Abstract
We study the 3-parametric family of vertex operator algebras based on the Grassmannian coset CFT \( \mathfrak{u} \)(M + N )k /(\( \mathfrak{u} \)(M )k × \( \mathfrak{u} \)(N )k ). This VOA serves as a basic building block for a large class of cosets and generalizes the \( {\mathcal{W}}_{\infty } \) algebra. We analyze representations and their characters in detail and find surprisingly simple character formulas for the representations in the generic parameter regime that admit an elegant combinatorial formulation. We also discuss truncations of the algebra and give a conjectural formula for the complete set of truncation curves. We develop a theory of gluing for these algebras in order to build more complicated coset and non-coset algebras. We demonstrate the power of this technology with some examples and show in particular that the \( \mathcal{N} \) = 2 supersymmetric Grassmannian can be obtained by gluing three bosonic Grassmannian algebras in a loop. We finally speculate about the tantalizing possibility that this algebra is a specialization of an even larger 4-parametric family of algebras exhibiting pentality symmetry. Specializations of this conjectural family should include both the unitary Grassmannian family as well as the Lagrangian Grassmannian family of VOAs which interpolates between the unitary and the orthosymplectic cosets.
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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, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
D. Maulik and A. Okounkov, Quantum Groups and Quantum Cohomology, arXiv:1211.1287 [INSPIRE].
O. Schiffmann and E. Vasserot, Cherednik algebras, \( \mathcal{W} \) -algebras and the equivariant cohomology of the moduli space of instantons on 𝔸2 , Publications mathématiques de l’IHÉS 118 (2013) 213 [arXiv:1202.2756].
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].
D. Gaiotto and M. Rapčák, Vertex Algebras at the Corner, JHEP 01 (2019) 160 [arXiv:1703.00982] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, An AdS3 Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Triality in Minimal Model Holography, JHEP 07 (2012) 127 [arXiv:1205.2472] [INSPIRE].
C. Candu, M.R. Gaberdiel, M. Kelm and C. Vollenweider, Even spin minimal model holography, JHEP 01 (2013) 185 [arXiv:1211.3113] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Higher Spins & Strings, JHEP 11 (2014) 044 [arXiv:1406.6103] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and I. Rienacker, Higher spin algebras and large \( \mathcal{N} \) = 4 holography, JHEP 03 (2018) 097 [arXiv:1801.00806] [INSPIRE].
K. Costello, Holography and Koszul duality: the example of the M 2 brane, arXiv:1705.02500 [INSPIRE].
D. Gaiotto and J. Oh, Aspects of Ω-deformed M-theory, arXiv:1907.06495 [INSPIRE].
A. Smirnov, On the Instanton R-matrix, Commun. Math. Phys. 345 (2016) 703 [arXiv:1302.0799] [INSPIRE].
A. Tsymbaliuk, The affine Yangian of \( {\mathfrak{gl}}_1 \) revisited, Adv. Math. 304 (2017) 583 [arXiv:1404.5240] [INSPIRE].
R.-D. Zhu and Y. Matsuo, Yangian associated with 2D\( \mathcal{N} \) = 1 SCFT, PTEP 2015 (2015) 093A01 [arXiv:1504.04150] [INSPIRE].
T. Procházka, Instanton R-matrix and \( \mathcal{W} \) -symmetry, JHEP 12 (2019) 099 [arXiv:1903.10372] [INSPIRE].
A. Neguţ, The R-matrix of the quantum toroidal algebra, arXiv:2005.14182 [INSPIRE].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The Topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].
P. Goddard, A. Kent and D.I. Olive, Virasoro Algebras and Coset Space Models, Phys. Lett. B 152 (1985) 88 [INSPIRE].
V. Drinfeld and V. Sokolov, Lie algebras and Korteweg-de Vries type equations, J. Soviet Math 30 (1985) 1975.
B. Feigin and E. Frenkel, Quantization of the Drinfeld-Sokolov reduction, Phys. Lett. B 246 (1990) 75 [INSPIRE].
L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds, Nucl. Phys. B 261 (1985) 678 [INSPIRE].
L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds. 2., Nucl. Phys. B 274 (1986) 285 [INSPIRE].
L.J. Dixon, D. Friedan, E.J. Martinec and S.H. Shenker, The Conformal Field Theory of Orbifolds, Nucl. Phys. B 282 (1987) 13 [INSPIRE].
