A Note On Lattice Definition of Gauge Theories With Chiral Fermions

Abstract: Because of the fermion doubling problem, it has been a longstanding challenge to describe a lattice algorithm for studying four-dimensional gauge theories coupled to chiral fermions. Recently, an algorithm to do so has been described by Craig, extending partial results in the previous literature. Here we describe a variant of this algorithm, based on interpolating a lattice gauge field over the continuum (as in Craig's work) and coupling the interpolated gauge field to continuum chiral fermions. The same variant was also proposed recently by Hsu. In any dimension and for any anomaly-free representation, this approach can be implemented on a computer in principle, but not in practice by Monte Carlo methods because the chiral fermion path integral is not positive.

• The paper presents a lattice definition for gauge theories with chiral fermions, addressing the fermion doubling problem by using continuum interpolated gauge fields.
• The method applies to any anomalous free fermions and compact gauge group, on any arbitrary compact Euclidean spin manifolds, permitting a broad construction.
• The approach involves defining the chiral determinant using the absolute value of the determinant and an eta-invariant phase, offering a regularization in principle but not necessarily a practical Monte Carlo formulation.

Problem and proposed construction

The paper addresses the longstanding problem of defining four-dimensional lattice gauge theories coupled to chiral fermions without reintroducing fermion doublers. The Nielsen–Ninomiya obstruction prevents a straightforward local lattice realization of a chiral spectrum, while overlap and Ginsparg–Wilson frameworks provide lattice chiral operators but do not, by themselves, supply an immediately usable fermion integration measure for general nonabelian chiral gauge theories. The paper proposes a variant of a recently developed construction: lattice gauge fields are restricted to an admissible sector, interpolated into continuum gauge fields, and coupled directly to continuum chiral fermions. The resulting fermion path integral is defined using the absolute value of the chiral determinant together with an Atiyah–Patodi–Singer eta-invariant phase (2611.05051).

The principal claim is substantially broader than a construction restricted to the Standard Model. The proposed definition applies, in principle, to any compact gauge group and any anomaly-free fermion representation, in arbitrary spacetime dimension, and on arbitrary compact Euclidean spin manifolds. The four-dimensional theory is therefore not tied to a toroidal lattice or to special representation-theoretic properties of the Standard Model. The generality depends on two mathematical inputs: the continuum fermion path integral must be well-defined in the absence of perturbative and global anomalies, and admissible lattice gauge fields must admit gauge-covariant continuum interpolations.

The construction is not presented as a practical Monte Carlo formulation. The paper explicitly concludes that the chiral fermion path integral is not positive even for anomaly-free theories, so standard importance-sampling methods do not apply. Its computational claim is weaker but precise: the regularization is computable in principle, although likely to be prohibitively inefficient with currently known methods.

Continuum chiral determinants and anomaly inflow

For a four-dimensional chiral theory on a spin manifold $W$, the absolute value of the chiral determinant is unproblematic. If $\mathcal{D}_W^+$ denotes the positive-chirality Dirac operator, one may define

$\left|\det \mathcal{D}_W^+\right|$

from the positive operator $\mathcal{D}_W^{+\dagger}\mathcal{D}_W^+$. The phase is the nontrivial part. It cannot generally be defined as a local functional of the four-dimensional gauge field because perturbative and global anomalies obstruct precisely such a definition.

The paper uses a five-dimensional manifold $Y$ with boundary $\partial Y=W$, together with extensions of the gauge field, metric, and spin structure. The chiral partition function is then written as

$$Z_{\mathrm{fermion}}(W) = \left|\det \mathcal{D}_W^+\right| \exp\!\left(-i\pi\eta(\mathcal{D}_Y)\right),$$

where $\eta(\mathcal{D}_Y)$ is the APS eta-invariant of an appropriate self-adjoint five-dimensional Dirac operator. This formula is not merely a convenient phase convention. It realizes anomaly inflow nonperturbatively: the eta-invariant contains the perturbative Chern–Simons contribution locally in field space and also detects global spectral asymmetry invisible to perturbative anomaly polynomials.

The key consistency criterion is

$\exp\!\left(-i\pi\eta(\mathcal{D}_Y)\right)=1$

for every closed five-manifold $Y$ equipped with the relevant spin and gauge data. The paper asserts that this condition is both necessary and sufficient for independence of the choice of bounding manifold and extension. Consequently, complete anomaly cancellation is equivalent to triviality of the corresponding eta-invariant character on the relevant cobordism group. This unifies ordinary triangle anomalies with global anomalies within one criterion.

The argument also clarifies the role of the APS index theorem. When the perturbative anomaly vanishes, the eta-invariant becomes a cobordism invariant. If $Y$ bounds a six-manifold $Z$, the APS index theorem relates $\eta(\mathcal{D}_Y)$ to the index on $Z$; vanishing of the six-dimensional anomaly polynomial then forces the exponentiated eta-invariant to be trivial. Thus the obstruction is naturally a homomorphism from the appropriate cobordism group to $U(1)$.

