Publications

Recent Papers and Books

1. Twisted intertwining operators and tensor products of (generalized) twisted modules, with J. Du, 65 pages, to appear.
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2. Lecture notes on vertex operator algebras and tensor categories, 46 pages on April 27, 2024, for my graduate course "Math 557: Vertex operator algebra theory: vertex operator algebras and tensor categories," Spring, 2024. These notes are not finished yet. Section 8 is still to be typed and more material will be added to Sections 6 and 7. They will be incorporated into my book-in-preparation "Two-dimensional conformal field theory."
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3. Modular invariance of (logarithmic) intertwining operators, Comm. Math. Phys. 405 (2024), article number 131,1-- 82.
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4. Weight-one elements of vertex operator algebras and automorphisms of categories of generalized twisted modules, with C. Sadowski, J. Alg. 628 (2023), 452--485.
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5. Convergence in conformal field theory, Chin. Ann. Math. B43 (2022), 1101--1124.
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6. Lecture notes on representation theory of vertex operator algebras and conformal field theory, 22 pages on March 29, 2022, for my graduate course "Math 557: Vertex operator algebra theory: Representation theory of vertex operator algebras and conformal field theory," Spring, 2022. These notes are still being updated every week during the semester. They will be incorporated into my book-in-preparation "Two-dimensional conformal field theory."
   pdf file

7. Associative algebras and intertwining operators, Comm. Math. Phys. 396 (2022), 1--44.
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8. Corrigendum to "Logarithmic intertwining operators and associative algebras" [J. Pure Appl. Alg. 216 (2012), 1467--1492], J. Pure Appl. Alg. (2021), 2 pages.
   pdf file
   (See also the pdf file of the corrected version of the original paper.)

9. Associative algebras and the representation theory of grading-restricted vertex algebras, Comm. Comtemp. Math. 26 (2024), paper no. 2350036 (46 pages).
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10. Lecture notes on vertex algebras and quantum vertex algebras, 96 pages on April 26, 2020, for my graduate course "Math 555: Selected Topics in Algebra: Vertex algebras and quantum vertex algebras," Spring, 2020. These notes will be updated frequently during the semester. They will be incorporated into my book-in-preparation "Two-dimensional conformal field theory."
    pdf file

11. Representation theory of vertex operator algebras and orbifold conformal field theory, in: Lie groups, number theory, and vertex algebras, ed. by D. Adamovic, A. Dujella, A. Milas and P. Pandzic, Contemp. Math., Vol. 768, Amer. Math. Soc., Providence, RI, 2021, 221–252.
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12. Lower-bounded and grading-restricted twisted modules for affine vertex (operator) algebras, 36 pages, J. Pure Appl. Alg.225 (2021), Paper no. 106618.
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13. Locally convex topological completions of modules for a vertex operator algebra, 3 pages, an extended abstract for the author's talk at the Oberwolfach Workshop ``Subfactors and Applications'' from October 27 to November 2, 2019.
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14. Generators, spanning sets and existence of twisted modules for a grading-restricted vertex (super)algebra, 41 pages, Selecta Math. 26 (2020), Paper no. 62.
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15. A construction of lower-bounded generalized twisted modules for a grading-restricted vertex (super)algebra, Comm. Math. Phys. 377 (2020), 909-945.
    pdf file

16. Twist vertex operators for twisted modules, J. Alg. 539 (2019), 53--83.
    pdf file

17. Affine Lie algebras and tensor categories, 14 pages, Proceedings of 10th CFT Seminar: A Conference on Vertex Algebras and Related Topics at RIMS, to appear.
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18. The first cohomology, derivations and the reductivity of a (meromorphic open-string) vertex algebra, with F. Qi, Trans. Amer. Math. Soc. 373 (2020), 7817--7868.
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Full List of Papers and Research Monographs

1. Ernst equation with cosmological constant, with Xu Chongming, Journal of Changsha Railway Institute 4 (1986), 1--3.

2. Bäcklund theorems in 3-dimensional Minkowski space and their higher-dimensional generalization, Acta Mathematica Sinica 29 (1986), 684--690.

