Tensor decomposition, parafermions, level-rank duality, and reciprocity law for vertex operator algebras

Jiang, Cuipo; Lin, Zongzhu

doi:10.1090/tran/8207

Tensor decomposition, parafermions, level-rank duality, and reciprocity law for vertex operator algebras
HTML articles powered by AMS MathViewer

by Cuipo Jiang and Zongzhu Lin PDF

Trans. Amer. Math. Soc. 375 (2022), 8325-8352 Request permission

Abstract:

For a semisimple Lie algebra $\frak {sl}_n$, the basic representation $L_{\widehat {\frak {sl}_{n}}}(1,0)$ of the affine Lie algebra $\widehat {\frak {sl}_{n}}$ is a lattice vertex operator algebra. The first main result of the paper is to prove that the commutant vertex operator algebra of $L_{\widehat {\frak {sl}_{n}}}(l,0)$ in the $l$-fold tensor product $L_{\widehat {\frak {sl}_{n}}}(1,0)^{\otimes l}$ is isomorphic to the parafermion vertex operator algebra $K(\frak {sl}_{l},n)$, which is the commutant of the Heisenberg vertex operator algebra $L_{\widehat {\frak {h}}}(n,0)$ in $L_{\widehat {\frak {sl}_l}}(n,0)$. The result provides a version of level-rank duality. The second main result of the paper is to prove more general version of the first result that the commutant of $L_{\widehat {\frak {sl}_{n}}}(l_1+\cdots +l_s, 0)$ in $L_{\widehat {\frak {sl}_{n}}}(l_1,0)\otimes \cdots \otimes L_{\widehat {\frak {sl}_{n}}}(l_s, 0)$ is isomorphic to the commutant of the vertex operator algebra generated by a Levi Lie subalgebra of $\frak {sl}_{l_1+\cdots +l_s}$ corresponding to the composition $(l_1, \cdots , l_s)$ in the rational vertex operator algebra $L_{\widehat {\frak {sl}}_{l_1+\cdots +l_s}}(n,0)$. This general version also resembles a version of reciprocity law discussed by Howe in the context of reductive Lie groups. In the course of the proof of the main results, certain Howe duality pairs also appear in the context of vertex operator algebras.