The discussion of the $SU(2)$ global anomaly illustrates the mechanism. For a pseudoreal half-integer-spin representation, nonzero eigenvalues of the five-dimensional Dirac operator occur in pairs of opposite sign, leaving only zero modes in the eta-invariant. Hence

$\exp\!\left(-i\pi\eta_Y\right)=(-1)^\zeta,$

where $\zeta$ is the number of zero modes. The familiar mod-two anomaly is therefore recovered as a special case of the general eta-invariant criterion. The paper further notes that the relevant five-dimensional spin-bordism group with an $SU(2)$ bundle is $\mathbb{Z}_2$, so the standard examples generate the complete obstruction rather than representing unrelated tests.

Treatment of zero modes and theta-angles

The simple determinant formula applies only when the chiral partition function is not identically zero. Instanton sectors require a more careful treatment because the index theorem can force fermion zero modes and make the partition function vanish. In such sectors, the theory must instead be specified through correlation functions with fermionic insertions that absorb all zero modes.

The paper constructs these correlation functions by imposing generalized APS boundary conditions that pair positive- and negative-chirality zero modes across disconnected boundary components. A small perturbation lifting the zero modes reduces the computation to the nonzero-mode determinant and then permits the perturbation parameter to be removed. The resulting expression has three components: a reduced determinant, the eta-invariant phase, and determinants of zero-mode wavefunctions evaluated at the insertion points. Schematically,

$$\langle \mathcal{O}\rangle \sim \prod_\alpha \left|\det{}' \mathcal{D}_{W_\alpha}^+\right| \exp\!\left(-i\pi\eta_Y\right) \times (\text{zero-mode wavefunction determinants}).$$

This construction is important because it avoids treating vanishing partition functions as singular exceptions to the definition of the theory. The fermion measure is instead specified by the complete set of nonzero correlation functions, with the propagator defined on the complement of the zero-mode subspace.

A second issue arises when $W$ is not null-cobordant. The paper resolves this using fiducial manifolds representing generators of the relevant cobordism group. For finite-order generators, taking the appropriate number of disjoint copies produces a boundary, and extracting a root of the resulting partition function introduces a discrete theta-angle. For infinite-order generators, the magnitude of the partition function is fixed by considering $W\sqcup(-W)$, while its phase remains arbitrary and is identified with a continuous theta-angle.

This yields a unified interpretation: continuous and discrete theta-angles are characters of the relevant cobordism group into $U(1)$. In four dimensions, the infinite-order components include ordinary gauge and gravitational instanton numbers, while finite-order components can arise from non-simply-connected or disconnected gauge groups. The paper emphasizes that some finite-order obstructions are not reflected in fermion zero modes or index formulas; they are nevertheless essential to defining the quantum theory on general manifolds.

Interpolation of admissible lattice gauge fields

The lattice construction begins with Lüscher-style admissibility. A lattice gauge field is restricted so that every plaquette holonomy $U_P$ lies sufficiently close to the identity,

$\lVert U_P-1\rVert<\epsilon.$

The admissibility condition prevents lattice instantons from shrinking through the lattice holes and disappearing continuously. It also ensures that lattice configurations possess well-defined topological sectors and can be interpreted as continuum gauge fields on nontrivial bundles.

The paper gives an inductive interpolation procedure for arbitrary compact gauge groups. On each cell, lattice gauge transformations first put the link variables close to the identity, allowing logarithms of the link holonomies to be defined. The gauge field is then extended successively from links to two-dimensional faces, higher-dimensional faces, and finally top-dimensional cells by solving small-field Yang–Mills Dirichlet problems in a fixed gauge. The smallness assumption controls the nonlinear terms and ensures existence and uniqueness of the required extensions.

On neighboring cells, the local interpolations are related by uniquely determined transition functions. Compatibility on triple overlaps follows from uniqueness of the Dirichlet problem, allowing the local data to be glued into a global $G$-bundle with connection. The resulting continuum field reproduces the lattice link holonomies exactly and has field strength of order $\epsilon$ in lattice units, or parametrically

$F\sim \frac{\epsilon}{a^2}$

when the lattice spacing is $a$.

Several properties are significant. The interpolation is gauge-covariant, preserves lattice symmetries, can reconstruct nontrivial bundle topology, and generalizes to arbitrary dimension. It is initially only piecewise smooth: the connection is continuous across cell boundaries, but its derivatives may jump. The paper argues that this regularity is sufficient for the determinant and eta-invariant formulas. It also describes smoothing alternatives, including short Yang–Mills gradient flow, which smooths modes below the lattice scale while preserving the relevant topological class. A further construction achieves smoothness while retaining exact link holonomies and locality within each cell.

The interpolation argument extends beyond hypercubic lattices. On a general compact smooth manifold, one can use triangulations with bounded flatness under repeated edgewise subdivision. The resulting simplicial lattice supports an analogous admissibility condition and Dirichlet-problem interpolation. This is what permits the proposed regularization to be formulated on arbitrary compact spin manifolds rather than only on periodic hypercubic lattices.

