Potential Toolkit to Attack Nonperturbative Aspects of QFT
－Resurgence and related topics－
9/7-9/25, 2020, YITP (Kyoto University), Zoom, and Mozilla hubs
Abstracts
Opening remarks
Tatsuhiro Misumi (Akita University)："Opening remarks & Brief introduction to resurgence"
In the opening remarks, we propose goals of the workshop, the way to achieve them, and technical details to attend lectures and talks by Zoom and to participate in poster sessions and discussion sessions by Mozilla Hubs. We also give a very brief introduction to resurgence theory in quantum physics, with mainly explaining the basic notions and terminologies.
Invited Lectures
Aleksey Cherman (University of Minnesota)："Large N, small N, and adiabatic continuity"
I'll review the idea of adiabatic continuity and its connection to large N volume independence. Adiabatic continuity constructions allow the use of semiclassics and resurgence techniques to study gauge theories and non-linear sigma models with finite number of "colors" N. In most theories of interest one can ensure adiabatic continuity even in the large N limit, but with the price that semiclassical techniques fail at large N. I'll also explain an unconventional large N limit of 4d gauge theory where semiclassical techniques continue to work, but the fate of adiabatic continuity is less clear.
Gerald Dunne (University of Connecticut)："Resurgence, Phase Transitions and Large N"
There are several important conceptual and computational questions concerning the Minkowski space path integral, which have recently been approached from a new perspective motivated by "resurgent asymptotics", which is a novel mathematical formalism that seeks to unify perturbative and non-perturbative physics. I will review the basic ideas behind the connections between resurgent asymptotics and physics, and report on some applications probing phase transitions. This requires an understanding of how a two-parameter trans-series changes its structure in different limits.
Mithat Ünsal (North Carolina State University)："Semi-classics, adiabatic continuity and resurgence in quantum theories"
More than 10 years ago, It is shown that non-supersymmetric QCD with adjoint fermions admits a (non-thermal) compactification on $R^3 \times S^1$ where non-perturbative gauge dynamics becomes calculable. The mass gap, linear confinement and discrete chiral symmetry breaking are sourced by magnetic bions, which are correlated monopole-instanton anti-instanton pairs which have non-vanishing magnetic charge but have zero topological charge. From more modern perspective, these are the dominant configurations living on the critical point at infinity.
QCD(adj) construction led to the idea of the double-trace deformation of pure Yang-Mills theory on $R^3 x S^1$, which is continuously connected to pure YM on $R^4$ in the sense of continuity of all gauge invariant order parameters. It turns out that all qualitative non-perturbative properties of deformed YM theory are in qualitative agreement with non-perturbative expectations concerning pure YM theory. Over the last year, numerical simulations have shown that the topological susceptibility of deformed YM on small $S^1 \times R^3$ is in precise agreement with the numerical results on large $S^1 \times R^3$. Therefore, it is very likely that there is more truth in deformed YM construction than that meets the eye. I will give a lecture style talk on these topics.
Invited Talks
Toshiaki Fujimori (Keio University)："Bions in large-N sigma models"
Bion is a saddle point solution consisting of a fractional instanton and anti-instanton pair. Unlike ordinary instantons with integer topological charges, non-perturbative effects of bions survive in the leading order large-N limit and hence they are expected to appear in the resurgence structure in the large-N limit. In this talk, I discuss bion contributions in the large-N sigma models.
Yasuyuki Hatsuda (Rikkyo University)："Spheroidal harmonics and Nekrasov's function"
The spheroidal harmonics appear in many branches of physics. Their generalizations play an important role in black hole physics. In this talk, I propose a new relation between eigenvalues of the (generalized) spheroidal harmonics and Nekrasov's function in N=2 supersymmetric gauge theories.
Masazumi Honda (YITP, Kyoto University)："Black hole microstate counting and Picard-Lefschetz theory"
In my talk, I will focus on superconformal index of 4d N=4 super Yang-Mills theory. Recent analyzes have shown that the index accounts for an entropy of supersymmetric black hole on the gravity dual. I will discuss this aspect from the viewpoint of Picard-Lefschetz theory.
Okuto Morikawa (Kyushu University)："Perturbative ambiguities in compactified spacetime and resurgence structure"
Recently, in the context of the resurgence program, it was conjectured that the perturbative ambiguity caused by the IR renormalon is canceled against the semi-classical object called bion. This conjecture requires the circle compactification with the Z_N twisted boundary condition, in which the bion solution is found. Contrary to this conjecture, we find that there is no IR renormalon in circle-compactified theories. We then argue that the bion cancels the perturbative ambiguity caused by the proliferation of Feynman diagrams, which are significantly affected by the compactification. These observations are helpful in giving a unified understanding on the resurgence structure.