P. Bowcock, B.L. Feigin, A.M. Semikhatov and A. Taormina, Affine sl(2—1) and affine D(2—1:alpha) as vertex operator extensions of dual affine sl(2) algebras, Commun. Math. Phys. 214 (2000) 495 [hep-th/9907171] [INSPIRE].
B. Feigin, Extensions of vertex algebras. Constructions and applications, Usp. Mat. Nauk 72 (2017) 131.
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].
W. Li and P. Longhi, Gluing two affine Yangians of \( {\mathfrak{gl}}_1 \) , JHEP 10 (2019) 131 [arXiv:1905.03076] [INSPIRE].
W. Li, Gluing affine Yangians with bi-fundamentals, JHEP 06 (2020) 182 [arXiv:1910.10129] [INSPIRE].
W. Li and M. Yamazaki, Quiver Yangian from Crystal Melting, arXiv:2003.08909 [INSPIRE].
F.A. Bais, P. Bouwknegt, M. Surridge and K. Schoutens, Coset Construction for Extended Virasoro Algebras, Nucl. Phys. B 304 (1988) 371 [INSPIRE].
T. Creutzig, B. Feigin and A.R. Linshaw, N = 4 superconformal algebras and diagonal cosets, arXiv:1910.01228 [INSPIRE].
T. Arakawa and A. Molev, Explicit generators in rectangular affine \( \mathcal{W} \) -algebras of type A, Lett. Math. Phys. 107 (2017) 47 [arXiv:1403.1017] [INSPIRE].
T. Creutzig and Y. Hikida, Rectangular W-algebras, extended higher spin gravity and dual coset CFTs, JHEP 02 (2019) 147 [arXiv:1812.07149] [INSPIRE].
T. Creutzig and Y. Hikida, Rectangular W algebras and superalgebras and their representations, Phys. Rev. D 100 (2019) 086008 [arXiv:1906.05868] [INSPIRE].
L. Eberhardt and T. Procházka, The matrix-extended W1+∞ algebra, JHEP 12 (2019) 175 [arXiv:1910.00041] [INSPIRE].
P. Cvitanovic, Group theory: Birdtracks, Lie’s and exceptional groups, (2008).
T. Procházka, Exploring \( {\mathcal{W}}_{\infty } \) in the quadratic basis, JHEP 09 (2015) 116 [arXiv:1411.7697] [INSPIRE].
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
N. Seiberg and E. Witten, The D1/D5 system and singular CFT, JHEP 04 (1999) 017 [hep-th/9903224] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and R. Gopakumar, The Worldsheet Dual of the Symmetric Product CFT, JHEP 04 (2019) 103 [arXiv:1812.01007] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and R. Gopakumar, Deriving the AdS3 /CFT2 correspondence, JHEP 02 (2020) 136 [arXiv:1911.00378] [INSPIRE].
L. Eberhardt, AdS3 /CFT2 at higher genus, JHEP 05 (2020) 150 [arXiv:2002.11729] [INSPIRE].
A.R. Linshaw, Universal two-parameter \( {\mathcal{W}}_{\infty } \) -algebra and vertex algebras of type \( \mathcal{W} \) (2, 3, . . . , N ), arXiv:1710.02275 [INSPIRE].
M. Rapčák, On extensions of \( \mathfrak{gl}\hat{\left(m\left|n\right.\right)} \) Kac-Moody algebras and Calabi-Yau singularities, JHEP 01 (2020) 042 [arXiv:1910.00031] [INSPIRE].
T. Procházka, On even spin \( {\mathcal{W}}_{\infty } \) , JHEP 06 (2020) 057 [arXiv:1910.07997] [INSPIRE].
F. Bais and P.G. Bouwknegt, A Classification of Subgroup Truncations of the Bosonic String, Nucl. Phys. B 279 (1987) 561 [INSPIRE].
D. Kumar and M. Sharma, Symmetry Algebras of Stringy Cosets, JHEP 08 (2019) 179 [arXiv:1812.11920] [INSPIRE].
P. Flajolet and M. Soria, The cycle construction, SIAM Journal on Discrete Mathematics 4 (1991) 58.
M.R. Gaberdiel and R. Gopakumar, String Theory as a Higher Spin Theory, JHEP 09 (2016) 085 [arXiv:1512.07237] [INSPIRE].
T. Procházka and M. Rapčák, \( \mathcal{W} \) -algebra modules, free fields, and Gukov-Witten defects, JHEP 05 (2019) 159 [arXiv:1808.08837] [INSPIRE].
