CFT and integrability in memorial of Alexei Zamolodchikov Sogan University, Seoul December 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
Introduction
4D/2D duality: Brief History In 1994, Seiberg and Witten found an interesting relation between 4D SUSY gauge theories and geometry of 2D Riemann surface. In 1995, DOZZ solved 3pt function of Liouville theory. In 2003, Nekrasov found the exact form of the instanton partition function by introducing Omega background. In 2007, Pestun found a closed formula of the partition function of SYM on S 4 by localization method. In 2009, Alday, Gaiotto and Tachikawa (AGT) conjectured a precise correspondence between partition function of SYM and correlation function of Liouville/Toda field theory.
AGT conjecture Alday-Gaiotto-Tachikawa (2009) N=2 SYM Toda Nekrasov partition function = Correlation function From the viewpoint of M5 brane, it may be understood as the consequence of two types of compactification, Riemann surface with punctures Four sphere Geometric Langrands program (Witten) N=2 SYM Toda/Liouville field theory Here we pursue more direct correspondence through the symmetry behind the scene.
Instanton moduli space and 2d CFT An Intuitive Picture (1) Let me start from rough illustration of the correspondence between the cohomology of instanton moduli space and 2D CFT We start from a simplified version of moduli space where we keep the location of instantons as a moduli: Hilbert scheme of k points on euclidian 4D space C 2. (C 2 ) k is divided by S k since these points are not distinguishable. S k : 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 over the moduli space of arbitrary number of instantons
More precise correspondence Actually the structure behind the scene is more complicated, Liouville/Toda Instanton moduli space W-algebra Nonlinear algebras! SH (spherical degenerate double affine Hecke algebra) It is recently recognized by mathematicians, the origin of the AGT correspondence is due to the coincidence of these two nonlinear symmetries. Schiffmann and Vassero( 12), Maulik and Okounkov ( 13) In this talk, we give an explicit realization of such correspondence in the form of recursion formulae for Nekrasov partition function
Some references on the proof of AGT relation Proof for pure super Yang-Mills Schiffmann and Vassero( 12), Maulik and Okounkov ( 13) Proof for N=2* Fateev-Litvinov ( 09) Proof for quiver gauge theories Alba-Fateev-Litvinov-Tarnopolskiy ( 10) Belavin-Belavin ( 11) Fateev-Litvinov ( 11) Morosov-Mironov-Shakirov ( 11), Morosov-Smirnov ( 13) Zhang-M ( 11), Kanno-M-Zhang ( 12, 13) The work by Alba et. al. give a proof of AGT with the same set-up as ours. Our emphasis here is to clarify the role of symmetry which was not understood by their work.
Recursion formula for Nekrasov partition function
AGT conjecture for quiver Alday-Gaiotto-Tachikawa,Wyllard,Mironov-Morozov Nekrasov Partition function for conformal inv. SU(N)x xsu(n) (L factors) quiver N N N N N N is identical to conformal block function of 2D CFT described by SU(N) Toda field theory with L+3 vertex operator insertions V V V with identification of parameters
Partition function takes the form of matrix multiplication N 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 Product of factors associated with each box of the Young diagram
Correlation function of Toda field theory V V μ V V V Pants decomposition V At the hole, we insert decomposition of 1 V The same summation shows up if we identify, : orthonormal basis Alba et. al. 2010 For an appropriate choice of, This is the key step to prove AGT.. How can we manage the descendant?
Recursion relation for Z In the following, we claim a recursion formulae in the following form, ±, is defined by adding/subtracting a box in Y and W. ±, is a polynomial of,. The structure constant which appears in ±, to the action of SH on the basis,. is mostly identical Equivalence between SH and W-algebra implies that above relation can be identified with the conformal Ward identities. We will later show it explicitly with the properties of the vertex operator.
Intermediate step for the proof: Variation formulae We evaluate the ratio of Nekrasov formula when we add/remove a box in Y and W. For example:
We combine them in the following form, with This factor is necessary to absorb extra terms from variations of vector multiplets.
