Abstract
We discuss a class of vertex operator algebras \( {\mathcal{W}}_{\left.m\right|n\kern0.33em \times \kern0.33em \infty } \) generated by a super- matrix of fields for each integral spin 1, 2, 3, . . . . The algebras admit a large family of truncations that are in correspondence with holomorphic functions on the Calabi-Yau singularity given by solutions to xy = zmwn. We propose a free-field realization of such truncations generalizing the Miura transformation for \( {\mathcal{W}}_N \) algebras. Relations in the ring of holomorphic functions lead to bosonization-like relations between different free-field realizations. The discussion provides a concrete example of a non-trivial interplay between vertex operator algebras, algebraic geometry and gauge theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.B. Zamolodchikov, Infinite additional symmetries in two-dimensional conformal quantum field theory, Theor. Math. Phys.65 (1985) 1205 [INSPIRE].
V.A. Fateev and S.L. Lukyanov, The models of two-dimensional conformal quantum field theory with Z (n) symmetry, Int. J. Mod. Phys.A 3 (1988) 507 [INSPIRE].
V.G. Drinfeld and V.V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, J. Sov. Math.30 (1984) 1975 [INSPIRE].
F.A. Bais, P. Bouwknegt, M. Surridge and K. Schoutens, Extensions of the Virasoro algebra constructed from Kac-Moody algebras using higher order Casimir invariants, Nucl. Phys.B 304 (1988) 348 [INSPIRE].
P. Goddard, A. Kent and D.I. Olive, Virasoro algebras and coset space models, Phys. Lett.B 152 (1985) 88.
P. Goddard, A. Kent and D.I. Olive, Unitary representations of the Virasoro and superVirasoro algebras, Commun. Math. Phys.103 (1986) 105 [INSPIRE].
F.A. Bais, P. Bouwknegt, M. Surridge and K. Schoutens, Coset construction for extended Virasoro algebras, Nucl. Phys.B 304 (1988) 371 [INSPIRE].
P. Bowcock, Quasi-primary fields and associativity of chiral algebras, Nucl. Phys.B 356 (1991) 367 [INSPIRE].
H.G. Kausch and G.M.T. Watts, A study of W algebras using Jacobi identities, Nucl. Phys.B 354 (1991) 740 [INSPIRE].
M. Bershadsky and H. Ooguri, Hidden SL(n) symmetry in conformal field theories, Commun. Math. Phys.126 (1989) 49 [INSPIRE].
B. Feigin and E. Frenkel, Quantization of the Drinfeld-Sokolov reduction, Phys. Lett.B 246 (1990) 75 [INSPIRE].
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].
N. Wyllard, A(N − 1) conformal Toda field theory correlation functions from conformal N = 2 SU(N ) quiver gauge theories, JHEP11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys.7 (2003) 831 [hep-th/0206161] [INSPIRE].
H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J.76 (1994) 365.
O. Schiffmann and E. Vasserot, Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on A2 , arXiv:1202.2756.
D. Maulik and A. Okounkov, Quantum groups and quantum cohomology, arXiv:1211.1287 [INSPIRE].
A. Braverman, M. Finkelberg and H. Nakajima, Instanton moduli spaces and W -algebras, arXiv:1406.2381 [INSPIRE].
F. Yu and Y.-S. Wu, Nonlinearly deformed W∞ algebra and second Hamiltonian structure of KP hierarchy, Nucl. Phys.B 373 (1992) 713 [INSPIRE].
J. de Boer, L. Feher and A. Honecker, A class of W algebras with infinitely generated classical limit, Nucl. Phys.B 420 (1994) 409 [hep-th/9312049] [INSPIRE].
B. Khesin and F. Malikov, Universal Drinfeld-Sokolov reduction and matrices of complex size, Commun. Math. Phys.175 (1996) 113 [hep-th/9405116] [INSPIRE].
