The resurgence structure of the 2d O(N) sigma model at large N is studied with a focus on an IR momentum cutoff scale a that regularizes IR singularities in the semiclassical expansion. Transseries expressions for condensates and correlators are derived as series of the dynamical scale Λ (nonperturbative exponential) and coupling λ  μ  renormalized at the momentum scale μ. While there is no ambiguity when a &gt; Λ, we find for a &lt; Λ that the nonperturbative sectors have new imaginary ambiguities besides the well-known renormalon ambiguity in the perturbative sector. These ambiguities arise as a result of an analytic continuation of transseries coefficients to small values of the IR cutoff a below the dynamical scale Λ. We find that the imaginary ambiguities are cancelled each other when we take all of them into account. By comparing the semiclassical expansion with the transseries for the exact large-N result, we find that some ambiguities vanish in the a → 0 limit and hence the resurgence structure changes when going from the semiclassical expansion to the exact result with no IR cutoff. An application of our approach to the ℂP  N−1 sigma model is also discussed. We find in the compactified model with the ℤ N  twisted boundary condition that the resurgence structure changes discontinuously as the compactification radius is varied.

Japan

Hiromichi Nishimura

Toshiaki Fujimori

Tatsuhiro Misumi

Muneto Nitta

Norisuke Sakai

Resurgence and semiclassical expansion in two-dimensional large-N sigma models

We discuss the deconfinement and the <math><mrow><mi>C</mi><mi>P</mi></mrow></math>-breaking phase transitions at <math><mi>θ</mi><mo>=</mo><mi>π</mi></math> in Yang-Mills theories. The ’t Hooft anomaly matching prohibits the confined phase with <math><mrow><mi>C</mi><mi>P</mi></mrow></math> symmetry and requires <math><msub><mi>T</mi><mi>dec</mi></msub><mo>(</mo><mi>θ</mi><mo>=</mo><mi>π</mi><mo>)</mo><mo>≤</mo><msub><mi>T</mi><mi>CP</mi></msub></math>, where <math><msub><mi>T</mi><mi>dec</mi></msub><mo>(</mo><mi>θ</mi><mo>=</mo><mi>π</mi><mo>)</mo></math> and <math><msub><mi>T</mi><mi>CP</mi></msub></math> denote the deconfinement and the <math><mrow><mi>C</mi><mi>P</mi></mrow></math>-restoration temperatures, respectively, at <math><mi>θ</mi><mo>=</mo><mi>π</mi></math>. We analytically study these two phase transitions in softly broken <math><mi>N</mi><mo>=</mo><mn>1</mn></math> supersymmetric Yang-Mills theories on small <math><msup><mi>R</mi><mn>3</mn></msup><mo>×</mo><msup><mi>S</mi><mn>1</mn></msup></math> with the periodic boundary condition for gluinos. For most gauge groups except SU(2) in this model, we find that the inequality is saturated, so deconfinement and <math><mrow><mi>C</mi><mi>P</mi></mrow></math> restoration occur simultaneously. We demonstrate special features of the SU(2) gauge theory: there is a finite window of two temperatures, <math><msub><mi>T</mi><mi>dec</mi></msub><mo>(</mo><mi>π</mi><mo>)</mo><mo>&lt;</mo><msub><mi>T</mi><mi>CP</mi></msub></math>, which indicates the existence of a novel <math><mrow><mi>C</mi><mi>P</mi></mrow></math>-broken deconfined phase. We also discuss an implication of the novel phase for domain walls and their junctions.

Shi Chen

Kenji Fukushima

Yuya Tanizaki

Deconfinement and <math><mrow><mi>C</mi><mi>P</mi></mrow></math> breaking at <math><mi>θ</mi><mo>=</mo><mi>π</mi></math> in Yang-Mills theories and a novel phase for SU(2)

At a nonzero temperature <math><mrow><mi>T</mi></mrow></math>, a constant field <math><mrow><msub><mrow><mover><mrow><mi>A</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mn>0</mn></mrow></msub><mo>∼</mo><mi>T</mi><mo>/</mo><mi>g</mi></mrow></math> generates nontrivial eigenvalues of the thermal Wilson line. We discuss contributions to the free energy of such a holonomous plasma when the coupling constant, <math><mi>g</mi></math>, is weak. We review the computation to <math><mrow><mo>∼</mo><msup><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math> by several alternate methods, and show that gauge invariant sources, which are nonlinear in the gauge potential <math><msub><mi>A</mi><mn>0</mn></msub></math>, generate novel contributions to the gluon self-energy at <math><mo>∼</mo><msup><mi>g</mi><mn>2</mn></msup></math>. These ensure the gluon self-energy remains transverse to <math><mo>∼</mo><msup><mi>g</mi><mn>2</mn></msup></math>, and are essential in computing contributions to the free energy at <math><mo>∼</mo><msup><mi>g</mi><mn>3</mn></msup></math> for small holonomy, <math><mrow><msub><mrow><mover><mrow><mi>A</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mn>0</mn></mrow></msub><mo>∼</mo><mi>T</mi></mrow></math>. We show that the contribution <math><mo>∼</mo><msup><mi>g</mi><mn>3</mn></msup></math> from off-diagonal gluons is discontinuous as the holonomy vanishes. The contribution from diagonal gluons is continuous as the holonomy vanishes, but sharply constrains the possible sources which generate nonzero holonomy, and must involve an infinite number of Polyakov loops.

