Feigin-Semikhatov conjecture and its applications

[ABD]

[Ad1]

[Ad2]

[Ad3]

[Ad4]

[ACGY]

[AKR]

[AP]

[AFO]

[ADJR]

[AC]

[AGT]

[Ar1]

[Ar2]

[Ar3]

[Ar4]

[Ar5]

[Ar6]

[Ar7]

[ACL1]

[ACL2]

[AKM]

[ALY]

[AM]

[AvE1]

[AvE2]

[ACKR]

[BZF]

[BE]

T. Abe, G. Buhl and C. Dong, Rationality, Regularity, and C2 -cofiniteness, Trans.

Amer. Math. Soc. 356, 2004, 3391-3402.

D. Adamovi´

c, Rationality of Neveu-Schwarz vertex operator superalgebras, Internat.

Math. Res. Notices, 1997, (17), 865–874.

D. Adamovi´

c, Representations of the N = 2 superconformal vertex algebra, Internat.

Math. Res. Notices, 1999, (2), 61–79.

D. Adamovi´

c, Vertex algebra approach to fusion rules for N = 2 superconformal minimal models, J. Algebra, 239, 2001, (2), 549–572.

D. Adamovi´

c, Classification of irreducible modules of certain subalgebras of free boson

vertex algebra, J. Algebra, 270, 2003, (1), 115–132.

D. Adamovic, and T. Creutzig, and N. Genra, and J. Yang, The vertex algebras R(p)

and V(p) , arXiv:2001.08048 [math.RT].

D. Adamovic, and K. Kawasetsu, and D. Ridout, A realisation of the Bershadsky–

Polyakov algebras and their relaxed modules, arXiv:2007.00396 [math.QA].

D. Adamovi´

c, and V. Pedi´

c, On fusion rules and intertwining operators for the Weyl

vertex algebra, J. Math. Phys., 60, 2019, (8), 081701–18.

M. Aganagic, and E. Frenkel, and A. Okounkov, Quantum q-Langlands correspondence,

Trans. Moscow Math. Soc., 79, 2018, 1–83.

C. Ai, and C. Dong, and Z. Jiao, and L. Ren, The irreducible modules and fusion rules

for the parafermion vertex operator algebras, Trans. Amer. Math. Soc., 370, 2018, (8),

5963–5981.

C. Alfes, and T. Creutzig, The mock modular data of a family of superalgebras, Proc.

Amer. Math. Soc., 142, 2014, (7), 2265–2280.

L. Alday, and D. Gaiotto, and Y. Tachikawa, Liouville correlation functions from fourdimensional gauge theories, Lett. Math. Phys., 91, 2010, (2), 167–197.

T. Arakawa, Representation theory of superconformal algebras and the Kac-RoanWakimoto conjecture, Duke Math. J., 130, 2005, (3), 435–478.

T. Arakawa, Representation theory of W-algebras, Invent. Math., 169, 2007, (2), 219–

320.

T. Arakawa, A remark on the C2 -cofiniteness condition on vertex algebras, Math. Z.,

270, 2012, 1-2, 559–575.

T. Arakawa, Rationality of Bershadsky-Polyakov vertex algebras, Comm. Math. Phys.,

323, 2013, (2), 627–633.

T. Arakawa, Rationality of W -algebras: principal nilpotent cases, Ann. of Math. (2),

182, 2015, (2), 565–604.

T. Arakawa, Associated varieties of modules over Kac-Moody algebras and C2 cofiniteness of W-algebras, Int. Math. Res. Not. IMRN, 201 5, (22), 11605–11666.

T. Arakawa, Introduction to W-algebras and their representation theory, Perspectives

in Lie theory, Springer INdAM Ser., 19, 179–250, Springer, Cham, 2017.

T. Arakawa, and T. Creutzig ,and A. Linshaw, Cosets of Bershadsky-Polyakov algebras

and rational W-algebras of type A, Selecta Math. (N.S.), 23, 2017, (4), 2369–2395.

T. Arakawa, and T. Creutzig, and A. Linshaw, W -algebras as coset vertex algebras,

Invent. Math., 218, 2019, (1), 145–195.

T. Arakawa, and T. Kuwabara, and F. Malikov, Localization of affine W-algebras,

Comm. Math. Phys., 335, 2015, (1), 143–182.

T. Arakawa, and C. Lam, and H. Yamada, Parafermion vertex operator algebras and

W-algebras, Trans. Amer. Math. Soc., 371, 2019, (6), 4277–4301.