The combination of variation reduces to the following simple expression: with the following replacements: This expression can be rewritten in the form: where q n is written by a generating functional: Due to a combinatorial identity: We arrive at This is one of the main claim of this talk.
SH, Calogero, Liouville/Toda and instantons
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 Dunkl operator transposition of z i and z j Hamiltonian is rewritten as, Higher conserved charges:
SH (Spherical DDAHA) DDAHA: generated by SH restriction on the symmetric part generators: reduces to if we replace Dunkl operator to derivative Algebra of SH
Double Affine Hecke Algebra (DAHA) Introduced by Cherednik to investigate properties of Macdonald polynomial DAHA for GL n Generators: Two parameters Braid-like Relations Cf. talks by Pestun, Pugai, Ravanini Reduction to DAHA: Generators are replaced by:
Free boson representation One may assign (in the large N limit) Representation of SH It is possible to generate all other generators from Awata, Odake, M, Shiraishi, 1995 by taking commutation relations
Action on instanton moduli space As explained in the introduction, there exists a free boson representation of the cohomology of the instanton moduli space. Action of product of c 1 in the cohomology of Hilb Morozov-Smirnov By introducing the equivariant cohomology with deformation parameter ε 1,2 the action of c 1 is generalized to, Interestingly, this coincides the Hamiltonian of Calogero-Sutherland model (D 0,2 ) after replacing p n by free boson oscillator In order to define the action on SU(n) instanton moduli, we need to find a representation with n bosons
Representation by n bosons We have derived a representation of SH on free boson Fock space Representation on the direct product is nontrivial since the algebra is nonlinear, i.e. the coproduct is nontrivial. In particular, the generator D rs is not sum of operators acting on each F This time the algebra SH has nontrivial central extension (SH c ) c l : central charges l=0,1,2,...
Representation by orthonormal basis We introduce an orthonormal basis for the representation of DDAHA which gives a representation of DDAHA with central charges
Comparison of the coefficients 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 δ 1,n is slightly shifted by ξ. This small mismatch will be explained by an exotic choice of vertex operator.
W algebra: Symmetry for Toda W-algebra Fateev-Zamolodchikov 87 Famous but complicated nonlinear algebra Direct comparison with SH seems far.
Comparison of two algebras s Definition of algebra plays a special role Hamiltonian for Calogero-Sutherland Virasoro Heisenberg -1 1 2 1 r From D 0,2 and D 1,0, D -1,0 one may reconstruct the action of other generators Definition of Heisenberg (U(1)) and Virasoro
Equivalence of SH c and W algebra Schifmann and Vasserot proved that the symmetric representation by n bosons are equivalent to the Hilbert space of W n algebra + U(1) factor. Expression for We made an additional confirmation of this fact by showing the null states in SH c is exactly the same as W n algebra This occurs if Y is an rectangular Young diagram with a restriction on a. Mimachi-Yamada Awata-M-Odake-Shiraishi 1995
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 ; These relations can be translated into the recursion formula for Z. One may establish that our recursion formula is identical to Ward identity. Nontriviality is the commutation relation of generators with the vertex operator.
Some comments on vertex operator The vertex operator is written as a product of U(1) (Heisenberg) part and W N 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
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 identities for ± and ± confirmed after lengthy computation. Those for L ±2 are recently confirmed by H. Zhang. These are enough to demonstrate the Ward identity for U(1) current and Virasoro generators. It completes an algebraic proof of AGT conjecture for SU(2) case.
Summary 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 SH (spherical Degenerate Double Affine Hecke Algebra) which is equivalent to W N algebra 3. They can be identified with the extended conformal Ward identities if we have the identification which gives a recursive proof of AGT for SU(2) gauge group
Future prospects General W algebra Ward Identities: Is our recursion relation enough to prove it? Connection between coproduct of SH and integrable models (Maulick-Okounkov): R-matrix and Yangian Generalization of SH for other gauge groups 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
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