INVARIANT SUBALGEBRAS OF THE SMALL �� = 4 SUPERCONFORMAL ALGEBRA

CREUTZIG, THOMAS; LINSHAW, ANDREW R.; RIEDLER, WOLFGANG

doi:10.1007/s00031-021-09652-1

Published: 27 April 2021

INVARIANT SUBALGEBRAS OF THE SMALL 𝒩 = 4 SUPERCONFORMAL ALGEBRA

THOMAS CREUTZIG¹,
ANDREW R. LINSHAW² &
WOLFGANG RIEDLER¹

Transformation Groups volume 27, pages 797–832 (2022)Cite this article

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Abstract

Various aspects of orbifolds and cosets of the small 𝒩 = 4 superconformal algebra are studied. First, we determine minimal strong generators for generic and specific levels. As a corollary, we obtain the vertex algebra of global sections of the chiral de Rham complex on any complex Enriques surface. We also identify orbifolds of cosets of the small 𝒩 = 4 superconformal algebra with Com(V^ℓ(𝔰𝔩₂); V^ℓ+1(𝔰𝔩₂) ⊗ 𝒲_–5/2(𝔰𝔩₄; f_rect)) and in addition at special levels with Grassmanian cosets and principal 𝒲-algebras of type A at degenerate admissible levels. These coincidences lead us to a novel level-rank duality involving Grassmannian supercosets.

