Modular Invariance for Twisted Modules over a Vertex Operator Superalgebra

Abstract

The purpose of this paper is to generalize Zhu's theorem about characters of modules over a vertex operator algebra graded by integer conformal weights, to the setting of a vertex operator superalgebra graded by rational conformal weights. To recover SL 2 (Z)invariance of the characters it turns out to be necessary to consider twisted modules alongside ordinary ones. It also turns out to be necessary, in describing the space of conformal blocks in the supersymmetric case, to include certain 'odd traces' on modules alongside traces and supertraces. We prove that the set of supertrace functions, thus supplemented, spans a finite dimensional SL 2 (Z)-invariant space. We close the paper with several examples. * email: jethrovanekeren@gmail.comIf Y (ω, z) = L(z) = n∈Z L n z −n−2 then the operators L n form a representation on V of the Virasoro algebra with some central charge c. The energy operator L 0 acts semisimply. The eigenvalue of an L 0 -eigenvector u ∈ V is called its conformal weight and is denoted ∆ u . We write V k for the set of vectors of conformal weight k.A module over a VOA V is a vector space M together with a fieldassigned to each u ∈ V . These data are to satisfy certain axioms, see Definition 2.2. A positive energy V -module is anWe say that V is rational if it has finitely many irreducible positive energy modules and every positive energy V -module is a direct sum of irreducible ones. Given V as above, Zhu introduced a second VOA structure Y [u, z] = n∈Z u ([n]) z −n−1 on V , and a new Virasoro elementω ∈ V (see Definition 2.3). Let L [0] be the new energy operator and V [k] the set of vectors of conformal weight k with respect toω.

Self-dual vertex operator superalgebras and superconformal field theory

“…Zhu proved [89] that certain trace functions on irreducible modules for suitable vertex operator algebras span representations of the modular group SL 2 (Z). More general modularity results that incorporate twisted modules have been obtained by Dong-Li-Mason [23], Dong-Zhao [28,29], and Van Ekeren [85]. We will use the extension of [89,23] to vertex operator superalgebras established in [28].…”

Modularity of Relatively Rational Vertex Algebras and Fusion Rules of Principal Affine W-Algebras

“…However ω(z) equips V with noninteger conformal weights, and Zhu's theorem actually fails in this case. This situation is rectified in the reference [28], where it is shown that modular transformations map the trace functions F M to trace functions on particular twisted modules. The task becomes to relate trace functions on twisted and untwisted V -modules.…”

“…This is achieved by the use of Li's shift operators ∆(u, z) (which appear explicitly in (1.2) above). The condition of relative cofiniteness is inspired by the work [7], and was used in [28].The transformation (1.2) was uncovered in the case of N = 2 superconformal vertex algebras in [13, Theorem 9.13 (b)], with h equal to the U (1) current of the N = 2 algebra. There the functions F M are shown to be flat sections of the bundle of conformal blocks over the universal elliptic curve, and (1.2) is derived from the geometry of this bundle.We also note that a result closely related to Theorem 1.2 was recently and independently obtained in [24] in the case of V rational and C 2 -cofinite (see also [26]).…”

The umbral moonshine module for the unique unimodular Niemeier root system

“…The importance of trace functions such as (1.14) within the broader context of modularity for super vertex operator algebras is analyzed in detail in [27]. (See also [28].…”