K. Hornfeck, W algebras of negative rank, Phys. Lett.B 343 (1995) 94 [hep-th/9410013] [INSPIRE].
R. Blumenhagen et al., Coset realization of unifying W algebras, Int. J. Mod. Phys.A 10 (1995) 2367 [hep-th/9406203] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Triality in minimal model holography, JHEP07 (2012) 127 [arXiv:1205.2472] [INSPIRE].
T. Procházka, Exploring W∞ in the quadratic basis, JHEP09 (2015) 116 [arXiv:1411.7697] [INSPIRE].
A.R. Linshaw, Universal two-parameter \( \mathcal{W} \)∞-algebra and vertex algebras of type \( \mathcal{W} \) (2, 3, . . . , N ), arXiv:1710.02275 [INSPIRE].
T. Procházka and M. Račák, Webs of W -algebras, JHEP11 (2018) 109 [arXiv:1711.06888] [INSPIRE].
T. Procházka, \( \mathcal{W} \)-symmetry, topological vertex and affine Yangian, JHEP10 (2016) 077 [arXiv:1512.07178] [INSPIRE].
M. Bershtein, B.L. Feigin and G. Merzon, Plane partitions with a “pit”: generating functions and representation theory, arXiv:1512.08779.
A. Litvinov and L. Spodyneiko, On W algebras commuting with a set of screenings, JHEP11 (2016) 138 [arXiv:1609.06271] [INSPIRE].
D. Gaiotto and M. Račák, Vertex algebras at the corner, JHEP01 (2019) 160 [arXiv:1703.00982] [INSPIRE].
N.C. Leung and C. Vafa, Branes and toric geometry, Adv. Theor. Math. Phys.2 (1998) 91 [hep-th/9711013] [INSPIRE].
N. Nekrasov and E. Witten, The Omega deformation, branes, integrability and Liouville theory, JHEP09 (2010) 092 [arXiv:1002.0888] [INSPIRE].
M. Rapcak, Y. Soibelman, Y. Yang and G. Zhao, Cohomological Hall algebras, vertex algebras and instantons, arXiv:1810.10402 [INSPIRE].
W.-y. Chuang, T. Creutzig, D.E. Diaconescu and Y. Soibelman, Hilbert schemes of nonreduced divisors in Calabi-Yau threefolds and W -algebras, arXiv:1907.13005 [INSPIRE].
P. Koroteev, On quiver W -algebras and defects from gauge origami, Phys. Lett.B 800 (2020) 135101 [arXiv:1908.04394] [INSPIRE].
M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, Commun. Num. Theor. Phys.5 (2011) 231 [arXiv:1006.2706] [INSPIRE].
N. Nekrasov, BPS/CFT correspondence II: instantons at crossroads, moduli and compactness theorem, Adv. Theor. Math. Phys.21 (2017) 503 [arXiv:1608.07272] [INSPIRE].
N. Nekrasov and N.S. Prabhakar, Spiked instantons from intersecting D-branes, Nucl. Phys.B 914 (2017) 257 [arXiv:1611.03478] [INSPIRE].
T. Procházka and M. Rapčák, W -algebra modules, free fields and Gukov-Witten defects, JHEP05 (2019) 159 [arXiv:1808.08837] [INSPIRE].
A. Negut, AGT relations for sheaves on surfaces, arXiv:1711.00390 [INSPIRE].
M. Dedushenko, S. Gukov and P. Putrov, Vertex algebras and 4-manifold invariants, in the proceedingso of the Nigel Hitchin’s 70thBirthday Conference. Geometry and Physics, September 55-16, Aarhus, Denmark, (2016), arXiv:1705.01645 [INSPIRE].
B. Feigin and S. Gukov, VOA[M4 ], arXiv:1806.02470 [INSPIRE].
M. Rapcak, The vertex algebra vertex, Ph.D. thesis, University Waterloo, Waterloo, Canada (2019).
M. Aganagic, D. Jafferis and N. Saulina, Branes, black holes and topological strings on toric Calabi-Yau manifolds, JHEP12 (2006) 018 [hep-th/0512245] [INSPIRE].
D. Jafferis, Crystals and intersecting branes, hep-th/0607032 [INSPIRE].
M. Aganagic and K. Schaeffer, Refined black hole ensembles and topological strings, JHEP01 (2013) 060 [arXiv:1210.1865] [INSPIRE].