T. Arakawa, and A. Moreau, Joseph ideals and lisse minimal W-algebras, J. Inst. Math.

Jussieu, 17, 2018, (2), 397–417.

T. Arakawa, and J. van Ekeren, Modularity of relatively rational vertex algebras and

fusion rules of principal affine W-algebras, Comm. Math. Phys., 370, 2019, (1), 205–

247.

T. Arakawa, and J. van Ekeren, Rationality and Fusion Rules of Exceptional WAlgebras, arXiv:1905.11473 [math.RT].

J. Auger, and T. Creutzig, and S. Kanade, and M. Rupert, Braided tensor categories

related to Bp vertex algebras, Comm. Math. Phys., 378, 2020, (1), 219–260.

D. Ben-Zvi, and E. Frenkel, Spectral curves, opers and integrable systems, Publ. Math.

Inst. Hautes Etudes

Sci., 94, 2001, 87–159.

J. Brundan, and A. Ellis, Monoidal supercategories. Comm. Math. Phys. 351, 2017,

(3), 1045–1089.

65

[BK]

B. Bakalov and A. Kirillov, Jr., Lectures on tensor categories and modular functors,

University Lecture Series, 21, American Mathematical Society, Providence, RI, 2001,

x+221.

[BMR]

C. Beem, and C. Meneghelli, and L. Rastelli, Free field realizations from the Higgs

branch, J. High Energy Phys., 2019, (9).

[BRvR]

C. Beem, and L. Rastelli, and B. van Rees, W-symmetry in six dimensions, J. High

Energy Phys., 2015, (5).

[BFM]

M. Bershtein, and B. Feigin, and G. Merzon, Plane partitions with a “pit”: generating

functions and representation theory, Selecta Math. (N.S.), 24, 2018, (1), 21–62.

[BFST]

P. Bowcock, and B. Feigin, and A. Semikhatov, and A. Taormina, sl(2|1)

and D(2|1;

α)

as vertex operator extensions of dual affine sl(2) algebras, Comm. Math. Phys., 214,

2000, (3), 495–545.

[BG]

J. Brundan, and S. Goodwin, Good grading polytopes, Proc. Lond. Math. Soc., (3),

94, 2007, (1), 155–180.

[Ca]

S. Carnahan, Building vertex algebras from parts, Comm. Math. Phys., 373, 2020, (1),

1–43.

[CaMi]

S. Carnahan, and M. Miyamoto, Regularity of fixed-point vertex operator subalgebras,

arXiv: 1603.05645 [math.RT].

[CCFGH] M. Cheng, and S. Chun, and F. Ferrari, and S. Gukov, and S. Harrison, 3d modularity,

J. High Energy Phys., 2019, (10).

[CW]

S. Cheng, and W. Wang, Dualities and representations of Lie superalgebras, Graduate

Studies in Mathematics, 144, American Mathematical Society, Providence, RI, 2012,

xviii+302.

[Cr]

T. Creutzig, Fusion categories for affine vertex algebras at admissible levels, Selecta

Math. (N.S.), 25, 2019, (2), Paper No. 27, 21.

[CG]

T. Creutzig, and D. Gaiotto, Vertex algebras for S-duality, Comm. Math. Phys., 379,

2020, (3), 785–845.

[CGN]

T. Creutzig, and N. Genra, and S. Nakatsuka, Duality of subregular W-algebras and

principal W-superalgebras,a arXiv:2005.10713 [math.QA].

[CGNS]

T. Creutzig, and N. Genra, and S. Nakatsuka, and R. Sato, to appear.

[CKL]

T. Creutzig, and S. Kanade, and A. R. Linshaw, Simple current extensions beyond

semi-simplicity, Commun. Contemp. Math., 22, 2020, (1), 1950001–49.

[CKLR] T. Creutzig, and S. Kanade, and A. Linshaw ,and D. Ridout, Schur-Weyl Duality for

Heisenberg Cosets, Transform. Groups, 24, 2019, (2), 301–354.

[CKM1] T. Creutzig, and S. Kanade, and R. McRae, Tensor categories for vertex operator

superalgebra extensions, arXiv:1705.05017 [math.QA].

[CKM2] T. Creutzig, and S. Kanade, and R. McRae, Glueing vertex algebras, arXiv:1906.00119

[math.QA].

[CL1]

T. Creutzig, and A. Linshaw, Orbifolds of symplectic fermion algebras, Trans. Amer.

Math. Soc., 369, 2017, (1), 467–494.

