Cosets from equivariant $\mathcal {W}$-algebras
HTML articles powered by AMS MathViewer
- by Thomas Creutzig and Shigenori Nakatsuka;
- Represent. Theory 27 (2023), 766-777
- DOI: https://doi.org/10.1090/ert/651
- Published electronically: September 11, 2023
- HTML | PDF | Request permission
Abstract:
The equivariant $\mathcal W$-algebra of a simple Lie algebra $\mathfrak {g}$ is a BRST reduction of the algebra of chiral differential operators on the Lie group of $\mathfrak {g}$. We construct a family of vertex algebras $A[\mathfrak {g}, \kappa , n]$ as subalgebras of the equivariant $\mathcal W$-algebra of $\mathfrak {g}$ tensored with the integrable affine vertex algebra $L_n(\check {\mathfrak {g}})$ of the Langlands dual Lie algebra $\check {\mathfrak {g}}$ at level $n\in \mathbb {Z}_{>0}$. They are conformal extensions of the tensor product of an affine vertex algebra and the principal $\mathcal {W}$-algebra whose levels satisfy a specific relation.
When $\mathfrak {g}$ is of type $ADE$, we identify $A[\mathfrak {g}, \kappa , 1]$ with the affine vertex algebra $V^{\kappa -1}(\mathfrak {g}) \otimes L_1(\mathfrak {g})$, giving a new and efficient proof of the coset realization of the principal $\mathcal W$-algebras of type $ADE$.
References
- T. Arakawa, Chiral algebras of class $\mathcal {S}$ and Moore–Tachikawa symplective varieties, arXiv:1811.01577 [math.RT], 2018.
- Tomoyuki Arakawa, Rationality of $W$-algebras: principal nilpotent cases, Ann. of Math. (2) 182 (2015), no. 2, 565–604. MR 3418525, DOI 10.4007/annals.2015.182.2.4
- Tomoyuki Arakawa, Associated varieties of modules over Kac-Moody algebras and $C_2$-cofiniteness of $W$-algebras, Int. Math. Res. Not. IMRN 22 (2015), 11605–11666. MR 3456698, DOI 10.1093/imrn/rnu277
- Tomoyuki Arakawa, Thomas Creutzig, and Boris Feigin, Urod algebras and translation of W-algebras, Forum Math. Sigma 10 (2022), Paper No. e33, 31. MR 4436591, DOI 10.1017/fms.2022.15
- T. Arakawa, T. Creutzig, and K. Kawsetsu. in preparation.
- Tomoyuki Arakawa, Thomas Creutzig, Kazuya Kawasetsu, and Andrew R. Linshaw, Orbifolds and cosets of minimal $\mathcal {W}$-algebras, Comm. Math. Phys. 355 (2017), no. 1, 339–372. MR 3670736, DOI 10.1007/s00220-017-2901-2
- Tomoyuki Arakawa, Thomas Creutzig, and Andrew R. Linshaw, $W$-algebras as coset vertex algebras, Invent. Math. 218 (2019), no. 1, 145–195. MR 3994588, DOI 10.1007/s00222-019-00884-3
- T.Arakawa, J. van Ekeren, and A. Moreau, Singularities of nilpotent slodowy slices and collapsing levels of W-algebras, 2021, arXiv:2102.13462 [math.RT].
- Tomoyuki Arakawa and Edward Frenkel, Quantum Langlands duality of representations of $\mathcal W$-algebras, Compos. Math. 155 (2019), no. 12, 2235–2262. MR 4016057, DOI 10.1112/s0010437x19007553
- S. Arkhipov and D. Gaitsgory, Differential operators on the loop group via chiral algebras, Int. Math. Res. Not. 4 (2002), 165–210. MR 1876958, DOI 10.1155/S1073792802102078
- Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima, Instanton moduli spaces and $\mathcal W$-algebras, Astérisque 385 (2016), vii+128 (English, with English and French summaries). MR 3592485
- T. Creutzig, T. Dimofte, N. Garner and N. Geer, A QFT for non-semisimple TQFT, arXiv:2112.01559 [hep-th], 2021.
- Thomas Creutzig, Shashank Kanade, and Robert McRae, Gluing vertex algebras, Adv. Math. 396 (2022), Paper No. 108174, 72. MR 4362778, DOI 10.1016/j.aim.2021.108174
- Thomas Creutzig and Davide Gaiotto, Vertex algebras for S-duality, Comm. Math. Phys. 379 (2020), no. 3, 785–845. MR 4163353, DOI 10.1007/s00220-020-03870-6
- T. Creutzig, N. Genra and A. Linshaw, Category $\mathcal O$ for vertex algebras of $\mathfrak {osp}_{1|2n}$, arXiv:2203.08188 [math.RT].