3. Global Bäcklund transformations in 3-dimensional Minkowski space, unpublished, 1985, 17 pages.

4. On the geometric interpretation of vertex operator algebras, Ph.D. thesis, Rutgers University, 1990, 131 pages.

5. Geometric interpretation of vertex operator algebras, Proc. Natl. Acad. Sci. USA 88 (1991), 9964--9968.

6. Toward a theory of tensor product for representations of a vertex operator algebra, with J. Lepowsky, in Proc. 20th Intl. Conference on Diff. Geom. Methods in Theoretical Physics, New York, 1991, ed. S. Catto and A. Rocha, World Scientific, Singapore, 1992, Vol. 1, 344--354.

7. Applications of the geometric interpretation of vertex operator algebras, in Proc. 20th Intl. Conference on Diff. Geom. Methods in Theoretical Physics, New York, 1991, ed. S. Catto and A. Rocha, World Scientific, Singapore, 1992, Vol. 1, 333--343.

8. Vertex operator algebras and conformal field theory, International Journal of Modern Physics A 7 (1992), 2109--2151.

9. On axiomatic approaches to vertex operator algebras and Modules, with I.B. Frenkel and J. Lepowsky, Memoirs Amer. Math. Soc. 104, No. 494 (1993), American Mathematical Society, Providence, 64 pages.

10. Vertex operator algebras and operads, with J. Lepowsky, The Gelfand Mathematical Seminars, 1990--1992, ed. L. Corwin, I. Gelfand and J. Lepowsky, Birkhäuser, Boston, 1993, 145--161.

11. A theory of tensor products for module categories for a vertex operator algebra, I, with J. Lepowsky, in: Geometric aspects of infinite integrable systems, Kyoto, 1993, RIMS Kokyuroku 883, RIMS, Kyoto, Japan, 1994, 148--203.

12. Binary trees and finite-dimensional Lie algebras, in Proc. AMS Summer Research Institute on Algebraic Groups and Their Generalizations, Pennsylvania State University, 1991, ed. W. J. Haboush and B. J. Parshall, American Mathematical Society, Providence, 1994, Vol. 2, 337--348.

13. Operadic formulation of the notion of vertex operator algebra, with J. Lepowsky, in: Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups, Proc. Joint Summer Research Conference, Mount Holyoke, 1992, ed. P. Sally, M. Flato, J. Lepowsky, N. Reshetikhin and G. Zuckerman, Contemporary Math., Vol. 175, Amer. Math. Soc., Providence, 1994, 131--148.

14. Operadic formulation of topological vertex algebras and Gerstenhaber or Batalin-Vilkovisky algebras, Comm. in Math. Phys. 164 (1994), 105--144.

15. A theory of tensor products for module categories for a vertex operator algebra, I, with J. Lepowsky, Selecta Math. 1 (1995), 699-756.

16. A theory of tensor products for module categories for a vertex operator algebra, II, with J. Lepowsky, Selecta Math. 1 (1995), 757--786.

17. Tensor products of modules for a vertex operator algebra and vertex tensor categories, with J. Lepowsky, in: Lie Theory and Geometry, in honor of Bertram Kostant, ed. R. Brylinski, J.-L. Brylinski, V. Guillemin, V. Kac, Birkhäuser, Boston, 1994, 349--383.

18. A theory of tensor products for module categories for a vertex operator algebra, III, with J. Lepowsky, J. Pure Appl. Alg. 100 (1995), 141--171.

19. A theory of tensor products for module categories for a vertex operator algebra, IV, J. Pure Appl. Alg. 100 (1995), 173--216.

20. Introduction to vertex operator algebras, III, in: Moonshine and vertex operator algebras, RIMS Kokyuroku 904, RIMS, Kyoto, Japan, 1995, 51--77.

21. A nonmeromorphic extension of the moonshine module vertex operator algebra, in: Moonshine, the Monster and related topics, Proc. Joint Summer Research Conference, Mount Holyoke, 1994, ed. C. Dong and G. Mason, Contemporary Math., Vol. 193, Amer. Math. Soc., Providence, 1996, 123--148.

22. Virasoro vertex operator algebras, (nonmeromorphic) operator product expansion and the tensor product theory, J. Alg. 182 (1996), 201--234.

