×

Instanton moduli spaces and bases in coset conformal field theory. (English) Zbl 1263.81252

Summary: The recently proposed relation between conformal field theories in two dimensions and supersymmetric gauge theories in four dimensions predicts the existence of the distinguished basis in the space of local fields in CFT. This basis has a number of remarkable properties: one of them is the complete factorization of the coefficients of the operator product expansion. We consider a particular case of the \(U(r)\) gauge theory on \(\mathbb{C}^{2}/\mathbb{Z}_{p}\) which corresponds to a certain coset conformal field theory and describe the properties of this basis. We argue that in the case \(p = 2, r = 2\) there exist different bases. We give an explicit construction of one of them. For another basis we propose the formula for matrix elements.

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
81T60 Supersymmetric field theories in quantum mechanics
22E70 Applications of Lie groups to the sciences; explicit representations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alday L.F., Gaiotto D., Tachikawa Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167-197 (2010) · Zbl 1185.81111 · doi:10.1007/s11005-010-0369-5
[2] Nekrasov N.A.: Seiberg-Witten Prepotential From Instanton Counting. Adv. Theor. Math. Phys. 7, 831-864 (2004) · Zbl 1056.81068
[3] Nakajima H.: Heisenberg algebra and Hilbert schemes of points on projective surfaces. Ann. of Math. 145, 379-388 (1997) · Zbl 0915.14001 · doi:10.2307/2951818
[4] Nakajima H.: Quiver varieties and Kac-Moody algebras. Duke Math. J. 91, 515-560 (1998) · Zbl 0970.17017 · doi:10.1215/S0012-7094-98-09120-7
[5] Atiyah M., Bott R.: The moment map and equivariant cohomology. Topology. 23, 1-28 (1984) · Zbl 0521.58025 · doi:10.1016/0040-9383(84)90021-1
[6] Belavin V., Feigin B.: Super Liouville conformal blocks from N = 2 SU(2) quiver gauge theories. JHEP. 1107, 079 (2011) · Zbl 1298.81154 · doi:10.1007/JHEP07(2011)079
[7] Goddard P., Kent A., Olive D.I.: Unitary representations of the Virasoro and Supervirasoro algebras. Commun. Math. Phys. 103, 105-119 (1986) · Zbl 0588.17014 · doi:10.1007/BF01464283
[8] Nakajima H.: Quiver varieties and finite dimensional representations of quantum affine algebras. J. Amer. Math. Soc. 14, 145-238 (2001) · Zbl 0981.17016 · doi:10.1090/S0894-0347-00-00353-2
[9] Miki K.: A(q,γ) analog of the W1+∞ algebra. J. Math. Phys. 48, 123520 (2007) · Zbl 1153.81405 · doi:10.1063/1.2823979
[10] Feigin B., Hoshino A., Shibahara J., Shiraishi J., Yanagida S.: Kernel function and quantum algebras. RIMS Kokyuroku. 1689, 133-152 (2010)
[11] Feigin B.: Unpublished
[12] Awata H., Feigin B., Hoshino A., Kanai M., Shiraishi J., Yanagida S.: Notes on Ding-Iohara algebra and AGT conjecture. http://arxiv.org/abs/1106.4088v3 [math-ph], 2011 · Zbl 1178.81241
[13] Li W.-P., Qin Z., Wang W.: The cohomology rings of Hilbert schemes via Jack polynomials. CRM proc. Lect. Notes 38, 249-258 (2004) · Zbl 1104.14002
[14] Carlsson, E., Okounkov, A.: Exts and vertex operators. http://arxiv.org/abs/0801.2565v2 [math.AG], 2009 · Zbl 1256.14010