[CL2]

T. Creutzig, and A. Linshaw, Cosets of the Wk (sl4 , fsubreg )-algebra, Vertex algebras

and geometry, Contemp. Math., 711, 105–117, Amer. Math. Soc., Providence, RI,

2018.

[CL3]

T. Creutzig, and A. Linshaw, Cosets of affine vertex algebras inside larger structures,

J. Algebra, 517, 2019, 396–438.

[CL4]

T. Creutzig, and A. Linshaw, Trialities of W-algebras, arXiv:2005.10234 [math.RT].

[CLRW] T. Creutzig, and T. Liu, and D. Ridout, and S. Wood, Unitary and non-unitary N = 2

minimal models, J. High Energy Phys., 2019, (6).

[CrMi]

T. Creutzig, and A. Milas, False theta functions and the Verlinde formula, Adv. Math.,

262, 2014, 520–545.

[CMY]

T. Creutzig, and R. McRae, and J. Yang, On ribbon categories for singlet vertex

algebras, 2020, arXiv.2007.12735 [math.QA].

[CR1]

T. Creutzig, and P. Rønne, The GL(1|1)-symplectic fermion correspondence, Nuclear

Phys. B, 815, 2009, (1)-(2), 95–124.

[CR2]

T. Creutzig, and D. Ridout, Relating the archetypes of logarithmic conformal field

theory, Nuclear Phys. B, 872, 2013, (3), 348–391.

[CR3]

T. Creutzig, and D. Ridout, W-algebras extending gl(1|1),

Lie theory and its applications in physics, Springer Proc. Math. Stat., 36, 349–367, Springer, Tokyo, 2013.

[CRW]

T. Creutzig, and D. Ridout, and S. Wood, Coset constructions of logarithmic (1, p)

models, Lett. Math. Phys., 104, 2014, (5), 553–583.

[DFMS] P. Di Francesco, and P. Mathieu, and D. S´

en´

echal, Conformal field theory, Graduate

Texts in Contemporary Physics, Springer-Verlag, New York, 1997, xxii+890.

66

[DVPYZ] P. Di Vecchia, and J. Petersen, and M. Yu, and H. Zheng, Explicit construction of

unitary representations of the N = 2 superconformal algebra, Phys. Lett. B, 174,

1986, (3), 280–284.

[DK]

A. De Sole, and V. Kac, Finite vs affine W -algebras, Jpn. J. Math., 1, 2006, (1),

137–261.

[DKV]

A. De Sole, and V. Kac and D. Valeri, Structure of classical (finite and affine) Walgebras, J. Eur. Math. Soc., 18, 2016, (9), 1873–1908.

[DLM1]

C. Dong, and H. Li, and G. Mason, Dong, Simple currents and extensions of vertex

operator algebras, Comm. Math. Phys., 180, 1996, (3), 671–707.

[DLM2]

C. Dong, and H. Li, and G. Mason, Regularity of rational vertex operator algebras,

Adv. Math., 132, 1997, (1), 148–166.

[DM1]

C. Dong, and G. Mason, Geoffrey, On quantum Galois theory, Duke Math. J., 86,

1997, (2), 305–321.

[DM2]

C. Dong, and G. Mason, Rational vertex operator algebras and the effective central

charge, Int. Math. Res. Not., 2004, 56, 2989–3008.

[DN]

C. Dong, and K. Nagatomo, Automorphism groups and twisted modules for lattice

operator algebras, Recent developments in quantum affine algebras and related topics

(Raleigh, NC, 1998), Contemp. Math., 248, 117–133, Amer. Math. Soc., Providence,

RI, 1999.

[DR]

C. Dong, and L. Ren, Representations of the parafermion vertex operator algebras,

Adv. Math, 315, 2017, 88–101.

[DW]

C. Dong, and Q. Wang, Quantum dimensions and fusion rules for parafermion vertex

operator algebras, Proc. Amer. Math. Soc., 144, 2016, (4), 1483–1492.

[DS]

V. Drinfel’d, and V. Sokolov, Lie algebras and equations of Korteweg-de Vries type,

Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, 81–180, Akad. Nauk

SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984.

[EK]

A. Elashvili and V. Kac, Classification of good gradings of simple Lie algebras,Lie

groups and invariant theory, Amer. Math. Soc. Transl. Ser. 2, 213, 85–104, Amer.

Math. Soc., Providence, RI, 2005.

