Instanton Density Operator in Lattice QCD from Higher Category Theory

Abstract

A natural definition for instanton density operator in lattice QCD has long been desired. We show this problem is, and has to be, solved by higher category theory. The problem is solved by refining at a conceptual level the Yang-Mills theory on lattice, in order to recover the homotopy information in the continuum, which would have been lost if we put the theory on lattice in the traditional way. The refinement needed is a generalization--through the lens of higher category theory--of the familiar process of Villainization that captures winding in lattice XY model and Dirac quantization in lattice Maxwell theory. The apparent difference is that Villainization is in the end described by principal bundles, hence familiar, but more general topological operators can only be captured on the lattice by more flexible structures beyond the usual group theory and fibre bundles, making the language of categories natural and necessary. The key structure we need for our particular problem is called multiplicative bundle gerbe, based upon which we can construct suitable structures to naturally define the 2d Wess-Zumino-Witten term, 3d skyrmion density operator and 4d hedgehog defect for lattice $S^3$ pion vacua non-linear sigma model, and the 3d Chern-Simons term, 4d instanton density operator and 5d Yang monopole defect for lattice $SU(N)$ Yang-Mills theory; the structures behind the non-linear sigma model and the Yang-Mills theory are related via an implicit Yang-Baxter equation. In a broader perspective, higher category theory enables us to rethink more systematically the relation between continuum quantum field theory and lattice quantum field theory. We sketch a proposal towards a general machinery that constructs the suitably refined lattice degrees of freedom for a given non-linear sigma model or gauge theory in the continuum, realizing the desired topological operators on the lattice.

144 pages. v3: Elaborations made in Section 5.5 on its relation to Section 4; minor reorganization and/or clarification of contents in Section 2.4, Section 4.1, Section 6, Section 7. v2: Added reference to the follow-up work on the physical model, 2411.07195; discussions updated, in particular separated the original Section 5.4 into 5.4 and 5.5, and revised the contents in Section 5.5

References in corpus (60)

• Fault-tolerant quantum computation by anyons
• String-net condensation: A physical mechanism for topological phases
• Instantons in QCD
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation
• Entanglement Renormalization and Holography
• Properties and uses of the Wilson flow in lattice QCD
• Knots and Particles
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• The index theorem in QCD with a finite cut-off
• Quasi-adiabatic Continuation of Quantum States: The Stability of Topological Ground State Degeneracy and Emergent Gauge Invariance
• Exploring 2-Group Global Symmetries
• On 2-Group Global Symmetries and Their Anomalies
• Higher-Dimensional Algebra V: 2-Groups
• Anomalies in the Space of Coupling Constants and Their Dynamical Applications I
• An Invitation to Higher Gauge Theory
• Static solitons with non-zero Hopf number
• Exact chiral symmetry, topological charge and related topics
• Spin TQFTs and fermionic phases of matter
• WZW branes and gerbes
• Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions
• A Modified Villain Formulation of Fractons and Other Exotic Theories
• Ground states of quantum antiferromagnets in two dimensions
• Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories
• Higher Yang-Mills Theory
• Abelian gauge theories on the lattice: $\theta$-terms and compact gauge theory with(out) monopoles
• Universality of the topological susceptibility in the SU(3) gauge theory
• Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics
• Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group
• Topology and Nematic Ordering I: A Gauge Theory
• Pauli stabilizer models of twisted quantum doubles
• Higher-dimensional generalizations of the Berry curvature
• Basic gerbe over non simply connected compact groups
• The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module
• Revisiting Wess-Zumino-Witten terms
• On Pure Lattice Chern-Simons Gauge Theories
• Higher symmetry and gapped phases of gauge theories
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Differentiable Cohomology of Gauge Groups
• A Construction of String 2-Group Models using a Transgression-Regression Technique
• Perfect Lattice Topology: The Quantum Rotor as a Test Case
• Modified Villain formulation of abelian Chern-Simons theory
• An invitation to topological orders and category theory
• Topology of $SU(N)$ lattice gauge theories coupled with ${\mathbb{Z}}_N$ $2$-form gauge fields
• Anomalous symmetries of quantum spin chains and a generalization of the Lieb-Schultz-Mattis theorem
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• Geometric Realizations of Tricategories
• Charting the space of ground states with tensor networks
• Fractional Hall Conductivity and Spin-c Structure in Solvable Lattice Hamiltonians
• Lattice Chern-Simons-Maxwell Theory and its Chirality
• Solvable Lattice Hamiltonians with Fractional Hall Conductivity
• Static Magnetic Response of Non-Fermi Liquid Density
• Pursuing Stacks
• Bundle gerbes: stable isomorphism and local theory
• A discrete formulation for three-dimensional winding number
• Abelian Topological Order on Lattice Enriched with Electromagnetic Background
• Phases of theories with ${\mathbb{Z}}_N$ 1-form symmetry and the roles of center vortices and magnetic monopoles
• Canonical quantization of lattice Chern-Simons theory
• An Explicit Categorical Construction of Instanton Density in Lattice Yang-Mills Theory
• Higher structures in matrix product states

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