Higher Gauss sums of modular categories. (English) Zbl 1430.18015
Quadratic Gauss sums are the sums of values of quadratic forms on finite abelian groups. As any such quadratic form determines a (pre)modular pointed braided tensor category (with tensor products given by the group multiplication), it is natural to generalize Gauss sums to any premodular tensor category. This paper introduces higher Gauss sums for any \(n\in\mathbb{Z}\) and premodular category \(\mathcal{C}\):
\[
\tau_n(\mathcal{C})=\sum_{X\in\mathcal{O}(\mathcal{C})} \theta_X^n \dim_\mathcal{C}(X)^2,
\]
where \(\mathcal{O}(\mathcal{C})\) is a set of representatives of the isomorphism classes of simple objects in \(\mathcal{C}\), \(\theta_X\) is the twist on the simple object \(X\), and \(\dim_\mathcal{C}(X)\) is the categorical dimension. The cases \(n=\pm 1\) are the Gauss sums of \(\mathcal{C}\), and \(n=0\) yields the global dimension of \(\mathcal{C}\). Then the \(n\)th (multiplicative) central charge of \(\mathcal{C}\) is defined to be \(\xi_n(\mathcal{C})=\tau_n(\mathcal{C})/\vert\tau_n(\mathcal{C})\vert\), if \(\tau_n(\mathcal{C})\neq 0\).
It is known that the Gauss sums of a modular tensor category are non-zero, but this is not always true for the higher Gauss sums. However, if \(\mathcal{C}\) is modular and \(n\) is relatively prime to the order of the \(T\)-matrix of \(\mathcal{C}\), the authors calculate \(\tau_n(\mathcal{C})\) in terms of the action of \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on the non-zero \(\tau_1(\mathcal{C})\) and thus show \(\tau_n(\mathcal{C})\neq 0\), so that \(\xi_n(\mathcal{C})\) is defined. They calculate the higher Gauss sums and central charges more explicitly for \(\mathcal{C}\) the Drinfeld double of a spherical fusion category \(\mathcal{D}\); for example, they show \(\xi_n(\mathcal{Z}(\mathcal{D}))=1\) for all \(n\) relatively prime to the order of the \(T\)-matrix of \(\mathcal{Z}(\mathcal{D})\).
A standard construction of a new (pre)modular category from an old one is the category \(\mathcal{C}^0_A\) of local modules for a ribbon (or rigid) algebra \(A\) in a premodular category \(\mathcal{C}\) [A. Kirillov jun. and V. Ostrik, Adv. Math. 171, No. 2, 183–227 (2002; Zbl 1024.17013)]. Here the authors compute higher Gauss sums and central charges of \(\mathcal{C}^0_A\) in terms of those for \(\mathcal{C}\). For \(\mathcal{C}\) modular and \(n\) relatively prime to the order of the \(T\)-matrix of \(\mathcal{C}\), \[ \tau_n(\mathcal{C}^0_A)=\frac{\sigma(\dim_\mathcal{C}(A))}{\dim_\mathcal{C}(A)^2}\tau_n(\mathcal{C}) \] for suitable \(\sigma\in\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\), and \(\xi_n(\mathcal{C}^0_A)=\xi_n(\mathcal{C})\). For \(\mathcal{C}\) premodular, they show the same relations hold at least when \(n=1\) or \(A\) is a simple current extension of the unit object in \(\mathcal{C}\), that is, \(A\) is the sum of invertible simple objects of \(\mathcal{C}\).
The main application of Gauss sums and central charges in this paper is to the Witt group of non-degenerate braided fusion categories of [A. Davydov et al., J. Reine Angew. Math. 677, 135–177 (2013; Zbl 1271.18008)]. The authors show that Witt-equivalent pseudo-unitary modular tensor categories have equal higher central charges, thus giving a sufficient condition for Witt non-equivalence. In an example, they show how higher central charges can be used to find generators of subgroups of the Witt group.
It is known that the Gauss sums of a modular tensor category are non-zero, but this is not always true for the higher Gauss sums. However, if \(\mathcal{C}\) is modular and \(n\) is relatively prime to the order of the \(T\)-matrix of \(\mathcal{C}\), the authors calculate \(\tau_n(\mathcal{C})\) in terms of the action of \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on the non-zero \(\tau_1(\mathcal{C})\) and thus show \(\tau_n(\mathcal{C})\neq 0\), so that \(\xi_n(\mathcal{C})\) is defined. They calculate the higher Gauss sums and central charges more explicitly for \(\mathcal{C}\) the Drinfeld double of a spherical fusion category \(\mathcal{D}\); for example, they show \(\xi_n(\mathcal{Z}(\mathcal{D}))=1\) for all \(n\) relatively prime to the order of the \(T\)-matrix of \(\mathcal{Z}(\mathcal{D})\).
A standard construction of a new (pre)modular category from an old one is the category \(\mathcal{C}^0_A\) of local modules for a ribbon (or rigid) algebra \(A\) in a premodular category \(\mathcal{C}\) [A. Kirillov jun. and V. Ostrik, Adv. Math. 171, No. 2, 183–227 (2002; Zbl 1024.17013)]. Here the authors compute higher Gauss sums and central charges of \(\mathcal{C}^0_A\) in terms of those for \(\mathcal{C}\). For \(\mathcal{C}\) modular and \(n\) relatively prime to the order of the \(T\)-matrix of \(\mathcal{C}\), \[ \tau_n(\mathcal{C}^0_A)=\frac{\sigma(\dim_\mathcal{C}(A))}{\dim_\mathcal{C}(A)^2}\tau_n(\mathcal{C}) \] for suitable \(\sigma\in\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\), and \(\xi_n(\mathcal{C}^0_A)=\xi_n(\mathcal{C})\). For \(\mathcal{C}\) premodular, they show the same relations hold at least when \(n=1\) or \(A\) is a simple current extension of the unit object in \(\mathcal{C}\), that is, \(A\) is the sum of invertible simple objects of \(\mathcal{C}\).
The main application of Gauss sums and central charges in this paper is to the Witt group of non-degenerate braided fusion categories of [A. Davydov et al., J. Reine Angew. Math. 677, 135–177 (2013; Zbl 1271.18008)]. The authors show that Witt-equivalent pseudo-unitary modular tensor categories have equal higher central charges, thus giving a sufficient condition for Witt non-equivalence. In an example, they show how higher central charges can be used to find generators of subgroups of the Witt group.
Reviewer: Robert McRae (Nashville)
MSC:
| 18M15 | Braided monoidal categories and ribbon categories |
| 57R56 | Topological quantum field theories (aspects of differential topology) |
| 11L05 | Gauss and Kloosterman sums; generalizations |
Keywords:
modular tensor categories; braided tensor categories; Gauss sums; central charges; Witt groupReferences:
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