[ERF]

S. Eswara Rao, and V. Futorny, Integrable modules for affine Lie superalgebras, Trans.

Amer. Math. Soc., 361, 2009, (10), 5435–5455.

[EGNO] P. Etingof, and S. Gelaki, and D. Nikshych, and V. Ostrik, Tensor categories, Mathematical Surveys and Monographs, 205, American Mathematical Society, Providence,

RI, 2015, xvi+343.

[FaZa]

V. Fateev, and A. Zamolodchikov, Conformal quantum field theory models in two

dimensions having Z3 symmetry, Nuclear Phys. B, 280, 1987, (4), 644–660.

[FL1]

V. Fateev, and S. Luk’yanov, Additional symmetries and exactly solvable models of

two-dimensional conformal field theory. Sov. Sci. Rev. A. Phys. 1990, 15, 1–117.

[FL2]

V. Fateev, and S. Luk’yanov, Poisson-Lie groups and classical W-algebras, Internat. J.

Modern Phys. A, 7, 1992, (5), 853–876.

[FKR]

Z. Fehily, and K. Kawasetsu, and D. Ridout, Classifying relaxed highest-weight modules

for admissible-level Bershadsky-Polyakov algebras, arXiv:2007.03917 [math.RT].

[Fei]

B. Feigin, Semi-infinite homology of Lie, Kac-Moody and Virasoro algebras, Uspekhi

Mat. Nauk, 39, 1984, (2), 236, 195–196.

[FFr1]

B. Feigin, and E. Frenkel, A family of representations of affine Lie algebras, Russ.

Math. Surv., 43, 1988, (5), 221-222.

[FFr2]

B. Feigin, and E. Frenkel, Quantization of the Drinfel’d-Sokolov reduction, Phys. Lett.

B, 246, 1990, (1)-(2), 75–81.

[FFr3]

B. Feigin, and E. Frenkel, Affine Kac-Moody algebras and semi-infinite flag manifolds,

Comm. Math. Phys., 128, 1990, (1), 161–189.

[FFr4]

B. Feigin, and E. Frenkel, Duality in W-algebras, Internat. Math. Res. Notices, 1991,

(6), 75–82.

[FFr5]

B. Feigin, and E. Frenkel, Affine Kac-Moody algebras at the critical level and GelfandDikii algebras, Infinite analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., 16,

World Sci. Publ., River Edge, NJ, 1992, 197–215.

[FFr6]

B. Feigin, and E. Frenkel, Kac-Moody groups and integrability of soliton equations,

Invent. Math., 120, 1995, (2), 379–408.

[FFr7]

B. Feigin, and E. Frenkel. Integrals of motion and quantum groups. In In- tegrable

systems and quantum groups (Montecatini Terme, 1993), volume 1620 of Lecture

Notes in Math., pages 349-418. Springer, Berlin, 1996.

[FFr8]

B. Feigin, and E. Frenkel, Integrable hierarchies and Wakimoto modules, Differential

topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl.

Ser. 2, 194, Amer. Math. Soc., Providence, RI, 1999, 27–60.

67

[FFu]

[FeGu]

[FS]

[FST]

[Fie]

[F1]

[F2]

[FBZ]

[FrGaio]

[FrGaits]

[FKW]

[Fr]

[FLM]

[FrZh]

[Fu]

[GR]

[GD]

[G1]

[G2]

[GL]

[Gep]

[GKO1]

[GKO2]

[GK]

[GS]

[H1]

[H2]

[H3]

B. Feigin, and D. Fuchs, Representations of the Virasoro algebra, Representation of

Lie groups and related topics, Adv. Stud. Contemp. Math., 7, 465–554.

B. Feigin, and S. Gukov, VOA[M4 ], J. Math. Phys., 61, 2020, (1).

(2)

B. L. Feigin, and A. Semikhatov, Wn algebras, Nuclear Phys. B, 698, 2004, (3),

409–449.

B. Feigin, and A. Semikhatov, and I. Tipunin, Equivalence between chain categories

of representations of affine sl(2) and N = 2 superconformal algebras, J. Math. Phys.,

39, 1998, (7), 3865–3905.

P. Fiebig, The combinatorics of category over symmetrizable Kac-Moody algebras,

Transform. Groups., 11 2006, 29–49.

E. Frenkel, Wakimoto modules, opers and the center at the critical level, Adv. Math.

195, 2005, (2), 297–404.

