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
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
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
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)
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)
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:
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
• 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.