Abstract
We associate vertex operator algebras to (p, q)-webs of interfaces in the topologically twisted \( \mathcal{N}=4 \) super Yang-Mills theory. Y-algebras associated to trivalent junctions are identified with truncations of \( \mathcal{W} \)1+∞ algebra. Starting with Y-algebras as atomic elements, we describe gluing of Y-algebras analogous to that of the topological vertex. At the level of characters, the construction matches the one of counting D0-D2-D4 bound states in toric Calabi-Yau threefolds. For some configurations of interfaces, we propose a BRST construction of the algebras and check in examples that both constructions agree. We define generalizations of \( \mathcal{W} \)1+∞ algebra and identify a large class of glued algebras with their truncations. The gluing construction sheds new light on the structure of vertex operator algebras conventionally constructed by BRST reductions or coset constructions and provides us with a way to construct new algebras. Many well-known vertex operator algebras, such as U(N)k affine Lie algebra, \( \mathcal{N}=2 \) superconformal algebra, \( \mathcal{N}=2 \) super-\( {\mathcal{W}}_{\infty } \), Bershadsky-Polyakov \( {\mathcal{W}}_3^{(2)} \), cosets and Drinfeld-Sokolov reductions of unitary groups can be obtained as special cases of this construction.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Gaiotto and M. Rapčák, Vertex algebras at the corner, arXiv:1703.00982 [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, JHEP 11 (2016) 138 [arXiv:1609.06271] [INSPIRE].
A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].
M. Aganagic, D. Jafferis and N. Saulina, Branes, black holes and topological strings on toric Calabi-Yau manifolds, JHEP 12 (2006) 018 [hep-th/0512245] [INSPIRE].
D. Jafferis, Crystals and intersecting branes, hep-th/0607032 [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Triality in minimal model holography, JHEP 07 (2012) 127 [arXiv:1205.2472] [INSPIRE].
K. Hornfeck, W algebras of negative rank, Phys. Lett. B 343 (1995) 94 [hep-th/9410013] [INSPIRE].
T. Procházka, Exploring W ∞ in the quadratic basis, JHEP 09 (2015) 116 [arXiv:1411.7697] [INSPIRE].
T. Procházka, W-symmetry, topological vertex and affine Yangian, JHEP 10 (2016) 077 [arXiv:1512.07178] [INSPIRE].
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].
O. Aharony, A. Hanany and B. Kol, Webs of (p, q) five-branes, five-dimensional field theories and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [INSPIRE].
V. Mikhaylov, Teichmüller TQFT vs. Chern-Simons theory, JHEP 04 (2018) 085 [arXiv:1710.04354] [INSPIRE].
E. Witten, A new look at the path integral of quantum mechanics, arXiv:1009.6032 [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].
D. Gaiotto and E. Witten, Janus configurations, Chern-Simons couplings, and the theta-angle in N = 4 super Yang-Mills theory, JHEP 06 (2010) 097 [arXiv:0804.2907] [INSPIRE].
D. Gaiotto and E. Witten, S-duality of boundary conditions in N = 4 super Yang-Mills theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
N. Nekrasov and E. Witten, The omega deformation, branes, integrability and Liouville theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [INSPIRE].
D. Gaiotto and E. Witten, Knot invariants from four-dimensional gauge theory, Adv. Theor. Math. Phys. 16 (2012) 935 [arXiv:1106.4789] [INSPIRE].
J. Yagi, Compactification on the Ω-background and the AGT correspondence, JHEP 09 (2012) 101 [arXiv:1205.6820] [INSPIRE].
K. Costello, M-theory in the Ω-background and 5-dimensional non-commutative gauge theory, arXiv:1610.04144 [INSPIRE].
C. Candu and M.R. Gaberdiel, Duality in N = 2 minimal model holography, JHEP 02 (2013) 070 [arXiv:1207.6646] [INSPIRE].
S. Lukyanov, Quantization of the Gel’fand-Dikii brackets, Funct. Anal. Appl. 22 (1988) 255.
V.S. Dotsenko and V.A. Fateev, Conformal algebra and multipoint correlation functions in two-dimensional statistical models, Nucl. Phys. B 240 (1984) 312 [INSPIRE].
