Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A 2
Publications Mathématiques de l'IHÉS, Tome 118 (2013), pp. 213-342.

We construct a representation of the affine W-algebra of 𝔤𝔩 r on the equivariant homology space of the moduli space of U r -instantons, and we identify the corresponding module. As a corollary, we give a proof of a version of the AGT conjecture concerning pure N=2 gauge theory for the group SU(r).

Our approach uses a deformation of the universal enveloping algebra of W 1+∞, which acts on the above homology space and which specializes to W(𝔤𝔩 r ) for all r. This deformation is constructed from a limit, as n tends to ∞, of the spherical degenerate double affine Hecke algebra of GL n .

DOI : 10.1007/s10240-013-0052-3
Schiffmann, O. 1 ; Vasserot, E. 2

1 Département de Mathématiques, Université de Paris-Sud Bâtiment 425, 91405, Orsay Cedex France
2 Département de Mathématiques, Université de Paris 7 175 rue du Chevaleret, 75013, Paris France
@article{PMIHES_2013__118__213_0,
     author = {Schiffmann, O. and Vasserot, E.},
     title = {Cherednik algebras, {W-algebras} and the equivariant cohomology of the moduli space of instantons on {<strong>A</strong>
\protect\textsuperscript{2}}},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {213--342},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {118},
     year = {2013},
     doi = {10.1007/s10240-013-0052-3},
     mrnumber = {3150250},
     zbl = {1284.14008},
     language = {en},
     url = {http://www.numdam.org/articles/10.1007/s10240-013-0052-3/}
}
TY  - JOUR
AU  - Schiffmann, O.
AU  - Vasserot, E.
TI  - Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A
2
JO  - Publications Mathématiques de l'IHÉS
PY  - 2013
SP  - 213
EP  - 342
VL  - 118
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - http://www.numdam.org/articles/10.1007/s10240-013-0052-3/
DO  - 10.1007/s10240-013-0052-3
LA  - en
ID  - PMIHES_2013__118__213_0
ER  - 
%0 Journal Article
%A Schiffmann, O.
%A Vasserot, E.
%T Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A
2
%J Publications Mathématiques de l'IHÉS
%D 2013
%P 213-342
%V 118
%I Springer Berlin Heidelberg
%C Berlin/Heidelberg
%U http://www.numdam.org/articles/10.1007/s10240-013-0052-3/
%R 10.1007/s10240-013-0052-3
%G en
%F PMIHES_2013__118__213_0
Schiffmann, O.; Vasserot, E. Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A
2. Publications Mathématiques de l'IHÉS, Tome 118 (2013), pp. 213-342. doi : 10.1007/s10240-013-0052-3. http://www.numdam.org/articles/10.1007/s10240-013-0052-3/

[1.] Alday, V. F.; Gaiotto, D.; Tachikawa, Y. Liouville correlation functions from four dimensional gauge theories, Lett. Math. Phys., Volume 91 (2010), pp. 167-197 | DOI | MR | Zbl

[2.] Arakawa, T. Representation theory of W-algebras, Invent. Math., Volume 169 (2007), pp. 219-320 | DOI | MR | Zbl

[3.] Baranovsky, V. Moduli of sheaves on surfaces and action of the oscillator algebra, J. Differ. Geom., Volume 55 (2000), pp. 193-227 | MR | Zbl

[4.] Berest, Y.; Etingof, P.; Ginzburg, V. Cherednik algebras and differential operators on quasi-invariants, Duke Math. J., Volume 118 (2003), pp. 279-337 | DOI | MR | Zbl

[5.] Bernstein, J.; Lunts, V. Equivariant Sheaves and Functors, Lecture Notes in Mathematics, 1578, Springer, Berlin, 1994 | MR | Zbl

[6.] Bilal, A. Introduction to W-algebras, String Theory and Quantum Gravity, World Scientific, River Edge (1992), pp. 245-280 | MR

[7.] Braverman, A.; Feigin, B.; Finkelberg, M.; Rybnikov, L. A finite analog of the AGT relation I: finite W-algebras and quasimaps’ spaces, Commun. Math. Phys., Volume 308 (2011), pp. 457-478 | DOI | MR | Zbl

[8.] Burban, I.; Schiffmann, O. On the Hall algebra of an elliptic curve, I, Duke Math. J., Volume 161 (2012), pp. 1171-1231 | DOI | MR | Zbl

[9.] Cheah, J. Cellular decompositions for nested Hilbert schemes of points, Pac. J. Math., Volume 183 (1998), pp. 39-90 | DOI | MR | Zbl

[10.] Cherednik, I. Double Affine Hecke Algebras, London Mathematical Society Lecture Note Series, 319, Cambridge University Press, Cambridge, 2005 | DOI | MR | Zbl

[11.] Chriss, N.; Ginzburg, V. Representation Theory and Complex Geometry, Birkhaüser, Basel, 1996 | MR | Zbl

[12.] Ellingsrud, G.; Stromme, S. A. On the homology of the Hilbert scheme of points in the plane, Invent. Math., Volume 87 (1987), pp. 343-352 | DOI | MR | Zbl

[13.] A. V. Fateev and V. A. Litvinov, Integrable structure, W-symmetry and AGT relation, preprint (2011). | arXiv | MR