[15] Alday L.F., Tachikawa Y.: Affine SL(2) conformal blocks from 4d gauge theories. Lett. Math. Phys. 94, 87-114 (2010) · Zbl 1198.81162
[16] Alba V.A., Fateev V.A., Litvinov A.V., Tarnopolsky G.M.: On combinatorial expansion of the conformal blocks arising from AGT conjecture. . Lett. Math. Phys. 98, 33-64 (2011) · Zbl 1242.81119 · doi:10.1007/s11005-011-0503-z
[17] Wyllard N.: AN-1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories. JHEP. 11, 002 (2009) · doi:10.1088/1126-6708/2009/11/002
[18] Mironov A., Morozov A.: On AGT relation in the case of U(3). Nucl. Phys. B825, 1-37 (2010) · Zbl 1196.81205 · doi:10.1016/j.nuclphysb.2009.09.011
[19] Fateev V.A., Litvinov A. V.: Integrable structure, W-symmetry and AGT relation. JHEP. 01, 051 (2012) · Zbl 1306.81100 · doi:10.1007/JHEP01(2012)051
[20] Belavin A., Belavin V., Bershtein M.: Instantons and 2d Superconformal field theory. JHEP. 1109, 117 (2011) · Zbl 1301.81102 · doi:10.1007/JHEP09(2011)117
[21] Ito Y.: Ramond sector of super Liouville theory from instantons on an ALE space. Nucl. Phys. B861, 387-402 (2012) · Zbl 1246.81135 · doi:10.1016/j.nuclphysb.2012.04.001
[22] Nishioka T., Tachikawa Y.: Para-Liouville/Toda central charges from M5-branes. Phys. Rev. D84, 046009 (2011)
[23] Wyllard N.: Coset conformal blocks andN = 2 gauge theories. http://arxiv.org/abs/1109.4264v1 [hep-th], 2011 · Zbl 1213.81173
[24] Alfimov M., Tarnopolsky G.: Parafermionic Liouville field theory and instantons on ALE spaces. JHEP. 1202, 036 (2012) · Zbl 1309.81141 · doi:10.1007/JHEP02(2012)036
[25] Bonelli G., Maruyoshi K., Tanzini A.: Instantons on ALE spaces and super Liouville conformal field theories. JHEP. 1108, 056 (2011) · Zbl 1298.81340 · doi:10.1007/JHEP08(2011)056
[26] Bonelli G., Maruyoshi K., Tanzini A.: Gauge Theories on ALE Space and Super Liouville Correlation Functions. http://arxiv.org/abs/1107.4609.v2[hep-th], 2012 · Zbl 1262.81156
[27] Argyres P.C., LeClair A., Tye S.H.H.: On the possibility of fractional superstrings. Phys. Lett. B253, 306-312 (1991)
[28] Fateev V.A., Zamolodchikov A.B.: Representations of the algebra of parafermion currents of spin 4/3 in two-dimensional conformal field theory. Minimal models and the tricritical Potts Z(3) model. Theor. Math. Phys. 71, 451-462 (1987)
[29] Pogosian R.G.: Operator algebra in two-dimensional conformal quantum field theory containing spin 4/3 parafermionic conserved currents. Int. J. Mod. Phys. A6, 2005-2023 (1991) · Zbl 0797.17021
[30] Nakajima, H.: Lectures on Hilbert schemes of points on surfaces. University Lecture Series. 18. Providence, RI: Amer. Math. Soc., 1999 · Zbl 0949.14001
[31] Nakajima H., Yoshioka K.: it Lectures on Instanton Counting. http://arxiv.org/abs/math/0311058v1 [math. AG], 2003 · Zbl 1080.14016
[32] Flume R., Poghossian R.: An algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential. Int. J. Mod. Phys. A18, 2541 (2003) · Zbl 1069.81569
[33] Nakajima H., Yoshioka K.: Instanton counting on blowup. I. 4-dimensional pure gauge theory. Invent. Math. 162, 313-355 (2005) · Zbl 1100.14009 · doi:10.1007/s00222-005-0444-1
[34] Fucito F., Morales J.F., Poghossian R.: Instantons on quivers and orientifolds. JHEP. 10, 037 (2004) · Zbl 1388.83243 · doi:10.1088/1126-6708/2004/10/037