V.S. Dotsenko and V.A. Fateev, Four point correlation functions and the operator algebra in the two-dimensional conformal invariant theories with the central charge c < 1, Nucl. Phys. B 251 (1985) 691 [INSPIRE].
V.S. Dotsenko and V.A. Fateev, Operator algebra of two-dimensional conformal theories with central charge c ≤ 1, Phys. Lett. B 154 (1985) 291 [INSPIRE].
T. Procházka and M. Rapčák, W-algebra modules, free fields and Gukov-Witten defects, arXiv:1808.08837 [INSPIRE].
R.E. Borcherds, Vertex algebras, Kac-Moody algebras and the monster, Proc. Nat. Acad. Sci. 83 (1986) 3068 [INSPIRE].
I. Frenkel, J. Lepowsky and A. Meurman, Vertex operator algebras and the monster, Elsevier, The Netherlands, (1988) [INSPIRE].
O. Schiffmann and E. Vasserot, Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on A 2, arXiv:1202.2756.
D. Maulik and A. Okounkov, Quantum groups and quantum cohomology, arXiv:1211.1287 [INSPIRE].
A. Tsymbaliuk, The affine Yangian of gl(1) revisited, Adv. Math. 304 (2017) 583.
N.C. Leung and C. Vafa, Branes and toric geometry, Adv. Theor. Math. Phys. 2 (1998) 91 [hep-th/9711013] [INSPIRE].
M. Aganagic and K. Schaeffer, Refined black hole ensembles and topological strings, JHEP 01 (2013) 060 [arXiv:1210.1865] [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].
E. Frenkel, V. Kac and M. Wakimoto, Characters and fusion rules for W algebras via quantized Drinfeld-Sokolov reductions, Commun. Math. Phys. 147 (1992) 295 [INSPIRE].
A. Okounkov, N. Reshetikhin and C. Vafa, Quantum Calabi-Yau and classical crystals, Prog. Math. 244 (2006) 597 [hep-th/0309208] [INSPIRE].
B. Feigin et al., Quantum toroidal \( \mathfrak{g}\mathfrak{l} \) 1 -algebra: plane partitions, Kyoto J. Math. 52 (2012) 621.
M. Bershtein, B. Feigin and G. Merzon, Plane partitions with a pit: generating functions and representation theory, arXiv:1512.08779.
S. Gukov and E. Witten, Gauge theory, ramification, and the geometric Langlands program, hep-th/0612073 [INSPIRE].
A.B. Zamolodchikov and V.A. Fateev, Disorder fields in two-dimensional conformal quantum field theory and N = 2 extended supersymmetry, Sov. Phys. JETP 63 (1986) 913 [Zh. Eksp. Teor. Fiz. 90 (1986) 1553] [INSPIRE].
W. Boucher, D. Friedan and A. Kent, Determinant formulae and unitarity for the N = 2 superconformal algebras in two-dimensions or exact results on string compactification, Phys. Lett. B 172 (1986) 316 [INSPIRE].
P. Di Vecchia, J.L. Petersen and H.B. Zheng, N = 2 extended superconformal theories in two-dimensions, Phys. Lett. B 162 (1985) 327 [INSPIRE].
P. Di Vecchia, J.L. Petersen and M. Yu, On the unitary representations of N = 2 superconformal theory, Phys. Lett. B 172 (1986) 211 [INSPIRE].
P. Di Vecchia, J.L. Petersen, M. Yu and H.B. Zheng, Explicit construction of unitary representations of the N = 2 superconformal algebra, Phys. Lett. B 174 (1986) 280 [INSPIRE].