Naohisa Sueishi (Nagoya University)："On exact-WKB analysis, resurgent structure, and quantization conditions"
There are two well-known approaches to studying nonperturbative aspects of quantum mechanical systems: Saddle point analysis of the partition functions in Euclidean path integral formulation and the exact-WKB analysis based on the wave functions in the Schrödinger equation. In this work, based on the quantization conditions obtained from the exact-WKB method, we determine the relations between the two formalism and in particular show how the two Stokes phenomena are connected to each other: the Stokes phenomenon leading to the ambiguous contribution of different sectors of the path integral formulation corresponds to the change of the “topology” of the Stoke curves in the exact-WKB analysis. We also clarify the equivalence of different quantization conditions including Bohr-Sommerfeld, path integral and Gutzwiller’s ones. In particular, by reorganizing the exact quantization condition, we improve Gutzwiller analysis in a crucial way by bion contributions (incorporating complex periodic paths) and turn it into an exact results. Furthermore, we argue the novel meaning of quasi-moduli integral and provide a relation between the Maslov index and the intersection number of Lefschetz thimbles.
Masahito Yamazaki (Kavli IPMU, University of Tokyo)："Large N and Small N in Yang-Mills"
I will discuss four-dimensional SU(N) Yang-Mills theories with topological theta-terms, highlighting similarities and differences between large and small values of N.
Poster Presentations
Ivan Arraut (The Open University of Hong Kong)："Black-Hole evaporation and Quantum-depletion in Bose-Einstein condensates"
Given the AdS/CFT correspondence, it has become interesting to analyze analogies between condensed matter systems and gravitational systems. Based on this, we study the analogy between the Hawking radiation in Black-Holes and the Quantum depletion process of a Bose-Einstein condensate by using the Bogoliubov transformations method. We find that the relation between the Bogoliubov coefficients is similar in both cases (in the appropriate regimes). We then connect the condensate variables with those associated to the Black-Hole, demonstrating then that the zero temperature regime of the condensate is equivalent to the existence of an event horizon in gravity.
Shi Chen (The University of Tokyo)："Deconfinement and CP-breaking at θ=π in a softly-broken N=1 SYM"
At θ=π in pure Yang-Mills theories, the CP-symmetry is spontaneously broken in the confined phase, justified by a mixed 't Hooft anomaly between center symmetry and CP-symmetry. We wanted to see whether CP-symmetry is restored at the same time when the deconfinement phase transition occurs. We deform the pure Yang-Mills theory to a softly-broken N=1 Super Yang-Mills theory (SYM). In this softly-broken SYM, both deconfinement and CP-restoration can occur in the weakly-coupled region where reliable evaluations can be made. We found that for gauge groups other than SU(2), these two transitions occur synchronously. For SU(2), the CP-restoration occurs strictly later than the deconfinement, and a CP-breaking deconfined phase appears between the two transitions.
Zhiwei Du (Fudan University)："Hydrodynamic attractors in viscous Hubble flow"
The gradient expansion solutions to hydrodynamic equations are found to be divergent in many cases. The application of resurgence theory rescues the divergent series solutions and reproduces hydrodynamic attractors in Bjorken flow[1,2] and Gubser flow[3,4]. In this poster, we consider a viscous fluid system undergoing Hubble flow at a controlled rate and study the associated attractor solution in Israel-Stewart fluid theory. We obtain the analytical attractor solution from perturbative series by Borel re-summation. It corresponds well to the late-time limit of the numerical solutions. Furthermore, with the tool of alien calculus[5], we demonstrate that only the 1st order non-hydrodynamic mode appears in the viscous Hubble flow.
[1] M. P. Heller and M. Spalínski,
Phys. Rev. Lett. 115, 072501 (2015).
[2] P. Romatschke,
Phys. Rev. Lett. 120, 012301 (2018).
[3] G. S. Denicol and J. Noronha, Phys. Rev. D 99, 116004 (2019).
[4] A. Behtash et al.,
Phys. Rev. D 97, 044041 (2018).
[5] I. Aniceto and R. Schiappa,
Commun. Math. Phys. 335.1, 183 (2015).
Zachary Harris (University of Connecticut)："Resurgence and the Two-Loop Euler-Heisenberg Effective Action"
We present a detailed Borel analysis of the one-loop and two-loop Euler-Heisenberg effective action, and show that the non-perturbative imaginary part with a background electric field can be accurately deduced from a finite-order perturbative expansion of the real effective action with just a magnetic field. The non-perturbative structure at two-loop level differs from that at one-loop in several interesting ways.
