A different approach to positive traces on generalized $q$-Weyl algebras

Let $\mathcal{A}$ be a generalized $q$-Weyl algebra, it is generated by $u,v,Z,Z^{-1}$ with relations $ZuZ^{-1}=q^{2}u$, $ZvZ^{-1}=q^{-2}v$, $uv=P(q^{-1}Z)$, $vu=P(qZ)$, where $P$ is a Laurent polynomial. Let $\rho$ be a conjugation on $\mathcal{A}$. We are interested in positive traces on $\mathcal{A}$. We assume that $0<q<1$.

Positive traces are defined as follows. Let $\mathcal{A}$ be a noncommutative algebra over $\mathbb{C}$, $\rho$ be an antilinear automorphism of $\mathcal{A}$. Let $g=\rho^{2}$. We say that a linear map $T\colon\mathcal{A}\to\mathbb{C}$ is a $g$-twisted trace if $T(ab)=T(bg(a))$ for all $a,b\in\mathcal{A}$. A $g$-twisted trace is positive if $T(a\rho(a))>0$ for all nonzero $a\in\mathcal{A}$.

Positive traces for $q$-Weyl algebras appear in the study of the Coulomb branch of 4-dimensional superconformal field theories DG ; GT .

We will consider $g$-twisted traces for automorphisms $g$ such that $g(Z)=Z$. In this case we have $T(ZaZ^{-1})=T(a)$ for all $a\in\mathcal{A}$. This means that $T$ is zero for any expression $u^{k}R(z)$ and $v^{k}R(z)$, where $R$ is a Laurent polynomial and $k$ is a positive integer. Hence $T$ is uniquely defined by its values on Laurent polynomials.

More precisely, we will consider $g=g_{k}$ such that $g(Z)=Z$, $g(u)=q^{k}Z^{-k}u$, $g(v)=q^{k}Z^{k}v$ for some integer $k$.

We will identify a trace $T$ on $\mathcal{A}$ with its restriction to $\mathbb{C}[Z,Z^{-1}]\subset\mathcal{A}$ below.

Let $P$ be a Laurent polynomial that has $n$ nonzero roots. For simplicity, we will consider the case when $P$ has no roots with absolute value $q^{\pm 1}$, but we expect that our methods could be modified to work in the general case.

Similarly to K , $T\colon\mathbb{C}[Z,Z^{-1}]\to\mathbb{C}$ is a $g_{k}$-twisted trace if and only if $T(uvR(q^{-1}Z)-vR(Z)q^{k}Z^{-k}u)=0$ for all $R\in\mathbb{C}[z,z^{-1}]$. We have

Hence $T$ is a twisted trace if and only if

for all Laurent polynomials $R$.

Consider the antilinear automorphism $\rho$ such that $\rho(Z)=Z^{-1}$, $\rho(u)=Z^{k}q^{k}v$, $\rho(v)=Z^{-k}q^{k}u$. Similarly to K $\rho$ is well-defined when $P(z)=\overline{P}(z^{-1})$. We have $\rho^{2}=g_{2k}$. Since we have a full classification for the case $k=0$, assume that $k\neq 0$. The case when $k=\frac{n}{2}$ is of particular interest, since it corresponds to the “standard” Schur index of a four-dimensionall gauge theory G .

As in K , Lemma 3.2, it is enough to check the positivity condition $T(a\rho(a))>0$ when $a\in\mathbb{C}[Z,Z^{-1}]$ or $a\in u\mathbb{C}[Z,Z^{-1}]$. The first condition translates to $T(R(z)\overline{R}(z^{-1}))>0$ for all nonzero Laurent polynomials $R$, whereas the second one translates to

which is equivalent to

Reasoning as in K , Theorem 3.3, this is equivalent to $w(z)$ and $z^{k}P(z)w(qz)$ being nonnegative on the unit circle $\lvert z\rvert=1$.

From (1) we have

Let $\vartheta(z)=\vartheta_{11}(\frac{\log z}{2\pi i})$, where $\vartheta_{11}$ is a Jacobi theta function for period $\tau=\frac{\log q}{\pi i}$. Hence $\vartheta(1)=0$. It is well-known that any meromorphic function satisfying (2) can be expressed as $w=cz^{l}\frac{\prod_{i}\vartheta(\frac{z}{\alpha_{i}})}{\prod_{j}\vartheta(\frac{z}{\beta_{j}})}$, where $\alpha_{1},\ldots,\alpha_{N}$ and $\beta_{1},\ldots,\beta_{M}$ are the zeroes and poles of $w$. Since $w(z)P(qz)$ is holomorphic on $1<\lvert z\rvert<q^{-2}$, elements $q\beta_{1},\ldots,q\beta_{M}$ are roots of $P$. Moreover, in order to cancel out poles, $q\beta_{1},\ldots,q\beta_{M}$ should satisfy $q<\lvert q\beta_{i}\rvert<q^{-1}$. Hence we can assume that $M$ equals to the number of zeroes of $P$ in the annulus $q<\lvert z\rvert<q^{-1}$.

We have $\vartheta(q^{2}z)=q^{-1}z^{-1}\vartheta(z)$. In particular, multiplying $\alpha_{1}$ or $\beta_{1}$ by a power of $q^{2}$, we can assume that $l=0$. Also

It follows that

Comparing with (2) we get $N=M-2k$, $\prod_{i}\alpha_{i}=\prod_{j}\beta_{j}q^{-N+M}$. Since $P(z)=\overline{P}(z^{-1})$ the product $\prod_{j}\beta_{j}$ has absolute value $q^{-M}$. Multiplying $Z$ by a complex number with absolute value one if necessary, we can assume that $\prod_{j}\beta_{j}=q^{-M}$. We get a classification similar to K : possible $w$ are parametrized by $n-2k$ parameters $\alpha_{i}$ with fixed product $q^{N}$, and $c$. Note that for $k=\frac{M}{2}$ we have no numerator and the function $w$ is unique up to scaling.

Similarly to K , Section 3.2 the denominator of $w$ is positive on $S^{1}$ and $qS^{1}$. Also reasoning similarly to K , Section 3.2, we see that when $c$ is positive and $\alpha_{1},\ldots,\alpha_{N}$ are divided into pairs $\alpha_{i}$ and $\alpha_{j}=q^{2}\overline{\alpha_{i}^{-1}}$, the numerator is positive on $S^{1}$ and $qS^{1}$. It follows that the cone of positive traces has maximal possible real dimension $N=M-2k$ (one if $N=0$).

Hence we obtain the following

In particular, we confirm a conjecture of Gaiotto and Teschner GT in the case of abelian gauge theories: if $A$ is a $K$-theoretic Coulomb branch and $\rho$ corresponds to the “standard” Schur index, then the positive trace on $A$, if it exists, is unique up to scaling.