We propose a new way to perform path integrals in quantum mechanics by using a quantum version of Hamilton-Jacobi (HJ) theory. In classical mechanics, Hamilton-Jacobi theory is a powerful formalism, however, its utility is not explored in quantum theory beyond approximation schemes. The canonical transformation enables one to set the new Hamiltonian to constant or zero, but keeps the information about solution in Hamilton’s characteristic function. To benefit from this in quantum theory, one must work with a formulation in which classical Hamiltonian is used. This uniquely points to phase space path integral. However, the main variable in HJ formalism is energy, not time. Thus, we are led to consider the Fourier transform of the path integral, the spectral path integral <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>Z</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></math>. The evaluation of path integrals reduces to determining the quantum Hamilton characteristic functions (which can be achieved via an asymptotic analysis) and a discrete sum over the quantum period lattice, generalizing Gutzwiller’s sum.

United States

Mustafa Türe

Mithat Ünsal

Quantum Hamilton-Jacobi theory, spectral path integrals, and exact WKB analysis

We analyze the phases of theories having a microscopic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">Z</mi><mi>N</mi></msub></math> 1-form symmetry, starting with a topological BF theory and deforming it so that only the microscopic symmetry is preserved. These theories have a well-defined notion of confinement, prototypical examples being pure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>SU</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">Z</mi><mi>N</mi></msub></math> gauge theories in the continuum and on the lattice. Our analysis shows that the generic phases are in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>=</mo><mn>2</mn></math>, only the confined phase; in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>=</mo><mn>3</mn></math>, both the confined phase and the topological BF phase; and in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>=</mo><mn>4</mn></math>, the confined phase, the topological BF phase, and a Coulomb phase. We construct a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">Z</mi><mi>N</mi></msub></math> lattice gauge theory with a deformation that, surprisingly, produces up to (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></math>) photons. We give an interpretation of these findings in terms of the behaviors of two competing drivers of confinement—magnetic monopoles and center vortices—and conclude that proliferation of center vortices is necessary but insufficient for confinement, while proliferation of magnetic monopoles is both necessary and sufficient.

Mendel Nguyen

United Kingdom

Tin Sulejmanpasic

Phases of Theories with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></math> 1-Form Symmetry, and the Roles of Center Vortices and Magnetic Monopoles

We address nonperturbative dynamics of the two-dimensional bosonic and supersymmetric −1 models for general by developing new tools directly on . The analysis starts with a new formulation of instantons that is consistent with the existence of the classical moduli space, classical dipole-dipole type interactions of instanton-anti-instanton pairs, and vanishing interaction of instanton-instanton pairs. The classical consistency is achieved via a representation of the instanton as a collection of pointlike constituents carrying pair of real and imaginary charges valued in the weight lattice of SU( ). The constituents interact via a generalized Coulomb interaction and do not violate the fact that instanton is a single lump with integer topological charge. By developing the appropriate Gibbs distribution, we show that the vacuum can be captured by a statistical field theory of these constituents, and their cluster expansion. Contrary to the common belief that instantons do not capture the vacuum structure and non-perturbation properties of such theories, our refined analysis is able to produce properties such as mass gap, theta dependence, and confinement of the theory on . In supersymmetric theory, our construction gives a new derivation of the mirror symmetry between the sigma model and the dual Landau-Ginzburg model by Hori and Vafa. Our construction also demonstrates that there is absolutely no conflict between large and instantons.

Refined instanton analysis of the 2D CP

There are two distinct regimes of Yang-Mills theory where we can demonstrate confinement, the existence of a mass gap, and the multi-branch structure of the effective potential as a function of the theta angle using a reliable semi-classical calculation. The two regimes are deformed Yang-Mills theory on $\textit{ℝ}$ × $\textit{S}$ , and Yang-Mills theory on $\textit{ℝ}$ × $\textit{T}$ where the torus is threaded by a ’t Hooft flux. The weak coupling regime is ensured by the small size of the circle or torus. In the first case the confinement mechanism is related to self-dual monopoles, whereas in the second case self-dual center-vortices play a crucial role. These two topological objects are distinct. In particular, they have different mutual statistics with Wilson loops. On the other hand, they carry the same topological charge and action. We consider the theory on $\textit{ℝ}$ × $\textit{T}$ × $\textit{S}$ and extrapolate both the monopole and vortex regimes to a quantum mechanical domain, where a cross-over takes place. Both sides of the cross-over are described by a deformed $\textit{ℤ}$ TQFT. On $\textit{ℝ}$ × $\textit{S}$ × $\textit{S}$ , we derive an effective field theory (EFT) of vortices from the EFT of monopoles in the presence of a ’t Hooft flux. This construction is based on a two-stage Higgs mechanism, reducing SU($\textit{N}$) to U(1) in 3d first, followed by reduction to a $\textit{ℤ}$ EFT in 2d in the second step. This result shows how monopoles transmute into center-vortices, and suggests adiabatic continuity between the two confinement mechanisms. The basic mechanism is flux fractionalization: the magnetic flux of the monopoles splits up and is collimated in such a way that 2d Wilson loops detect it as a center vortex.

Canberk Güvendik

Thomas Schaefer

The metamorphosis of semi-classical mechanisms of confinement: from monopoles on $\textit{ℝ}$

We present our investigations of SU(N) adjoint QCD in two dimensions with one Majorana fermion on the lattice. We determine the relevant parameter range for the simulations with Wilson fermions and present results for Polyakov loop, chiral condensate, and string tension. In the theory with massive fermions, all observables we checked show qualitative agreement between numerical lattice data and theory, while the massless limit is more subtle since chiral and non-invertible symmetry of the continuum theory are explicitly broken by lattice regularization. In thermal compactification, we observe N perturbative vacua for the holonomy potential at high-T with instanton events connecting them, and a unique vacuum at low-T. At finite-N, this is a cross-over and it turns to a phase transition at large-N thermodynamic limit. In circle compactification with periodic boundary conditions, we observe a unique center-symmetric minimum at any radius. In continuum, the instantons in the thermal case carry zero modes (for even N) and indeed, in the lattice simulations, we observe that chiral condensate is dominated by instanton centers, where zero modes are localized. We present lattice results on the issue of confinement vs. screening in the theory and comment on the roles of chiral symmetry and non-invertible symmetry.

