Abstract
We show that given a Frobenius algebra there is a unique notion of its second quantization, which is the sum over all symmetric group quotients of nth tensor powers, where the quotients are given by symmetric group twisted Frobenius algebras. To this end, we consider the setting of Frobenius algebras given by functors from geometric categories whose objects are endowed with geometric group actions and prove structural results, which in turn yield a constructive realization in the case of nth tensor powers and the natural permutation action. We also show that naturally graded symmetric group twisted Frobenius algebras have a unique algebra structure already determined by their underlying additive data together with a choice of super–grading. Furthermore we discuss several notions of discrete torsion and show that indeed a non–trivial discrete torsion leads to a non–trivial super structure on the second quantization.
Similar content being viewed by others
References
Atiyah, M., Segal, G.: On equivariant Euler characteristics. J. Geom. Phys. 6, 671–677 (1989)
Adem, A., Ruan, Y.: Twisted Orbifold K-Theory. Commun. Math. Phys. 237, 533–556 (2003)
Batyrev, V., Borisov, L.: Mirror duality and string-theoretic Hodge numbers. Invent. Math. 126(1), 183–203 (1996)
Chen, W., Ruan, Y.: A New Cohomology Theory for Orbifold. Preprint, math.AG/0004129 and Orbifold Quantum Cohomology. Preprint, math.AG/0005198
Dijkgraaf, R.: Fields, strings, matrices and symmetric products. In: Moduli of curves and abelian varieties, Aspects Math., E33, Braunschweig: Vieweg, 1999, pp. 151–199
Dijkgraaf, R.: Discrete Torsion and Symmetric Products. Preprint, hep-th/9912101
Dijkgraaf, R., Moore, G., Verlinde, E., Verlinde, H.: Elliptic Genera of Symmetric Products and Second Quantized Strings. Commun. Math. Phys. 185, 197–209 (1997)
Fantechi, B., Goettsche, L.: Orbifold cohomology for global quotients. Duke Math. J. 117, 197–227 (2003)
Jarvis, T., Kaufmann, R., Kimura, T.: Pointed admissible G-covers and G-cohomologic field theories, Preprint MPI 2003-51, IHES M/03/22, math. AG/0302316
Kaufmann, R.: The tensor Product in the Theory of Frobenius manifolds. Int. J. Math. 10, 159–206 (1999)
Kaufmann, R.: Orbifolding Frobenius algebras. Talk at WAGP2000 conference at SISSA Trieste, October 2000
Kaufmann, R.: Orbifolding Frobenius algebras. Internat. J. of Math. 14, 573–619 (2003)
Kaufmann, R.M.: The algebra of discrete torsion. Preprint, MPI 2002-112, math.AG/0208081, p 23
Kaufmann, R.: Discrete torsion, symmetric products and the Hilbert scheme. Preprint,. 2002. To appear In: Proceedings of the conference in honor of Yuri Ivanovich Manin’s 65th birthday.
Karpilovsky, G.: The Schur multiplier. Oxford New York: Clarendon Press, Oxford University Press, 1987
Lehn, M., Sorger, C.: The cup product of the Hilbert scheme for K3 surfaces. Invent. Math. 152, 305–329 (2003)
Qin, Z., Wang, W.: Hilbert schemes and symmetric products: a dictionary. In: Orbifolds in Mathematics and Physics, Contemp. Math. (310). Providence, RI: Amer. Math. Soc., 2002, pp. 233–257
Satake, I.: The Gauss-Bonnet theorem for V-manifolds. J. Math. Soc. Japan 9, 464–492 (1957)
Toen, B.: Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford. K-Theory 18, 33–76 (1999)
Uribe, B.: Orbifold Cohomology of the Symmetric Product. Preprint, math.AT/0109125
Weiqiang Wang, W., Zhou, J.: Orbifold Hodge numbers of the wreath product orbifolds. J. Geom. Phys. 38, 152–169 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
This work was partially supported by NSF grant #0070681.
Rights and permissions
About this article
Cite this article
Kaufmann, R. Second Quantized Frobenius Algebras. Commun. Math. Phys. 248, 33–83 (2004). https://doi.org/10.1007/s00220-004-1090-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-004-1090-y