RIMS-iTHEMS International Workshop on Resurgence Theory

chaired by Tetsuo Hatsuda (iTHEMS/Nishina Center), Tatsuhiro Misumi (Akita U. / Keio U. / iTHEMS)
from to (Asia/Tokyo)
at RIKEN Kobe ( Integrated Innovation Building )
6-7-1 Minatojima-minamimachi, Chuo-ku, Kobe, Hyogo 650-0047
Dates: September 6-8, 2017
Place: RIKEN Kobe Campus, Integrated Innovation Building (IIB)
Travel to Kobe: all about Kobe 

Resurgence theory and other related methods have recently attracted a great deal of attention in mathematics and theoretical physics. The purpose of this workshop is to bring both mathematicians and physicists together to accelerate progress in non-perturbative quantum analyses such as the resurgence theory. The topics to be discussed at the workshop include -Resurgence theory -Exact WKB analysis -Wall-crossing (Stokes) phenomena -Lefschetz thimble -Perturbative calculation for non-pert. physics -Functional renormalization group -Other non-perturbative methods. Although the workshop mainly consists of invited talks, we will have relatively long discussion time each day and welcome lots of people. We hope that this workshop will advance the research on non-perturbative quantum analysis and produce significant works.

Invited Speakers (in alphabetical order):
 Takashi Aoki (Kindai U.)
 Hideaki Aoyama (Kyoto U.)
 Carl Bender (Washington U., St.Louis)
 Aleksey Cherman (U. of Washington)
 Gerald Dunne (U. of Connecticut)
 Toshiaki Fujimori (Keio U.)
 Yasuyuki Hatsuda (Rikkyo U.)
 Masazumi Honda (Weizmann)
 Shingo Kamimoto (Hiroshima U.)
 Tatsuya Koike (Kobe U.)
 Teiji Kunihiro (Kyoto U.)
 Tsunehide Kuroki (Kagawa College)
 Ricardo Schiappa (IST, U. of Lisbon)
 Tin Sulejmanpasic (ENS, Paris)
 Hiroshi Suzuki (Kyushu U.)
 Yuya Tanizaki (RIKEN BNL)
 Mithat Unsal (North Carolina State U.)
   Jean Zinn-Justin (CEA, Saclay)

Hosted by
 Kyoto University RIMS & RIKEN iTHEMS

Organizing Committee:
 Tetsuo Hatsuda (RIKEN iTHEMS / Nishina Center)
 Yoshimasa Hidaka (RIKEN Nishina Center)
 Tatsuhiro Misumi (Akita U. / Keio U. / iTHEMS)
 Yoshitsugu Takei (Doshisha U.)

