Abstract
We consider genus one n-point functions for a vertex operator superalgebra with a real grading. We compute all n-point functions for rank one and rank two fermion vertex operator superalgebras. In the rank two fermion case, we obtain all orbifold n-point functions for a twisted module associated with a continuous automorphism generated by a Heisenberg bosonic state. The modular properties of these orbifold n-point functions are given and we describe a generalization of Fay’s trisecant identity for elliptic functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Borcherds R.: Vertex algebras, Kac-Moody algebras and the Monster. Proc. Natl. Acad. Sci. U.S.A. 83, 3068–3071 (1986)
Dong C., Li H., Mason G.: Modular-invariance of trace functions in orbifold theory and generalized moonshine. Commun. Math. Phys. 214, 1–56 (2000)
Dong, C., Lin, Z., Mason, G.: On Vertex operator algebras as sl 2-modules. Arasu, K. T. (ed.) et al., In: Groups, difference sets, and the Monster. Proceedings of a special research quarter, Columbus, OH, USA, Spring 1993. Ohio State Univ. Math. Res. Inst. Publ. 4, Berlin: Walter de Gruyter, pp. 349–362 (1996)
Dong C., Mason G.: Shifted Vertex Operator Algebras. Math. Proc. Cambridge Philos. Soc. 141, 67–80 (2006)
Dong C., Zhao Z.: Modularity in orbifold theory for vertex operator superalgebras. Commun. Math. Phys. 260, 227–256 (2005)
Dong, C., Zhao, Z.: Modularity of trace functions in orbifold theory for \({\mathbb{Z}}\) -graded vertex operator superalgebras. http://arxiv.org/list/math.QA/0601571, 2006
Eguchi T., Ooguri H.: Chiral bosonization on a Riemann surface. Phys. Lett. B187, 127–134 (1987)
Fay, J.D.: Theta functions on Riemann surfaces. Lecture Notes on Mathematics 352, New York:Springer-Verlag, 1973
Feingold, A.J., Frenkel, I.B., Reis, J.F.X.: Spinor construction of vertex operator algebras and \({E_{8}^{(1)}}\). Contemp. Math. 121, Providence, RI: Amer. Math. Soc., 1991
Frenkel, I., Huang, Y-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104(494), Providence, RI: Amer. Math. Soc., 1993
Farkas, H., Kra, I.M.: Theta constants, Riemann surfaces and the modular group. Graduate Studies in Mathematics, 37. Providence, RI: Ameri. Math. Soc., 2001
Frenkel I., Lepowsky J., Meurman A.: Vertex operator algebras and the Monster. Academic Press, New York (1988)
Di Francesco Ph., Mathieu P., Sénéchal D.: Conformal field theory Graduate Texts in Contemporary Physics. Springer-Verlag, New York (1997)
Kac, V.: Vertex Operator Algebras for Beginners. University Lecture Series, Vol. 10, Providence, RI: Ameri. Math. Soc., 1998
Kaneko, M., Zagier, D.: A generalized Jacobi theta function and quasimodular forms, The Moduli Space of Curves (Texel Island, 1994), Progr. in Math. 129, Boston: Birkhauser, 1995
Lang, S.: Elliptic functions. With an appendix by J. Tate. Second edition. Graduate Texts in Mathematics, 112. New York: Springer-Verlag, 1987
Li, H.: Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules, Contemp. Math. AMS. 193, Providence, RI: Ameri. Math. Soc., 1996, pp. 203–236
Matsuo, A., Nagatomo, K.: Axioms for a Vertex Algebra and the locality of quantum fields. Math. Soc. Japan Memoirs 4, Tokyo: Math. Soc. of Japan, 1999
Matsuo A., Nagatomo K.: A note on free bosonic vertex algebras and its conformal vector. J. Alg. 212, 395–418 (1999)
Mason G., Tuite M.P.: Torus chiral n-point functions for free boson and lattice vertex operator algebras. Commun. Math. Phys. 235, 47–68 (2003)
Mason G., Tuite M.P.: On genus two Riemann surfaces formed from sewn tori. Commun. Math. Phys. 270, 587–648 (2007)
Mason, G., Tuite, M.P.: The genus two partition function for free bosonic and lattice vertex operator algebras. http://arxiv.org/list/math.QA/07120628, 2007
Mumford D.: Tata lectures on Theta I. Birkhäuser, Boston (1983)
Polchinski J.: String Theory, Volumes I and II. Cambridge University Press, Cambridge (1998)
Raina A. K.: Fay’s trisecant identity and conformal field theory. Commun. Math. Phys. 122, 625–641 (1989)
Raina A.K.: An algebraic geometry study of the b-c system with arbitrary twist fields and arbitrary statistics. Commun. Math. Phys. 140, 373–397 (1991)
Sen S., Raina A.K.: Grassmannians, multiplicative Ward identities and theta function identities. Phys. Lett. B 203(3), 256–262 (1988)
Serre J-P.: A course in arithmetic. Springer-Verlag, Berlin (1978)
Tuite M.P.: Genus two meromorphic conformal field theory. CRM Proceedings and Lecture Notes 30, 231–251 (2001)
Tsuchiya A., Ueno K., Yamada Y.: Conformal field theory on universal family of stable curves with gauge symmetries. Adv. Stud. Pure. Math. 19, 459–566 (1989)
Tuite, M.P., Zuevsky, A.: Shifting, twisting and intertwining Heisenberg modules. To appear
Zhu Y: Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9, 237–302 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Partial support provided by NSF, NSA and the Committee on Research, University of California, Santa Cruz.
Supported by a Science Foundation Ireland Frontiers of Research Grant, and by Max-Planck Institut für Mathematik, Bonn.
Rights and permissions
About this article
Cite this article
Mason, G., Tuite, M.P. & Zuevsky, A. Torus n-Point Functions for \({\mathbb{R}}\) -graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds. Commun. Math. Phys. 283, 305–342 (2008). https://doi.org/10.1007/s00220-008-0510-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-008-0510-9