France

Chris P. Korthals Altes

United States

Robert D. Pisarski

Vladimir V. Skokov

Free energy of a holonomous plasma

We compute the term <math><mo>∼</mo><msup><mrow><mi>g</mi></mrow><mrow><mn>3</mn></mrow></msup></math> in the free energy for a <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> gauge theory with nonzero holonomy at nonzero temperature. If the holonomy is generated kinematically by the introduction of gauge invariant sources coupled to Polyakov loops, the contribution of charged (off-diagonal) gluons to the free energy at order <math><msup><mrow><mi>g</mi></mrow><mrow><mn>3</mn></mrow></msup></math>, <math><msup><mrow><mi>F</mi></mrow><mrow><mo>(</mo><mn>3</mn><mrow><mo>:</mo><mi>c</mi><mo>.</mo><mi>g</mi><mo>.</mo></mrow><mo>)</mo></mrow></msup></math>, is singular: <math><msup><mrow><mi>F</mi></mrow><mrow><mo>(</mo><mn>3</mn><mrow><mo>:</mo><mi>c</mi><mo>.</mo><mi>g</mi><mo>.</mo></mrow><mo>)</mo></mrow></msup><mo>≠</mo><mn>0</mn></math> without holonomy, but <math><msup><mrow><mi>F</mi></mrow><mrow><mo>(</mo><mn>3</mn><mrow><mo>:</mo><mi>c</mi><mo>.</mo><mi>g</mi><mo>.</mo></mrow><mo>)</mo></mrow></msup><mo>=</mo><mn>0</mn></math> when the holonomy is nonzero, even infinitesimally. We show that the absence of the charged gluon contribution is required by gauge invariance alone and is therefore a universal feature.

Netherlands

Christiaan P. Korthals Altes

Conundrum for the free energy of a holonomous gluonic plasma at cubic order

We study QCD with massless quarks on ℝ3 × S 1 under symmetry-twisted boundary conditions with small compactification radius, i.e. at high temperatures. Under suitable boundary conditions, the theory acquires a part of the center symmetry and it is spontaneously broken at high temperatures. We show that these vacua at high temperatures can be regarded as different symmetry-protected topological orders, and the domain walls between them support nontrivial massless gauge theories as a consequence of anomaly-inflow mechanism. At sufficiently high temperatures, we can perform the semi-classical analysis to obtain the domain-wall theory, and 2d U(N c − 1) gauge theories with massless fermions match the ’t Hooft anomaly. We perform these analysis for the high-temperature domain wall of   <math> <msub> <mi>ℤ</mi> <msub> <mi>N</mi> <mi>c</mi> </msub> </msub> </math>  $$ {\mathbb{Z}}_{N_{\mathrm{c}}} $$  -QCD and also of Roberge-Weiss phase transitions.

High-temperature domain walls of QCD with imaginary chemical potentials

We study phase transitions in <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>∞</mi><mo>)</mo></math> gauge theories at nonzero temperature using matrix models. Our basic assumption is that the effective potential is dominated by double trace terms for the Polyakov loops. As a function of the various parameters, related to terms linear, quadratic, and quartic in the Polyakov loop, the phase diagram exhibits a universal structure. In a large region of this parameter space, there is a continuous phase transition whose order is larger than second. This is a generalization of the phase transition of Gross, Witten, and Wadia. Depending upon the detailed form of the matrix model, the eigenvalue density and the behavior of the specific heat near the transition differ drastically. We speculate that in the pure gauge theory, although the deconfining transition is thermodynamically of first order, it can be nevertheless conformally symmetric at infinite <math><mi>N</mi></math>.

Finite-temperature phase transitions of third and higher order in gauge theories at large <math><mi>N</mi></math>

We propose new gradient flows that define Lefschetz thimbles and do not blow up in a finite flow time. We study analytic properties of these gradient flows, and confirm them by numerical tests in simple examples.

Jacobus Verbaarschot

Gradient flows without blow-up for Lefschetz thimbles

We study the dynamics of gravitationally collapsing massive shells in AdS spacetime, and show in detail how one can determine extremal surfaces traversing them. The results are used to solve the time evolution of the holographic entanglement entropy in a strongly coupled dual conformal gauge theory, which is is seen to exhibit a regime of linear growth independent of the shape of the boundary entangling region and the equation of state of the shell. Our exact results are finally compared to those of two commonly used approximation schemes, the Vaidya metric and the quasistatic limit, whose respective regions of validity we quantitatively determine.

United Kingdom

Ville Keränen

Germany

Austria

Stefan Stricker

Olli Taanila

Finland

Aleksi Vuorinen

Gravitational collapse of thin shells: time evolution of the holographic entanglement entropy