R. Blumenhagen, W. Eholzer, A. Honecker, K. Hornfeck and R. Hubel, Coset realization of unifying W algebras, Int. J. Mod. Phys. A 10 (1995) 2367 [hep-th/9406203] [INSPIRE].
M. Bershadsky, Conformal field theories via Hamiltonian reduction, Commun. Math. Phys. 139 (1991) 71 [INSPIRE].
A.M. Polyakov, Gauge transformations and diffeomorphisms, Int. J. Mod. Phys. A 5 (1990) 833 [INSPIRE].
T. Creutzig and D. Gaiotto, Vertex algebras for S-duality, arXiv:1708.00875 [INSPIRE].
V.G. Kac, S. Shyr Roan and M. Wakimoto, Quantum reduction for affine superalgebras, Commun. Math. Phys. 241 (2003) 307 [math-ph/0302015].
T. Nishioka and Y. Tachikawa, Central charges of para-Liouville and Toda theories from M5-branes, Phys. Rev. D 84 (2011) 046009 [arXiv:1106.1172] [INSPIRE].
V. Belavin and B. Feigin, Super Liouville conformal blocks from N = 2 SU(2) quiver gauge theories, JHEP 07 (2011) 079 [arXiv:1105.5800] [INSPIRE].
A. Belavin, V. Belavin and M. Bershtein, Instantons and 2d superconformal field theory, JHEP 09 (2011) 117 [arXiv:1106.4001] [INSPIRE].
A.A. Belavin, M.A. Bershtein, B.L. Feigin, A.V. Litvinov and G.M. Tarnopolsky, Instanton moduli spaces and bases in coset conformal field theory, Commun. Math. Phys. 319 (2013) 269 [arXiv:1111.2803] [INSPIRE].
M.N. Alfimov, A.A. Belavin and G.M. Tarnopolsky, Coset conformal field theory and instanton counting on C 2 /Z p, JHEP 08 (2013) 134 [arXiv:1306.3938] [INSPIRE].
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
P. Sulkowski, Wall-crossing, free fermions and crystal melting, Commun. Math. Phys. 301 (2011) 517 [arXiv:0910.5485] [INSPIRE].
E. Bergshoeff, C.N. Pope, L.J. Romans, E. Sezgin and X. Shen, The super W ∞ algebra, Phys. Lett. B 245 (1990) 447 [INSPIRE].
L.J. Romans, The N = 2 super W 3 algebra, Nucl. Phys. B 369 (1992) 403 [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 N = 2 affine Yangian, arXiv:1807.11304 [INSPIRE].
V.A. Fateev and A.V. Litvinov, Correlation functions in conformal Toda field theory. I, JHEP 11 (2007) 002 [arXiv:0709.3806] [INSPIRE].
V.A. Fateev and A.V. Litvinov, Correlation functions in conformal Toda field theory. II, JHEP 01 (2009) 033 [arXiv:0810.3020] [INSPIRE].
V. Belavin, B. Estienne, O. Foda and R. Santachiara, Correlation functions with fusion-channel multiplicity in W 3 Toda field theory, JHEP 06 (2016) 137 [arXiv:1602.03870] [INSPIRE].
V. Belavin, X. Cao, B. Estienne and R. Santachiara, Second level semi-degenerate fields in W 3 Toda theory: matrix element and differential equation, JHEP 03 (2017) 008 [arXiv:1610.07993] [INSPIRE].
C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees, Infinite chiral symmetry in four dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
C. Beem, W. Peelaers, L. Rastelli and B.C. van Rees, Chiral algebras of class S, JHEP 05 (2015) 020 [arXiv:1408.6522] [INSPIRE].