[14.] Feigin, B.; Frenkel, E. Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras, Int. J. Mod. Phys., Volume A7 (1992), pp. 197-215 | DOI | MR | Zbl

[15.] Feigin, B.; Frenkel, E. Integrals of motion and quantum groups, Proceedings of the C.I.M.E. School Integrable Systems and Quantum Groups (Lect. Notes in Math., 1620), Springer, Berlin (1995), pp. 349-418 | DOI | MR | Zbl

[16.] Frenkel, E.; Ben Zvi, D. Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs, Am. Math. Soc., Providence, 2004 | MR | Zbl

[17.] Frenkel, E.; Kac, V.; Radul, A.; Wang, W. W 1+∞ and W(𝔤𝔩 N ) with central charge N , Commun. Math. Phys., Volume 170 (1995), pp. 337-357 | DOI | MR | Zbl

[18.] D. Gaiotto, Asymptotically free N=2 theories and irregular conformal blocks, (2009). | arXiv

[19.] Goresky, M.; Kottwitz, R.; MacPherson, R. Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math., Volume 131 (1998), pp. 25-83 | DOI | MR | Zbl

[20.] Grojnowski, I. Instantons and affine algebras. I. The Hilbert scheme and vertex operators, Math. Res. Lett., Volume 3 (1996), pp. 275-291 | DOI | MR | Zbl

[21.] Kac, V. Vertex Algebras for Beginners, University Lecture Series, 10, Am. Math. Soc., Providence, 1998 | MR | Zbl

[22.] Kapranov, M. Eisenstein series and quantum affine algebras. Algebraic geometry, 7, J. Math. Sci. (N.Y.), Volume 84 (1997), pp. 1311-1360 | DOI | MR | Zbl

[23.] Kassel, C. Quantum Groups, Graduate Texts in Mathematics, 155, Springer, New York, 1995 | DOI | MR | Zbl

[24.] Licata, A.; Savage, A. Vertex operators and the geometry of moduli spaces of framed torsion-free sheaves, Sel. Math., Volume 16 (2010), pp. 201-240 | DOI | MR | Zbl

[25.] Macdonald, I. G. Symmetric Functions and Hall Polynomials, Oxford Math. Mon., 1995 | MR | Zbl

[26.] Malkin, A. Tensor product varieties and crystals: The ADE case, Duke Math. J., Volume 116 (2003), pp. 477-524 | DOI | MR | Zbl

[27.] Matsuo, A.; Nagatomo, K.; Tsuchiya, A. Quasi-Finite Algebras Graded by Hamiltonian and Vertex Operator Algebras, Moonshine: The First Quarter Century and Beyond, London Math. Soc. Lecture Note Ser., 372, Cambridge Univ. Press, Cambridge, 2010, pp. 282-329 | MR | Zbl

[28.] D. Maulik and A. Okounkov, Quantum cohomology and quantum groups, (2012). | arXiv

[29.] Nakajima, H. Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. (2), Volume 145 (1997), pp. 379-388 | DOI | MR | Zbl

[30.] Nakajima, H. Quiver varieties and tensor products, Invent. Math., Volume 146 (2001), pp. 399-449 | DOI | MR | Zbl

[31.] Nakajima, H.; Yoshioka, K. Instanton counting on blowup. I. 4-Dimensional pure gauge theory, Invent. Math., Volume 162 (2005), pp. 313-355 | DOI | MR | Zbl

[32.] Schiffmann, O.; Vasserot, E. The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compos. Math., Volume 147 (2011), pp. 188-234 | DOI | MR | Zbl

[33.] Schiffmann, O.; Vasserot, E. The elliptic Hall algebra and the K-theory of the Hilbert scheme of A 2 , Duke Math J., Volume 162 (2013), pp. 279-366 | DOI | MR | Zbl

[34.] Sekiguchi, J. Zonal spherical functions on some symmetric spaces, Publ. Res. Inst. Math. Sci., Volume 12 (1977), pp. 455-459 | DOI | MR | Zbl

[35.] Stanley, R. Some combinatorial properties of Jack symmetric functions, Adv. Math., Volume 77 (1989), pp. 76-115 | DOI | MR | Zbl

[36.] Suzuki, T. Rational and trigonometric degeneration of the double affine Hecke algebra of type A , Int. Math. Res. Not., Volume 37 (2005), pp. 2249-2262 | DOI | MR | Zbl

[37.] Varagnolo, M.; Vasserot, E. On the K-theory of the cyclic quiver variety, Int. Math. Res. Not., Volume 18 (1999), pp. 1005-1028 | DOI | MR | Zbl

[38.] Varagnolo, M.; Vasserot, E. Standard modules of quantum affine algebras, Duke Math. J., Volume 111 (2002), pp. 509-533 | DOI | MR | Zbl

[39.] Varagnolo, M.; Vasserot, E. Finite dimensional representations of DAHA and affine Springer fibers: the spherical case, Duke Math. J., Volume 147 (2007), pp. 439-540 | DOI | MR | Zbl

[40.] Vasserot, E. Sur l’anneau de cohomologie du schéma de Hilbert de C 2 , C. R. Acad. Sci. Paris Sér. I Math., Volume 332 (2001), pp. 7-12 | DOI | MR | Zbl

[41.] Weyl, H. The Classical Groups, Their Invariants and Representations, Princeton University Press, Princeton, 1949 | JFM | MR | Zbl

Cité par Sources :