Higher Genus Characters for Vertex Operator Superalgebras on Sewn Riemann Surfaces

Author(s)
Alexander Zuevsky

ABSTRACT

We review our recent results on computation of the higher genus characters for vertex operator superalgebras modules. The vertex operator formal parameters are associated to local parameters on Riemann surfaces formed in one of two schemes of (self- or tori- ) sewing of lower genus Riemann surfaces. For the free fermion vertex operator superalgebra we present a closed formula for the genus two continuous orbifold partition functions (in either sewings) in terms of an infinite dimensional determinant with entries arising from the original torus Szeg? kernel. This partition function is holomorphic in the sewing parameters on a given suitable domain and possesses natural modular properties. Several higher genus generalizations of classical (including Fay’s and Jacobi triple product) identities show up in a natural way in the vertex operator algebra approach.

Cite this paper

A. Zuevsky, "Higher Genus Characters for Vertex Operator Superalgebras on Sewn Riemann Surfaces,"*Applied Mathematics*, Vol. 4 No. 10, 2013, pp. 33-52. doi: 10.4236/am.2013.410A3006.

A. Zuevsky, "Higher Genus Characters for Vertex Operator Superalgebras on Sewn Riemann Surfaces,"

References

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[2] M. P. Tuite and A. Zuevsky, “Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I,” Communications in Mathematical Physics, Vol. 306, No. 2, 2011, pp. 419-447. http://dx.doi.org/10.1007/s00220-011-1258-1

[3] M. P. Tuite and A. Zuevsky, “A Generalized Vertex Operator Algebra for Heisenberg Intertwiners,” Journal of Pure and Applied Algebra, Vol. 216, No. 6, 2012, pp. 1253-1492.

http://dx.doi.org/10.1016/j.jpaa.2011.10.025

[4] M. P. Tuite and A. Zuevsky, “Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras II,” to Appear, 2013.

[5] M. P. Tuite and A. Zuevsky, “The Bosonic Vertex Operator Algebra on a Genus g Riemann Surface,” RIMS Kokyuroko, Vol. 1756, No. 9, 2011, pp. 81-93.

[6] R. E. Borcherds, “Vertex Algebras, Kac-Moody Algebras and the Monster,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 83, No. 10, 1986, pp. 3068-3071. http://dx.doi.org/10.1073/pnas.83.10.3068

[7] C. Dong and J. Lepowsky, “Generalized Vertex Algebras and Relative Vertex Operators,” Birkhauser, Boston, 1993. http://dx.doi.org/10.1007/978-1-4612-0353-7

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[11] C. Dong, H. Li and G. Mason, “Twisted Representation of Vertex Operator Algebras,” Mathematische Annalen, Vol. 310, No. 3, 1998, pp. 571-600. http://dx.doi.org/10.1007/s002080050161

[12] C. Dong, H. Li and G. Mason, “Simple Currents and Extensions of Vertex Operator Algebras,” Communications in Mathematical Physics, Vol. 180, No. 3, 1996, pp. 671707.

http://dx.doi.org/10.1007/BF02099628

[13] H. Li, “Symmetric Invariant Bilinear Forms on Vertex Operator Algebras,” Journal of Pure and Applied Algebra, Vol. 96, No. 3, 1994, pp. 279-297. http://dx.doi.org/10.1016/0022-4049(94)90104-X

[14] N. Scheithauer, “Vertex Algebras, Lie Algebras and Superstrings,” Journal of Algebra, Vol. 200, No. 2, 1998, pp. 363-403. http://dx.doi.org/10.1006/jabr.1997.7235

[15] H. M. Farkas and I. Kra, “Theta Constants, Riemann Surfaces and the Modular Group,” Graduate Studies in Mathematics, AMS, Providence, 2001.

[16] R. C. Gunning, “Lectures on Riemann Surfaces,” Princeton University Press, Princeton, 1966.

[17] A. Yamada, “Precise Variational Formulas for Abelian Differentials,” Kodai Mathematical Journal, Vol. 3, No. 1, 1980, pp. 114-143. http://dx.doi.org/10.2996/kmj/1138036124

[18] D. Mumford, “Tata Lectures on Theta I and II,’’ Birkhauser, Boston, 1983.

