Can we categorify the formula for the quadratic Gauss sum?

Question

Background

Fix an odd prime $p$ and set $\zeta=e^{2\pi i/p}$. We define the quadratic Gauss sum as

$$g=\sum_{n=0}^{p-1} \zeta^{n^2}.$$

It's pretty easy to show that

$$g^2= \begin{cases} p & \textrm{if } p\equiv 1 \mod 4 \\\ -p & \textrm{if } p\equiv 3 \mod 4, \end{cases}$$

and from this we can deduce quadratic reciprocity; it's harder to determine the modulus. We can actually find an explicit formula for $g$, namely:

$$g= \begin{cases} \sqrt{p} & \textrm{if } p\equiv 1 \mod 4 \\\ i\sqrt{p} & \textrm{if } p\equiv 3 \mod 4. \end{cases}$$

This is the result I refer to for the remainder of the question.

Question

Can we categorify this result?

By categorification, I mean the opposite of decategorification, and by decategorification, I mean the process of removing structure by e.g. taking the cardinality of a set or the dimension of a vector space. (Thus an example of categorification would be interpreting some combinatorial identity of positive integers as a bijection between sets.) This is intentionally vague, because there are plenty of people who have a much better idea of what constitutes categorification than I do, so feel free to interpret "categorification" liberally.

Motivation

Gauss's original proof of our result uses q-binomial coefficients. (A modern exposition of this proof can be found in "The determination of Gauss sums" by Bruce C. Berndt and Ronald J. Evans.)

Now, $q$-binomial coefficients can be categorified by Grassmannian varieties. What I mean by that is: the $q$-binomial coefficient $\binom{n}{k}_q$ is the number of $k$-dimensional subspaces of an $n$-dimensional vector space over the finite field $\mathbb{F}_q$, i.e. the cardinality of the Grassmannian $\textrm{Gr}(n,k)$. Basically, I'm wondering if there is some way this can be connected to the formula for the quadratic Gauss sum, seeing as how the formula is clearly related to the properties of $q$-binomial coefficients.

Answer 1

Since you are using an vague definition of categorification, the following maybe relevant. Gauss sums appear in the theory of (pre)-modular categories. The Gauss sum can be viewed as the sum of values of the quadratic form $$\zeta\colon \mathbb{Z}/p\mathbb{Z}\longrightarrow \mathbb{C}^\times, \quad n\longmapsto \zeta^{n^2}.$$ According to [1], "Premodular or ribbon categories are categorical generalizations of quadratic forms of finite abelian groups". As a reference for this analogy, see for example Example 8.13.5 and Section 8.4 of [2]. Any quadratic form $\omega\colon G\to \mathbb{C}^\times$ of an abelian group $G$ gives rise to a tensor category $\mathcal{C}(G,\omega)$ with simple objects $X_g$ corresponding to the elements $g$ of $G$.

A pre-modular category is, a tensor category with duals, a braiding, which is a collection of functorial isomorphisms $c_{X,Y}\colon X\otimes Y\to Y\otimes X$, and other favourable properties like having a finite set of simple objects. To any pre-modular category one can associate the datum of an $S$-matrix which has a motivation from Physics which is a driving force behing the theory of modular categories. The $S$-matrix of the category $\mathcal{C}(G,\omega)$ now corresponds to $\left(b(g,h)\right)_{g,h\in G}$, where $$b(g,h)=\frac{\omega(gh)}{\omega(g)\omega(h)}.$$ In general, the $S$-matrix is defined by $S_{X,Y}=\operatorname{Tr}(c_{Y,X}c_{X,Y})$ for representatives of simple objects $X,Y$ in a pre-modular category $\mathcal{C}$. This used the categorical trace $\operatorname{Tr}$ of $\mathcal{C}$ which is defined using duality. A pre-modular category is modular if the $S$-matrix is non-degenerate. For $\mathcal{C}(G,\omega)$, this condition is equivalent to non-negeneracy of $\omega$.

Therefore, picking $G=\mathbb{Z}/p\mathbb{Z}$ with the above pairing $\omega=\zeta$ gives a modular tensor category categorifying (the group algebra of) $\mathbb{Z}/p\mathbb{Z}$, which is its Grothendieck ring, together with the Gauss sum, which is given by the sum $$g=\sum_{g\in G} \theta_{X_g}\dim (X_g)^2,$$ where $\dim(X_g)$ is the categorical dimension of an object $X_g$, and $\theta_{X_g}$ is the scalar defining the so-called twist isomorphism of $X_g$. In the case of $\mathcal{C}(G,\omega)$, $\theta_{X_g}=b(g,g)$. Gauss sums an invariant for general pre-modular categories.

[1] S.-H. Ng, A. Schopieray, and Y. Wang: Higher Gauss Sums of Modular Categories. https://arxiv.org/pdf/1812.11234.pdf

[2] P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik: Tensor Categories, Mathematical Surveys and Monographs, AMS