Abstract
We construct a quadratic basis of generators of matrix-extended \( {\mathcal{W}}_{1+\infty } \) using a generalization of the Miura transformation. This makes it possible to conjecture a closed-form formula for the operator product expansions defining the algebra. We study truncations of the algebra. An explicit calculation at low levels shows that these are parametrized in a way consistent with the gluing description of the algebra. It is perhaps surprising that in spite of the fact that the algebras are rather complicated and non-linear, the structure of their truncations follows very simple gluing rules.
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References
A.B. Zamolodchikov, Infinite additional symmetries in two-dimensional conformal quantum field theory, Theor. Math. Phys.65 (1985) 1205 [INSPIRE].
P. Bouwknegt and K. Schoutens, W symmetry in conformal field theory, Phys. Rept.223 (1993) 183 [hep-th/9210010] [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 − 1conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
C. Beem et al., Infinite chiral symmetry in four dimensions, Commun. Math. Phys.336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, \( \mathcal{W} \)symmetry in six dimensions, JHEP05 (2015) 017 [arXiv:1404.1079] [INSPIRE].
M.R. Gaberdiel and T. Hartman, Symmetries of holographic minimal models, JHEP05 (2011) 031 [arXiv:1101.2910] [INSPIRE].
C.N. Pope, L.J. Romans and X. Shen, The complete structure of W ∞ , Phys. Lett.B 236 (1990) 173 [INSPIRE].
C.N. Pope, L.J. Romans and X. Shen, W ∞and the Racah-Wigner algebra, Nucl. Phys.B 339 (1990) 191 [INSPIRE].
C.N. Pope, L.J. Romans and X. Shen, A new higher spin algebra and the lone star product, Phys. Lett.B 242 (1990) 401 [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Triality in minimal model holography, JHEP07 (2012) 127 [arXiv:1205.2472] [INSPIRE].
T. Procházka, Exploring \( {\mathcal{W}}_{\infty } \)in the quadratic basis, JHEP09 (2015) 116 [arXiv:1411.7697] [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].
A. Tsymbaliuk, The affine Yangian of \( \mathfrak{g}{\mathfrak{l}}_1 \)revisited, Adv. Math.304 (2017) 583 [arXiv:1404.5240] [INSPIRE].
T. Procházka, \( \mathcal{W} \)-symmetry, topological vertex and affine Yangian, JHEP10 (2016) 077 [arXiv:1512.07178] [INSPIRE].
A. Negut, The q-AGT-W relations via shuffle algebras, Commun. Math. Phys.358 (2018) 101 [arXiv:1608.08613] [INSPIRE].
M.R. Gaberdiel, R. Gopakumar, W. Li and C. Peng, Higher spins and Yangian symmetries, JHEP04 (2017) 152 [arXiv:1702.05100] [INSPIRE].
D. Gaiotto and M. Rapčák, Vertex algebras at the corner, JHEP01 (2019) 160 [arXiv:1703.00982] [INSPIRE].
T. Procházka and M. Rapčák, Webs of W-algebras, JHEP11 (2018) 109 [arXiv:1711.06888] [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].
S. Odake and T. Sano, \( {\mathcal{W}}_{1+\infty } \)and super \( {\mathcal{W}}_{\infty } \)algebras with SU(N) symmetry, Phys. Lett.B 258 (1991) 369.
K. Costello, Holography and Koszul duality: the example of the M2 brane, arXiv:1705.02500 [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and I. Rienacker, Higher spin algebras and large \( \mathcal{N} \) = 4 holography, JHEP03 (2018) 097 [arXiv:1801.00806] [INSPIRE].
T. Creutzig and Y. Hikida, Rectangular W-algebras, extended higher spin gravity and dual coset CFTs, JHEP02 (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].
T. Creutzig, Y. Hikida and T. Uetoko, Rectangular W-algebras of types SO(M) and sp(2M) and dual coset CFTs, JHEP10 (2019) 023 [arXiv:1906.05872] [INSPIRE].
M. Rapčák, On extensions of \( \mathfrak{g}\hat{\mathfrak{l}\left(m|n\right)} \)Kac-Moody algebras and Calabi-Yau Singularities, arXiv:1910.00031 [INSPIRE].
S. Lukyanov, Quantization of the Gel’fand-Dikii brackets, Funct. Anal. Appl.22 (1988) 255.
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. Procházka, Instanton R-matrix and W-symmetry, JHEP12 (2019) 099 [arXiv:1903.10372] [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].
T. Procházka and M. Rapčák, \( \mathcal{W} \)-algebra modules, free fields and Gukov-Witten defects, JHEP05 (2019) 159 [arXiv:1808.08837] [INSPIRE].
C. Candu and M.R. Gaberdiel, Duality in N = 2 minimal model holography, JHEP02 (2013) 070 [arXiv:1207.6646] [INSPIRE].
C. Candu, M.R. Gaberdiel, M. Kelm and C. Vollenweider, Even spin minimal model holography, JHEP01 (2013) 185 [arXiv:1211.3113] [INSPIRE].
M. Beccaria, C. Candu, M.R. Gaberdiel and M. Groher, N = 1 extension of minimal model holography, JHEP07 (2013) 174 [arXiv:1305.1048] [INSPIRE].
M. Beccaria, C. Candu and M.R. Gaberdiel, The large N = 4 superconformal W ∞algebra, JHEP06 (2014) 117 [arXiv:1404.1694] [INSPIRE].
K. Thielemans, A Mathematica package for computing operator product expansions, Int. J. Mod. Phys.C 2 (1991) 787 [INSPIRE].
A. Pressley and G. Segal, Loop Groups, Clarendon Press, U.K. (1988).
J. Fuchs, A. Ganchev and P. Vecsernyes, Simple WZW superselection sectors, Lett. Math. Phys.28 (1993) 31 [INSPIRE].
T. Arakawa, T. Creutzig and A.R. Linshaw, W-algebras as coset vertex algebras, arXiv:1801.03822 [INSPIRE].
D. Maulik and A. Okounkov, Quantum groups and quantum cohomology, arXiv:1211.1287 [INSPIRE].
R.-D. Zhu and Y. Matsuo, Yangian associated with 2D \( \mathcal{N} \) = 1 SCFT, PTEP2015 (2015) 093A01 [arXiv:1504.04150] [INSPIRE].
O. Schiffmann and E. Vasserot, Cherednik algebras, \( \mathcal{W} \)-algebras and the equivariant cohomology of the moduli space of instantons on A 2 , Publ. Math. IHÉS118 (2013) 213.
A.V. Litvinov, On spectrum of ILW hierarchy in conformal field theory, JHEP11 (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, JHEP02 (2015) 150 [arXiv:1411.3313] [INSPIRE].
S. Datta, M.R. Gaberdiel, W. Li and C. Peng, Twisted sectors from plane partitions, JHEP09 (2016) 138 [arXiv:1606.07070] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, String theory as a higher spin theory, JHEP09 (2016) 085 [arXiv:1512.07237] [INSPIRE].
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ArXiv ePrint: 1910.00041
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Eberhardt, L., Procházka, T. The matrix-extended \( {\mathcal{W}}_{1+\infty } \) algebra. J. High Energ. Phys. 2019, 175 (2019). https://doi.org/10.1007/JHEP12(2019)175
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DOI: https://doi.org/10.1007/JHEP12(2019)175