[19] G. Mason, M. P. Tuite and A. Zuevsky, “Torus N-Point Functions for -Graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds,” Communications in Mathematical Physics, Vol. 283, No. 2, 2008, pp. 305342. http://dx.doi.org/10.1007/s00220-008-0510-9

[20] C. Dong, H. Li and G. Mason, “Modular-Invariance of Trace Functions in Orbifold Theory and Generalized Moonshine,” Communications in Mathematical Physics, Vol. 214, No. 1, 2000, pp. 1-56. http://dx.doi.org/10.1007/s002200000242

[21] J. D. Fay, “Theta Functions on Riemann Surfaces,” Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1973.

[22] J. D. Fay, “Kernel Functions, Analytic Torsion and Moduli Spaces,” American Mathematical Society, Providence, Rhode Island, 1992.

[23] G. Mason and M. P. Tuite, “On Genus Two Riemann Surfaces Formed from Sewn Tori,” Communications in Mathematical Physics, Vol. 270, No. 3, 2007, pp. 587634.

http://dx.doi.org/10.1007/s00220-006-0163-5

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[25] A. Matsuo and K. Nagatomo, “Axioms for a Vertex Algebra and the Locality of Quantum Fields,” Mathematical Society of Japan, Hongo, Bunkyo-ku, Tokio, 1999.

[26] G. Mason and M. P. Tuite, “Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I”, Communications in Mathematical Physics, Vol. 300, No. 3, 2010, pp. 673-713.

http://dx.doi.org/10.1007/s00220-010-1126-4

[27] G. Mason and M. P. Tuite, “Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces II,” arXiv:1111.2264v1.

[28] G. Mason and M. P. Tuite, “Chiral N-Point Functions for Free Boson and Lattice Vertex Operator Algebras,” Communications in Mathematical Physics, Vol. 235, No. 1, 2003, pp. 47-68.

http://dx.doi.org/10.1007/s00220-002-0772-6

[29] Y. Zhu, “Modular Invariance of Characters of Vertex Operator Algebras,” Journal of the American Mathematical Society, Vol. 9, 1996, pp. 237-302. http://dx.doi.org/10.1090/S0894-0347-96-00182-8

[30] P. di Vecchia, K. Hornfeck, M. Frau, A. Lerda and S. Sciuto, “N-String, G-Loop Vertex for the Fermionic String,” Physics Letter B, Vol. 211, No. 3, 1988, pp. 301307.

http://dx.doi.org/10.1016/0370-2693(88)90907-0

[31] T. Eguchi and H. Ooguri, “Chiral Bosonization on a Riemann Surface,” Physics Letter B, Vol. 187, No. 1-2, 1987, pp. 127-134. http://dx.doi.org/10.1016/0370-2693(87)90084-0

[32] D. Freidan and S. Shenker, “The Analytic Geometry of Two Dimensional Conformal Field Theory,” Nuclear Physics B, Vol. 281, No. 3-4, 1987, pp. 509-545.

http://dx.doi.org/10.1016/0550-3213(87)90418-4

[33] M. R. Gaberdiel, Ch. A. Keller and R. Volpato, “Genus Two Partition Functions of Chiral Conformal Field Theories,” arXiv:1002.3371, 2010.

[34] M. R. Gaberdiel and R. Volpato, ‘‘Higher Genus Partition Functions of Meromorphic Conformal Field Theories,” Journal of High Energy Physics, Vol. 9, No. 6, 2009, p. 48.

[35] N. Kawamoto, Y. Namikawa, A. Tsuchiya and Y. Yamada, “Geometric Realization of Conformal Field Theory on Riemann Surfaces,” Communications in Mathematical Physics, Vol. 116, No. 2, 1988, pp. 247-308.

[36] F. Pezzella, “g-Loop Vertices for Free Fermions and Bosons,” Physics Letter B, Vol. 220, No. 4, 1989, pp. 544550. http://dx.doi.org/10.1016/0370-2693(89)90784-3

[37] A. K. Raina, “Fay’s Trisecant Identity and Conformal Field Theory,” Communications in Mathematical Physics, Vol. 122, No. 4, 1989, pp. 625-641. http://dx.doi.org/10.1007/BF01256498

[38] A. Tsuchiya, K. Ueno and Y. Yamada, “Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries,” Academic Press, Boston, 1989.

[39] K. Ueno, “Introduction to Conformal Field Theory with Gauge Symmetries,” Geometry and Physics, Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 1997, pp. 603-745.

