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Counting gauge-invariant states with matter fields and finite gauge groups

Alessandro Mariani

TL;DR

This work provides a comprehensive framework for counting gauge-invariant states in lattice gauge theories with finite gauge groups and arbitrary matter, extending previous pure-gauge results to include scalar and fermionic fields and various boundary conditions. The authors derive a general master counting formula encoded in conjugacy-class data and matter-representation characters, and then instantiate it for scalars and fermions, including twisted boundaries and charge-conjugation symmetries. The results yield explicit expressions for the number of physical states across nontrivial Gauss-law sectors, boundary twists, and different fermion formulations (naive, Wilson, staggered), with clear implications for resource estimation and cross-checking gauge-invariant bases. The methodology highlights the role of group-theoretical structure (characters, centralizers, and automorphisms) in quantifying gauge-invariant content and provides a toolkit applicable to quantum simulation and related areas in condensed matter and quantum gravity.

Abstract

Gauge theories with finite gauge groups have applications to quantum simulation and quantum gravity. Recently, the exact number of gauge-invariant states was computed for pure gauge theories on arbitrary lattices. In this work, we generalize this counting to include the case of scalar and fermionic matter, as well as various kinds of boundary conditions. As a byproduct, we consider several related questions, such as the implementation of charge conjugation for a generic finite group. These results are relevant for resource estimation and also as a crosscheck when working in a gauge-invariant basis.

Counting gauge-invariant states with matter fields and finite gauge groups

TL;DR

This work provides a comprehensive framework for counting gauge-invariant states in lattice gauge theories with finite gauge groups and arbitrary matter, extending previous pure-gauge results to include scalar and fermionic fields and various boundary conditions. The authors derive a general master counting formula encoded in conjugacy-class data and matter-representation characters, and then instantiate it for scalars and fermions, including twisted boundaries and charge-conjugation symmetries. The results yield explicit expressions for the number of physical states across nontrivial Gauss-law sectors, boundary twists, and different fermion formulations (naive, Wilson, staggered), with clear implications for resource estimation and cross-checking gauge-invariant bases. The methodology highlights the role of group-theoretical structure (characters, centralizers, and automorphisms) in quantifying gauge-invariant content and provides a toolkit applicable to quantum simulation and related areas in condensed matter and quantum gravity.

Abstract

Gauge theories with finite gauge groups have applications to quantum simulation and quantum gravity. Recently, the exact number of gauge-invariant states was computed for pure gauge theories on arbitrary lattices. In this work, we generalize this counting to include the case of scalar and fermionic matter, as well as various kinds of boundary conditions. As a byproduct, we consider several related questions, such as the implementation of charge conjugation for a generic finite group. These results are relevant for resource estimation and also as a crosscheck when working in a gauge-invariant basis.

Paper Structure

This paper contains 20 sections, 6 theorems, 119 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. General construction
  3. The pure gauge sector
  4. Master counting formula
  5. Non-trivial Gauss law sectors
  6. Twisted boundary conditions
  7. Charged states and twisted boundaries
  8. Physical symmetries and charge conjugation
  9. Scalar fields
  10. Quantization of scalars
  11. Examples
  12. Counting gauge-invariant states
  13. Fermion fields
  14. Quantization of fermions
  15. Gauge group representation
  16. ...and 5 more sections

Key Result

Theorem 1

Sharp All compact connected simple Lie groups (with the exception of $E_6$) are quasi-ambivalent. More precisely:

Figures (2)

  • Figure 1: A two dimensional periodic square lattice. The dashed links are the boundary links. The labelled link $l$ connects sites $x$ and $y$, but from the point of view of $l$, site $y$ belongs to a periodic copy. Removing all boundary links keeps the lattice connected: for example, the yellow path connects $x$ and $y$ without crossing the boundary.
  • Figure 2: A two dimensional square lattice with open boundaries. Some links are "dangling", i.e. they are connected to the rest of the lattice via only one site. Every link carries a gauge field, but on dangling links it is not dynamical. The matter field is present on all black sites, but not on white sites. Gauge transformations are imposed on black sites, but not on white sites. This figure is a two-dimensional version of Fig.2 in Chung:2024hsq.

Theorems & Definitions (10)

  • Definition 1
  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof