In this paper we produce evidence that confinement of colour is due to dual superconductivity of QCD vacuum. To do that we put together results of old numerical simulations and results of more recent investigations. The starting point is the expectation that gauge theories admit a dual description in terms of monopoles. The strategy is then to construct the creation operator μ of a monopole and to compute its vacuum expectation value ⟨ μ ⟩, the disorder parameter which indicates dual superconductivity. The mechanism of confinement is dual superconductivity of vacuum if ⟨ μ ⟩ ≠ 0 in the confined phase, and 〈 μ 〉 = 0 in the deconfined phase. Confinement has to be certified by independent methods. This procedure works well in compact U (1) gauge theory, as proved by numerical results on the lattice and by analytic treatments. In QCD and more generically in SU ( N ) gauge theories with N > 1 there are infinitely many monopoles, one for each SU (2) subgroup of the gauge group (Abelian Projection), and for decades it has not been clear which ones would condense to make the vacuum a dual superconductor. Attempts based on technically convenient assumptions, like equivalence of all abelian projections, led to pathologies like spurious infrared divergences spoiling the existence of the thermodynamical limit. Only recently it was realised that gauge invariance forbids most of the choices for the abelian projections and fixes it up to a global group transformation. The field strengths are replaced in this approach by gauge invariant field strengths, which are their parallel transports to infinity. The resulting disorder parameter can be expressed in terms of a sum of correlation functions of gauge invariant field strength, and its behaviour understood by use of existing lattice data of two-point gauge invariant correlation functions. As a byproduct an apparent existing inconsistency, the lack of preferred orientation in colour space of the chromo-electric field inside confining flux tubes, is resolved.

Understanding the approach to thermalization in a non-Abelian gauge theory

A bstract Open effective field theories provide a systematic framework for describing systems coupled to an environment, where dissipation, noise, and modified conservation laws naturally arise. Working within the Schwinger-Keldysh formalism, we examine open extensions of three well-studied theories: the superfluid, Maxwell theory, and Einstein gravity. In gauge and gravitational theories, open terms that break advanced symmetries while preserving physical ones are not automatically consistent; they are allowed only if they lead to deformed identities among the equations of motion. We explicitly construct such a term in open gravity and show that it leads to a consistent deformation of the diffeomorphism identities.

A bstract Symmetric product orbifolds provide a controlled environment to explore generic features of gauge theory and holography. The tractability of these theories lies in the complete characterisation of their gauge structure through holomorphic covering maps. In this paper, we introduce a novel class of generalised covering maps, which define a universal family of interfaces between symmetric product orbifolds. These interfaces coincide with the holographic interfaces that were recently proposed as duals to AdS 2 branes in pure NSNS AdS 3 backgrounds. The new covering-map description enables efficient evaluation of interface correlators via a generalisation of the Lunin-Mathur method. To organise these computations, we derive a generalised Riemann-Hurwitz formula for interface coverings and introduce novel diagrammatic rules that systematically classify these maps. The new framework allows us to define a concrete grand-canonical ensemble that has the correct properties to compute correlation functions dual to open string scattering amplitudes. Using the generalised Riemann-Hurwitz formula, we explicitly show that the correlators of the ensemble structurally match string perturbation theory to all orders in the string coupling.

The KP equation is perhaps the most famous example of a three-dimensional integrable system. Here we show that a non-commutative five-dimensional Chern-Simons theory living on the projective spinor bundle of three-dimensional space-time compactifies to a Lagrangian formulation of the KP equation. Essential to the definition of the theory is a 2-form pulled back from minitwistor space. The dispersionless limit of the KP equation is similarly described by Poisson-Chern-Simons theory. We further show that, consistent with integrability, all tree level amplitudes vanish. The universal vertex algebra living on a two-dimensional surface defect in $5d$ is $W_{1+\infty}$, and its operator products coincide with collinear splitting functions on space-time. Taking the dispersionless limit contracts the vertex algebra to $w_{1+\infty}$.

We investigate how isolated quantum many-body systems dynamically equilibrate under non-Abelian gauge-symmetry constraints. By encoding gauge superselection sectors into static SU(2) background charges, we map out the dynamical phase diagram of a (1+1)D SU(2) lattice gauge theory with dynamical matter. We uncover three distinct regimes: (i)an ergodic phase, (ii)a fragmented phase that is nonthermal but delocalized, and (iii)a disorder-free many-body localized regime. In the latter, a superposition of gauge superselection sectors preserves spatial matter inhomogeneities in time, as evidenced by distinct temporal scalings of entropy. We highlight the non-Abelian nature of these phases and argue for potential realizations on qudit processors.