E. Frenkel, Langlands Correspondence for Loop Groups, Cambridge Studies in Advanced Mathematics 103, Cambridge University Press, 2007.

E. Frenkel, and D. Ben-Zvi, Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, 88, American Mathematical Society, Providence, RI, second

edition, 2004.

E. Frenkel, and D. Gaiotto, Quantum Langlands dualities of boundary conditions,

D-modules, and conformal blocks, Commun. Number Theory Phys., 14, 2020, (2),

199–313.

E. Frenkel, and D. Gaitsgory, Local geometric Langlands correspondence and affine

Kac-Moody algebras, Algebraic geometry and number theory, Progr. Math., 253, 69–

260, Birkh¨

auser Boston, Boston, MA, 2006.

E. Frenkel, and V. Kac, and M. Wakimoto, Characters and fusion rules for W-algebras

via quantized Drinfel’d-Sokolov reduction, Comm. Math. Phys., 147, 1992, (2), 295–

328.

I. Frenkel, Representations of affine Lie algebras, Hecke modular forms and Kortewegde Vries type equations, Lie algebras and related topics (New Brunswick, N.J., 981),

Lecture Notes in Math., 933, 1982, 71–110.

I. Frenkel, J. Lepowsky, and A. Meurman, Vertex operator algebras and the Monster,

Pure and Applied Mathematics, 134, Academic Press, Inc., Boston, MA, 1988, liv+508.

I. Frenkel, and Y. Zhu, Vertex operator algebras associated to representations of affine

and Virasoro algebras, Duke Math. J., 66, 1992, (1), 123–168.

J. Fuchs, Simple WZW currents, Comm. Math. Phys., 136, 1991, (2), 345–356.

D. Gaiotto, and M. Rapˇ

c´

ak, Vertex algebras at the corner, J. High Energy Phys., 2019,

(1).

I. Gel’fand, and L. Diki˘ı, Asymptotic properties of the resolvent of Sturm-Liouville

equations, and the algebra of Korteweg-de Vries equations, Uspehi Mat. Nauk, 30,

1975, (5), 67–100.

N. Genra, Screening operators for W-algebras, Sel. Math. New. Ser., 23, 2017, (3),

2157–2202.

N. Genra, Screening operators and parabolic inductions for affine W-algebras, Adv.

Math., 369, 2020, with an appendix by S. Nakatsuka.

N. Genra, and A. Linshaw, Ito’s conjecture and the coset construction for Wk (sl(3|2)),

arXiv:1901.02397 [math.RT].

D. Gepner, Fusion rings and geometry, Comm. Math. Phys., 141, 1991, (2), 381–411.

P. Goddard, and A. Kent, and D. Olive, Virasoro algebras and coset space models,

Phys. Lett. B, 152, 1985, (1)-(2), 88–92.

P. Goddard, and A. Kent, and D. Olive, Unitary representations of the Virasoro and

super-Virasoro algebras, Comm. Math. Phys., 103, 1986, (1), 105–119.

M. Gorelik, and V. Kac, Characters of (relatively) integrable modules over affine Lie

superalgebras, Jpn. J. Math., 10, 2015, (2), 135–235

M. Gorelik, and V. Serganova, Integrable modules over affine Lie superalgebras

sl(1|n)(1) , Comm. Math. Phys., 364, 2018, (2), 635–654.

Y. Huang, A nonmeromorphic extension of the Moonshine module vertex operator algebra, Moonshine, the Monster, and related topics (South Hadley, MA, 1994), Contemp.

Math., 193, 123–148, Amer. Math. Soc., Providence, RI, 1996.

Y. Huang, A theory of tensor products for module categories for a vertex operator

algebra. IV, J. Pure Appl. Algebra, 100, 1995, (1)–(3), 173–216.

Y. Huang, Differential equations and intertwining operators, Commun. Contemp.

Math., 7, 2005, (3), 375–400.

68

[H4]

[H5]

[H6]

[H7]

[HKL]

[HL1]

[HL2]

[HL3]

[HLZ1]

[HLZ2]

[HLZ3]

[HLZ4]

[HLZ5]

[HLZ6]

[HLZ7]

[HLZ8]

[Hum]

[IK1]

[IK2]

[IMP1]

[IMP2]

[Kac1]

[Kac2]

[Kac3]

[KRW]

[KW1]

[KW2]

Y. Huang, Differential equations, duality and modular invariance, Commun. Contemp.

Math., 7, 2005, (5), 649–706.

