Abstract
We analyze the asymptotic symmetry of higher spin gravity with M × M matrix valued fields, which is given by rectangular W-algebras with su(M) symmetry. The matrix valued extension is expected to be useful for the relation between higher spin gravity and string theory. With the truncation of spin as s = 2, 3,…, n, we evaluate the central charge c of the algebra and the level k of the affine currents with finite c, k. For the simplest case with n = 2, we obtain the operator product expansions among generators by requiring their associativity. We conjecture that the symmetry is the same as that of Grassmannian-like coset based on our proposal of higher spin holography. Comparing c, k from the both theories, we obtain the map of parameters. We explicitly construct low spin generators from the coset theory, and, in particular, we reproduce the operator product expansions of the rectangular W-algebra for n = 2. We interpret the map of parameters by decomposing the algebra in the coset description.
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References
D.J. Gross, High-energy symmetries of string theory, Phys. Rev. Lett. 60 (1988) 1229 [INSPIRE].
M.A. Vasiliev, Nonlinear equations for symmetric massless higher spin fields in (A)dS(d), Phys. Lett. B 567 (2003) 139 [hep-th/0304049] [INSPIRE].
M.A. Vasiliev, From Coxeter higher-spin theories to strings and tensor models, JHEP 08 (2018) 051 [arXiv:1804.06520] [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].
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].
O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].
T. Creutzig, Y. Hikida and P.B. Ronne, Extended higher spin holography and Grassmannian models, JHEP 11 (2013) 038 [arXiv:1306.0466] [INSPIRE].
S.F. Prokushkin and M.A. Vasiliev, Higher spin gauge interactions for massive matter fields in 3 − D AdS space-time, Nucl. Phys. B 545 (1999) 385 [hep-th/9806236] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and I. Rienacker, Higher spin algebras and large N = 4 holography, JHEP 03 (2018) 097 [arXiv:1801.00806] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, An AdS 3 dual for minimal model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
T. Creutzig, Y. Hikida and P.B. Ronne, Higher spin AdS 3 supergravity and its dual CFT, JHEP 02 (2012) 109 [arXiv:1111.2139] [INSPIRE].
E. Joung, J. Kim, J. Kim and S.-J. Rey, Asymptotic symmetries of colored gravity in three dimensions, JHEP 03 (2018) 104 [arXiv:1712.07744] [INSPIRE].
S. Gwak, E. Joung, K. Mkrtchyan and S.-J. Rey, Rainbow valley of colored (Anti-)de sitter gravity in three dimensions, JHEP 04 (2016) 055 [arXiv:1511.05220] [INSPIRE].
S. Gwak, E. Joung, K. Mkrtchyan and S.-J. Rey, Rainbow vacua of colored higher-spin (A)dS 3 gravity, JHEP 05 (2016) 150 [arXiv:1511.05975] [INSPIRE].
M. Henneaux and S.-J. Rey, Nonlinear W ∞ as asymptotic symmetry of three-dimensional higher spin Anti-de Sitter gravity, JHEP 12 (2010) 007 [arXiv:1008.4579] [INSPIRE].
A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP 11 (2010) 007 [arXiv:1008.4744] [INSPIRE].
M.R. Gaberdiel and T. Hartman, Symmetries of holographic minimal models, JHEP 05 (2011) 031 [arXiv:1101.2910] [INSPIRE].
A. Campoleoni, S. Fredenhagen and S. Pfenninger, Asymptotic W-symmetries in three-dimensional higher-spin gauge theories, JHEP 09 (2011) 113 [arXiv:1107.0290] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Triality in minimal model holography, JHEP 07 (2012) 127 [arXiv:1205.2472] [INSPIRE].
T. Procházka, Exploring \( \mathcal{W} \) ∞ in the quadratic basis, JHEP 09 (2015) 116 [arXiv:1411.7697] [INSPIRE].
C. Candu and M.R. Gaberdiel, Duality in N = 2 minimal model holography, JHEP 02 (2013) 070 [arXiv:1207.6646] [INSPIRE].
D. Gaiotto and M. Rapčák, Vertex algebras at the corner, JHEP 01 (2019) 160 [arXiv:1703.00982] [INSPIRE].
T. Creutzig and D. Gaiotto, Vertex algebras for S-duality, arXiv:1708.00875 [INSPIRE].
T. Procházka and M. Rapčák, Webs of W-algebras, JHEP 11 (2018) 109 [arXiv:1711.06888] [INSPIRE].
T. Procházka and M. Rapčák, \( \mathcal{W} \) -algebra modules, free fields and Gukov-Witten defects, arXiv:1808.08837 [INSPIRE].