Germany

Georg Bergner

Stefano Piemonte

Investigating two-dimensional adjoint QCD on the lattice

While the θ dependence of field theories is 2π periodic, the ground-state wavefunctions at θ and θ + 2π often belong to different classes of symmetry-protected topological states. When this is the case, a continuous change of the θ parameter can introduce an interface that supports a nontrivial field theory localized on the wall. We consider the 2d ℂP  N−1 sigma model as an example and construct a weak-coupling setup of this interface theory by considering the small S 1 compactification with nonzero winding θ parameter and a suitable symmetry-twisted boundary condition. This system has N classical vacua connected by fractional instantons, but the anomaly constraint tells us that the fractional-instanton amplitudes should vanish completely to have N-fold degeneracy at the quantum level. We show how this happens in this purely bosonic system, uncovering that the integration over the zero modes annihilates the fractional instanton amplitudes, in sharp contrast to what happens when the θ angle is constant. Moreover, we provide another explanation of this selection rule by showing that the N perturbative vacua acquire different charges under the global symmetry with the activation of the winding θ angle. We also demonstrate a similar destructive interference between instanton effects in the ℂP  N−1 quantum mechanics with the Berry phase.

Japan

Yuya Tanizaki

Winding θ and destructive interference of instantons

To study gapped phases of 4d gauge theories, we introduce the temporal gauging of ℤ N 1-form symmetry in 4d quantum field theories (QFTs), thereby defining effective 3d QFTs with <math> <mover> <mi>ℤ</mi> <mo>~</mo> </mover> </math> $$ \overset{\sim }{\mathbb{Z}} $$ N × ℤ N 1-form symmetry. In this way, spatial fundamental Wilson and ’t Hooft loops are simultaneously genuine line operators. Assuming a mass gap and Lorentz invariant vacuum of the 4d QFT, the <math> <mover> <mi>ℤ</mi> <mo>~</mo> </mover> </math> $$ \overset{\sim }{\mathbb{Z}} $$ N × ℤ N symmetry must be spontaneously broken to an order-N subgroup H, and we can classify the 4d gapped phases by specifying H. This establishes the 1-to-1 correspondence between the two classification schemes for gapped phases of 4d gauge theories: one is the conventional Wilson-’t Hooft classification, and the other is the modern classification using the spontaneous breaking of 4d 1-form symmetry enriched with symmetry-protected topological states.

Study of gapped phases of 4d gauge theories using temporal gauging of the ℤ N  1-form symmetry

In the UY reconfined phase on a lattice the thermal trace is over states transforming in all <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo><mo>/</mo><mi>Z</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> irreducible representations as opposed to only over <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> singlets in the standard formulation. As <math><mi>N</mi><mo>→</mo><mo>∞</mo></math>, on a finite lattice, the usual Hilbert space becomes orthogonal to the deformed one. Concerns about the extended UY proposal for large N Eguchi-Kawai reduction are raised.

Herbert Neuberger

Deconstructing Ünsal-Yaffe reconfinement

Quantum deformed potentials arise naturally in quantum mechanical systems of one bosonic coordinate coupled to <math><msub><mi>N</mi><mi>f</mi></msub></math> Grassmann valued fermionic coordinates, or to a topological Wess-Zumino term. These systems decompose into sectors with a classical potential plus a quantum deformation. Using exact Wentzel-Kramers-Brillouin, we derive exact quantization condition and its median resummation. The solution of median resummed form gives physical Borel-Ecalle resummed results, as we show explicitly in quantum deformed double- and triple-well potentials. Despite the fact that instantons have finite actions, for generic quantum deformation, they do not contribute to the energy spectrum at leading order in semiclassics. For certain quantized quantum deformations, where the alignment of levels to all order in perturbation theory occurs, instantons contribute to the spectrum. If deformation parameter is not properly quantized, their effect disappears, but higher-order effects in semiclassics survive. In this sense, we classify saddle contributions as fading and robust. Finally, for quantum deformed triple-well potential, we demonstrate the perturbative/nonperturbative relation, by computing period integrals and Mellin transform.

Poland

Syo Kamata

Tatsuhiro Misumi

Naohisa Sueishi

Exact WKB analysis for SUSY and quantum deformed potentials: Quantum mechanics with Grassmann fields and Wess-Zumino terms

We consider the massless charge-N Schwinger model and its deformation with two four-fermion operators. Without the deformations, this model exhibits chiral symmetry breaking without confinement. It is usually asserted that the massless Schwinger model is always deconfined and a string tension emerges only when a mass for the fermion field is turned on. We show that in the presence of these four-fermion operators, the massless theory can in fact confine. One of the four-fermion deformations is chirally neutral, and is a marginal deformation. The other operator can be relevant or irrelevant, and respects a ℤ2 subgroup of chiral symmetry for even N, hence forbidding a mass term. When it is relevant, even the exactly massless theory exhibits both confinement and spontaneous chiral symmetry breaking. The construction is analogous to QCD(adj) in 2d. While the theory without four-fermion deformations is deconfined, the theory with these deformations is generically in a confining phase. We study the model on ℝ2 using bosonization, and also analyze the mechanism of confinement on ℝ × S 1, where we find that confinement is driven by fractional instantons.