We investigate the long-distance behavior of dyonic loop operators in 4d SU($\textit{N}$) gauge theories on ℝ × $\textit{S}$ using the 3d monopole semiclassics. If we employ the naive definition of the ’t Hooft loop in the Abelianized regime, the dyonic loop operators do not admit the well-defined computations within the effective field theory. Moreover, if one forcibly proceeds with the computations of their expectation values, all the dyonic loops turn out to show the area law, which contradicts the prediction of the Wilson-’t Hooft classification. In this paper, we resolve this puzzle by employing the notion of screening for line operators, and we argue that the dyonic loops are screened by a defect known as the twist vortex, which is non-dynamical in the infrared effective theory but is dynamical in the original ultraviolet theory. The dyonic loops properly dressed by twist vortices admit the well-defined computations within the effective field theory, and we reproduce the kinematic prediction of the Wilson-’t Hooft classification using the 3d monopole semiclassics. Furthermore, we apply our framework to the thermal deconfined phase to evaluate the dual string tension, elucidating the topological nature of ℤ domain walls. We confirm that the domain-wall state has the phase transition at $\textit{θ}$ = $\textit{π}$ in the thermal deconfined phase despite the fact that the bulk state is smooth there.

Japan

Yui Hayashi

Yuya Tanizaki

Wilson-’t Hooft classification and the perimeter law for dyonic loops in 3d monopole semiclassics

<p>The locality of field theories strongly constrains the possible behaviors of symmetry-twisted partition functions, and thus they serve as order parameters to detect low-energy realizations of global symmetries, such as spontaneous symmetry breaking (SSB). We demonstrate that the tensor renormalization group (TRG) offers an efficient framework to compute the symmetry-twisted partition functions, which enables us to detect the symmetry-breaking transition and also to study associated critical phenomena. As concrete examples of SSB, we investigate the two-dimensional (2D) classical Ising model and the three-dimensional (3D) classical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></math> nonlinear sigma model, and we identify their critical points solely from the twisted partition function. By employing the finite-size scaling argument, we find the critical temperature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mi>c</mi></msub><mo>=</mo><mn>2.2017</mn><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></math> with the critical exponent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ν</mi><mo>=</mo><mn>0.663</mn><mo stretchy="false">(</mo><mn>33</mn><mo stretchy="false">)</mo></math> for the 3D <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></math> model. In addition, we also study the Berezinskii–Kosterlitz–Thouless (BKT) criticality of the 2D classical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></math> model by extracting the helicity modulus from the twisted partition functions, and we obtain the BKT transition temperature, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mi>BKT</mi></msub><mo>=</mo><mn>0.8928</mn><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></math>.</p>

Shinichiro Akiyama

United States

Raghav G. Jha

Jun Maeda

Judah Unmuth-Yockey

Tensor renormalization group approach to critical phenomena via symmetry-twisted partition functions

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.

Tatsuhiro Misumi

Muneto Nitta

Keisuke Ohashi

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

We consider the semiclassical description of confinement for 4d SU($\textit{N}$) Yang-Mills theory on small $\textit{ℝ}$ × $\textit{T}$ with non-minimal ’t Hooft twist $\textit{p}$ with gcd($\textit{N}$, $\textit{p}$) = 1. For this purpose, we construct the self-dual center vortex for non-minimal ’t Hooft twists from the Kraan-van Baal-Lee-Lu-Yi (KvBLLY) monopoles by using the 3d Abelianized description of SU($\textit{N}$) gauge fields on $\textit{ℝ}$ × $\textit{S}$ with nontrivial holonomy backgrounds. This construction shows the self-dual vortex has (1) the fractional magnetic charge $\textit{q}$/$\textit{N}$ with $\textit{pq}$ = 1 mod $\textit{N}$, (2) the fractional topological charge 1/$\textit{N}$, and (3) the fractional instanton action $\textit{S}$ = 8$\textit{π}$ /($\textit{Ng}$ ). The confinement vacua for $\textit{NL}$Λ ≪ 1 can be described by the dilute gas approximation of center vortices, and we give the semiclassical formula for the $\textit{θ}$ dependence and confining string tensions. We apply this result to understand the suitable choice of the twist $\textit{p}$ for center stabilization at large $\textit{N}$. In particular, we test the proposal using the Fibonacci sequence, $\textit{N}$ = $\textit{F}$ and $\textit{p}$ = $\textit{F}$ , suggested in studies of the twisted Eguchi-Kawai model, from the viewpoint of the 1-form and 0-form center symmetries.