23. On the D-module and formal variable approaches to vertex algebras, with J. Lepowsky, in: Topics in Geometry: In Memory of Joseph D'Atri, ed. S. Gindikin, Progress in Nonlinear Differential Equations, Vol. 20, Birkhäuser, Boston, 1996, 175--202.

24. Intertwining operator algebras, genus-zero modular functors and genus-zero conformal field theories, in: Operads: Proceedings of Renaissance Conferences, ed. J.-L. Loday, J. Stasheff, and A. A. Voronov, Contemporary Math., Vol. 202, Amer. Math. Soc., Providence, 1997, 335--355.

25. Two-dimensional conformal geometry and vertex operator algebras, Progress in Mathematics, Vol. 148, 1997, Birkhäuser, Boston, 280 pages.

26. Genus-zero modular functors and intertwining operator algebras, Internat. J. Math. 9 (1998), 845--863.

27. Intertwining operator algebras and vertex tensor categories for affine Lie algebras, with J. Lepowsky, Duke Math. J. 99 (1999), 113--134.
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28. A functional-analytic theory of vertex (operator) algebras, I, Comm. Math. Phys. 204 (1999), 61--84.
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29. Generalized rationality and a Jacobi identity for intertwining operator algebras, Selecta Math. 6 (2000), 225--267.
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30. Semi-infinite forms and topological vertex operator algebras>, with W. Zhao, Comm. Contemp. Math., 2 (2000), 191--241.

31. Factorization of formal exponential and uniformization, with K. Barron and J. Lepowsky, J. Alg. 228 (2000), 551--579.
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32. Vertex operators, with J. Lepowsky, article in: Encyclopaedia of Mathematics, Supplement III, ed. by M. Hazewinkel, Kluwer Academic Publishers, 2001.

33. Vertex operator algebras, with J. Lepowsky, article in: Encyclopaedia of Mathematics, Supplement III, ed. by M. Hazewinkel, Kluwer Academic Publishers, 2001.

34. Intertwining operator algebras and vertex tensor categories for superconformal algebras, I, with A. Milas, Comm. Contemp. Math. 4 (2002), 327--355.

35. Intertwining operator algebras and vertex tensor categories for superconformal algebras, II, with A. Milas, Trans. Amer. Math. Soc. 354 (2002), 363--385.
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36. Review of Vertex Algebras and Algebraic Curves by E. Frenkel and D. Ben-Zvi, Bull. Amer. Math. Soc. 39 (2002), 585--591.
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37. Riemann surfaces with boundaries and the theory of vertex operator algebras, in: Vertex Operator Algebras in Mathematics and Physics, ed. S. Berman, Y. Billig, Y.-Z. Huang and J. Lepowsky, Fields Institute Communications, Vol. 39, Amer. Math. Soc., Providence, 2003, 109--125.
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38. A functional-analytic theory of vertex (operator) algebras, II, Comm. Math. Phys. 242 (2003), 425--444.
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39. Differential equations and conformal field theories, in: Nonlinear Evolution Equations and Dynamical Systems, Proc. ICM2002 Satellite Conference, Yellow Mountains, 2002, ed. by Y. Cheng, S. Hu, Y. Li and C. Peng, World Scientific, Singapore, 2003, 61--71.
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40. Conformal-field-theoretic analogues of codes and lattices, in: Kac-Moody Lie Algebras and Related Topics, Proc. Ramanujan International Symposium on Kac-Moody Lie algebras and applications, ed. N. Sthanumoorthy and K. C. Misra, Contemp. Math., Vol. 343, Amer. Math. Soc., Providence, 2004, 131--145.
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41. Open-string vertex algebras, tensor categories and operads, with L. Kong, Comm. Math. Phys. 250 (2004), 433--471.
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42. Vertex operator algebras, the Verlinde conjecture and modular tensor categories, Proc. Natl. Acad. Sci. USA 102 (2005), 5352--5356.
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    (A commentary on this paper by J. Lepowsky has also appeared in the same issue.)