References

Toshiyuki Abe, Fusion rules for the charge conjugation orbifold, J. Algebra 242 (2001), no. 2, 624–655. MR 1848962, DOI 10.1006/jabr.2001.8838
Toshiyuki Abe, Chongying Dong, and Haisheng Li, Fusion rules for the vertex operator algebra $M(1)$ and $V^+_L$, Comm. Math. Phys. 253 (2005), no. 1, 171–219. MR 2105641, DOI 10.1007/s00220-004-1132-5
Chunrui Ai, Chongying Dong, Xiangyu Jiao, and Li Ren, The irreducible modules and fusion rules for the parafermion vertex operator algebras, Trans. Amer. Math. Soc. 370 (2018), no. 8, 5963–5981. MR 3812115, DOI 10.1090/tran/7302
Daniel Altschüler, Michel Bauer, and Claude Itzykson, The branching rules of conformal embeddings, Comm. Math. Phys. 132 (1990), no. 2, 349–364. MR 1069826, DOI 10.1007/BF02096653
Daniel Altschüler, Michel Bauer, and Hubert Saleur, Level-rank duality in nonunitary coset theories, J. Phys. A 23 (1990), no. 16, L789–L793. MR 1069465
H. Saleur and D. Altschüler, Level-rank duality in quantum groups, Nuclear Phys. B 354 (1991), no. 2-3, 579–613. MR 1103065, DOI 10.1016/0550-3213(91)90367-7
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, 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
Tomoyuki Arakawa and Cuipo Jiang, Coset vertex operator algebras and $\mathcal W$-algebras of $A$-type, Sci. China Math. 61 (2018), no. 2, 191–206. MR 3755775, DOI 10.1007/s11425-017-9161-7
Tomoyuki Arakawa, Ching Hung Lam, and Hiromichi Yamada, Zhu’s algebra, $C_2$-algebra and $C_2$-cofiniteness of parafermion vertex operator algebras, Adv. Math. 264 (2014), 261–295. MR 3250285, DOI 10.1016/j.aim.2014.07.021
Tomoyuki Arakawa, Ching Hung Lam, and Hiromichi Yamada, Parafermion vertex operator algebras and $W$-algebras, Trans. Amer. Math. Soc. 371 (2019), no. 6, 4277–4301. MR 3917223, DOI 10.1090/tran/7547
Anne-Marie Aubert, Jean Michel, and Raphaël Rouquier, Correspondance de Howe pour les groupes réductifs sur les corps finis, Duke Math. J. 83 (1996), no. 2, 353–397 (French). MR 1390651, DOI 10.1215/S0012-7094-96-08312-X
R. Blumenhagen, W. Eholzer, A. Honecker, R. Hübel, and K. Hornfeck, Coset realization of unifying $\scr W$ algebras, Internat. J. Modern Phys. A 10 (1995), no. 16, 2367–2430. MR 1334477, DOI 10.1142/S0217751X95001157
Richard E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 10, 3068–3071. MR 843307, DOI 10.1073/pnas.83.10.3068
Andrea Cappelli, Lachezar S. Georgiev, and Ivan T. Todorov, Parafermion Hall states from coset projections of abelian conformal theories, Nuclear Phys. B 599 (2001), no. 3, 499–530. MR 1825204, DOI 10.1016/S0550-3213(00)00774-4
Tung-Shyan Chen and Ching Hung Lam, Extension of the tensor product of unitary Virasoro vertex operator algebra, Comm. Algebra 35 (2007), no. 8, 2487–2505. MR 2345796, DOI 10.1080/00927870701326338
Shun-Jen Cheng, Ngau Lam, and Weiqiang Wang, Super duality and irreducible characters of ortho-symplectic Lie superalgebras, Invent. Math. 183 (2011), no. 1, 189–224. MR 2755062, DOI 10.1007/s00222-010-0277-4
Yanjun Chu and Zongzhu Lin, The varieties of semi-conformal vectors of affine vertex operator algebras, J. Algebra 515 (2018), 77–101. MR 3859961, DOI 10.1016/j.jalgebra.2018.08.016
YanJun Chu and ZongZhu Lin, The varieties of Heisenberg vertex operator algebras, Sci. China Math. 60 (2017), no. 3, 379–400. MR 3600931, DOI 10.1007/s11425-015-0778-8
T. Creutzig, S. Kanade, A. R. Linshaw, and D. Ridout, Schur-Weyl duality for Heisenberg cosets, Transform. Groups 24 (2019), no. 2, 301–354. MR 3948937, DOI 10.1007/s00031-018-9497-2
T. Creutzig, S. Kanade, and R. McRae, Tensor categories for vertex operator superalgebra extensions, Adv. Math. 396 (2022).
T. Creutzig, S. Kanade, and R. McRae, Glueing vertex operator algebras, arXiv:190600119.
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
Chongying Dong, Representations of the moonshine module vertex operator algebra, Mathematical aspects of conformal and topological field theories and quantum groups (South Hadley, MA, 1992) Contemp. Math., vol. 175, Amer. Math. Soc., Providence, RI, 1994, pp. 27–36. MR 1302010, DOI 10.1090/conm/175/01835