K. Harada and Y. Matsuo, Plane partition realization of (web of) W-algebra minimal models, JHEP 02 (2019) 050 [arXiv:1810.08512] [INSPIRE].
A. Achucarro and P.K. Townsend, A Chern-Simons action for three-dimensional Anti-de Sitter supergravity theories, Phys. Lett. B 180 (1986) 89 [INSPIRE].
E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
M.P. Blencowe, A consistent interacting massless higher spin field theory in D = (2 + 1), Class. Quant. Grav. 6 (1989) 443 [INSPIRE].
C.N. Pope, L.J. Romans and X. Shen, W ∞ and the Racah-Wigner algebra, Nucl. Phys. B 339 (1990) 191 [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Large N = 4 holography, JHEP 09 (2013) 036 [arXiv:1305.4181] [INSPIRE].
A. Castro, R. Gopakumar, M. Gutperle and J. Raeymaekers, Conical defects in higher spin theories, JHEP 02 (2012) 096 [arXiv:1111.3381] [INSPIRE].
J. Balog et al., Toda theory and W algebra from a gauged WZNW point of view, Annals Phys. 203 (1990) 76 [INSPIRE].
A. Campoleoni, S. Fredenhagen and J. Raeymaekers, Quantizing higher-spin gravity in free-field variables, JHEP 02 (2018) 126 [arXiv:1712.08078] [INSPIRE].
M. Henneaux, L. Maoz and A. Schwimmer, Asymptotic dynamics and asymptotic symmetries of three-dimensional extended AdS supergravity, Annals Phys. 282 (2000) 31 [hep-th/9910013] [INSPIRE].
T. Arakawa, Introduction to W-algebras and their representation theory, arXiv:1605.00138 [INSPIRE].
T. Creutzig and A.R. Linshaw, The super W 1+∞ algebra with integral central charge, Trans. Amer. Math.Soc. 367 (2015) 5521 [arXiv:1209.6032].
T. Creutzig and A.R. Linshaw, Cosets of affine vertex algebras inside larger structures, J. Algebra 517 (2019) 396 [arXiv:1407.8512] [INSPIRE].
V.G. Kac and M. Wakimoto, Quantum reduction and representation theory of superconformal algebras, math-ph/0304011 [INSPIRE].
N. Genra, Screening operators for W-algebras, arXiv:1606.00966 [INSPIRE].
T. Arakawa, T. Creutzig and A.R. Linshaw, W-algebras as coset vertex algebras, arXiv:1801.03822 [INSPIRE].
A.R. Linshaw, Universal two-parameter \( \mathcal{W} \) ∞ -algebra and vertex algebras of type \( \mathcal{W} \)(2, 3,…, N), arXiv:1710.02275 [INSPIRE].
S. Kanade and A.R. Linshaw, Universal two-parameter even spin \( \mathcal{W} \) ∞ -algebra, arXiv:1805.11031 [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, Y. Hikida and P.B. Rønne, Correspondences between WZNW models and CFTs with W-algebra symmetry, JHEP 02 (2016) 048 [arXiv:1509.07516] [INSPIRE].
Y. Hikida and V. Schomerus, H +3 WZNW model from Liouville field theory, JHEP 10 (2007) 064 [arXiv:0706.1030] [INSPIRE].
Y. Hikida and V. Schomerus, Structure constants of the OSp(1|2) WZNW model, JHEP 12 (2007) 100 [arXiv:0711.0338] [INSPIRE].
T. Creutzig, Y. Hikida and P.B. Ronne, Supergroup-extended super Liouville correspondence, JHEP 06 (2011) 063 [arXiv:1103.5753] [INSPIRE].
I. Bakas and E. Kiritsis, Grassmannian coset models and unitary representations of W ∞, Mod. Phys. Lett. A 5 (1990) 2039 [INSPIRE].
S. Odake and T. Sano, W 1+∞ and super W ∞ algebras with SU(N) symmetry, Phys. Lett. B 258 (1991) 369.
K. Thielemans, A Mathematica package for computing operator product expansions, Int. J. Mod. Phys. C 2 (1991) 787 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer, Germany (1997).
T. Creutzig, S. Kanade, A.R. Linshaw and D. Ridout, Schur-Weyl duality for Heisenberg cosets, arXiv:1611.00305 [INSPIRE].
T. Creutzig, W-algebras for Argyres-Douglas theories, arXiv:1701.05926 [INSPIRE].
D. Adamovic and A. Milas, On some vertex algebras related to V −1(\( \mathfrak{s}\mathfrak{l} \)(n)) and their characters, arXiv:1805.09771.