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References

D. Adamović, V. G. Kac, P. Moseneder Frajria, P. Papi, O. Perše, Finite vs infinite decompositions in conformal embeddings, Comm. Math. Phys. 348 (2016), 445–473.
Article MathSciNet MATH Google Scholar
T. Arakawa, T. Creutzig, K. Kawasetsu, A. R. Linshaw, Orbifolds and cosets of minimal 𝒲-algebras, Comm. Math. Phys. 355 (2017), 339–372.
Article MathSciNet MATH Google Scholar
T. Arakawa, T. Creutzig, A. R. Linshaw, Cosets of Bershadsky–Polyakov algebras and rational W-algebras of type A, Sel. Math. New Ser. 23 (2017), 2369–2395.
Article MathSciNet MATH Google Scholar
T. Arakawa, T. Creutzig, A. R. Linshaw, W-algebras as coset vertex algebras, Invent. Math. 218 (2019), no. 1, 145–195.
Article MathSciNet MATH Google Scholar
D. Adamović, Representations of the N = 2 superconformal vertex algebra, Int. Math. Res. Not. 1999 (1999), no. 2, 61–79.
Article MathSciNet MATH Google Scholar
D. Adamović, A realization of certain modules for the N = 4 superconformal algebra and the affine Lie algebra A₂⁽¹⁾, Transform. Groups 21 (2016), no. 2, 299–327.
Article MathSciNet MATH Google Scholar
M. Al-Ali, The ℤ₂-orbifold of the universal affine vertex algebra, J. Pure Appl. Algebra 223 (2019), no. 12, 5430–5443.
Article MathSciNet MATH Google Scholar
M. Al-Ali, A. R. Linshaw, The ℤ₂-orbifold of the 𝒲₃-algebra, Comm. Math. Phys. 353 (2017), no. 3, 1129–1150.
Article MathSciNet MATH Google Scholar
L. A. Borisov, A. Libgober, Elliptic genera of toric varieties and applications to mirror symmetry, Invent. Math. 140 (2000), no. 2, 453–485.
Article MathSciNet MATH Google Scholar
F. Bonetti, C. Meneghelli, L. Rastelli, VOAs labelled by complex reflection groups and 4d SCFTs, JHEP 05 (2019), 155.
Article MathSciNet Google Scholar
R. E. Borcherds, Vertex algebras, Kac–Moody algebras, and the monster, Proc. Nat. Acad. Sci. 83 (1986), 3068–3071.
Article MathSciNet MATH Google Scholar
L. A. Borisov, Vertex algebras and mirror symmetry, Comm. Math. Phys. 215 (2001), 517–557.
Article MathSciNet MATH Google Scholar
B. H. Lian, G. J. Zuckerman, Commutative quantum operator algebras, J. Pure Appl. Algebra 100 (1995), no. 1, 117–139.
Article MathSciNet MATH Google Scholar
T. Creutzig, B. Feigin, A. R. Linshaw, N = 4 superconformal algebras and diagonal cosets, Int. Math. Res. Not. 2020 (2020), doi.org/10.1093/imrn/rnaa078.
T. Creutzig, D. Gaiotto, Vertex algebras for S-duality, Comm. Math. Phys. 379 (2020), no. 3, 785–845.
Article MathSciNet MATH Google Scholar
T. Creutzig, D. Gaiotto, A. R. Linshaw, S-duality for the large N = 4 superconformal algebra, Comm. Math. Phys. 374 (2020), no. 3, 1787–1808.
Article MathSciNet MATH Google Scholar
T. Creutzig, G. Höhn, Mathieu moonshine and the geometry of K3 surfaces, Comm. Num. Theor. Phys. 08 (2014), 295–328.
Article MathSciNet MATH Google Scholar
T. Creutzig, Y. Hikida, Rectangular W-algebras and superalgebras and their representations, Phys. Rev. D 100 (2019) no. 8, 086008.
T. Creutzig, S. Kanade, A. R. Linshaw, Simple current extensions beyond semi-simplicity, Comm. Contemp. Math. 22 (2020), no. 1, 1950001, 49 pp.
T. Creutzig, S. Kanade, A. R. Linshaw, D. Ridout, Schur–Weyl duality for Heisenberg cosets, Transform. Groups 24 (2019), no. 2, 301–354.
Article MathSciNet MATH Google Scholar
T. Creutzig, A. R. Linshaw, Cosets of the W^k(𝔰𝔩₄; f_subreg)-algebra, Contemp. Math. 711 (2018), 105–117.
Article MATH Google Scholar
T. Creutzig, A. R. Linshaw, Cosets of affine vertex algebras inside larger structures, J. Algebra 517 (2019), 396–438.
Article MathSciNet MATH Google Scholar
T. Creutzig, A. R. Linshaw, Trialities of W-algebras, arXiv:2005.10234 (2020).
T. Creutzig, W-algebras for Argyres–Douglas theories, Europ. J. Math. 3 (2017), no. 3, 659–690.
Article MathSciNet MATH Google Scholar
C. Dong, H. Li, G. Mason, Compact automorphism groups of vertex operator algebras, Int. Math. Res. Not. 1996 (1996), no. 18, 913–921.
Article MathSciNet MATH Google Scholar
C. Dong, C. H. Lam, Q. Wang, H. Yamada, The structure of parafermion vertex operator algebras, J. Algebra 323 (2010), no. 2, 371–381.
Article MathSciNet MATH Google Scholar
T. Eguchi, H. Ooguri, Y. Tachikawa, Notes on the K3 surface and the Mathieu group M₂₄, Exper. Math. 20 (2011), 91–96.
Article MATH Google Scholar
T. Eguchi, A. Taormina, On the unitary representations of N = 2 and N = 4 superconformal algebras, Phys. Lett. B210 (1988), 125–132.