Contact: resurgence@ml.riken.jp
Registration Want to participate? Apply here
Go to day
  • Wednesday, 6 September 2017
    • 09:50 - 10:00 Opening remarks 10'
    • 10:00 - 11:00 From multi-instantons to exact results 1h0'
      We review conjectures about the exact semi-classical expansion of low lying energy levels for analytic potentials with degenerate minima. They take the form of generalized Bohr–Sommerfeld quantization formulae. They were initially motivated by semi-classical evaluations of the partition function based on the path integral formalism (instanton calculus) and have later been proven, to some extent, using the theory of resurgent functions. We explain their relation with the corresponding complex WKB expansion of the solutions of the Schr¨odinger equation, or alternatively of the Fredholm determinant det(H − E). Finally, we recall how these conjectures emerge from a leading order summation of multi-instanton contributions to the path integral representation of the partition function, because the same strategy could result in new conjectures for problems where our present understanding is more limited.
      Speaker: Prof. Jean Zinn-Justin (CEA Saclay)
    • 11:00 - 12:00 TBA 1h0'
      Speaker: Prof. Gerald Dunne (U. of Connecticut)
    • 12:00 - 12:20 Coffee Break
    • 12:20 - 13:10 TBA 50'
      Speaker: Prof. Toshiaki Fujimori (Keio U.)
    • 13:10 - 14:40 Lunch time
    • 14:40 - 15:30 Exact WKB analysis of the Gauss hypergeometric differential equation 50'
      The Gauss hyperometric differential equation is investigated from the viewpoint of exact WKB analysis. There are three parameters α, β and γ in the equation. We consider the case where these parameters are linear func- tions of a large parameter. Then the equation can be analyzed by using the standard theory of exact WKB analysis. We can construct WKB solutions of the equation and take the Borel sum of them. Thus obtained analytic solutions can be related to the Gauss hypergeometric function with the large parameter. As an application, asymptotic expansion formulas for the hyper- geometric function in terms of WKB solutions are obtained. These formulas contain asymptotic formulas for the Jacobi polynomials as a special case.
      Speaker: Prof. Takashi Aoki (Kindai U.)
    • 15:30 - 15:50 Coffee Break
    • 15:50 - 16:40 TBA 50'
      Speaker: Prof. Tatsuya Koike (Kobe U.)
    • 16:40 - 17:30 Iterated convolution and resurgence 50'
      We introduce the notion of $\Omega$-resurgence associated with a discrete filtered set $\Omega$ following Candelpergher-Nosmas-Pham. We describe the singularity structure of iterated convolution products of  such resurgent functions. We further discuss the resurgence of formal series solutions of nonlinear ODE. This talk is partially based on a joint work with David Sauzin.
      Speaker: Prof. Shingo Kamimoto (Hiroshima U.)
  • Thursday, 7 September 2017
    • 09:00 - 10:00 TBA 1h0'
    • 10:00 - 11:00 Resurgent Asymptotics in String Theory 1h0'
       I will begin with a light introduction to resurgent asymptotics. These techniques will then be explored (again in the spirit of a light introduction) within the context of transseries solutions to topological and non-critical string theories; themselves obtained via a nonperturbative completion of the holomorphic anomaly equations, or a nonperturbative string equation, respectively.
      Speaker: Prof. Ricardo Schiappa (IST, U. of Lisbon)
    • 11:00 - 11:30 Coffee break
    • 11:30 - 12:20 TBA 50'
      Speaker: Prof. Yasuyuki Hatsuda (Rikkyo U.)
    • 12:20 - 13:10 Nonperturbative ambiguity in double-well type matrix models 50'
      We observe how nonperturbative ambiguities become absent in SUSY/non-SUSY matrix models with double-well type potential, which provide nonperturbative formulations of SUSY/non-SUSY noncritical string theories. We find resurgence structure is quite different in SUSY/non-SUSY correlation functions, even in the same model.
      Speaker: Prof. Tsunehide Kuroki (Kagawa College)
    • 13:10 - 14:40 Lunch time
    • 14:40 - 15:30 Borel resummation and Exact results in supersymmetric gauge theories 50'
