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Register a new userLog-modular quantum groups at even roots of unity and the quantum Frobenius I
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Cris Negron
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We construct log-modular quantum groups at even order roots of unity, both as finite-dimensional ribbon quasi-Hopf algebras and as finite ribbon tensor categories, via a de-equivariantization procedure. The existence of such quantum groups had been predicted by certain conformal field theory considerations, but constructions had not appeared until recently. We show that our quantum groups can be identified with those of Creutzig-Gainutdinov-Runkel in type A_1, and Gainutdinov-Lentner-Ohrmann in arbitrary Dynkin type. We discuss conjectural relations with vertex operator algebras at (1,p)-central charge. For example, we explain how one can (conjecturally) employ known linear equivalences between the triplet vertex algebra and quantum sl_2, in conjunction with a natural PSL_2-action on quantum sl_2 provided by our de-equivariantization construction, in order to deduce linear equivalences between extended quantum groups, the singlet vertex operator algebra, and the (1,p)-Virasoro logarithmic minimal model. We assume some restrictions on the order of our root of unity outside of type A_1, which we intend to eliminate in a subsequent paper.
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In 2007, G.E. Andrews introduced the $(n+1)$-variable combinatorial generating function $R_n(x_1,x_2,cdots,x_n;q)$ for ranks of $n$-marked Durfee symbols, an $(n+1)$-dimensional multisum, as a vast generalization to the ordinary two-variable partition rank generating function. Since then, it has been a problem of interest to understand the automorphic properties of this function; in special cases and under suitable specializations of parameters, $R_n$ has been shown to possess modular, quasimodular, and mock modular properties when viewed as a function on the upper half complex plane $mathbb H$, in work of Bringmann, Folsom, Garvan, Kimport, Mahlburg, and Ono. Quantum modular forms, defined by Zagier in 2010, are similar to modular or mock modular forms but are defined on the rationals $mathbb Q$ as opposed to $mathbb H$, and exhibit modular transformations there up to suitably analytic error functions in $mathbb R$; in general, they have been related to diverse areas including number theory, topology, and representation theory. Here, we establish quantum modular properties of $R_n$.
Understanding the relationship between mock modular forms and quantum modular forms is a problem of current interest. Both mock and quantum modular forms exhibit modular-like transformation properties under suitable subgroups of $rm{SL}_2(mathbb Z)$, up to nontrivial error terms; however, their domains (the upper half-plane $mathbb H$, and the rationals $mathbb Q$, respectively) are notably different. Quantum modular forms, originally defined by Zagier in 2010, have also been shown to be related to the diverse areas of colored Jones polynomials, meromorphic Jacobi forms, partial theta functions, vertex algebras, and more. In this paper we study the $(n+1)$-variable combinatorial rank generating function $R_n(x_1,x_2,dots,x_n;q)$ for $n$-marked Durfee symbols. These are $n+1$ dimensional multisums for $n>1$, and specialize to the ordinary two-variable partition rank generating function when $n=1$. The mock modular properties of $R_n$ when viewed as a function of $tauinmathbb H$, with $q=e^{2pi i tau}$, for various $n$ and fixed parameters $x_1, x_2, cdots, x_n$, have been studied in a series of papers. Namely, by Bringmann and Ono when $n=1$ and $x_1$ a root of unity; by Bringmann when $n=2$ and $x_1=x_2=1$; by Bringmann, Garvan, and Mahlburg for $ngeq 2$ and $x_1=x_2=dots=x_n=1$; and by the first and third authors for $ngeq 2$ and the $x_j$ suitable roots of unity ($1leq j leq n$). The quantum modular properties of $R_1$ readily follow from existing results. Here, we focus our attention on the case $ngeq 2$, and prove for any $ngeq 2$ that the combinatorial generating function $R_n$ is a quantum modular form when viewed as a function of $x in mathbb Q$, where $q=e^{2pi i x}$, and the $x_j$ are suitable distinct roots of unity.
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