43. Differential equations and intertwining operators, Comm. Contemp. Math. 7 (2005), 375--400.
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44. Differential equations, duality and modular invariance, Comm. Contemp. Math. 7 (2005), 649--706.
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45. On the concepts of intertwining operator and tensor product module in vertex operator algebra theory, with J. Lepowsky, H. Li and L. Zhang, Jour. Pure Appl. Alg. 204 (2005), 507--535.
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46. Vertex operator algebras, fusion rules and modular transformations, in: Non-commutative Geometry and Representation Theory in Mathematical Physics, ed. J. Fuchs, J. Mickelsson, G. Rozenblioum and A. Stolin, Contemporary Math. Vol. 391, Amer. Math. Soc., Providence, 2005, 135--148.
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47. A logarithmic generalization of tensor product theory for modules for a vertex operator algebra, with J. Lepowsky and L. Zhang, Internat. J. Math. 17 (2006), 975--1012.
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48. Full field algebras, with L. Kong, Comm. Math. Phys. 272 (2007), 345--396.
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49. An equivalence of two constructions of permutation-twisted modules for lattice vertex operator algebras, with K. Barron and J. Lepowsky, Jour. Pure Appl. Alg. 210 (2007), 797--826.
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50. Vertex operator algebras and the Verlinde conjecture, Comm. Contemp. Math. 10 (2008), 103--154.
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51. Rigidity and modularity of vertex tensor categories, Comm. Contemp. Math. 10 (2008), 871--911.
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52. Cofiniteness conditions, projective covers and the logarithmic tensor product theory, J. Pure Appl. Alg. 213 (2009), 458--475.
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53. Representations of vertex operator algebras and braided finite tensor categories, in: Vertex Operator Algebras and Related Topics, An International Conference in Honor of Geoffery Mason's 60th Birthday, ed. M. Bergvelt, G. Yamskulna and W. Zhao, Contemporary Math., Vol. 497, Amer. Math. Soc., Providence, 2009, 97--111.
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54. Modular invariance for conformal full field algebras, with L. Kong, Trans. Amer. Math. Soc. 362 (2010), 3027--3067.
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55. Generalized twisted modules associated to general automorphisms of a vertex operator algebra, Comm. Math. Phys. 298 (2010), 265--292.
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56. Logarithmic intertwining operators and associative algebras, with J. Yang, J. Pure Appl. Alg. 216 (2011), 1467--1492.
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    (The pdf file here is a minor revision of the published version. The statement of Lemma 5.5 in the published version is wrong because it is not compatible with the definitions of AN(W) and the right action of V on W in the original version. These definitions are modified. The proofs of Lemma 4.4, Lemma 4.6, Theorem 4.7 are also modified accordingly. Lemma 5.5 is correct with the modified definitions.)

57. Meromorphic open-string vertex algebras, J. Math. Phys. 54 (2013), 051702.
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58. Tensor categories and the mathematics of rational and logarithmic conformal field theory, with J. Lepowsky, J. Phys. A 46 (2013), 494009.
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59. Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules, with J, Lepowsky and L. Zhang, in: Conformal Field Theories and Tensor Categories, Proceedings of a Workshop Held at Beijing International Center for Mathematics Research, ed. C. Bai, J. Fuchs, Y.-Z. Huang, L. Kong, I. Runkel and C. Schweigert, Mathematical Lectures from Beijing University, Vol. 2, Springer, New York, 2014, 169--248.
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60. A cohomology theory of grading-restricted vertex algebras, Comm. Math. Phys. 327 (2014), 279-307.
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61. First and second cohomologies of grading-restricted vertex algebras, Comm. Math. Phys. 327 (2014), 261-278.
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    Addentum to the proof of Proposition 1.4 in this paper.
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62. On functors between module categories for associative algebras and for $\N$-graded vertex algebras, with J. Yang, J. Alg. 409 (2014), 344-361.
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63. Braided tensor categories and extensions of vertex operator algebras, with A. Kirillov, Jr. and J. Lepowsky, Comm. Math. Phys. 337 (2015), 1143-1159.
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64. Two constructions of grading-restricted vertex (super)algebras, J. Pure Appl. Alg. 220 (2016), 3628-3649.
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65. Some open probelms in mathematical two-dimensional conformal field theory, in: Proceedings of the Conference on Lie Algebras, Vertex Operator Algebras, and Related Topics, held at University of Notre Dame, Notre Dame, Indiana, August 14-18, 2015, ed. K. Barron, E. Jurisich, H. Li, A. Milas, K. C. Misra, Contemp. Math, Vol. 695, American Mathematical Society, Providence, RI, 2017, 123--138.
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66. Two-dimensional conformal field theory: Definition and the method of the representation theory of vertex operator algebras (in Chinese), in: Institute Talk 2015, ed. N. Xi, X. Zhang, B. Fu, Y. Wang, Science Press, Beijing, 2018, 54--91.
    pdf file in Chinese
    (This is a survey in Chinese. It will be translated into English and posted here and in the archive. The last section is based on the paper "Some open probelms in mathematical two-dimensional conformal field theory" above. The latter paper is in turn based on my slides prepared for the problem sessions of the conference.)