      In the past decade, there appeared many exact results in supersymmetric gauge theories thanks to localization method. These exact results are useful to study properties of perturbative series in quantum field theory (QFT) especially in the following two sense. First we can systematically analyze perturbative series around (non-)trivial saddle points in QFT. Second when we manage to obtain resummation of perturbative series without ambiguities in someway (e.g. Borel resummation and resurgence), one can explicitly see how the resummation is related to the exact result including non-perturbative corrections. In my talk I will demonstrate this in 3d N=2, 4d N=2 and 5d N=1 supersymmetric gauge theories on sphere. I will first discuss that we can show Borel summability of perturbative series by Yang-Mills coupling along real positive axis in 4d N=2 and 5d N=1 supersymmetric gauge theories with Lagrangians for various observables. It turns out that exact results in these theories can be obtained by summing over the Borel resummations with every instanton number. I will also discuss perturbative series in general 3d N=2 supersymmetric Chern-Simons matter theory, which is given by a power series expansion of inverse Chern-Simons levels. For this case we can prove that the perturbative series are always Borel summable along imaginary axis while it is not Borel summable along real positive axis often. It turns out that the Borel resummations along this direction are the same as exact results. I will also give physical interpretations of infinite singularities in Borel plane for this class of theories. [PRL116,no.21,211601(2016), PRD94, no.2, 025039 (2016) and upcoming paper(s)]
      Speaker: Dr. Masazumi Honda (Weizmann)
    • 15:30 - 15:50 Coffee break
    • 15:50 - 16:40 Global symmetries and continuity: weakly coupled road to strongly coupled theories 50'
      I will discuss the connection between volume independence and twisted boundary conditions exploiting global symmetries of the theory in question. Such twist serve as projectors to a reduced Hilbert space, and are potentially capable of eliminating phase transitions as one of the space-time radii are shrunk. Such twist have been argued to reduce several  small-circle theories to weakly coupled quantum mechanics with resurgent structure. I will demonstrate on the example of CP(N-1) nonlinear sigma model in 1+1D how such twist eliminate such a phase transition at infinite N, otherwise known to exist. If time permits I will discuss potential application of these ideas to pure Yang-Mills theory and twisted Eguchi-Kawai.
      Speaker: Dr. Tin Sulejmanpasic (ENS, Paris)
    • 16:40 - 17:30 The vacuum structure of the O(3)sigma model at \theta=\pi 50'
      The two-dimensional O(3) non-linear sigma model has a mass gap at \theta = 0, but at \theta = \pi it is widely expected that at long distances it flows to a gapless fixed point.  This is difficult to show from first principles directly in the continuum limit due to strong coupling problems.  We propose a theoretically controlled approach to the problem by examining the spectrum of the O(3) sigma model $\mathbb{R} \times S^1$ to all orders in the semiclassical expansion, using certain `center symmetric' twisted boundary conditions on $S^1$.  These boundary conditions are uniquely determined by leveraging recent insights on discrete 't Hooft anomalies.  Our results are consistent with gaplessness of the theory on $\mathbb{R}^2$. 
      Speaker: Dr. Aleksey Cherman (U. of Washington)
    • 18:00 - 20:00 Banquet
  • Friday, 8 September 2017
    • 09:00 - 10:00 PT-symmetry, nonlinear eigenvalue problems and Painleve transcendents 1h0'
      Semiclassical (WKB) techniques are commonly used to find the large-energy behavior of the eigenvalues of linear time-independent Schr\"odinger equations. In this talk we generalize the concept of an eigenvalue problem to nonlinear differential equations. The role of an eigenfunction is now played by a separatrix curve, and the special initial condition that gives rise to the separatrix curve is the eigenvalue. The Painlev\'e transcendents are examples of nonlinear eigenvalue problems, and semiclassical techniques are devised to calculate the behavior of the large eigenvalues. This behavior is found by reducing the Painlev\'e equation to the linear Schr\"odinger equation associated with a non-Hermitian $\cPT$-symmetric Hamiltonian. The concept of a nonlinear eigenvalue problem extends far beyond the Painlev\'e equations to huge classes of nonlinear differential equations.
      Speaker: Prof. Carl Bender (Washington U., St.Louis)
    • 10:00 - 11:00 TBA 1h0'
      Speaker: Prof. Mithat Unsal (North Carolina State U.)
    • 11:00 - 11:30 Coffee break
    • 11:30 - 12:20 TBA 50'
      Speaker: Prof. Teiji Kunihiro (Kyoto U.)
    • 12:20 - 13:10 Reconstruction of the tunneling amplitude from the perturbation series 50'
      I give a retrospective review on how we arrived at a formula which extracts the tunneling rate from a non Borel summable perturbation series in a quantum mechanical example. The formula gives rise to a very accurate tunneling rate, especially in the strong coupling regime. The keywords are: the (scaled) delta expansion, the Borel singularity, and the conformal mapping on the Borel plane.
      Speaker: Prof. Hiroshi Suzuki (Kyushu U.)
    • 13:10 - 14:40 Lunch time
    • 14:40 - 15:30 TBA 50'
      Speaker: Prof. Hideaki Aoyama (Kyoto U.)
    • 15:30 - 16:20 TBA 50'
      Speaker: Dr. Yuya Tanizaki (RIKEN BNL)
    • 16:20 - 16:30 Closing remarks 10'