Date of Award

8-22-2025

Date Published

September 2025

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Physics

Advisor(s)

Simon Catterall

Second Advisor

G. Scott Watson

Subject Categories

Physical Sciences and Mathematics | Physics

Abstract

In this thesis, we explore anomalies both in the continuum and on the lattice, and attempt to establish a connection between the two frameworks. In the continuum limit, we work with Kähler-Dirac fields, while in the discrete setting, we focus on staggered fermions. To facilitate insights relevant to the growing field of quantum computing, we consider a Hamiltonian formalism in addition to the path integral. Anomalies offer a concrete bridge between ultraviolet (UV) and infrared (IR) physics, understanding how anomalies manifest on the lattice versus in the continuum provides a powerful predictive framework. By numerically studying lattice gauge theories, we can impose non-perturbative constraints on the continuum theory. We show that massless Kähler-Dirac fermions exhibit a mixed gravitational anomaly involving a U (1) symmetry which is unique to Kähler-Dirac fields. Under this U (1) symmetry, the partition function transforms by a phase depending only on the Euler character of the background space. Compactifying flat space to a sphere, we find that the anomaly vanishes in odd dimensions but breaks the symmetry to Z4 in even dimensions. This Z4 prohibits bilinear terms in the fermionic effective action. Four-fermion terms are allowed but require multiples of two flavors of Kähler-Dirac fields. In four-dimensional flat space, each Kähler- Dirac field can be decomposed into four Dirac spinors, and hence these anomaly constraints ensure that eight Dirac fermions—or, for real representations, six- teen Majorana fermions—are needed for a consistent interacting theory. These constraints on fermion number agree with results for topological insulators and discrete anomalies rooted in the Dai-Freed theorem. Our work suggests that Kähler-Dirac fermions may offer an independent path to understanding these constraints. We point out that this anomaly survives intact under discretization and is therefore relevant to recent numerical results on lattice models possessing massive symmetric phases. We further show that the effective action resulting from integrating out mas- sive Kähler-Dirac fermions propagating on a curved three-dimensional space is a topological gravity theory of Chern-Simons type. In the presence of a domain wall, massless, two-dimensional Kähler-Dirac fermions appear, localized to the wall. Potential gravitational anomalies arising for these domain wall fermions are canceled via anomaly inflow from the bulk gravitational theory. We also study the invariance of the theory under large gauge transformations. The analysis and conclusions generalize straightforwardly to higher dimensions. Staggered fermions can be thought of as discretized Kähler-Dirac fermions, their shift symmetries satisfy an algebra and, in four Euclidean dimensions, can be re- lated to a discrete subgroup of an SU(4) flavor symmetry. This connection plays a crucial role in showing that staggered fermions lead to a theory of four degener- ate Dirac fermions in the continuum limit. These symmetries are associated with the appearance of certain Z2-valued global parameters. Lattice anomalies can be thought of as obstructions to gauging on the lattice. We propose a strategy to partially gauge these translation symmetries by allowing these parameters to vary locally on the lattice. To maintain invariance of the action under such local variations requires the introduction of Z2-valued higher-form lattice gauge fields. An analogous procedure can also be carried out for reduced staggered fermions, where the shifts correspond to a discrete subgroup of an SO(4) flavor symmetry. We then review the shift and time reversal symmetries of Hamiltonian staggered fermions and their connection to continuum symmetries, concentrating in par- ticular on the case of massless fermions in (3 + 1) dimensions. We construct operators using the staggered fields that implement these symmetries on the lat- tice. We show that the elementary shift symmetry of a single staggered field is tied to a Z4 subgroup of an additional U(1) phase symmetry and anti-commutes with time reversal. This latter property implies that time reversal symmetry will be broken if this phase symmetry is gauged—a mixed ’t Hooft anomaly. How- ever, this anomaly can be canceled for multiples of four staggered fields. Finally, we observe that the naive continuum limit of the minimal anomaly-free lattice model possesses the symmetries and matter representations characteristic of the Pati–Salam Grand Unified Theory (GUT).

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