Article MathSciNet Google Scholar
E. Frenkel, D. Ben-Zvi, Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs, Vol. 88, American Mathematical Society, Providence, RI, 2001.
E. Frenkel, M. Szczesny, Chiral de Rham complex and orbifolds, J. Algebraic Geom. 16 (2007), 599–624.
Article MathSciNet MATH Google Scholar
B. L. Feigin, A. M. Semikhatov, I. Yu. Tipunin, Equivalence between chain categories of representations of affine (2) and N = 2 superconformal algebras, J. Math. Phys. 39 (1998), 3865–3905.
Article MathSciNet MATH Google Scholar
R. Heluani, Supersymmetry of the chiral de Rham complex 2: Commuting sectors, Int. Math. Res. Not. 2009 (2009), no. 6, 953–987.
Article MathSciNet MATH Google Scholar
V. Kac, Vertex Algebras for Beginners, University Lecture Series, Vol. 10, American Mathematical Society, Providence, RI, 1998.
MATH Google Scholar
A. Kapustin, Chiral de Rham complex and the half-twisted sigma-model, arXiv:hep-th/0504074 (2005).
V. Kac, A. Radul, Representation theory of the vertex algebra W_1+∞, Transform. Groups 1 (1996), no. 1, 41–70.
Article MathSciNet MATH Google Scholar
H. Li, Vertex algebras and vertex Poisson algebras, Comm. Contemp. Math. 06 (2004), no. 01, 61–110.
Article MathSciNet MATH Google Scholar
A. R. Linshaw, Universal two-parameter 𝒲_∞-algebra and vertex algebras of type (2, 3, ..., N), Compos. Math. 157 (2021), no. 1, 12–82.
Article MathSciNet MATH Google Scholar
B. H. Lian, A. R. Linshaw, Howe pairs in the theory of vertex Algebras, J. Algebra 317 (2007), 111–152.
Article MathSciNet MATH Google Scholar
A. R. Linshaw, G. Schwarz, B. Song, Arc spaces and the vertex algebra commutant problem, Adv. Math. 277 (2015), 338–364.
Article MathSciNet MATH Google Scholar
J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999), 1113–1133.
Article MathSciNet MATH Google Scholar
F. Malikov, V. Schechtman, Chiral de Rham complex. II, Amer. Math. Soc. Transl. 194 (1999), 149–188.
MathSciNet MATH Google Scholar
F. Malikov, V. Schechtman, Chiral Poincarà duality, Math. Res. Lett. 6 (1999), no. 5-6, 533–546.
Article MathSciNet MATH Google Scholar
F. Malikov, V. Schechtman, Deformations of vertex algebras, quantum cohomology of toric varieties, and elliptic genus, Comm. Math. Phys. 234 (2003), no. 1, 77–100.
Article MathSciNet MATH Google Scholar
F. Malikov, V. Schechtman, A. Vaintrob, Chiral de Rham complex, Comm. Math. Phys. 204 (1999), 439–473.
Article MathSciNet MATH Google Scholar
V. Ostrik, M. Sun, Level-rank duality via tensor categories, Comm. Math. Phys. 326 (2014), no. 1, 49–61.
Article MathSciNet MATH Google Scholar
B. Song, The global sections of the chiral de Rham complex on a Kummer surface, Int. Math. Res. Not. 2016 (2016), no. 14, 4271–4296.
Article MathSciNet MATH Google Scholar
B. Song, Vector bundles induced from jet schemes, Trans. Amer. Math. Soc. 374 (2021), no. 4, 2661–2685.
Article MathSciNet MATH Google Scholar
B. Song, The global sections of chiral de Rham complexes on compact Ricci-flat Kähler manifolds, Comm. Math. Phys. 382 (2021), no. 1, 351–379.
Article MathSciNet MATH Google Scholar
K. Thielemans, A Mathematica^TM package for computing operator product expansions, Int. J. Modern Phys. C 02 (1991), 787.
Article MathSciNet MATH Google Scholar

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Department of Mathematics, University of Alberta, Edmonton, AB, Canada
THOMAS CREUTZIG & WOLFGANG RIEDLER
Department of Mathematics, University of Denver, Denver, CO, USA
ANDREW R. LINSHAW

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THOMAS CREUTZIG
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ANDREW R. LINSHAW
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WOLFGANG RIEDLER
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Correspondence to THOMAS CREUTZIG.

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Thomas Creutzig is supported by NSERC Discovery Grant RES0048511.

Andrew R. Linshaw is supported by Simons Foundation Grant 635650 and NSF Grant DMS 2001484.

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CREUTZIG, T., LINSHAW, A.R. & RIEDLER, W. INVARIANT SUBALGEBRAS OF THE SMALL 𝒩 = 4 SUPERCONFORMAL ALGEBRA. Transformation Groups 27, 797–832 (2022). https://doi.org/10.1007/s00031-021-09652-1

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Received: 04 October 2019
Accepted: 16 November 2020
Published: 27 April 2021
Issue Date: September 2022
DOI: https://doi.org/10.1007/s00031-021-09652-1