[40] G. Mason and M. P. Tuite, “Vertex Operators and Modular Forms,” In: K. Kirsten and F. Williams, Eds., A Window into Zeta and Modular Physics, Cambridge University Press, Cambridge, 2010, pp. 183-278.

[1] M. P. Tuite and A. Zuevsky, “The Szego Kernel on a Sewn Riemann Surface,” Communications in Mathematical Physics, Vol. 306, No. 3, 2011, pp. 617-645.

[2] M. P. Tuite and A. Zuevsky, “Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I,” Communications in Mathematical Physics, Vol. 306, No. 2, 2011, pp. 419-447. http://dx.doi.org/10.1007/s00220-011-1258-1

[3] M. P. Tuite and A. Zuevsky, “A Generalized Vertex Operator Algebra for Heisenberg Intertwiners,” Journal of Pure and Applied Algebra, Vol. 216, No. 6, 2012, pp. 1253-1492.

http://dx.doi.org/10.1016/j.jpaa.2011.10.025

[4] M. P. Tuite and A. Zuevsky, “Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras II,” to Appear, 2013.

[5] M. P. Tuite and A. Zuevsky, “The Bosonic Vertex Operator Algebra on a Genus g Riemann Surface,” RIMS Kokyuroko, Vol. 1756, No. 9, 2011, pp. 81-93.

[6] R. E. Borcherds, “Vertex Algebras, Kac-Moody Algebras and the Monster,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 83, No. 10, 1986, pp. 3068-3071. http://dx.doi.org/10.1073/pnas.83.10.3068

[7] C. Dong and J. Lepowsky, “Generalized Vertex Algebras and Relative Vertex Operators,” Birkhauser, Boston, 1993. http://dx.doi.org/10.1007/978-1-4612-0353-7

[8] I. Frenkel, Y. Huang and J. Lepowsky, “On Axiomatic Approaches to Vertex Operator Algebras and Modules,” American Mathematical Society, Providence, Rhode Island, 1993.

[9] I. Frenkel, J. Lepowsky and A. Meurman, “Vertex Operator Algebras and the Monster,” Academic Press, New York, 1988.

[10] V. Kac, “Vertex Operator Algebras for Beginners,” University Lecture Series, AMS, Providence, 1998.

[11] C. Dong, H. Li and G. Mason, “Twisted Representation of Vertex Operator Algebras,” Mathematische Annalen, Vol. 310, No. 3, 1998, pp. 571-600. http://dx.doi.org/10.1007/s002080050161

[12] C. Dong, H. Li and G. Mason, “Simple Currents and Extensions of Vertex Operator Algebras,” Communications in Mathematical Physics, Vol. 180, No. 3, 1996, pp. 671707.

http://dx.doi.org/10.1007/BF02099628

[13] H. Li, “Symmetric Invariant Bilinear Forms on Vertex Operator Algebras,” Journal of Pure and Applied Algebra, Vol. 96, No. 3, 1994, pp. 279-297. http://dx.doi.org/10.1016/0022-4049(94)90104-X

[14] N. Scheithauer, “Vertex Algebras, Lie Algebras and Superstrings,” Journal of Algebra, Vol. 200, No. 2, 1998, pp. 363-403. http://dx.doi.org/10.1006/jabr.1997.7235

[15] H. M. Farkas and I. Kra, “Theta Constants, Riemann Surfaces and the Modular Group,” Graduate Studies in Mathematics, AMS, Providence, 2001.

[16] R. C. Gunning, “Lectures on Riemann Surfaces,” Princeton University Press, Princeton, 1966.

[17] A. Yamada, “Precise Variational Formulas for Abelian Differentials,” Kodai Mathematical Journal, Vol. 3, No. 1, 1980, pp. 114-143. http://dx.doi.org/10.2996/kmj/1138036124

[18] D. Mumford, “Tata Lectures on Theta I and II,’’ Birkhauser, Boston, 1983.

[19] G. Mason, M. P. Tuite and A. Zuevsky, “Torus N-Point Functions for -Graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds,” Communications in Mathematical Physics, Vol. 283, No. 2, 2008, pp. 305342. http://dx.doi.org/10.1007/s00220-008-0510-9

[20] C. Dong, H. Li and G. Mason, “Modular-Invariance of Trace Functions in Orbifold Theory and Generalized Moonshine,” Communications in Mathematical Physics, Vol. 214, No. 1, 2000, pp. 1-56. http://dx.doi.org/10.1007/s002200000242

[21] J. D. Fay, “Theta Functions on Riemann Surfaces,” Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1973.