G. James and A. Kerber, The representation theory of the symmetric group, Cambridge University Press (2009).
Y. Kazama and H. Suzuki, New N = 2 Superconformal Field Theories and Superstring Compactification, Nucl. Phys. B 321 (1989) 232 [INSPIRE].
Y. Kazama and H. Suzuki, Characterization of N = 2 Superconformal Models Generated by Coset Space Method, Phys. Lett. B 216 (1989) 112 [INSPIRE].
M. Blau, F. Hussain and G. Thompson, Grassmannian topological Kazama-Suzuki models and cohomology, Nucl. Phys. B 488 (1997) 599 [hep-th/9510194] [INSPIRE].
S.G. Naculich and H.J. Schnitzer, Superconformal coset equivalence from level rank duality, Nucl. Phys. B 505 (1997) 727 [hep-th/9705149] [INSPIRE].
T. Ali, Level rank duality in Kazama-Suzuki models, hep-th/0201214 [INSPIRE].
A. Sevrin, W. Troost and A. Van Proeyen, Superconformal Algebras in Two-Dimensions with N = 4, Phys. Lett. B 208 (1988) 447 [INSPIRE].
J. de Boer, A. Pasquinucci and K. Skenderis, AdS/CFT dualities involving large 2 – D N = 4 superconformal symmetry, Adv. Theor. Math. Phys. 3 (1999) 577 [hep-th/9904073] [INSPIRE].
S. Gukov, E. Martinec, G.W. Moore and A. Strominger, The Search for a holographic dual to AdS3 × S3 × S3 × S1 , Adv. Theor. Math. Phys. 9 (2005) 435 [hep-th/0403090] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel, R. Gopakumar and W. Li, BPS spectrum on AdS3 ×S3 ×S3 ×S1 , JHEP 03 (2017) 124 [arXiv:1701.03552] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and W. Li, A holographic dual for string theory on AdS3 × S3 × S3 × S1 , JHEP 08 (2017) 111 [arXiv:1707.02705] [INSPIRE].
L. Eberhardt and M.R. Gaberdiel, Strings on AdS3 × S3 × S3 × S1 , JHEP 06 (2019) 035 [arXiv:1904.01585] [INSPIRE].
K. Thielemans, A Mathematica package for computing operator product expansions, Int. J. Mod. Phys. C 2 (1991) 787 [INSPIRE].
V.G. Kac and M. Wakimoto, Modular invariant representations of infinite dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci. 85 (1988) 4956 [INSPIRE].
T. Creutzig and A.R. Linshaw, Trialities of \( \mathcal{W} \) -algebras, arXiv:2005.10234 [INSPIRE].
K. Hornfeck, W algebras with set of primary fields of dimensions (3, 4, 5) and (3, 4, 5, 6), Nucl. Phys. B 407 (1993) 237 [hep-th/9212104] [INSPIRE].
T. Procházka, \( \mathcal{W} \) -symmetry, topological vertex and affine Yangian, JHEP 10 (2016) 077 [arXiv:1512.07178] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
C.-M. Chang, S. Minwalla, T. Sharma and X. Yin, ABJ Triality: from Higher Spin Fields to Strings, J. Phys. A 46 (2013) 214009 [arXiv:1207.4485] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Large N =4 Holography, JHEP 09 (2013) 036 [arXiv:1305.4181] [INSPIRE].
T. Creutzig, Y. Hikida and P.B. Ronne, Extended higher spin holography and Grassmannian models, JHEP 11 (2013) 038 [arXiv:1306.0466] [INSPIRE].
C. Candu and C. Vollenweider, On the coset duals of extended higher spin theories, JHEP 04 (2014) 145 [arXiv:1312.5240] [INSPIRE].
A. Belin, N. Benjamin, A. Castro, S.M. Harrison and C.A. Keller, \( \mathcal{N} \) = 2 Minimal Models: A Holographic Needle in a Symmetric Orbifold Haystack, SciPost Phys. 8 (2020) 084 [arXiv:2002.07819] [INSPIRE].
E. Witten, On the Landau-Ginzburg description of N = 2 minimal models, Int. J. Mod. Phys. A 9 (1994) 4783 [hep-th/9304026] [INSPIRE].
P. Di Francesco and S. Yankielowicz, Ramond sector characters and N = 2 Landau-Ginzburg models, Nucl. Phys. B 409 (1993) 186 [hep-th/9305037] [INSPIRE].
S. Kanade and A.R. Linshaw, Universal two-parameter even spin \( {\mathcal{W}}_{\infty } \) -algebra, Adv. Math. 355 (2019) 106774 [arXiv:1805.11031] [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].