- Thomas Creutzig, Naoki Genra, and Shigenori Nakatsuka, Duality of subregular $\mathcal W$-algebras and principal $\mathcal W$-superalgebras, Adv. Math. 383 (2021), Paper No. 107685, 52. MR 4232554, DOI 10.1016/j.aim.2021.107685
- Thomas Creutzig and Andrew R. Linshaw, Cosets of affine vertex algebras inside larger structures, J. Algebra 517 (2019), 396–438. MR 3869280, DOI 10.1016/j.jalgebra.2018.10.007
- Thomas Creutzig and Andrew R. Linshaw, Trialities of $\mathcal W$-algebras, Camb. J. Math. 10 (2022), no. 1, 69–194. MR 4445343, DOI 10.4310/CJM.2022.v10.n1.a2
- Thomas Creutzig and Andrew R. Linshaw, Trialities of orthosymplectic $\mathcal {W}$-algebras. part B, Adv. Math. 409 (2022), no. part B, Paper No. 108678, 79. MR 4481140, DOI 10.1016/j.aim.2022.108678
- Thomas Creutzig and Andrew R. Linshaw, The super $\mathcal {W}_{1+\infty }$ algebra with integral central charge, Trans. Amer. Math. Soc. 367 (2015), no. 8, 5521–5551. MR 3347182, DOI 10.1090/S0002-9947-2015-06214-X
- T. Creutzig, A. R. Linshaw, S. Nakatsuka and R. Sato, Duality via convolution of W-algebras, arXiv:2203.01843 [math.QA].
- Jürgen Fuchs, Affine Lie algebras and quantum groups, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1995. An introduction, with applications in conformal field theory; Corrected reprint of the 1992 original. MR 1337497
- Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, 2nd ed., Mathematical Surveys and Monographs, vol. 88, American Mathematical Society, Providence, RI, 2004. MR 2082709, DOI 10.1090/surv/088
- Boris Feigin and Edward Frenkel, Duality in $W$-algebras, Internat. Math. Res. Notices 6 (1991), 75–82. MR 1136408, DOI 10.1155/S1073792891000119
- Edward Frenkel and Davide Gaiotto, Quantum Langlands dualities of boundary conditions, $D$-modules, and conformal blocks, Commun. Number Theory Phys. 14 (2020), no. 2, 199–313. MR 4084137, DOI 10.4310/CNTP.2020.v14.n2.a1
- B. L. Feigin and I. Y. Tipunin, Logarithmic CFTs connected with simple Lie algebras, arXiv:1002.5047 [math.QA], 2010.
- P. Goddard, A. Kent, and D. Olive, Unitary representations of the Virasoro and super-Virasoro algebras, Comm. Math. Phys. 103 (1986), no. 1, 105–119. MR 826859, DOI 10.1007/BF01464283
- Vassily Gorbounov, Fyodor Malikov, and Vadim Schechtman, Gerbes of chiral differential operators, Math. Res. Lett. 7 (2000), no. 1, 55–66. MR 1748287, DOI 10.4310/MRL.2000.v7.n1.a5
- Vassily Gorbounov, Fyodor Malikov, and Vadim Schechtman, On chiral differential operators over homogeneous spaces, Int. J. Math. Math. Sci. 26 (2001), no. 2, 83–106. MR 1836785, DOI 10.1155/S0161171201020051
- Davide Gaiotto and Miroslav Rapčák, Vertex algebras at the corner, J. High Energy Phys. 1 (2019), 160, front matter+85. MR 3919335, DOI 10.1007/jhep01(2019)160
- P. J. Hilton and U. Stammbach, A course in homological algebra, 2nd ed., Graduate Texts in Mathematics, vol. 4, Springer-Verlag, New York, 1997. MR 1438546, DOI 10.1007/978-1-4419-8566-8
- Hai Sheng Li, Symmetric invariant bilinear forms on vertex operator algebras, J. Pure Appl. Algebra 96 (1994), no. 3, 279–297. MR 1303287, DOI 10.1016/0022-4049(94)90104-X
- Yuto Moriwaki, Quantum coordinate ring in WZW model and affine vertex algebra extensions, Selecta Math. (N.S.) 28 (2022), no. 4, Paper No. 68, 49. MR 4439907, DOI 10.1007/s00029-022-00782-2
- Robert McRae, On semisimplicity of module categories for finite non-zero index vertex operator subalgebras, Lett. Math. Phys. 112 (2022), no. 2, Paper No. 25, 28. MR 4395119, DOI 10.1007/s11005-022-01523-4
- V. G. Kac and M. Wakimoto, Branching functions for winding subalgebras and tensor products, Acta Appl. Math. 21 (1990), no. 1-2, 3–39. MR 1085771, DOI 10.1007/BF00053290
Bibliographic Information
- Thomas Creutzig
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, 632 CAB, Edmonton, Alberta T6G 2G1, Canada
- MR Author ID: 832147
- ORCID: 0000-0002-7004-6472
- Email: creutzig@ualberta.ca
- Shigenori Nakatsuka
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, 632 CAB, Edmonton, Alberta T6G 2G1, Canada
- MR Author ID: 1429269
- Email: shigenori.nakatsuka@ualberta.ca
- Received by editor(s): June 17, 2022
- Received by editor(s) in revised form: December 20, 2022, February 2, 2023, and April 18, 2023
- Published electronically: September 11, 2023
- Additional Notes: The work of the first author is supported by NSERC Grant Number RES0048511 and the work of the second author is supported by JSPS Overseas Research Fellowships Grant Number 202260077.
- © Copyright 2023 American Mathematical Society
- Journal: Represent. Theory 27 (2023), 766-777
- MSC (2020): Primary 17B45, 17B68, 17B69
- DOI: https://doi.org/10.1090/ert/651
- MathSciNet review: 4640149