67. Intertwining operators among twisted modules associated to not-necessarily-commuting automorphisms, J. Alg. 493 (2018), 346--380.
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68. Braided tensor categories of admissible modules for affine Lie algebras, with T. Creutzig and J. Yang, Comm. Math. Phys. 362 (2018), 827--854.
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69. Associative algebras for (logarithmic) twisted modules for a vertex operator algebra, with J. Yang, Trans. Amer. Math. Soc., 371 (2019), 3747--3786.
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70. The first cohomology, derivations and the reductivity of a (meromorphic open-string) vertex algebra, with F. Qi, Trans. Amer. Math. Soc. 373 (2020), 7817--7868.
    pdf file

71. Affine Lie algebras and tensor categories, 14 pages, Proceedings of 10th CFT Seminar: A Conference on Vertex Algebras and Related Topics at RIMS, to appear.
    pdf file

72. Twist vertex operators for twisted modules, J. Alg., 539 (2019), 53--83.
    pdf file

73. Locally convex topological completions of modules for a vertex operator algebra, 3 pages, an extended abstract for the author's talk at the Oberwolfach Workshop ``Subfactors and Applications'' from October 27 to November 2, 2019.
    pdf file

74. A construction of lower-bounded generalized twisted modules for a grading-restricted vertex (super)algebra, 39 pages, Comm. Math. Phys. 377 (2020), 909-945.
    pdf file

75. Generators, spanning sets and existence of twisted modules for a grading-restricted vertex (super)algebra, 41 pages, Selecta Math. 26 (2020), Paper no. 62.
    pdf file

76. Lower-bounded and grading-restricted twisted modules for affine vertex (operator) algebras, 36 pages, J. Pure Appl. Alg.225 (2021), Paper no. 106618.
    pdf file

77. Representation theory of vertex operator algebras and orbifold conformal field theory, in: Lie groups, number theory, and vertex algebras, ed. by D. Adamovic, A. Dujella, A. Milas and P. Pandzic, Contemp. Math., Vol. 768, Amer. Math. Soc., Providence, RI, 2021, 221–252.
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78. Associative algebras and the representation theory of grading-restricted vertex algebras, 43 pages, Comm. Comtemp. Math., to appear.
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79. Corrigendum to "Logarithmic intertwining operators and associative algebras" [J. Pure Appl. Alg. 216 (2012), 1467--1492], J. Pure Appl. Alg. (2021), 2 pages.
    pdf file
    (See also the pdf file of the corrected version of the original paper.)

80. Associative algebras and intertwining operators, Comm. Math. Phys. 396 (2022), 1--44.
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81. Convergence in conformal field theory, Chinese Ann. Math. Ser. B, 27 pages, to appear.
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82. Weight-one elements of vertex operator algebras and automorphisms of categories of generalized twisted modules, with C. Sadowski, J. Alg. 628 (2023), 452--485.
    pdf file

