Abstract
Recently, Gaiotto and Rapčák (GR) proposed a new family of the vertex operator algebra (VOA) as the symmetry appearing at an intersection of five-branes to which they refer as Y algebra. Procházka and Rapčák, then proposed to interpret Y algebra as a truncation of affine Yangian whose module is directly connected to plane partitions (PP). They also developed GR’s idea to generate a new VOA by connecting plane partitions through an infinite leg shared by them and referred it as the web of W-algebra (WoW). In this paper, we demonstrate that double truncation of PP gives the minimal models of such VOAs. For a single PP, it generates all the minimal model irreducible representations of W-algebra. We find that the rule connecting two PPs is more involved than those in the literature when the U(1) charge connecting two PPs is negative. For the simplest nontrivial WoW, \( \mathcal{N} \) = 2 superconformal algebra, we demonstrate that the improved rule precisely reproduces the known character of the minimal models.
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References
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Graduate Texts in Contemporary Physics. Springer, Germany (1997).
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
O. Schiffmann and E. Vasserot, Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A 2, Publ. Math. IHES 118 (2013) 213.
D. Maulik and A. Okounkov, Quantum groups and quantum cohomology, arXiv:1211.1287 [INSPIRE].
B. Feigin et al., Quantum continuous gl(∞): Semiinfinite construction of representations, Kyoto J. Math. 51 (2011) 337 [arXiv:1002.3100].
A. Tsymbaliuk, The affine Yangian of gl1 revisited, Adv. Math. 304 (2017) 583.
M.R. Gaberdiel and R. Gopakumar, Triality in minimal model holography, JHEP 07 (2012) 127 [arXiv:1205.2472] [INSPIRE].
M.R. Gaberdiel and T. Hartman, Symmetries of holographic minimal models, JHEP 05 (2011) 031 [arXiv:1101.2910] [INSPIRE].
D. Gaiotto and M. Rapčák, Vertex algebras at the corner, JHEP 01 (2019) 160 [arXiv:1703.00982] [INSPIRE].
T. Procházka and M. Rapčák, Webs of W-algebras, JHEP 11 (2018) 109 [arXiv:1711.06888] [INSPIRE].
M.R. Gaberdiel, W. Li, C. Peng and H. Zhang, The supersymmetric affine Yangian, JHEP 05 (2018) 200 [arXiv:1711.07449] [INSPIRE].
M.R. Gaberdiel, W. Li and C. Peng, Twin-plane-partitions and \( \mathcal{N} \) = 2 affine Yangian, JHEP 11 (2018) 192 [arXiv:1807.11304] [INSPIRE].
M. Bershtein, B. Feigin and G. Merzon, Plane partitions with a “pit”: generating functions and representation theory, Selecta Math. 24 (2018) 21.
W. Burge, Restricted partition pairs, J. Comb. Theory A 63 (1993) 210.
V. Belavin, O. Foda and R. Santachiara, AGT, N-Burge partitions and \( \mathcal{W} \) N minimal models, JHEP 10 (2015) 073 [arXiv:1507.03540] [INSPIRE].
K.B. Alkalaev and V.A. Belavin, Conformal blocks of W N minimal models and AGT correspondence, JHEP 07 (2014) 024 [arXiv:1404.7094] [INSPIRE].
F. Ravanini and S.-K. Yang, Modular invariance in N = 2 superconformal field theories, Phys. Lett. B 195 (1987) 202 [INSPIRE].
Y. Matsuo, Character formula of C < 1 unitary representation of N = 2 superconformal algebra, Prog. Theor. Phys. 77 (1987) 793 [INSPIRE].
V.K. Dobrev, Characters of the unitarizable highest weight modules over the N = 2 superconformal algebras, Phys. Lett. B 186 (1987) 43 [INSPIRE].
E. Kiritsis, Character formulae and the structure of the representations of the N = 1, N = 2 superconformal algebras, Int. J. Mod. Phys. A 3 (1988) 1871 [INSPIRE].