The lattice definition

Let $\mathcal{A}$ denote the space of admissible lattice gauge fields. For each lattice configuration, choose its continuum interpolation and couple continuum chiral fermions to the resulting gauge field. In the zero-instanton sector, the lattice partition function takes the form

$$Z = \int_{\mathcal{A}} dU\, e^{-I[U]}\, \left|\det \mathcal{D}_W^+\right| \exp\!\left(-i\pi\eta(\mathcal{D}_Y)\right).$$

Here $I[U]$ is the lattice gauge action, while both the four-dimensional determinant and the five-dimensional eta-invariant are computed from the interpolated continuum field. Nonzero topological sectors are incorporated through the zero-mode and theta-angle machinery described above.

This differs conceptually from overlap-based formulations. No four-dimensional lattice chiral fermion operator is introduced as the fundamental fermionic variable, so the measure problem is not solved by constructing a lattice determinant with a chiral phase. Instead, the lattice supplies a finite-dimensional integration domain for gauge fields, while the fermion factor is defined through continuum spectral data associated with each interpolated configuration.

The construction therefore avoids the fermion doubling problem at the level of the definition. It does not claim that a local, positive, efficient lattice fermion action has been found. Rather, it changes the division of labor: lattice variables regulate the gauge-field integration, and continuum anomaly theory supplies the chiral fermion measure.

Numerical implementability

The final section distinguishes formal definability from practical simulation. At fixed lattice spacing and finite volume, the admissible gauge-field space contains finitely many link variables and is compact. If the fermion factor can be evaluated effectively, the full path integral is, in principle, a finite-dimensional numerical integration problem.

The interpolation itself is not regarded as a fundamental computational obstacle. It reduces to a finite sequence of numerical elliptic boundary-value problems. An extension over a five-manifold can likewise be constructed, and anomaly freedom guarantees that the final result is independent of the chosen extension.

The paper’s more substantive computational observation is that the fermion-doubling problem need not reappear when evaluating the spectral quantities. The modulus of the chiral determinant can be obtained from the Laplace-like operator

$\mathcal{D}_W^{+\dagger}\mathcal{D}_W^+,$

which is nonchiral and can be discretized without the usual obstruction. For the eta-invariant, one need not discretize the first-order five-dimensional operator directly. Instead, one computes the spectrum of its square, with boundary conditions inherited from APS boundary conditions, and determines the signs of the corresponding eigenvalues from matrix elements of the original first-order operator. Degenerate eigenspaces can be treated by diagonalizing this operator within each degenerate subspace.

This argument supports the paper’s in-principle computability claim: numerical evaluation can be organized around Laplace-type operators rather than a naive lattice Weyl operator. The claim is nevertheless conditional on controlling spectral truncation, boundary-condition discretization, interpolation errors, and the phase accuracy of the eta-invariant. No complexity estimate, convergence theorem for a complete numerical algorithm, or demonstration on a nontrivial gauge theory is supplied.

Limitations and open questions

The most direct limitation is the absence of a positive measure. Even with complete anomaly cancellation, the factor $\exp(-i\pi\eta)$ is generally complex, and the fermion path integral lacks the positivity required by standard Monte Carlo methods. The proposed regularization is therefore not an immediately practical nonperturbative simulation algorithm.

The approach also depends on admissibility. The paper argues, following the standard asymptotic-freedom rationale, that the plaquette constraint should become irrelevant near the continuum limit because typical plaquette deviations decrease with the bare coupling. This is a physical expectation rather than a demonstrated result for the full chiral construction. Establishing universality and continuum-limit equivalence for interacting nonabelian chiral theories remains an open technical question.

A further issue is algorithmic efficiency. Although the determinant modulus and eta-invariant can be reduced to nonchiral spectral problems, their numerical evaluation may require resolving extensive ultraviolet spectra and tracking spectral asymmetry under changes in the gauge field. The paper establishes that doubling is not an unavoidable formal obstruction, but it does not show that the resulting computation is polynomially efficient or suitable for large-volume simulations.

Finally, the treatment of general cobordism sectors requires choices of fiducial manifolds, fiducial observables, and renormalized theta-angles. The construction is mathematically coherent once these data are fixed, but the relation between these choices and a conventional local lattice action is not made explicit. The paper leaves open how best to formulate these sectors in an implementation that simultaneously preserves locality, smooth dependence on lattice variables, and numerical tractability.

Conclusion

The paper presents a continuum-interpolation definition of lattice gauge theories with anomaly-free chiral fermions. Its central formula combines the absolute chiral determinant with the APS eta-invariant, thereby incorporating both perturbative and global anomaly information. Admissibility permits lattice gauge fields to be promoted to continuum connections on potentially nontrivial bundles, while zero modes, cobordism obstructions, and theta-angles are treated within the same framework.

The result is a broad formal regularization applicable to arbitrary compact gauge groups, anomaly-free representations, dimensions, and compact spin manifolds. Its principal limitation is practical rather than definitional: the resulting fermion measure is nonpositive, and the paper establishes computability in principle rather than an efficient Monte Carlo method.