Y. Huang, Vertex operator algebras and the Verlinde conjecture, Commun. Contemp.

Math., 10, 2008, (1), 103–154.

Y. Huang, Rigidity and modularity of vertex tensor categories, Commun. Contemp.

Math., 10, 2008, suppl. (1), 871–911.

Y. Huang, Cofiniteness conditions, projective covers and the logarithmic tensor product

theory, J. Pure Appl. Algebra, 213, 2009, (4), 458–475.

Y. Huang, and A. Kirillov, and J. Lepowsky, Braided tensor categories and extensions

of vertex operator algebras, Comm. Math. Phys., 337, 2015, (3), 1143–1159.

Y. Huang, and J. Lepowsky, A theory of tensor products for module categories for a

vertex operator algebra. I, Selecta Math. (N.S.) 1, 1995, (4), 699–756.

Y. Huang, and J. Lepowsky, A theory of tensor products for module categories for a

vertex operator algebra. I, Selecta Math. (N.S.) 1, 1995, (4), 757–786.

Y. Huang, and J. Lepowsky, A theory of tensor products for module categories for a

vertex operator algebra. III, J. Pure Appl. Algebra, 100, 1995, (1)–(3), 141–171.

Y. Huang, and J. Lepowsky, and L. Zhang, Logarithmic tensor category theory for

generalized modules for a conformal vertex algebra, I: introduction and strongly graded

algebras and their generalized modules, Conformal field theories and tensor categories,

Math. Lect. Peking Univ., 169–248, Springer, Heidelberg, 2014.

Y. Huang, and J. Lepowsky, and L. Zhang, Logarithmic tensor category theory,

II: Logarithmic formal calculus and properties of logarithmic intertwining operators,

arXiv:1012.4196 [math.QA].

Y. Huang, and J. Lepowsky, and L. Zhang, Logarithmic tensor category theory, III:

Intertwining maps and tensor product bifunctors, arXiv:1012.4197 [math.QA].

Y. Huang, and J. Lepowsky, and L. Zhang, Logarithmic tensor category theory,

IV: Constructions of tensor product bifunctors and the compatibility conditions,

arXiv:1012.4198 [math.QA].

Y. Huang, and J. Lepowsky, and L. Zhang, Logarithmic tensor category theory, V:

Convergence condition for intertwining maps and the corresponding compatibility condition, arXiv:1012.4199 [math.QA].

Y. Huang, and J. Lepowsky, and L. Zhang, Logarithmic tensor category theory, VI:

Expansion condition, associativity of logarithmic intertwining operators, and the associativity isomorphisms, arXiv:10212.4202 [math.QA].

Y. Huang, and J. Lepowsky, and L. Zhang, Logarithmic tensor category theory, VII:

Convergence and extension properties and applications to expansion for intertwining

maps, arXiv:1110.1929 [math.QA].

Y. Huang, and J. Lepowsky, and L. Zhang, Logarithmic tensor category theory, VIII:

Braided tensor category structure on categories of generalized modules for a conformal

vertex algebra, arXiv:1110.1931 [math.QA].

J. Humphreys, Representations of semisimple Lie algebras in the BGG category O,

Graduate Studies in Mathematics, 94, American Mathematical Society, Providence,

RI, 2008, xvi+289.

K. Iohara, and Y. Koga, Wakimoto modules for the affine Lie superalgebras A(m −

1, n − 1)(1) and D(2, 1, a)(1) , Math. Proc. Cambridge Philos. Soc., 132, 2002, (3),

419–433.

K. Iohara, and Y. Koga Iohara, Representation theory of the Virasoro algebra, Springer

Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2011, xviii+474.

K. Ito, and J. Madsen, and J. Petersen, Free field representations of extended superconformal algebras, Nuclear Phys. B, 398, 1993, (2), 425–458.

K. Ito, and J. Madsen, and J. Petersen, Extended superconformal algebras and free

field realizations from Hamiltonian reduction, Phys. Lett. B, 318, 1993, (2), 315–322.

V. Kac, Lie superalgebra, Adv. Math. 26, 1977, (1), 8–96.

V. Kac, Infinite-dimensional Lie algebras, Third Edition, Cambridge University Press,

Cambridge, 1990, xxii+400.

V. Kac, Vertex algebras for beginners, University Lecture Series, 10, Second, American

Mathematical Society, Providence, RI, 1998, vi+201.

V. Kac, and S. Roan, and M. Wakimoto, Quantum reduction for affine superalgebras,

Comm. Math. Phys., 241, 2003, (2)-(3), 307–342.