V.G. Kac and M. Wakimoto, A remark on boundary level admissible representations, arXiv:1612.07423.
T. Creutzig and D. Ridout, Logarithmic conformal field theory: beyond an introduction, J. Phys. A 46 (2013) 4006 [arXiv:1303.0847] [INSPIRE].
D. Adamović and A. Milas, Vertex operator algebras associated to modular invariant representations for A (1)1 , Math. Res. Lett. 2 (1995) 563.
T. Creutzig and D. Ridout, Modular data and Verlinde formulae for fractional level WZW models II, Nucl. Phys. B 875 (2013) 423.
D. Ridout and S. Wood, Relaxed singular vectors, Jack symmetric functions and fractional level \( \widehat{sl}(2) \) models, Nucl. Phys. B 894 (2015) 621.
V.G. Kac and M. Wakimoto, Classification of modular invariant representations of affine algebras, in Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), V.G. Kac ed., Advanced Series Mathematical Physics volume 7, World Scientific, Singapore (1989).
T. Arakawa, Rationality of admissible affine vertex algebras in the category \( \mathcal{O} \), Duke Math. J. 165 (2016) 67 [arXiv:1207.4857] [INSPIRE].
V.G. Kac and M. Wakimoto, Modular invariant representations of infinite-dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci. U.S.A. 85 (1988) 4956.
T. Creutzig, Y.-Z. Huang and J. Yang, Braided tensor categories of admissible modules for affine Lie algebras, Commun. Math. Phys. 362 (2018) 827 [arXiv:1709.01865] [INSPIRE].
T. Creutzig, Fusion categories for affine vertex algebras at admissible levels, arXiv:1807.00415.
K. Bringmann, T. Creutzig and L. Rolen, Negative index Jacobi forms and quantum modular forms, Res. Math. Sci. 1 (2014) 11.
K. Bringmann, L. Rolen and S. Zwegers, On the fourier coefficients of negative index meromorphic jacobi forms, Res. Math. Sci. 3 (2016) 5.
C. Candu and C. Vollenweider, On the coset duals of extended higher spin theories, JHEP 04 (2014) 145 [arXiv:1312.5240] [INSPIRE].
T. Creutzig, Y. Hikida and P.B. Ronne, Higher spin AdS 3 holography with extended supersymmetry, JHEP 10 (2014) 163 [arXiv:1406.1521] [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].
O. Tsymbaliuk, The affine Yangian of gl 1 , and the infnitesimal Cherednik algebras, Ph.D. thesis, MIT, Department of Mathematics, Cambridge, U.S.A. (2014).
T. Procházka, \( \mathcal{W} \) -symmetry, topological vertex and affine Yangian, JHEP 10 (2016) 077 [arXiv:1512.07178] [INSPIRE].
M.R. Gaberdiel, R. Gopakumar, W. Li and C. Peng, Higher spins and Yangian symmetries, JHEP 04 (2017) 152 [arXiv:1702.05100] [INSPIRE].
D. Altschuler, M. Bauer and C. Itzykson, The branching rules of conformal embeddings, Commun. Math. Phys. 132 (1990) 349 [INSPIRE].
M.A. Walton, Conformal branching rules and modular invariants, Nucl. Phys. B 322 (1989) 775 [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, JHEP 11 (2018) 192 [arXiv:1807.11304] [INSPIRE].
E. Perlmutter, T. Prochazka and J. Raeymaekers, The semiclassical limit of W N CFTs and Vasiliev theory, JHEP 05 (2013) 007 [arXiv:1210.8452] [INSPIRE].
Y. Hikida, Conical defects and N = 2 higher spin holography, JHEP 08 (2013) 127 [arXiv:1212.4124] [INSPIRE].
Y. Hikida and P.B. Rønne, Marginal deformations and the Higgs phenomenon in higher spin AdS 3 holography, JHEP 07 (2015) 125 [arXiv:1503.03870] [INSPIRE].
M. Henneaux, G. Lucena Gómez, J. Park and S.-J. Rey, Super-W ∞ asymptotic symmetry of higher-spin AdS 3 supergravity, JHEP 06 (2012) 037 [arXiv:1203.5152] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Higher spins & strings, JHEP 11 (2014) 044 [arXiv:1406.6103] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Stringy symmetries and the higher spin square, J. Phys. A 48 (2015) 185402 [arXiv:1501.07236] [INSPIRE].
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Creutzig, T., Hikida, Y. Rectangular W-algebras, extended higher spin gravity and dual coset CFTs. J. High Energ. Phys. 2019, 147 (2019). https://doi.org/10.1007/JHEP02(2019)147
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DOI: https://doi.org/10.1007/JHEP02(2019)147