83. Modular invariance of (logarithmic) intertwining operators, Comm. Math. Phys., 88 pages, to appear.
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84. Logarithmic tensor category theory, II: Logarithmic formal calculus and properties of logarithmic intertwining operators, with J, Lepowsky and L. Zhang, 40 pages, to appear.
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85. Logarithmic tensor category theory, III: Intertwining maps and tensor product bifunctors, with J, Lepowsky and L. Zhang, 36 pages, to appear.
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86. Logarithmic tensor category theory, IV: Constructions of tensor product bifunctors and the compatibility conditions, with J, Lepowsky and L. Zhang, 94 pages, to appear.
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87. Logarithmic tensor category theory, V: Convergence condition for intertwining maps and the corresponding compatibility condition, with J, Lepowsky and L. Zhang, 50 pages, to appear.
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88. Logarithmic tensor category theory, VI: Expansion condition, associativity of logarithmic intertwining operators, and the associativity isomorphisms, with J, Lepowsky and L. Zhang, 108 pages, to appear.
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89. Logarithmic tensor category theory, VII: Convergence and extension properties and applications to expansion for intertwining maps, with J, Lepowsky and L. Zhang, 20 pages, to appear.
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90. Logarithmic tensor category theory, VIII: Braided tensor category structure on categories of generalized modules for a conformal vertex algebra, with J, Lepowsky and L. Zhang, 36 pages, to appear.
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91. Meromorphic open-string vertex algebras and Riemannian manifolds, 23 pages, to appear.
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92. On the applicability of logarithmic tensor category theory, 9 pages, to appear.
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93. Lecture notes on vertex algebras and quantum vertex algebras, 96 pages on April 26, 2020, for my graduate course "Math 555: Selected Topics in Algebra: Vertex algebras and quantum vertex algebras," Spring, 2020. These notes will be updated frequently during the semester. They will be incorporated into my book-in-preparation "Two-dimensional conformal field theory."
    pdf file

94. Lecture notes on representation theory of vertex operator algebras and conformal field theory, 22 pages on March 29, 2020, for my graduate course "Math 557: Vertex operator algebra theory: Representation theory of vertex operator algebras and conformal field theory," Spring, 2022. These notes are still being updated every week during the semester. They will be incorporated into my book-in-preparation "Two-dimensional conformal field theory."
    pdf file

95. Lecture notes on vertex operator algebras and tensor categories, 46 pages on April 27, 2024, for my graduate course "Math 557: Vertex operator algebra theory: vertex operator algebras and tensor categories," Spring, 2024. These notes are not finished yet. Section 8 is still to be typed and more material will be added to Sections 6 and 7. They will be incorporated into my book-in-preparation "Two-dimensional conformal field theory."
    pdf file

96. Twisted intertwining operators and tensor products of (generalized) twisted modules, with J. Du, 65 pages, to appear.
    pdf file

97. Two-dimensional conformal field theory, a book in preparation.
    Tentative table of contents.
    Lecture notes on vertex algebras and quantum vertex algebras (for my graduate course "Math 555: Selected Topics in Algebra: Vertex algebras and quantum vertex algebras," Spring, 2020). These notes will be part of this book.

98. A theory of tensor products for module categories for a vertex operator algebra, V, with J. Lepowsky, 50 pages, to appear.

99. Fractional quantum Hall states, vertex operator superalgebras and topological orders, 45 pages, in preparation.

100. New examples of meromorphic open-string vertex algebras, in preparation.

101. A construction of conformal intertwining algebras, in preparation.

102. Intertwining algebras, in preparation.

Books Edited

1. Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory, Proceedings of an International Conference on Infinite-Dimensional Lie Theory and Conformal Field Theory, Charlottesville, 2000, with S. Berman, P. Fendley, K. Misra and B. Parshall, Contemporary Mathematics, Vol. 297, Amer. Math. Soc., Providence, 2002.

2. Vertex Operator Algebras in Mathematics and Physics, with S. Berman, Y. Billig and J. Lepowsky, Fields Institute Communications, Vol. 39, Amer. Math. Soc., Providence, 2003.

3. Lie Algebras, Vertex Operator Algebras and Their Applications, Proceedings of a Conference in Honor of James Lepowsky and Robert Wilson, 2005, with K. Misra, Contemporary Mathematics, Vol. 442, Amer. Math. Soc., Providence, 2007.

4. Conformal Field Theories and Tensor Categories, Proceedings of a Workshop Held at Beijing International Center for Mathematical Research, 2011, with C. Bai, J. Fuchs, L. Kong, I. Runkel and C. Schweigert, Mathematical Lectures from Peking University, Vol. 2, Springer, New York, 2014.

Notes of lectures and talks

• Moved to Notes of lectures and talks.