×

zbMATH — the first resource for mathematics

Free bosonic vertex operator algebras on genus two Riemann surfaces. I. (English) Zbl 1226.17024
One feature of chiral conformal field theory is the natural occurrence of elliptic functions and modular forms in the \(n\)-point correlation functions. In string theory their occurrence has been known for a long time and has inspired mathematicians and physicists alike. In the present paper the authors define the partition function and the \(n\)-point functions for a vertex operator algebra (VOA) on a genus-2 Riemann surface obtained by sewing two genus-1 tori together. They compute closed formulas for the partition function for the rank one Heisenberg VOA and for any pair of simple Heisenberg modules. They prove that the partition function is holomorphic in the sewing parameters within some complex domain and describe its modular properties for the Heisenberg VOA and the lattice VOA. They also describe a continuous orbifolding of the fermion vertex operator super algebra of rank two, compute the \(n\)-point function, and show that the Virasoro vector 1-point function satisfies a genus-2 Ward identity.

MSC:
17B69 Vertex operators; vertex operator algebras and related structures
81T10 Model quantum field theories
81T45 Topological field theories in quantum mechanics
58D99 Spaces and manifolds of mappings (including nonlinear versions of 46Exx)
57R56 Topological quantum field theories (aspects of differential topology)
53Z05 Applications of differential geometry to physics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Borcherds R.E.: Vertex algebras, Kac-Moody algebras and the Monster. Proc. Nat. Acad. Sc. 83, 3068–3071 (1986) · Zbl 0613.17012
[2] Borcherds R.E.: Monstrous moonshine and monstrous Lie superalgebras. Inv. Math. 109, 405–444 (1992) · Zbl 0799.17014
[3] Belavin A., Knizhnik V.: Algebraic geometry and the geometry of strings. Phys. Lett. 168B, 201–206 (1986) · Zbl 0693.58043
[4] Belavin A., Polyakov A., Zamolodchikov A.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B241, 333–380 (1984) · Zbl 0661.17013
[5] Conway J.H., Norton S.P.: Monstrous moonshine. Bull. Lond. Math. Soc. 12, 308–339 (1979) · Zbl 0424.20010
[6] Di Francesco P., Mathieu P., Senechal D.: Conformal Field Theory. Springer-Verlag, New York (1997)
[7] Dolan L., Goddard P., Montague P.: Conformal field theories, representations and lattice constructions. Commun. Math. Phys. 179, 61–120 (1996) · Zbl 0878.17025
[8] Dong C., Mason G.: Shifted vertex operator algebras. Proc. Camb. Phil. Math. Soc. 141, 67–80 (2006) · Zbl 1141.17020
[9] D’Hoker E., Phong D.H.: The geometry of string perturbation theory. Rev. Mod. Phys. 60, 917–1065 (1988)
[10] Eguchi T., Ooguri H.: Conformal and current algebras on a general Riemann surface. Nucl. Phys. B 282, 308–328 (1987)
[11] Fay, J.: Theta functions on Riemann surfaces, Lecture Notes in Mathematics 352, Berlin-New York: Springer-Verlag, 1973 · Zbl 0281.30013
[12] Frenkel, I., Huang, Y., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104 (1993) · Zbl 0789.17022
[13] Farkas H.M., Kra I.: Riemann surfaces. Springer-Verlag, New York (1980) · Zbl 0475.30001
[14] Frenkel I., Lepowsky J., Meurman A.: Vertex operator algebras and the Monster. Academic Press, New York (1988) · Zbl 0674.17001
[15] Freitag E.: Siegelische modulfunktionen. Springer-Verlag, Berlin and New York (1983) · Zbl 0498.10016
[16] Freidan D., Shenker S.: The analytic geometry of two dimensional conformal field theory. Nucl. Phys. B281, 509–545 (1987)
[17] Gaberdiel M., Goddard P.: Axiomatic conformal field theory. Commun. Math. Phys. 209, 549–594 (2000) · Zbl 0976.81091
