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Positive traces on quantized abelian Coulomb branches

Daniil Klyuev

TL;DR

The paper classifies positive traces on quantized abelian Coulomb branches $A_{G,N}$ with $G=(\mathbb{C}^*)^d$ (hypertoric envelopes). It develops an analytic framework: traces are encoded by exponential generating functions $u_T$, shown to be meromorphic with a universal denominator, and every trace admits a representation as a sum of integrals over subspaces with exponentially decaying weights. Positivity imposes strong structural constraints: $u_T$ is bounded, traces decompose into real-subspace integrals, and in the unimodular setting positivity reduces to nonnegativity of certain polynomials $Q$; the cone of positive traces has a combinatorial dimension given by counting lattice points in symmetrized weight polytopes $\mathcal{P}_U^+$. These results advance the understanding of traces on Coulomb branches and their connections to representation theory and symplectic duality, providing explicit criteria and dimension counts for the positive-trace space.

Abstract

Let $A=A_{G,N}^{\hbar=1}$ be a quantized Coulomb branch with an antilinear automorphism $ρ$. A map $T\colon A\to\mathbb{C}$ is called a positive trace if $T(aρ(a))>0$ for all nonzero $a\in A$. Positive traces on Coulomb branches appear in the study of supersymmetric gauge theories. We classify positive traces on all abelian Coulomb branches, meaning $G=(\mathbb{C}^{\times})^d$ is a torus.

Positive traces on quantized abelian Coulomb branches

TL;DR

The paper classifies positive traces on quantized abelian Coulomb branches with (hypertoric envelopes). It develops an analytic framework: traces are encoded by exponential generating functions , shown to be meromorphic with a universal denominator, and every trace admits a representation as a sum of integrals over subspaces with exponentially decaying weights. Positivity imposes strong structural constraints: is bounded, traces decompose into real-subspace integrals, and in the unimodular setting positivity reduces to nonnegativity of certain polynomials ; the cone of positive traces has a combinatorial dimension given by counting lattice points in symmetrized weight polytopes . These results advance the understanding of traces on Coulomb branches and their connections to representation theory and symplectic duality, providing explicit criteria and dimension counts for the positive-trace space.

Abstract

Let be a quantized Coulomb branch with an antilinear automorphism . A map is called a positive trace if for all nonzero . Positive traces on Coulomb branches appear in the study of supersymmetric gauge theories. We classify positive traces on all abelian Coulomb branches, meaning is a torus.

Paper Structure

This paper contains 13 sections, 23 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Setup, notation, analytic formula
  3. Quantized abelian Coulomb branches
  4. Trace condition
  5. Analytic formulas for traces
  6. Exponential generating function of a trace
  7. Definition and basic properties
  8. Example: $T=\mathbb{C}^{\times}$.
  9. Generating function is meromorphic
  10. Traces as a sum of integrals
  11. Positive traces
  12. Exponential generating function of a positive trace
  13. Positivity in terms of weights

Key Result

Proposition 1

Let $T$ be the trace constructed in Proposition PropTraceAsAnIntegral with weight $w(x)=e^{2\pi\zeta}\frac{Q(e^{2\pi z_1},\ldots, e^{2\pi z_d})}{\prod_{j=1}^n (e^{2\pi \xi_j}+e^{2\pi \mathrm{i} b_j})}$. Then $T$ is positive if and only if $Q\neq 0$ is a polynomial that has nonnegative values on $\ma

Theorems & Definitions (44)

  • Proposition : \ref{['PropPositivityOnTermsOfWeight']}
  • Theorem : \ref{['ThrFourierTransformIsMeromorphic']}
  • Corollary : \ref{['CorTraceIsAnIntegral']}
  • Theorem : \ref{['ThrAnswer']}
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • ...and 34 more