Topological strings in SU(3) gauge theory at finite temperature

The cuprate pseudogap phase displays Fermi arc spectral weight in photoemission and scanning tunneling microscopy, while recent magnetotransport observations yield evidence for the existence of hole pockets of fractional area p/8, where p is the doping density. We present a Monte Carlo study of a thermal SU(2) lattice gauge theory which can reconcile these observations. Our simulation includes the SU(2) gauge field U of a π-flux spin liquid, and a SU(2) fundamental charge e Higgs boson B. There is a Yukawa coupling between B, the fermionic spinons of the spin liquid, and the hole pockets of a fractionalized Fermi liquid. At the higher temperatures of the pseudogap, the finite-doping sign problem is evaded by including only thermal fluctuations of B and U, while the fermions are diagonalized exactly for each boson background. Our study also yields a fractionalized description of intertwined orders at lower temperatures, including the onset of d-wave superconductivity by the expulsion of vortices with flux [Formula: see text], each with charge-order halos. We discuss conditions under which quantum oscillations in the density of states from hole pockets of area [Formula: see text] could be observable in clean under-hole-doped cuprates.

Abstract Finding a complete and yet minimal on-shell basis of operators of a given mass-dimension that are compatible with a specific set of transformation properties is the first step in any Effective Field Theory description. This step is the main bottleneck for systematic studies of leading logarithmic corrections to integer-power lattice artifacts in Symanzik Effective Field Theory targeting various local fields and lattice actions. The focus on discrete symmetry transformations in lattice field theory, especially reduced hypercubic spacetime symmetry with Euclidean signature, complicates the use of standard continuum field theory tools. Here, a new Python package is being presented that targets the typical lattice field-theorist’s use cases. While the main target lies on continuum EFTs describing 4D non-Abelian lattice gauge theories, the applicability can be extended beyond Effective Field Theories. New discrete symmetries, twisted masses, or the introduction of boosts are just a few examples of possible extensions that can be easily implemented by the user. This should allow for a wider range of theories and applications beyond the initial focus of this package. The general functionality of the package is explained along the lines of three examples: The $$\mathop {\textrm{O}}(a)$$ O ( a ) operator basis of the axial-vector in Wilson QCD, operator bases compatible with the symmetries of unrooted Staggered quarks as well as a pedestrian derivation of a $$B^*(\textbf{p})\pi (-\textbf{p})$$ B ∗ ( p ) π ( - p ) operator with pseudo-scalar quantum numbers. Each example makes use of an increasing range of features and requires user-defined extensions show-casing the versatility of the package.

Remarks on Constraint Analysis and Vacuum Persistence Amplitudes in Gauge Theories

A bstract We construct dimer graphs for relativistic Toda chains associated with classical untwisted Lie algebras of A, B, C 0 , C π , D types and twisted A, D types. We show that the Seiberg-Witten curve of 5d $$ \mathcal{N} $$ N = 1 pure supersymmetric gauge theory of gauge group G is a spectral curve of the relativistic Toda chain of the dual group G ∨ .

Quantum spin liquids give rise to exotic emergent particles by weaving intricate entanglement patterns in the underlying electrons. Bipartite measures between subregions can detect the presence of anyons, but little is known about the full entanglement structure of these phases. Here, we show that the multiparty entanglement of quantum spin liquids can be resolved via entanglement microscopy. We find that in contrast to conventional matter, the genuine multiparty entanglement between spins is absent in the smallest subregions, a phenomenon we call entanglement frustration. Instead, multiparty entanglement is more collective and arises solely in loops. By exploiting exact results and large-scale numerics, we confirm these properties in various gapped and gapless quantum spin liquids on the honeycomb and Kagome lattices, as well as with string-net wavefunctions hosting abelian or non-abelian anyons. Our results provide a framework for understanding fractionalization and the means by which gauge bosons encode quantum information, suggesting that entanglement loops are a universal property of quantum gauge theories.

We investigate the ( 2 + 1 )-dimensional q -deformed SU ( N ) k Yang-Mills theory in the lattice Hamiltonian formalism, which is characterized by three parameters: the number of colors N , the coupling constant g , and the level k . By treating these as tunable parameters, we explore how key properties of the theory, such as confinement and topological order, emerge in different regimes. Employing a variational mean-field analysis that interpolates between the strong- and weak-coupling regimes, we determine the large- N phase structure in terms of the ’t Hooft coupling λ tH = g 2 N and the ratio k / N . We find that the topologically ordered phase remains robust at large N under appropriate scalings of these parameters. This result indicates that the continuum limit of large- N gauge theory may be more intricate than naively expected, and it motivates studies beyond the mean-field theory, both to achieve a further understanding of confinement in gauge theories and to guide quantum simulations of large- N gauge theories.