M. Dedushenko, S. Gukov and P. Putrov, Vertex algebras and 4-manifold invariants, arXiv:1705.01645 [INSPIRE].
A. Braverman, M. Finkelberg and H. Nakajima, Instanton moduli spaces and W-algebras, arXiv:1406.2381 [INSPIRE].
T. Nishinaka and S. Yamaguchi, Affine SU(N) algebra from wall-crossings, JHEP 07 (2014) 030 [arXiv:1107.4762] [INSPIRE].
T. Kimura and V. Pestun, Quiver W -algebras, Lett. Math. Phys. 108 (2018) 1351 [arXiv:1512.08533] [INSPIRE].
J.-E. Bourgine, Y. Matsuo and H. Zhang, Holomorphic field realization of SH c and quantum geometry of quiver gauge theories, JHEP 04 (2016) 167 [arXiv:1512.02492] [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, Instanton counting, Macdonald functions and the moduli space of D-branes, JHEP 05 (2005) 039 [hep-th/0502061] [INSPIRE].
O. Foda and J.-F. Wu, A Macdonald refined topological vertex, J. Phys. A 50 (2017) 294003 [arXiv:1701.08541] [INSPIRE].
J. Song, Macdonald index and chiral algebra, JHEP 08 (2017) 044 [arXiv:1612.08956] [INSPIRE].
H. Awata, B. Feigin and J. Shiraishi, Quantum algebraic approach to refined topological vertex, JHEP 03 (2012) 041 [arXiv:1112.6074] [INSPIRE].
B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuous gl ∞ : semiinfinite construction of representations, Kyoto J. Math. 51 (2011) 337 [arXiv:1002.3100].
B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuous gl ∞ : tensor products of Fock modules and W n characters, Kyoto J. Math. 51 (2011) 365 [arXiv:1002.3113] [INSPIRE].
A. Mironov, A. Morozov and Y. Zenkevich, Ding-Iohara-Miki symmetry of network matrix models, Phys. Lett. B 762 (2016) 196 [arXiv:1603.05467] [INSPIRE].
H. Awata et al., Explicit examples of DIM constraints for network matrix models, JHEP 07 (2016) 103 [arXiv:1604.08366] [INSPIRE].
H. Awata et al., Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations, JHEP 10 (2016) 047 [arXiv:1608.05351] [INSPIRE].
H. Awata et al., Anomaly in RTT relation for DIM algebra and network matrix models, Nucl. Phys. B 918 (2017) 358 [arXiv:1611.07304] [INSPIRE].
J.-E. Bourgine, M. Fukuda, K. Harada, Y. Matsuo and R.-D. Zhu, (p, q)-webs of DIM representations, 5d N = 1 instanton partition functions and qq-characters, JHEP 11 (2017) 034 [arXiv:1703.10759] [INSPIRE].
M. Bershtein and A. Tsymbaliuk, Homomorphisms between different quantum toroidal and affine Yangian algebras, J. Pure Appl. Alg. 223 (2019) 867 [arXiv:1512.09109].
R.-D. Zhu and Y. Matsuo, Yangian associated with 2D N = 1 SCFT, PTEP 2015 (2015) 093A01 [arXiv:1504.04150] [INSPIRE].
M. Fukuda, K. Harada, Y. Matsuo and R.-D. Zhu, The Maulik-Okounkov R-matrix from the Ding-Iohara-Miki algebra, PTEP 2017 (2017) 093A01 [arXiv:1705.02941] [INSPIRE].
C. Cordova and S.-H. Shao, Schur indices, BPS particles and Argyres-Douglas theories, JHEP 01 (2016) 040 [arXiv:1506.00265] [INSPIRE].
M. Lemos and W. Peelaers, Chiral algebras for trinion theories, JHEP 02 (2015) 113 [arXiv:1411.3252] [INSPIRE].
N. Arbesfeld and O. Schiffmann, A presentation of the deformed W 1+∞ algebra, in Symmetries, integrable systems and representations, Springer, London, U.K., (2013), pg. 1.
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].
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: 1711.06888
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Procházka, T., Rapčák, M. Webs of W-algebras. J. High Energ. Phys. 2018, 109 (2018). https://doi.org/10.1007/JHEP11(2018)109
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2018)109