[18] Green M., Schwartz J., Witten E.: Superstring theory Vol. 1. Cambridge University Press, Cambridge (1987)
[19] Gunning R.C.: Lectures on Riemann surfaces. Princeton Univ. Press, Princeton, NJ (1966) · Zbl 0175.36801
[20] Kac, V.: Vertex operator algebras for beginners. University Lecture Series, Vol. 10, Providence, RI: Amer. Math. Soc., 1998 · Zbl 0924.17023
[21] Knizhnik V.G.: Multiloop amplitudes in the theory of quantum strings and complex geometry. Sov. Phys. Usp. 32, 945–971 (1989)
[22] Kawamoto N., Namikawa Y., Tsuchiya A., Yamada Y.: Geometric realization of conformal field theory on Riemann surfaces. Commun. Math. Phys. 116, 247–308 (1988) · Zbl 0648.35080
[23] Kaneko, M., Zagier, D.: A generalized Jacobi theta function and quasimodular forms. In: The Moduli Space of Curves (Texel Island, 1994), Progr. in Math. 129, Boston: Birkhauser, 1995
[24] Li H.: Symmetric invariant bilinear forms on vertex operator algebras. J. Pure. Appl. Alg. 96, 279–297 (1994) · Zbl 0813.17020
[25] Lepowsky J., Li H.: Introduction to vertex operator algebras and their representations. Birkhäuser, Boston (2004) · Zbl 1055.17001
[26] Matsuo, A., Nagatomo, K,: Axioms for a vertex algebra and the locality of quantum fields. Math. Soc. Jap. Mem. 4 (1999) · Zbl 0928.17025
[27] 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) · Zbl 1020.17020
[28] Mason G., Tuite M.P.: On genus two Riemann surfaces formed from sewn tori. Commun. Math. Phys. 270, 587–634 (2007) · Zbl 1116.30027
[29] Mason, G., Tuite, M.P.: Partition functions and chiral algebras. In: Lie algebras, vertex operator algebras and their applications (in honor of Jim Lepowsky and Robert L. Wilson), Contemporary Mathematics 442, Providence, RI: Amer. Math. Soc., 2007, pp. 401–410 · Zbl 1141.82003
[30] Mason, G., Tuite, M.P.: Free bosonic vertex operator algebras on genus two Riemann surfaces II. to appear · Zbl 1226.17024
[31] Mason, G., Tuite, M.P.: In preparation
[32] Mason, G., Tuite, M.P.: Vertex operators and modular forms. In: Kirsten, K., Williams, F. (eds.) A Window into Zeta and Modular Physics, vol. 57, pp. 183–278. MSRI Publications, Cambridge University Press, Cambridge (2010) · Zbl 1231.17021
[33] 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) · Zbl 1211.17026
[34] Mumford D.: Tata lectures on Theta I and II. Birkhäuser, Boston (1983) · Zbl 0509.14049
[35] Polchinski J.: String theory. Volume I. Cambridge University Press, Cambridge (1998) · Zbl 1006.81521
[36] Serre J-P.: A course in arithmetic. Springer-Verlag, Berlin (1978)
[37] Sonoda H.: Sewing conformal field theories I. Nucl. Phys. B311, 401–416 (1988)
[38] Sonoda H.: Sewing conformal field theories II. Nucl. Phys. B311, 417–432 (1988)
[39] Tuite, M.P.: Genus two meromorphic conformal field theory. CRM Proceedings and Lecture Notes 30, Providence, RI: Amer. Math. Soc., 2001, pp. 231–251 · Zbl 1028.81051
[40] 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) · Zbl 0696.17010
[41] Tuite, M.P., Zuevsky, A.: Genus two partition and correlation functions for fermionic vertex operator superalgebras I, arXiv:1007.5203 · Zbl 1254.17024
[42] Ueno, K.: Introduction to conformal field theory with gauge symmetries. In: Geometry and Physics - Proceedings of the conference at Aarhus Univeristy, Aaarhus, Denmark, New York: Marcel Dekker, 1997 · Zbl 0873.32022
[43] Yamada A.: Precise variational formulas for abelian differentials. Kodai Math. J. 3, 114–143 (1980) · Zbl 0451.30039
[44] Zhu Y.: Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9, 237–302 (1996) · Zbl 0854.17034
[45] Zhu Y.: Global vertex operators on Riemann surfaces. Commun. Math. Phys. 165, 485–531 (1994) · Zbl 0819.17019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.