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 \( \mathrm{\mathcal{R}}\mathcal{TT} \) 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].
A. Negu¸t, The q-AGT-W relations via shuffle algebras, Commun. Math. Phys. 358 (2018) 101 [arXiv:1608.08613] [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].
H. Awata, H. Kanno, A. Mironov, A. Morozov, K. Suetake and Y. Zenkevich, The MacMahon R-matrix, JHEP 04 (2019) 097 [arXiv:1810.07676] [INSPIRE].
J.-E. Bourgine, M. Fukuda, K. Harada, Y. Matsuo and R.-D. Zhu, (p, q)-webs of DIM representations, 5d \( \mathcal{N} \) = 1 instanton partition functions and qq-characters, JHEP 11 (2017) 034 [arXiv:1703.10759] [INSPIRE].
J.E. Bourgine and K. Zhang, A note on the algebraic engineering of 4D \( \mathcal{N} \) = 2 super Yang-Mills theories, Phys. Lett. B 789 (2019) 610 [arXiv:1809.08861] [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].
M. Rapcak, Y. Soibelman, Y. Yang and G. Zhao, Cohomological Hall algebras, vertex algebras and instantons, Commun. Math. Phys. 376 (2019) 1803 [arXiv:1810.10402] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Stringy Symmetries and the Higher Spin Square, J. Phys. A 48 (2015) 185402 [arXiv:1501.07236] [INSPIRE].
A.V. Litvinov, On spectrum of ILW hierarchy in conformal field theory, JHEP 11 (2013) 155 [arXiv:1307.8094] [INSPIRE].
M.N. Alfimov and A.V. Litvinov, On spectrum of ILW hierarchy in conformal field theory II: coset CFT’s, JHEP 02 (2015) 150 [arXiv:1411.3313] [INSPIRE].
R. Mkrtchyan, A. Sergeev and A. Veselov, Casimir eigenvalues for universal lie algebra, J. Math. Phys. 53 (2012) 102106.
C. Beem and L. Rastelli, Vertex operator algebras, Higgs branches, and modular differential equations, JHEP 08 (2018) 114 [arXiv:1707.07679] [INSPIRE].
O. Chacaltana and J. Distler, Tinkertoys for Gaiotto Duality, JHEP 11 (2010) 099 [arXiv:1008.5203] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
M. Lemos and W. Peelaers, Chiral Algebras for Trinion Theories, JHEP 02 (2015) 113 [arXiv:1411.3252] [INSPIRE].
T. Nishioka and Y. Tachikawa, Central charges of para-Liouville and Toda theories from M-5-branes, Phys. Rev. D 84 (2011) 046009 [arXiv:1106.1172] [INSPIRE].
N. Wyllard, Coset conformal blocks and N = 2 gauge theories, arXiv:1109.4264 [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 and G.M. Tarnopolsky, Parafermionic Liouville field theory and instantons on ALE spaces, JHEP 02 (2012) 036 [arXiv:1110.5628] [INSPIRE].
O. Foda, N. Macleod, M. Manabe and T. Welsh, \( \hat{\mathfrak{sl}}{(n)}_N \) WZW conformal blocks from SU(N ) instanton partition functions on ℂ2 /ℤn , Nucl. Phys. B 956 (2020) 115038 [arXiv:1912.04407] [INSPIRE].
M. Manabe, n-th parafermion \( {\mathcal{W}}_N \) characters from U (N ) instanton counting on ℂ2 /ℤn , JHEP 06 (2020) 112 [arXiv:2004.13960] [INSPIRE].
O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, The Hagedorn - deconfinement phase transition in weakly coupled large N gauge theories, Adv. Theor. Math. Phys. 8 (2004) 603 [hep-th/0310285] [INSPIRE].
J. Ben Geloun and S. Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, Ann. Inst. H. Poincaŕe Comb. Phys. Interact. 1 (2014) 77 [arXiv:1307.6490] [INSPIRE].
J.-B. Bae, E. Joung and S. Lal, Exploring Free Matrix CFT Holographies at One-Loop, Universe 3 (2017) 77 [arXiv:1708.04644] [INSPIRE].
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Eberhardt, L., Procházka, T. The Grassmannian VOA. J. High Energ. Phys. 2020, 150 (2020). https://doi.org/10.1007/JHEP09(2020)150
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DOI: https://doi.org/10.1007/JHEP09(2020)150