T. Procházka, \( \mathcal{W} \) -symmetry, topological vertex and affine Yangian, JHEP 10 (2016) 077 [arXiv:1512.07178] [INSPIRE].
T. Procházka, Exploring \( \mathcal{W} \) ∞ in the quadratic basis, JHEP 09 (2015) 116 [arXiv:1411.7697] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].
A. Iqbal, C. Kozcaz and C. Vafa, The Refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].
H. Awata and H. Kanno, Refined BPS state counting from Nekrasov’s formula and Macdonald functions, Int. J. Mod. Phys. A 24 (2009) 2253 [arXiv:0805.0191] [INSPIRE].
H. Awata, B. Feigin and J. Shiraishi, Quantum algebraic approach to refined topological vertex, JHEP 03 (2012) 041 [arXiv:1112.6074] [INSPIRE].
N. Wyllard, A N − 1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
V.A. Alba, V.A. Fateev, A.V. Litvinov and G.M. Tarnopolskiy, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 (2011) 33 [arXiv:1012.1312] [INSPIRE].
A. Okounkov, N. Reshetikhin and C. Vafa, Quantum Calabi-Yau and classical crystals, Prog. Math. 244 (2006) 597 [hep-th/0309208] [INSPIRE].
D. Friedan, E.J. Martinec and S.H. Shenker, Conformal invariance, supersymmetry and string theory, Nucl. Phys. B 271 (1986) 93 [INSPIRE].
T. Procházka and M. Rapčák, \( \mathcal{W} \) -algebra modules, free fields and Gukov-Witten defects, arXiv:1808.08837 [INSPIRE].
C.N. Pope, L.J. Romans and X. Shen, The complete structure of W(∞), Phys. Lett. B 236 (1990) 173 [INSPIRE].
V. Kac and A. Radul, Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Commun. Math. Phys. 157 (1993) 429 [hep-th/9308153] [INSPIRE].
H. Awata, M. Fukuma, Y. Matsuo and S. Odake, Representation theory of the W(1 + ∞) algebra, Prog. Theor. Phys. Suppl. 118 (1995) 343 [hep-th/9408158] [INSPIRE].
H. Awata, M. Fukuma, Y. Matsuo and S. Odake, Representation theory of W 1+∞ algebra, in the proceedings of the 40th Yamada Conference, 20th International Colloquium, July 4–9, Toyonaka, Japan (1994).
H. Awata, M. Fukuma, Y. Matsuo and S. Odake, Quasifinite highest weight modules over the super W(1 + ∞) algebra, Comm. Math. Phys. 170 (1995) 151.
M. Fukuda, S. Nakamura, Y. Matsuo and R.-D. Zhu, SH c realization of minimal model CFT: triality, poset and Burge condition, JHEP 11 (2015) 168 [arXiv:1509.01000] [INSPIRE].
F.J. Narganes Quijano, On the parafermionic W N algebra, Int. J. Mod. Phys. A 6 (1991) 2611.
F.J. Narganes-Quijano, Bosonization of parafermions and related conformal models: W(N) algebras, Annals Phys. 206 (1991) 494 [INSPIRE].
T. Kimura and V. Pestun, Quiver W-algebras, Lett. Math. Phys. 108 (2018) 1351 [arXiv:1512.08533] [INSPIRE].
J.-E. Bourgine et al., (p, q)-webs of DIM representations, 5d \( \mathcal{N} \) = 1 instanton partition functions and qq-characters, JHEP 11 (2017) 034 [arXiv:1703.10759] [INSPIRE].
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Harada, K., Matsuo, Y. Plane partition realization of (web of) \( \mathcal{W} \)-algebra minimal models. J. High Energ. Phys. 2019, 50 (2019). https://doi.org/10.1007/JHEP02(2019)050
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DOI: https://doi.org/10.1007/JHEP02(2019)050