L. Eberhardt and T. Procházka, The matrix-extended \( \mathcal{W} \)1+∞algebra, to appear.
T. Creutzig and Y. Hikida, Rectangular W -algebras, extended higher spin gravity and dual coset CFTs, JHEP02 (2019) 147 [arXiv:1812.07149] [INSPIRE].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys.254 (2005) 425 [hep-th/0305132] [INSPIRE].
A. Linshaw and F. Malikov, One example of a chiral Lie group, arXiv:1902.07414 [INSPIRE].
T. Procházka, Instanton R-matrix and W-symmetry, JHEP12 (2019) 099 [arXiv:1903.10372] [INSPIRE].
K. Costello, M-theory in the Omega-background and 5-dimensional non-commutative gauge theory, arXiv:1610.04144 [INSPIRE].
T. Creutzig and Y. Hikida, Rectangular W algebras and superalgebras and their representations, Phys. Rev.D 100 (2019) 086008 [arXiv:1906.05868] [INSPIRE].
B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Branching rules for quantum toroidal gln , Adv. Math.300 (2016) 229 [arXiv:1309.2147] [INSPIRE].
M. Bershtein and A. Tsymbaliuk, Homomorphisms between different quantum toroidal and affine Yangian algebras, J. Pure Appl. Algebra223 (2019) 867 [INSPIRE].
M.R. Gaberdiel, W. Li, C. Peng and H. Zhang, The supersymmetric affine Yangian, JHEP05 (2018) 200 [arXiv:1711.07449] [INSPIRE].
M.R. Gaberdiel, W. Li and C. Peng, Twin-plane-partitions and \( \mathcal{N}=2 \)affine Yangian, JHEP11 (2018) 192 [arXiv:1807.11304] [INSPIRE].
H. Awata et al., (q, t)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces, JHEP03 (2018) 192 [arXiv:1712.08016] [INSPIRE].
W. Chaimanowong and O. Foda, Coloured refined topological vertices and parafermion conformal field theories, arXiv:1811.03024 [INSPIRE].
R. Dijkgraaf, B. Heidenreich, P. Jefferson and C. Vafa, Negative branes, supergroups and the signature of spacetime, JHEP02 (2018) 050 [arXiv:1603.05665] [INSPIRE].
T. Kimura and V. Petsun, Super instanton counting and localization, arXiv:1905.01513 [INSPIRE].
W. Li and P. Longhi, Gluing two affine Yangians of \( \mathfrak{g}{\mathfrak{l}}_1 \), JHEP10 (2019) 131 [arXiv:1905.03076] [INSPIRE].
E. Witten, Fivebranes and knots, arXiv:1101.3216 [INSPIRE].
V. Mikhaylov and E. Witten, Branes and supergroups, Commun. Math. Phys.340 (2015) 699 [arXiv:1410.1175] [INSPIRE].
D. Gaiotto and E. Witten, Supersymmetric boundary conditions in N = 4 super Yang-Mills theory, J. Statist. Phys.135 (2009) 789 [arXiv:0804.2902] [INSPIRE].
M. Wakimoto, Fock representations of the affine lie algebra A1 (1), Commun. Math. Phys.104 (1986) 605 [INSPIRE].
B.L. Feigin and E.V. Frenkel, Representations of affine Kac-Moody algebras, bosonization and resolutions, Lett. Math. Phys.19 (1990) 307 [INSPIRE].
W.-L. Yang, Y.-Z. Zhang and X. Liu, Free field realization of current superalgebra gl(M|N)(k), J. Math. Phys.48 (2007) 053514 [arXiv:0806.0190] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1910.00031
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Rapčák, M. On extensions of \( \mathfrak{gl}\widehat{\left(\left.m\right|n\right)} \) Kac-Moody algebras and Calabi-Yau singularities. J. High Energ. Phys. 2020, 42 (2020). https://doi.org/10.1007/JHEP01(2020)042
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2020)042