[35] Shadchin S.: Cubic curves from instanton counting. JHEP. 03, 046 (2006) · Zbl 1226.81273 · doi:10.1088/1126-6708/2006/03/046
[36] Macdonald, I.G.: Symmetric functions and Hall polynomials. Oxford: Oxford University Press, 1995 · Zbl 0824.05059
[37] Belavin A., Belavin V.: AGT conjecture and Integrable structure of Conformal field theory for c = 1. Nucl. Phys. B850, 199-213 (2011) · Zbl 1215.81096 · doi:10.1016/j.nuclphysb.2011.04.014
[38] Estienne B., Pasquier V., Santachiara R., Serban D.: Conformal blocks in Virasoro and W theories: Duality and the Calogero-Sutherland model. Nucl. Phys. B860, 377-420 (2012) · Zbl 1246.81323 · doi:10.1016/j.nuclphysb.2012.03.007
[39] Zamolodchikov A.B., Zamolodchikov Al.B.: Structure constants and conformal bootstrap in Liouville field theory. Nucl. Phys. B477, 577-605 (1996) · Zbl 0925.81301 · doi:10.1016/0550-3213(96)00351-3
[40] Nakajima H.: Sheaves on ALE spaces and quiver varieties. Mosc. Math. J. 7, 699-722 (2007) · Zbl 1166.14007
[41] Bruzzo U., Poghossian R., Tanzini A.: Poincaré Polynomial of Moduli Spaces of Framed Sheaves on (Stacky) Hirzebruch Surfaces. Commun. Math. Phys. 304, 395-409 (2011) · Zbl 1216.81114 · doi:10.1007/s00220-011-1231-z
[42] Crnkovic C., Sotkov G., Stanishkov M.: Renormalization group flow for general SU(2) coset models. Phys. Lett. B226, 297 (1989)
[43] Crnkovic C., Paunov R., Sotkov G., Stanishkov M.: Fusions of conformal models. Nucl. Phys. B336, 637 (1990) · doi:10.1016/0550-3213(90)90445-J
[44] Lashkevich M.: Superconformal 2-D minimal models and an unusual coset construction. Mod. Phys. Lett. A8, 851-860 (1993) · Zbl 1015.81562
[45] Fucito F., Morales J.F., Poghossian R.: Multi instanton calculus on ALE spaces. Nucl. Phys. B703, 518-536 (2004) · Zbl 1198.53024 · doi:10.1016/j.nuclphysb.2004.09.014
[46] Fucito F., Morales J.F., Poghossian R.: Instanton on toric singularities and black hole countings. JHEP. 12, 073 (2006) · Zbl 1226.81262 · doi:10.1088/1126-6708/2006/12/073
[47] Nagao K.: Quiver varieties and Frenkel-Kac construction. J. Algebra. 321, 3764-3789 (2007) · Zbl 1196.17021 · doi:10.1016/j.jalgebra.2009.03.012
[48] Poghossian, R.: Unpublished · Zbl 1166.14007
[49] Zamolodchikov Al.B.: Three-point function in the minimal Liouville gravity. Theor. Math. Phys. 142, 183-196 (2005) · Zbl 1178.81241
[50] Rashkov R.C., Stanishkov M.: Three-point correlation functions in N = 1 Super Lioville Theory. Phys. Lett. B380, 49-58 (1996)
[51] Poghosian R.H.: Structure constants in the N = 1 super-Liouville field theory. Nucl. Phys. B496, 451-464 (1997) · Zbl 0935.81063 · doi:10.1016/S0550-3213(97)00218-6
[52] Bershtein M.A., Fateev V.A., Litvinov A.V.: Parafermionic polynomials, Selberg integrals and three- point correlation function in parafermionic Liouville field theory. Nucl. Phys. B847, 413-459 (2011) · Zbl 1208.81177 · doi:10.1016/j.nuclphysb.2011.01.035
[53] Fateev V.A.: The sigma model (dual) representation for a two-parameter family of integrable quantum field theories. Nucl. Phys. B473, 509-538 (1996) · Zbl 0925.81297 · doi:10.1016/0550-3213(96)00256-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.