V. Kac, and M. Wakimoto, Integrable highest weight modules over affine superalgebras

and Appell’s function, Comm. Math. Phys., 215, 2001, (3), 631–682.

V. Kac, and M. Wakimoto, Quantum reduction and representation theory of superconformal algebras, Adv. Math., 185, 2004, (2), 400–458.

69

[KW3]

[KW4]

[KW5]

[Kas]

[Kaw]

[KaSu]

[KO]

[KoSa]

[Kos]

[Ku]

[La]

[LS]

[LLi]

[Li1]

[Li2]

[Li3]

[Lin]

[Mas]

[MN]

[Mat]

[Mi1]

[Mi2]

[Mo]

[N1]

[N2]

[N3]

[OS]

[PR]

[RSYZ]

[R]

[Sa1]

V. Kac, and M. Wakimoto, Representations of affine superalgebras and mock theta

functions, Transform. Groups, 19, 2014,(2), 383–455.

V. Kac, and M. Wakimoto, Representations of affine superalgebras and mock theta

functions II, Adv. Math., 300, 2016, 17–70.

V. Kac, and M. Wakimoto, Representations of affine superalgebras and mock theta

functions III, Izv. Ross. Akad. Nauk Ser. Mat., 80, 2016, (4), 65–122.

C. Kassel, Christian, Quantum groups, Graduate Texts in Mathematics, 155, SpringerVerlag, New York, 1995, xii+531.

K. Kawasetsu, W-algebras with non-admissible levels and the Deligne exceptional series, Int. Math. Res. Not. IMRN, 2018, (3), 641–676.

Y. Kazama, and H. Suzuki, New N = 2 superconformal field theories and superstring

compactification, Nuclear Phys. B, 321, 1989, (1), 232–268.

A. Kirillov, Jr., and V. Ostrik, On a q-analogue of the McKay correspondence and the

ADE classification of sl2 conformal field theories, Adv. Math., 171, 2002, (2), 183–227.

S. Koshida, and R. Sato, On resolution of highest weight modules over the N = 2

superconformal algebra, arXiv:1810.13147 [math.QA].

B. Kostant, Verma modules and the existence of quasi-invariant differential operators,

Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1974), 101–128.

Lecture Notes in Math., Vol. 466, 1975.

S. Kumar, Kac-Moody groups, their flag varieties and representation theory, Progress

in Mathematics, 204, Birkh¨

auser Boston, Inc., Boston, MA, 2002

C. Lam, Induced modules for orbifold vertex operator algebras, J. Math. Soc. Japan,

53, 2001, (3), 541–557.

C. Lam, and H. Shimakura, Classification of holomorphic framed vertex operator algebras of central charge 24, Amer. J. Math., 137, 2015, (1), 111–137.

J. Lepowsky and H. Li, Introduction to vertex operator algebras and their representations, Progress in Mathematics, 227, Birkh¨

auser Boston, Inc., Boston, MA, 2004,

xiv+318.

H. Li, Extension of vertex operator algebras by a self-dual simple modules, J.Algebra,

187, 1997, (1), 236–267.

H. Li, On abelian coset generalized vertex algebras, Commun. Contemp. Math., 3,

2001, (2), 287–340.

H. Li, Abelianizing vertex algebras, Comm. Math. Phys., 259, 2005, (2), 391–411.

A. Linshaw, Universal two-parameter W∞ -algebra and vertex algebras of type

W(2, 3, . . . , N ), arXiv:1710.02275 [math.RT].

G. Mason, Lattice subalgebras of strongly regular vertex operator algebras, Conformal

field theory, automorphic forms and related topics, Contrib. Math. Comput. Sci., 8,

31–53.

A. Matsuo, and K. Nagatomo, A note on free bosonic vertex algebra and its conformal

vectors, J. Algebra, 212, 1999, (2), 395–418.

Y. Matsuo, Character formula of c < 1 unitary representation of N = 2 superconformal

algebra, Progr. Theoret. Phys., 77, 1987, (4), 793–797.

M. Miyamoto, Griess algebras and conformal vectors in vertex operator algebras, J.

Algebra, 179, 1996, (2), 523–548.

M. Miyamoto, C2 -cofiniteness of cyclic-orbifold models, Comm. Math. Phys., 335,

2015, (3), 1279–1286.

Y. Moriwaki, On classification of conformal vectors in vertex operator algebra and the

vertex algebra automorphism group, J. Algebra, 546, 2020, 689–702.