Keita Imaizumi (Tokyo institute of technology)："Exact WKB analysis and TBA equations for the Mathieu equation"
We derive Thermodynamic Bethe Ansatz (TBA) equations governing the exact WKB periods of the Mathieu equation in the weak coupling region. The TBA equations provide a way to calculate the quantum corrections to the WKB periods, which are regarded as the quantum periods of N = 2 SU(2) super Yang-Mills theory at strong coupling. We calculate the effective central charge of the TBA equations, which is found to be proportional to the coefficient of the one-loop beta function of the 4d gauge theory. We also study the spectral problem for the Mathieu equation based on the TBA equations numerically.
Etsuko Itou (Keio University)："Fractional instanton of SU(3) gauge theory on the lattice"
According to recent studies on resurgence scenario of quantum systems, some topological objects with fractional charges play an important role to see the resurgence structure. In this talk, we report a numerical evidence of the fractional-instantons of the SU(3) gauge theory. The fractional-instanton appears in a weak coupling regime, if the theory is regularized by an infrared (IR) cutoff via the 1-form twisted boundary conditions. The Polyakov loop is also measured to investigate the center symmetry and confinement. The fractional-instanton corresponds to a solution linking two of degenerate Z3-broken vacua in the deconfinement phase. This talk is based on arXiv:1811.05708.
Takuya Shimazaki (The University of Tokyo)："Lefschetz thimble application to the Schwinger mechanism"
Dykhne–Davis–Pechukas (DDP) method is a common approximation scheme for the transition probability in two-level quantum systems, as realized in the Landau–Zener effect, leading to an exponentially damping form comparable to the Schwinger pair production rate. We analyze the foundation of the DDP method using a modern complex technique inspired by the Lefschetz-thimble method. We derive an alternative and more adaptive formula that is useful even when the DDP method is inapplicable. As a benchmark, we study the modified Landau–Zener model and compare results from the DDP and our methods. We then revisit a derivation of the Schwinger Mechanism of particle production under electric fields using the DDP and our methods. We find that the DDP method gets worse for the Sauter type of short-lived electric pulse, while our method is still a reasonable approximation. We also study the Dynamically Assisted Schwinger Mechanism in two methods. Ref. K. Fukushima, T. Shimazaki,
Annals Phys. 415 (2020) 168111
Yuya Tanizaki (YITP, Kyoto University)："Flag-manifold sigma model: Phase structure, Anomaly, and Semiclassics"
It has been known that 2d $\mathbb{C}P^1$ sigma model gives an effective description of quantum anti-ferromagnetic spin chains, and this connection by Haldane leads to a very fruitful insight on both high-energy and condensed matter physics. We discuss an application of anomaly matching to study the phase structure of SU(N) spin chain. Anomaly matching is a field-theoretic avatar of Lieb-Schultz-Mattis theorem, and gives a useful tool to understand nonperturbative aspects of strongly-coupled systems. Low-energy effective theory of $SU(N)$ anti-ferromagnetic Heisenberg chain of $p$-box representation is described by $SU(N)/U(1)^{N-1}$ nonlinear sigma model with the specific theta angles, and we show that its phase diagram in terms of theta angles is almost completely determined only by symmetry, anomaly, and global inconsistency. We consider the symmetry-twisted $S^1$ compactification of this system, and show the semiclassical ground-state energies reproduce the conjectured phase diagram.
Shoichiro Tsutsui (RIKEN Nishina Center)："On the fastest apparent convergence condition in optimized perturbation theory"
We discuss fundamental properties of the fastest apparent convergence (FAC) condition which is used as a variational criterion in optimized perturbation theory (OPT). We examine an integral representation of the FAC condition and a distribution of the zeros of the integral in a complex artificial parameter space on the basis of theory of Lefschetz thimbles. We find that the zeros accumulate on a certain line segment so-called anti-Stokes line in the limit K→\infty, where K is a truncation order of a perturbation series. This phenomenon gives an underlying mechanism that physical quantities calculated by OPT can be insensitive to the choice of the artificial parameter.
Takuya Yoda (Kyoto University)："Quantum phase transition in 3dim SQED and Lefschetz thimble analysis"
Recently, it has been reported that three dimensional N=4 SQED with multiple hypermultiplets undergoes a quantum phase transition in the large-flavor limit. In my talk, I am going to discuss interpretations of the phase transition in terms of resurgence. In particular we describe the phase transition from the viewpoint of the Lefschetz thimble analysis. We also discuss resurgence structure of the large flavor expansion and find that Borel transformation of the expansion has infinitely many singularities along imaginary axis. We provide some evidence that this class of Borel singularities should universally appear in various classes of large-flavor gauge theories. This talk is based on a joint work in progress with Toshiaki Fujimori, Masazumi Honda, Syo Kamata, Tatsuhiro Misumi and Norisuke Sakai.