Orbifolding Frobenius algebras. (English) Zbl 1083.57037

It is well-known that \(1+1\)-dimensional topological field theories are parametrized by commutative Frobenius algebras [see M. Atiyah, Publ. Math., Inst. Hautes Étud. Sci. 68, 175–186 (1988; Zbl 0692.53053), R. Dijkgraaf, PhD Thesis, Utrecht (1989), B. Dubrovin, Lect. Notes Math. 1620, 120–348 (1996; Zbl 0841.58065)]. The purpose of this paper is to extend this result to orbifolds. More precisely, let \(G\) be a finite group. First, the author introduces the notion of \(G\)-Frobenius algebras together with graded and super versions. Essentially a \(G\)-Frobenius algebra (with respect to a character \(\chi\) of \(G\)) is an algebra \(A\) in the category of Yetter-Drinfeld modules over the group algebra of \(G\) together with an associative bilinear form on \(A\) (so that \(A\) is a Frobenius algebra), subject to a number of axioms, including braided commutativity and twisted invariance of the bilinear form with respect to \(\chi\). Then the author proves that \(G\)-Frobenius algebras classify functors from some geometric cobordism theories to the category of vector spaces over the base field. Finally particular families of \(G\)-Frobenius algebras are studied in detail; for instance, special \(G\)-Frobenius algebras – those with \(\dim A_g = 1\) for all \(g\in G\). These are in particular strongly \(G\)-graded algebras. The author provides an explicit construction of special \(G\)-Frobenius algebras in terms of 2-cocycles, generalizing the well-known description of strongly graded algebras.


57R56 Topological quantum field theories (aspects of differential topology)
14N99 Projective and enumerative algebraic geometry
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
58D29 Moduli problems for topological structures
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16W50 Graded rings and modules (associative rings and algebras)
16L60 Quasi-Frobenius rings
Full Text: DOI arXiv


[1] DOI: 10.1007/BF02698547 · Zbl 0692.53053
[2] DOI: 10.1007/BFb0094793
[3] DOI: 10.1007/BF01238812 · Zbl 0674.46051
[4] DOI: 10.1007/BF02096988 · Zbl 0703.58011
[5] DOI: 10.1007/BF02096860 · Zbl 0788.58013
[6] DOI: 10.1016/0550-3213(90)90622-K
[7] DOI: 10.1016/0040-9383(80)90009-9 · Zbl 0447.57005
[8] DOI: 10.1016/0550-3213(90)90535-L
[9] Jarvis T., Compositio Math. 126
[10] DOI: 10.1142/S0129167X99000070 · Zbl 0982.53072
[11] Steenrod N., Princeton Mathematical Series 14, in: The Topology of Fibre Bundles (1951) · Zbl 0054.07103
[12] DOI: 10.1142/S0217732389001350
[13] DOI: 10.1112/blms/12.3.169 · Zbl 0427.32010
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.