[22] J. D. Fay, “Kernel Functions, Analytic Torsion and Moduli Spaces,” American Mathematical Society, Providence, Rhode Island, 1992.

[23] G. Mason and M. P. Tuite, “On Genus Two Riemann Surfaces Formed from Sewn Tori,” Communications in Mathematical Physics, Vol. 270, No. 3, 2007, pp. 587634.

http://dx.doi.org/10.1007/s00220-006-0163-5

[24] Y. Huang, “Two-Dimensional Conformal Geometry and Vertex Operator Algebras,” Birkhauser, Boston, 1997.

[25] A. Matsuo and K. Nagatomo, “Axioms for a Vertex Algebra and the Locality of Quantum Fields,” Mathematical Society of Japan, Hongo, Bunkyo-ku, Tokio, 1999.

[26] G. Mason and M. P. Tuite, “Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I”, Communications in Mathematical Physics, Vol. 300, No. 3, 2010, pp. 673-713.

http://dx.doi.org/10.1007/s00220-010-1126-4

[27] G. Mason and M. P. Tuite, “Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces II,” arXiv:1111.2264v1.

[28] G. Mason and M. P. Tuite, “Chiral N-Point Functions for Free Boson and Lattice Vertex Operator Algebras,” Communications in Mathematical Physics, Vol. 235, No. 1, 2003, pp. 47-68.

http://dx.doi.org/10.1007/s00220-002-0772-6

[29] Y. Zhu, “Modular Invariance of Characters of Vertex Operator Algebras,” Journal of the American Mathematical Society, Vol. 9, 1996, pp. 237-302. http://dx.doi.org/10.1090/S0894-0347-96-00182-8

[30] P. di Vecchia, K. Hornfeck, M. Frau, A. Lerda and S. Sciuto, “N-String, G-Loop Vertex for the Fermionic String,” Physics Letter B, Vol. 211, No. 3, 1988, pp. 301307.

http://dx.doi.org/10.1016/0370-2693(88)90907-0

[31] T. Eguchi and H. Ooguri, “Chiral Bosonization on a Riemann Surface,” Physics Letter B, Vol. 187, No. 1-2, 1987, pp. 127-134. http://dx.doi.org/10.1016/0370-2693(87)90084-0

[32] D. Freidan and S. Shenker, “The Analytic Geometry of Two Dimensional Conformal Field Theory,” Nuclear Physics B, Vol. 281, No. 3-4, 1987, pp. 509-545.

http://dx.doi.org/10.1016/0550-3213(87)90418-4

[33] M. R. Gaberdiel, Ch. A. Keller and R. Volpato, “Genus Two Partition Functions of Chiral Conformal Field Theories,” arXiv:1002.3371, 2010.

[34] M. R. Gaberdiel and R. Volpato, ‘‘Higher Genus Partition Functions of Meromorphic Conformal Field Theories,” Journal of High Energy Physics, Vol. 9, No. 6, 2009, p. 48.

[35] N. Kawamoto, Y. Namikawa, A. Tsuchiya and Y. Yamada, “Geometric Realization of Conformal Field Theory on Riemann Surfaces,” Communications in Mathematical Physics, Vol. 116, No. 2, 1988, pp. 247-308.

[36] F. Pezzella, “g-Loop Vertices for Free Fermions and Bosons,” Physics Letter B, Vol. 220, No. 4, 1989, pp. 544550. http://dx.doi.org/10.1016/0370-2693(89)90784-3

[37] A. K. Raina, “Fay’s Trisecant Identity and Conformal Field Theory,” Communications in Mathematical Physics, Vol. 122, No. 4, 1989, pp. 625-641. http://dx.doi.org/10.1007/BF01256498

[38] A. Tsuchiya, K. Ueno and Y. Yamada, “Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries,” Academic Press, Boston, 1989.

[39] K. Ueno, “Introduction to Conformal Field Theory with Gauge Symmetries,” Geometry and Physics, Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 1997, pp. 603-745.

[40] G. Mason and M. P. Tuite, “Vertex Operators and Modular Forms,” In: K. Kirsten and F. Williams, Eds., A Window into Zeta and Modular Physics, Cambridge University Press, Cambridge, 2010, pp. 183-278.