We propose a novel, machine-learning-based framework for constructing lattice fermions using physics-informed neural networks (PINNs). Our approach treats the formulation of the Dirac operator as an optimization problem guided by physical requirements, such as symmetries, locality and doubler-decoupling conditions. We first demonstrate that, when trained to satisfy the Ginsparg-Wilson (GW) relation as a soft constraint, a neural network reproduces the overlap fermion operator to high numerical accuracy and learns an effective sign-function mapping without explicitly using a prescribed polynomial or rational approximation. Second, we extend the framework from operator construction to machine-assisted algebraic discovery. Within a generalized polynomial ansatz, the network autonomously drives higher-order terms to zero and recovers the standard Ginsparg-Wilson relation. Remarkably, by changing the initial search bias, the same framework also finds a distinct solution corresponding to a Fujikawa-type generalized GW relation.

Japan

Tatsuhiro Misumi

Lattice fermion formulation via physics-informed neural networks: Ginsparg-Wilson relation and overlap fermions

We discuss the exact quantization of general one-dimensional potentials in view of the exact-WKB formalism. Building on our previous work, we perform analytic continuations across different sectors via the complexification to the spectral (energy) parameter $\textit{u}$ and identify continuous and discontinuous transitions of the exact spectrum for generic potentials. When the transition is discontinuous, it is characterized by the Stokes phenomena, inducing different exact (median) quantization conditions, thereby distinct trans-series structures valid in different sectors. We analyze two illustrative examples, namely asymmetric triple-well (ATW) and tilted double-well (TDW), and verify the general qualitative analysis by deriving exact (median) quantization conditions in each sector. Moreover, by obtaining the trans-series solutions for each system, we identify bion/bounce configurations and show that the trans-series of ATW is organized in accordance with the cluster expansion of the bion gas and there should exist a previously neglected complex saddle in the TDW system. These identifications further strengthen the link between path integral and exact-WKB formalisms, while also demonstrating the predictive power of the latter. In parallel, for the P-NP relations of genus-1 systems, we derive transformation rules between any perturbative and non-perturbative pair of WKB-cycles. Our results show that the entire resurgence data of a genus-1 system transforms only by the change of classical parameters, i.e. frequencies and bion/bounce actions, and the perturbative energy series. This also reveals the underlying reasons of the previously found $\textit{S}$-duality transformations.

Cihan Pazarbaşı

Exact WKB in all sectors. Part II. Potentials with non-degenerate saddles

We develop a systematic Hamiltonian formulation of minimally doubled lattice fermions in (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>3</mn><mo>+</mo><mn>1</mn></mrow></math>) dimensions, derive their nodal structures (structures of zeros), and classify their symmetry patterns for both four-component Dirac and two-component Weyl constructions. Motivated by recent single-Weyl proposals based on Bogoliubov-de Gennes representation, we argue that the corresponding single-Weyl Hamiltonians are obtained from the minimal-doubling Hamiltonians supplemented by an appropriate species-splitting mass term, and we reexamine the non-on-site symmetry protecting the physical Weyl node in terms of a Ginsparg-Wilson-type relation. We then construct a one-parameter family of deformations that preserves all the symmetries and demonstrate that, once the parameter exceeds a critical value, additional Weyl nodes emerge and the system exits the single-node regime. This indicates that in interacting theories radiative corrections can generate symmetry-allowed counterterms, so maintaining the desired single-Weyl phase generically requires “moderate” parameter tuning.

Minimal-doubling and single-Weyl fermion Hamiltonians

We derive the explicit formula for fractional BPS lumps (or fractional instantons) in the $\textit{ℂP}$ nonlinear sigma model on a two-dimensional torus under various shift-clock twisted boundary conditions. After regularizing the $\textit{ℂP}$ model by an $\textit{N}$-component Abelian-Higgs model, those twisted boundary conditions introduce nontrivial ’t Hooft fluxes $\textit{p}$/$\textit{N}$ for the U(1) gauge field, and the topological charge becomes fractionalized as $\textit{k}$ + $\textit{p}$/$\textit{N}$ ∈ $\textit{ℤ}$ + $\textit{p}$/$\textit{N}$. The moduli space is globally determined as the $\textit{ℂP}$ -fiber bundle on a 2-torus, which is a Kähler manifold of complex dimension $\textit{Nk}$ + $\textit{p}$ as predicted by the index theorem. We present two different parametrizations of the moduli space: one of them immediately identifies the small-lump singularity appearing in the $\textit{ℂP}$ limit, while the other makes the modular invariance manifest.