Chongying Dong, Robert L. Griess Jr., and Gerald Höhn, Framed vertex operator algebras, codes and the Moonshine module, Comm. Math. Phys. 193 (1998), no. 2, 407–448. MR 1618135, DOI 10.1007/s002200050335
Chongying Dong, Cuipo Jiang, and Xingjun Lin, Rationality of vertex operator algebra $V^+_L$: higher rank, Proc. Lond. Math. Soc. (3) 104 (2012), no. 4, 799–826. MR 2908783, DOI 10.1112/plms/pdr055
Chongying Dong, Ching Hung Lam, Qing Wang, and Hiromichi Yamada, The structure of parafermion vertex operator algebras, J. Algebra 323 (2010), no. 2, 371–381. MR 2564844, DOI 10.1016/j.jalgebra.2009.08.003
Chongying Dong, Ching Hung Lam, and Hiromichi Yamada, Decomposition of the vertex operator algebra $V_{\sqrt {2}A_3}$, J. Algebra 222 (1999), no. 2, 500–510. MR 1734233, DOI 10.1006/jabr.1999.8019
Chongying Dong, Ching Hung Lam, and Hiromichi Yamada, $W$-algebras related to parafermion algebras, J. Algebra 322 (2009), no. 7, 2366–2403. MR 2553685, DOI 10.1016/j.jalgebra.2009.03.034
Chongying Dong and James Lepowsky, Generalized vertex algebras and relative vertex operators, Progress in Mathematics, vol. 112, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1233387, DOI 10.1007/978-1-4612-0353-7
Chongying Dong, Haisheng Li, and Geoffrey Mason, Regularity of rational vertex operator algebras, Adv. Math. 132 (1997), no. 1, 148–166. MR 1488241, DOI 10.1006/aima.1997.1681
Chongying Dong, Haisheng Li, and Geoffrey Mason, Simple currents and extensions of vertex operator algebras, Comm. Math. Phys. 180 (1996), no. 3, 671–707. MR 1408523, DOI 10.1007/BF02099628
Chongying Dong, Haisheng Li, and Geoffrey Mason, Twisted representations of vertex operator algebras, Math. Ann. 310 (1998), no. 3, 571–600. MR 1615132, DOI 10.1007/s002080050161
Chongying Dong, Haisheng Li, and Geoffrey Mason, Modular-invariance of trace functions in orbifold theory and generalized Moonshine, Comm. Math. Phys. 214 (2000), no. 1, 1–56. MR 1794264, DOI 10.1007/s002200000242
Chongying Dong, Geoffrey Mason, and Yongchang Zhu, Discrete series of the Virasoro algebra and the moonshine module, Algebraic groups and their generalizations: quantum and infinite-dimensional methods (University Park, PA, 1991) Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 295–316. MR 1278737
Chongying Dong and Li Ren, Representations of the parafermion vertex operator algebras, Adv. Math. 315 (2017), 88–101. MR 3667581, DOI 10.1016/j.aim.2017.05.016
Chongying Dong and Qing Wang, The structure of parafermion vertex operator algebras: general case, Comm. Math. Phys. 299 (2010), no. 3, 783–792. MR 2718932, DOI 10.1007/s00220-010-1114-8
Chongying Dong and Qing Wang, On $C_2$-cofiniteness of parafermion vertex operator algebras, J. Algebra 328 (2011), 420–431. MR 2745574, DOI 10.1016/j.jalgebra.2010.10.015
Chongying Dong and Qing Wang, Quantum dimensions and fusion rules for parafermion vertex operator algebras, Proc. Amer. Math. Soc. 144 (2016), no. 4, 1483–1492. MR 3451226, DOI 10.1090/proc/12838
I. B. Frenkel, Representations of affine Lie algebras, Hecke modular forms and Korteweg-de Vries type equations, Lie algebras and related topics (New Brunswick, N.J., 1981) Lecture Notes in Math., vol. 933, Springer, Berlin-New York, 1982, pp. 71–110. MR 675108
Igor B. Frenkel, Yi-Zhi Huang, and James Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993), no. 494, viii+64. MR 1142494, DOI 10.1090/memo/0494
Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, MA, 1988. MR 996026
Igor B. Frenkel and Yongchang Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), no. 1, 123–168. MR 1159433, DOI 10.1215/S0012-7094-92-06604-X
P. Goddard, A. Kent, and D. Olive, Virasoro algebras and coset space models, Phys. Lett. B 152 (1985), no. 1-2, 88–92. MR 778819, DOI 10.1016/0370-2693(85)91145-1
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
Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1–182. MR 1321638
Roger Howe, Reciprocity laws in the theory of dual pairs, Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 159–175. MR 733812