Mithat Ünsal

Center-vortex semiclassics with non-minimal ’t Hooft fluxes on $\textit{ℝ}$ 2 × $\textit{T}$ 2 and center stabilization at large $\textit{N}$

The Cardy-Rabinovici model is a toy model of the lattice U(1) gauge theories to study various oblique confinement states associated with the nonzero $\textit{θ}$ angles. We reformulate the 2d version of this model using the modified Villain lattice formalism, and we establish the exact $\textit{θ}$ periodicity for the Witten effect and the strong-weak duality at the finite lattice spacings. We then study the phase structure of this model based on the duality, symmetry and anomaly, and the perturbative renormalization group.

Nagare Katayama

2d Cardy-Rabinovici model with the modified Villain lattice: exact dualities and symmetries

For quantum field theories with global symmetry, we can study the behavior of the partition function with the background gauge field to diagnose different quantum phases. For the case of discrete symmetries, we find that the symmetry-twisted partition function works as an order parameter that discriminates spontaneous symmetry breaking (SSB), symmetry-protected topological (SPT) states, and symmetry-enriched topological (SET) states. We then consider its application to the case of 4d Yang-Mills theory with adjoint matters to understand the relation between the twisted partition function and the Wilson-’t Hooft classification. We also study its behavior for the spontaneously broken U(1) symmetry and interpret the result from the viewpoint of the mixed anomaly with the emergent solitonic symmetry.

Twisted partition functions as order parameters

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 investigate the stability of topological charge under gradient flow taking the admissibility condition into account. For the SU(2) Wilson gauge theory with $\textit{β}$ = 2$\textit{.}$45 and $\textit{L}$ = 12 , we numerically show that the gradient flows with the Iwasaki and DBW2 gauge actions stabilize the topological sectors significantly, and they have qualitatively different behaviors compared with the Wilson and tree-level Symanzik flows. By considering the classical continuum limit of the flow actions, we discuss that the coefficient of dimension-6 operators has to be positive for stabilizing the one-instanton configuration, and the Iwasaki and DBW2 actions satisfy this criterion while the Wilson and Symanzik actions do not. Moreover, we observe that the DBW2 flow stabilizes the topological sectors at the very early stage of the flow ( $$ \hat{t} $$ ≈ 0$\textit{.}$5–1), suggesting that a further systematic investigation of the DBW2 flow is warranted to confirm its computational efficiency in determining the gauge topology.

Akio Tomiya

Hiromasa Watanabe

Lattice gradient flows (de-)stabilizing topological sectors

Magnetic excitations play a crucial role in understanding the color confinement of 4D Yang-Mills theory, and we have the monopole and the center vortex as plausible candidates to explain its mechanism. Under suitable compactified setups of 4D Yang-Mills theory, we can achieve different weakly coupled descriptions of confinement phenomena: The monopole mechanism takes place on <math><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></mrow></math> with the double-trace deformation, and the center-vortex mechanism is effective on <math><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math> with the ’t Hooft flux. We unify these two semiclassical descriptions by showing the explicit relation between the monopole and center vortex.

Unifying Monopole and Center Vortex as the Semiclassical Confinement Mechanism

We study the θ-dependent mass spectrum of the massive 2-flavor Schwinger model in the Hamiltonian formalism using the density-matrix renormalization group (DMRG). The masses of the composite particles, the pion and sigma meson, are computed by two independent methods. One is the improved one-point-function scheme, where we measure the local meson operator coupled to the boundary state and extract the mass from its exponential decay. Since the θ term causes a nontrivial operator mixing, we unravel it by diagonalizing the correlation matrix to define the meson operator. The other is the dispersion-relation scheme, a heuristic approach specific to Hamiltonian formalism. We obtain the dispersion relation directly by measuring the energy and momentum of the excited states. The sign problem is circumvented in these methods, and their results agree with each other even for large θ. We reveal that the θ-dependence of the pion mass at m/g = 0.1 is consistent with the prediction by the bosonized model. We also find that the mass of the sigma meson satisfies the semi-classical formula, M  σ  /M  π  =   <math> <msqrt> <mn>3</mn> </msqrt> </math>  $$ \sqrt{3} $$ , for almost all region of θ. While the sigma meson is a stable particle thanks to this relation, the eta meson is no longer protected by the G-parity and becomes unstable for θ ≠ 0.