Yuya Tanizaki

A bstract: Yang-Mills theories at θ and θ +2 π are unitarily equivalent, but their 2 π periodicity has a nontrivial realization. Recent developments in generalized symmetries rigorously prove that confinement vacua at θ = 0 and 2 π should belong to different symmetry-protected topological (SPT) states with the 1-form center symmetry. For its examination, we measure the Wilson-’t Hooft loop operators at θ = 2 π for the SU(2) Wilson lattice gauge action and discuss their long-distance behaviors. This requires us to identify the gauge topological charge in the presence of defects, and we employ the 1-form covariant DBW2 gradient flow to smear lattice gauge fields. We find a clear perimeter-law signal for the dyonic Wilson-’t Hooft loop at θ = 2 π , providing numerical evidence in the pure SU(2) Yang-Mills theory for the theoretically expected dyon condensation at θ = 2 π .

We propose gauge-invariant criteria for center-vortex condensation and monopole condensation using the $\mathbb {Z}_N^{[1]}$-symmetry twisted partition functions: the torus twisted partition function characterizes the center-vortex condensation, and the lens-space twisted partition function characterizes the monopole condensation. To justify our proposal, we study how these twisted partition functions behave in the adjoint Higgs phase and show that their leading nontrivial contributions come from the center vortex and monopole, respectively. Using the techniques of topological field theories, we uncover the relationship between the center-vortex and monopole condensations, and, in particular, we prove that the gapped phase with the center-vortex condensation necessarily shows the monopole condensation, too. We then study a center-vortex model with monopoles as an illustrative example, and the higher-charge monopole condensation gives an example of symmetry fractionalization, which goes beyond the conventional Wilson–’t Hooft classification.

A bstract: We investigate the long-distance behavior of dyonic loop operators in 4d SU( N ) gauge theories on ℝ ³ × S ¹ 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 _N domain walls. We confirm that the domain-wall state has the phase transition at θ = π in the thermal deconfined phase despite the fact that the bulk state is smooth there.

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 O ( 2 ) 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 T c = 2.2017 ( 2 ) with the critical exponent ν = 0.663 ( 33 ) for the 3D O ( 2 ) model. In addition, we also study the Berezinskii–Kosterlitz–Thouless (BKT) criticality of the 2D classical O ( 2 ) model by extracting the helicity modulus from the twisted partition functions, and we obtain the BKT transition temperature, T BKT = 0.8928 ( 2 ) .

A bstract: We derive the explicit formula for fractional BPS lumps (or fractional instantons) in the ℂP ^{N  − 1} nonlinear sigma model on a two-dimensional torus under various shift-clock twisted boundary conditions. After regularizing the ℂP ^{N  − 1} model by an N -component Abelian-Higgs model, those twisted boundary conditions introduce nontrivial ’t Hooft fluxes p / N for the U(1) gauge field, and the topological charge becomes fractionalized as k + p / N ∈ ℤ + p / N . The moduli space is globally determined as the ℂP ^{Nk  +  p  − 1} -fiber bundle on a 2-torus, which is a Kähler manifold of complex dimension Nk + 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 ℂP ^{N  − 1} limit, while the other makes the modular invariance manifest. We also discuss the implications of our finding for the 4d SU( N ) Yang-Mills theory on the 4-torus with ’t Hooft twists. By tuning the aspect ratio of the 4-torus, fractional instantons in the ℂP ^{N  − 1} model with a non-Fubini-Study metric are obtained through the dimensional reduction of 4d Yang-Mills theory, whose moduli space coincides with the one obtained for the standard ℂP ^{N  − 1} model as complex manifolds.

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

We study the [ ] super Yang–Mills (SYM) theory on [ ] with ’t Hooft twist. This setup allows us to interpolate between the monopole and center-vortex semiclassical frameworks, which have been proposed for understanding confinement. Based on this unified perspective, we revisit the semiclassical analysis of [ ] super Yang–Mills theory and elucidate aspects not fully understood previously. Among a few puzzles, we address the underlying mechanism for the switching of the Wilson loop behavior: area law in 3d and perimeter law in 2d. This proceedings paper is based on Ref. 1 .

A bstract: 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 θ angles. We reformulate the 2d version of this model using the modified Villain lattice formalism, and we establish the exact 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.

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.