Roger E. Howe, Eng-Chye Tan, and Jeb F. Willenbring, Reciprocity algebras and branching for classical symmetric pairs, Groups and analysis, London Math. Soc. Lecture Note Ser., vol. 354, Cambridge Univ. Press, Cambridge, 2008, pp. 191–231. MR 2528468, DOI 10.1017/CBO9780511721410.011
Cuipo Jiang and Ching Hung Lam, Level-rank duality for vertex operator algebras of types $B$ and $D$, Bull. Inst. Math. Acad. Sin. (N.S.) 14 (2019), no. 1, 31–54. MR 3888282
Cuipo Jiang and Zongzhu Lin, The commutant of $L_{\widehat {\mathfrak {sl_2}}}(n,0)$ in the vertex operator algebra $L_{\widehat {\mathfrak {sl}_2}}(1,0)^{\otimes n}$, Adv. Math. 301 (2016), 227–257. MR 3539374, DOI 10.1016/j.aim.2016.06.010
C. Jiang and Z. Lin, Categories and functors in the representation theory of vertex operator algebras (in Chinese). Sci. Sin. Math. 47 (2017), 1579-1594.
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Masaaki Kitazume, Ching Hung Lam, and Hiromichi Yamada, Decomposition of the Moonshine vertex operator algebra as Virasoro modules, J. Algebra 226 (2000), no. 2, 893–919. MR 1752768, DOI 10.1006/jabr.1999.8206
Stephen S. Kudla, Seesaw dual reductive pairs, Automorphic forms of several variables (Katata, 1983) Progr. Math., vol. 46, Birkhäuser Boston, Boston, MA, 1984, pp. 244–268. MR 763017
James Lepowsky and Haisheng Li, Introduction to vertex operator algebras and their representations, Progress in Mathematics, vol. 227, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2023933, DOI 10.1007/978-0-8176-8186-9
Ching Hung Lam, A level-rank duality for parafermion vertex operator algebras of type A, Proc. Amer. Math. Soc. 142 (2014), no. 12, 4133–4142. MR 3266984, DOI 10.1090/S0002-9939-2014-12167-8
Ching-Hung Lam and Shinya Sakuma, On a class of vertex operator algebras having a faithful $S_{n+1}$-action, Taiwanese J. Math. 12 (2008), no. 9, 2465–2488. MR 2479066, DOI 10.11650/twjm/1500405190
Ching Hung Lam and Hiromichi Yamada, Decomposition of the lattice vertex operator algebra $V_{\sqrt {2}A_l}$, J. Algebra 272 (2004), no. 2, 614–624. MR 2028073, DOI 10.1016/S0021-8693(03)00507-6
Gustav Lehrer and Ruibin Zhang, The second fundamental theorem of invariant theory for the orthogonal group, Ann. of Math. (2) 176 (2012), no. 3, 2031–2054. MR 2979865, DOI 10.4007/annals.2012.176.3.12
Haisheng Li, Some finiteness properties of regular vertex operator algebras, J. Algebra 212 (1999), no. 2, 495–514. MR 1676852, DOI 10.1006/jabr.1998.7654
Andrew R. Linshaw, Universal two-parameter $\mathcal W_\infty$-algebra and vertex algebras of type $\mathcal W(2, 3, \ldots , N)$, Compos. Math. 157 (2021), no. 1, 12–82. MR 4215650, DOI 10.1112/s0010437x20007514
Alina Marian and Dragos Oprea, The level-rank duality for non-abelian theta functions, Invent. Math. 168 (2007), no. 2, 225–247. MR 2289865, DOI 10.1007/s00222-006-0032-z
Swarnava Mukhopadhyay, Rank-level duality and conformal block divisors, Adv. Math. 287 (2016), 389–411. MR 3422680, DOI 10.1016/j.aim.2015.09.020
Stephen G. Naculich and Howard J. Schnitzer, Level-rank duality of the $\textrm {U}(N)$ WZW model, Chern-Simons theory, and 2d $q$YM theory, J. High Energy Phys. 6 (2007), 023, 19. MR 2326628, DOI 10.1088/1126-6708/2007/06/023
Victor Ostrik and Michael Sun, Level-rank duality via tensor categories, Comm. Math. Phys. 326 (2014), no. 1, 49–61. MR 3162483, DOI 10.1007/s00220-013-1869-9
Feng Xu, Algebraic coset conformal field theories. II, Publ. Res. Inst. Math. Sci. 35 (1999), no. 5, 795–824. MR 1739703, DOI 10.2977/prims/1195143424
Feng Xu, Algebraic coset conformal field theories, Comm. Math. Phys. 211 (2000), no. 1, 1–43. MR 1757004, DOI 10.1007/s002200050800
A. B. Zamolodchikov and V. A. Fateev, Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in $Z_N$-symmetric statistical systems, Zh. Èksper. Teoret. Fiz. 89 (1985), no. 2, 380–399 (Russian); English transl., Soviet Phys. JETP 62 (1985), no. 2, 215–225 (1986). MR 830910
Weiqiang Wang, Rationality of Virasoro vertex operator algebras, Internat. Math. Res. Notices 7 (1993), 197–211. MR 1230296, DOI 10.1155/S1073792893000212
Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158
Yongchang Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), no. 1, 237–302. MR 1317233, DOI 10.1090/S0894-0347-96-00182-8