S. Nakatsuka, Miura maps and parabolic Wakimoto resolutions, appendix to Screening

operators and parabolic inductions for affine W-algebras, Adv. Math., 369, 2020.

S. Nakatsuka, On Miura maps for W-superalgebras, arXiv:2005.10472, [math.QA].

S. Nakatsuka, A geometric construction of integrable Hamiltonian hierarchies associated with the classical affine W-algebras, arXiv:2006.00302, [math.QA].

V. Ostrik, and M. Sun, Level-rank duality via tensor categories, Comm. Math. Phys.,

326, 2014, (1), 49–61.

T. Proch´

azka, and M. Rapˇ

c´

ak, W-algebra modules, free fields, and Gukov-Witten

defects, J. High Energy Phys., 2019, (5)

M. Rapˇ

c´

ak, and Y. Soibelman, and Y. Yang, and G. Zhao, Cohomological Hall algebras,

vertex algebras and instantons, Comm. Math. Phys., 376, 2020, (3), 1803–1873.

J. Rasmussen, Free field realizations of affine current superalgebras, screening currents

and primary fields, Nuclear Phys. B, 510, (1998), (3), 688–720.

R. Sato, Equivalences between weight modules via N = 2 coset constructions,

arXiv:1605.02343 [math.RT].

70

[Sa2]

[Sa3]

[SV]

[Shi]

[Su]

[T]

[TW]

[V1]

[V2]

[Wak1]

[Wak2]

[Wal]

[W1]

[W2]

[X]

[YY]

[Y]

[Za]

[Zh]

R. Sato, Kazama–Suzuki coset construction and its inverse, arXiv:1907.02377

[math.QA].

R. Sato, Modular invariant representations of the N = 2 superconformal algebra, Int.

Math. Res. Not. IMRN, 2019, 24, 7659–7690.

O. Schiffmann, and E. Vasserot, Cherednik algebras, W-algebras and the equivariant

cohomology of the moduli space of instantons on A2 , Publ. Math. Inst. Hautes Etudes

Sci., 118, 2013, 213–342.

H. Shimakura, Lifts of automorphisms of vertex operator algebras in simple current

extensions, Math. Z., 256, 2007, (3), 491–508.

S. Sugimoto, On the Feigin-Tipunin conjecture, arXiv:2004.05769, [math.RT]

C. Taubes, Differential geometry, Oxford Graduate Texts in Mathematics, 23, Oxford

University Press, Oxford, 2011, xiv+298.

A. Tsuchiya, and S. Wood, The tensor structure on the representation category of the

Wp triplet algebra, J. Phys. A, 46, 2013, (44), 445203–40.

A. Voronov, Semi-infinite homological algebra, Invent. Math., 113, 1993, (1), 103–146.

A. Voronov, Semi-infinite induction and Wakimoto modules, Amer. J. Math., 121,

1999, (5), 1079–1094.

(1)

M. Wakimoto, Fock representations of the affine Lie algebra A1 , Comm. Math. Phys.,

104, 1986, (4), 605–609.

M. Wakimoto, Fusion rules for N = 2 superconformal modules, arXiv:9807144 [hep-th]

N. Wallack, Real reductive groups. I, Pure and Applied Mathematics, 132, Academic

Press, Inc., Boston, MA, 1988, xx+412.

E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys., 121,

1989, (3), 351–399

E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, Geometry,

topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, 357–422, Int.

Press, Cambridge, MA, 1995.

X. Xu, Introduction to vertex operator superalgebras and their modules, Mathematics

and its Applications, 456, Kluwer Academic Publishers, Dordrecht, 1998, xvi+356.

H. Yamada, and H. Yamauchi, Simple Current Extensions of Tensor Products of Vertex Operator Algebras, Internat. Math. Res. Notices, rnaa107,

https://doi.org/10.1093/imrn/rnaa107.

H. Yamauchi, Module categories of simple current extensions of vertex operator algebras, J. Pure Appl. Algebra, 189, 2004, (1)-(3), 315–328.

A. Zamolodchikov, Infinite extra symmetries in two-dimensional conformal quantum

field theory, Teoret. Mat. Fiz., 65, 1985, (3), 347–359.

Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math.

Soc., 9, 1996, (1), 237–302.

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba,

Tokyo, Japan 153-8914

Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa,

Chiba 277-8583, Japan

Email address: nakatuka@ms.u-tokyo.ac.jp

71

...