXiv:1105.1667 )

For general β, the algebra becomes much more complicated.

Central charge of DDAHA

This is identical to that for U(1) current+W algebra with background Q It implies that DDAHA and W algebra has very similar structure. Indeed they are the same!

One may evaluate the “central charge” of Virasoro algebra by the commutation relation of

The evaluation of the algebra is involved since it has degree 2 generators. In the end, we found

Correspondence with WN algebra By comparing the central charge, it is obvious that DDAHA represented by N Young diagrams is directly related to WN algebra. Representation theory of WN algebra is described by Kac formula which represents the structure of degenerate states. For DDAHA, the null state condition is given by

to be zero for any l. This implies that

For N=2 case, it is satisfied by a state of the form,

which implies there exists null state at level mn when

This coincides with the null state structure of Virasoro module.

Construction of basis At this moment, the explicit construction of all the basis is difficult since the problem reduces to diagonalization of Hamiltonian.

from Morozov-Smirnov 1307.2576

The diagonal part is equivalent to Calogero-Sutherland Hamiltonian. The Jack polynomial gives a basis for such Hamiltonian. The second part gives the interaction. So far, the explicit form of the eigenstate is difficult while it is known to be integrable (Maulik-Okounkov).

There is, however, a special situation where such basis are available

In this case, the off-diagonal term vanishes and the eigenstate reduces to Jack polynomial. This was observed in different method by Alba et. al. 1012.1312. They went further to give an argument that (in our language)

In the language of DDAHA, one may confirm it by computing the eigenvalues of both hand sides for infinite number of commuting charges El. We confirmed they agree. So at least for such special value of a, the basis proposed by Alba et. al. and DDAHA coincides.

Some explicit computation

where

Factor Ω takes simple form when Y is a rectangle Young diagram

For general Y with rectangle decomposition, it takes the form

With this formula, for the Young diagram in the previous page, we have

which implies the identification of state:

DERIVATION OF CONFORMAL WARD IDENTITIES

Recursion formula and DDAHA In the following, we would like to establish the identification,

by showing the recursion formulae we obtained can be identified with the conformal Ward identity. In the language of Fock space, it is written as “obvious relations”;

While the action of DDAHA on the bra and ket basis are known, the commutation relation with the vertex operator remains mysterious.

Free boson representation We understood that DDAHA gives the Heisenberg + WN algebra with central charge

We introduce free bosons which describe such system

where

The generators of WN algebra is given in terms of them by quantum Miura transf.

Some comments on vertex operator

We need modify the U(1) part of the 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,

Carlsson-Okounkov

It correctly reproduce the correlation function of U(1) part:

But the conformal property of the vertex becomes much more complicated:

The third term represent the anomaly due to the modification of the vertex. It has the effect of correcting the shift of a coefficient in recursion relation. Ward identity for these operators is confirmed after lengthy computation. Those for L±2 are recently confirmed by H. Zhang.

We have confirmed the Ward identities for

This will be sufficient to generate the Ward identities for Virasoro and U(1) current which seems to prove AGT conjecture for SU(2) by repetedly use the commutation relations:

For higher generators, it will be difficult to evaluate the commutator with vertex operator.

The existence of recursion relation for D±1,n is encouraging enough to speculate if it is just sufficient to prove SU(N) case.

The coincidence of basis with Alba et. al. seems to be a good sign.

SUMMARY

Summary

• We found “a proof” for AGT conjecture through the recursion formulae.

• The algebraic structure of the recursion strongly reflect the cohomology of instantons.

• The connection with W-algebra is illuminated by the null states.

• The instanton moduli space has an interesting structure of integrable models which generalizes the Calogero-Sutherland models.

Future prospects

• The role of (modified) vertex operator should be understood to give the interpretation of the recursion formula.

• The integrable structure for generalized Jack polynomial may provide a good hint to derive various new integrable hierarchies.

• Nekrasov-Shatashvili limit may serve a good starting point for such researches.

Thank you!