Yui Hayashi

Muneto Nitta

Keisuke Ohashi

Yuya Tanizaki

Fractional instantons in 2d $\textit{ℂP}$ $\textit{N}$ − 1 model and 4d Yang-Mills theory with ’t Hooft twists

We investigate the twisted Eguchi-Kawai (TEK) reduced model of four-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> gauge theory in the presence of two-flavor adjoint fermions (adjoint TEK model). Using Monte Carlo simulations with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>=</mo><mn>121</mn></math>, twist parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi><mo>=</mo><mn>1</mn></math>, hopping parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>κ</mi><mo>=</mo><mn>0.01</mn><mi>–</mi><mn>0.03</mn></mrow></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>≪</mo><msub><mi>κ</mi><mi>c</mi></msub></math>) and inverse ’t Hooft coupling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mo>=</mo><mn>0.30</mn><mi>–</mi><mn>0.45</mn></math>, we show that heavy adjoint fermions stabilize the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>N</mi></msub><msup><mo stretchy="false">)</mo><mn>4</mn></msup></math> center-symmetric vacuum even for the minimal twist satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi><mo>/</mo><msqrt><mi>N</mi></msqrt><mo>&lt;</mo><mn>1</mn><mo>/</mo><mn>9</mn></math>, where the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>N</mi></msub><msup><mo stretchy="false">)</mo><mn>4</mn></msup></math> symmetry is spontaneously broken in the absence of adjoint fermions. This result also suggests that the adjoint TEK model with the minimal twist is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> gauge theory over a broader <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><mi>κ</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></math> parameter region than the adjoint EK model without twist. We further extend our analysis to a partially reduced model to realize a geometry akin to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup><mo>×</mo><msup><mi>S</mi><mn>1</mn></msup></math> and study the theory under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>S</mi><mn>1</mn></msup></math> compactification with periodic adjoint fermions. Numerical simulations with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>=</mo><mn>16</mn><mi>–</mi><mn>49</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mo>=</mo><mn>0.30</mn><mi>–</mi><mn>0.46</mn></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>κ</mi><mo>=</mo><mn>0.03</mn><mi>–</mi><mn>0.16</mn></math> are consistent with the adiabatic continuity conjecture; with periodic adjoint fermions, the theory remains in a center-symmetric (confined) phase as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>S</mi><mn>1</mn></msup></math> circle size is reduced, in contrast to the deconfining transition observed in the pure TEK model or in the TEK model with antiperiodic adjoint fermions. We present the Polyakov loop measurements and consistency checks supporting these findings.

Yudai Hamada

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">Z</mi><mi>N</mi></msub></math> stability and continuity in the twisted Eguchi-Kawai model with two-flavor adjoint fermions

We explore the application of the exact Wentzel–Kramers–Brillouin (WKB) analysis to two-level Floquet systems and establish a systematic procedure to calculate the quasi-energy and Floquet effective Hamiltonian. We show that, in the exact-WKB analysis, the quasi-energy and Floquet effective Hamiltonian can be expressed in terms of cycle integrals (Voros symbol), which characterize monodromy matrices for Schrödinger-type differential equations governing two-level Floquet systems. We analytically evaluate the cycle integrals using the low-frequency expansion and derive both perturbative and nonperturbative corrections to the quasi-energy and Floquet effective Hamiltonian. To verify the accuracy of our results, we compare them with numerical calculations and analyze resonant oscillations, which reveal nonperturbative features that cannot be captured by the perturbative expansion.