Duality between correlation functions and Wilson loops in gauge theory from effective field theory

The unique off-shell fermionic gauge invariance of a vector-spinor field theory is found, and the invariant action is derived. The latter is Weyl invariant in any dimension in the massless limit, and it coincides with the singular point of the one-parameter family of Rarita-Schwinger Lagrangians, in agreement with previous findings in flat space. Pure gauge configurations are represented by gamma-trace vector-spinors, which can be gauged away in a global fashion. Previous claims that this theory is classically inconsistent are shown to be flawed, and the Velo-Zwanziger instability is proved to be absent. The theory propagates a massive spin- 3 2 particle together with a spin- 1 2 state whose mass is twice that of the j = 3 2 mode. The causal construction of the quantum field is consistent with the field equations in that the ratio of the masses is the same, while it shows that the lower-spin component is a negative-norm state. The conformal anomaly is derived using known results for the heat kernel of nonminimal second-order operators, and the resulting a charge is negative consistently with the Hofman-Maldacena bound, which applies only to unitary theories.

The Wilson formulation of fermions in lattice gauge theory provides a unified description of the chiral anomalies in the standard model. The discrete Dirac operator diagonalizes into a series of 2 × 2 blocks. In each block the possible eigenvalues either form a complex pair or separate into two real eigenvalues that have specific chirality. The collision of these pairs of eigenvalues occurs outside the perturbative region and provides a path between topological sectors.

A bstract We study the dynamics and phase structure of Abelian gauge theories in d = 1 + 1 dimensions. These include U(1) gauge theory coupled to a scalar and a fermion, as well as the two-flavour Schwinger model with different charges. Both theories exhibit a surprisingly rich phase diagram as masses are varied, with both c = 1 and c = 1/2 critical lines or points. We build up to the study of 2d chiral gauge theories, which hold particular interest because they provide a mechanism for symmetric mass generation, a phenomenon in which fermions become gapped without breaking chiral symmetries.

We test the eigenstate thermalization hypothesis (ETH) in 1 + 1 -dimensional SU(2) lattice gauge theory (LGT) with one flavor of dynamical fermions. Using the loop-string-hadron framework of the LGT with a bosonic cutoff, we exactly diagonalize the Hamiltonian for finite size systems and calculate matrix elements (MEs) in the eigenbasis for both local and nonlocal operators. We analyze different indicators to identify the parameter space for quantum chaos at finite lattice sizes and investigate how the ETH behavior emerges in both the diagonal and off-diagonal MEs. Our investigations allow us to study various timescales of thermalization and the emergence of random matrix behavior, and highlight the interplays of the several diagnostics with each other. Furthermore, from the off-diagonal MEs, we extract a smooth function that is closely related to the spectral function for both local and nonlocal operators. We find numerical evidence of the spectral gap and the memory peak in the nonlocal operator case. Finally, we investigate aspects of subsystem ETH in the lattice gauge theory and identify certain features in the subsystem reduced density matrix that are unique to gauge theories.

Generalized parton distribution functions (GPDs) are off-diagonal light-cone matrix elements that encode the internal structure of hadrons in terms of quark and gluon degrees of freedom. In this work, we present the first nonperturbative study of quasi-GPDs in the massive Schwinger model, quantum electrodynamics in 1 + 1 dimensions ( QED 2 ), within the Hamiltonian formulation of lattice field theory. Quasidistributions are spatial correlation functions of boosted states, which approach the relevant light-cone distributions in the luminal limit. Using tensor networks, we prepare the first excited state in the strongly coupled regime and boost it to close to the light-cone on lattices of up to 400 lattice sites. We compute both quasiparton distribution functions and, for the first time, quasi-GPDs, and study their convergence for increasingly boosted states. In addition, we perform analytic calculations of GPDs in the two-particle Fock-space approximation and in the Reggeized limit, providing qualitative benchmarks for the tensor network results. Our analysis establishes computational benchmarks for accessing partonic observables in low-dimensional gauge theories, offering a starting point for future extensions to higher dimensions, non-Abelian theories, and quantum simulations.

Abstract The universal phase i D +2 of the Euclidean de Sitter path integral obstructs a straightforward state-counting interpretation of the Gibbons–Hawking entropy. Building on Maldacena’s proposal that specific black-hole observers can reorganize this phase, we derive a general constraint on when such 'real observers' can succeed. By distinguishing gravitational observers from topological spectators , we show that any sector whose infrared effective action is metric independent at the de Sitter saddle factorizes in the path integral, Z tot = Z grav ( obs ) Z top , so the imaginary phase persists regardless of the sector’s information-processing capabilities. Using confining SU(3) gauge theory and topological orders as examples, we demonstrate that an information-bearing clock is necessary but insufficient: only observers whose fluctuations share the negative modes of the conformal factor belong to the special class that can remove the de Sitter phase.