Toshiaki Fujimori

Syo Kamata

Naohisa Sueishi

Hidetoshi Taya

Exact Wentzel–Kramers–Brillouin (WKB) Analysis of Two-Level Floquet Systems

We study $$ \mathcal{N} $$ = 1 SU($\textit{N}$) super Yang-Mills (SYM) theory on $\textit{ℝ}$ × ($\textit{S}$ ) × ($\textit{S}$ ) with the ’t Hooft twist. The theory becomes weakly coupled if the length $\textit{L}$ of ($\textit{S}$ ) is sufficiently small, $\textit{NL}$ Λ ≪ 1. We explore the nonperturbative dynamics at the weak-coupling regime by changing the size of $\textit{L}$ and uncover how 3d monopole/bion-based effective theory for $\textit{L}$ ≫ $\textit{L}$ is related to the 2d vortex-based theory for $\textit{L}$ ≈ $\textit{L}$ . The highlights of our results are (1) the smooth “weak-weak” continuity of the vacuum structure and gluino condensate during the 3d-2d dimensional reduction, (2) the switching of Wilson loop behavior from the area law in 3d to the perimeter law in 2d via a “double-string” picture, (3) the role of mass deformation in breaking discrete chiral symmetry and restoring the area law in 2d, and (4) the microscopic investigation of bions during the reduction from 3d to 2d and the cancellation of the vacuum energy due to the hidden topological angle. We also discuss the generalization of our results for (1)–(3) from $$ \mathcal{N} $$ = 1 SYM to QCD with adjoint quarks.

Monopole-vortex continuity of $$ \mathcal{N} $$ = 1 super Yang-Mills theory on $\textit{ℝ}$

We explore the exact-WKB (EWKB) method through the analysis of Airy and Weber types, with an emphasis on the exact quantization of locally harmonic potentials in multiple sectors. The core innovation of our work lies in introducing a novel complexification approach to the energy parameter $\textit{u}$, distinct from the common complexification of the (semi-classical) expansion parameter used in Borel summability. This new technique allows for continuous analytical continuation across different sectors of a potential while maintaining the exact quantization condition, even before median summation. By redefining the $\textit{A}$-cycle above the potential barrier top, we ensure the quantization condition remains real and, by use of the Stokes automorphism and the median resummation, show that the resurgence structure is preserved across transitions between sectors. Furthermore, we discuss the Weber-type exact-WKB method, offering exact estimates for quantum actions around all types of saddle points, generalizing previous results. Through the analysis of these quantum actions, we reveal the presence of $\textit{S}$-duality, facilitating the exchange between perturbative and non-perturbative behaviors, and we conjecture the mapping of the P-NP relations between dual theories. Our study encompasses periodic and symmetric double-well potentials, demonstrating that the exact-WKB method captures intricate structures in quantum systems in all sectors, including multi-instanton contributions and the resurgence of quantum actions.

Exact WKB in all sectors. Part I. Potentials with degenerate saddles

Abstract We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix called an “antisymmetrized adjacency matrix”, specifically tailored for cycle digraphs (T$^{1}$ lattice) and simple directed paths (B$^{1}$ lattice). The nontrivial relationship between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the antisymmetrized adjacency matrix, along with its extensions to higher dimensions, is equivalent to naive lattice Dirac operators. Building upon these connections, we provide rigorous proofs for two key assertions: (i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice). (ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the D-dimensional graph results from a Cartesian product of cycle digraphs (T$^{1}$ lattice) and simple directed paths (B$^{1}$ lattice).

Jun Yumoto

Equivalence of lattice operators and graph matrices Quarks and strings on a lattice

Abstract We propose a new conjecture on the relation between the exact Dirac zero-modes of free and massless lattice fermions and the topology of the manifold on which the fermion action is defined. Our conjecture claims that the maximal number of exact Dirac zero-modes of fermions on finite-volume and finite-spacing lattices defined by a discretizing torus, hyperball, their direct-product space, and hypersphere is equal to the summation of the Betti numbers of their manifolds if several specific conditions on lattice formulations are satisfied. We start with reconsidering exact Dirac zero-modes of naive fermions on the lattices whose topologies are a torus, hyperball, and their direct-product space (T$^{D}$ × B$^{d}$). We find that the maximal number of exact zero-modes of free Dirac fermions is in exact agreement with the sum of Betti numbers  for these manifolds. Indeed, the 4D lattice fermion on a torus has up to 16 zero-modes while the sum of Betti numbers of T$^{4}$ is 16. This coincidence holds also for the D-dimensional hyperball and their direct-product space T$^{D}$ × B$^{d}$. We study several examples of lattice fermions defined on a certain discretized hypersphere (S$^{D}$), and find that it has up to two exact zero-modes, which is the same number as the sum of Betti numbers of S$^{D}$. From these facts, we conjecture the equivalence of the maximal number of exact Dirac zero-modes and the summation of Betti numbers under specific conditions. We discuss a program for proof of the conjecture in terms of Hodge theory and spectral graph theory.