Resurgence in Gauge and String Theories

Instituto Superior Técnico, Lisbon, Portugal. July 18-22, 2016.

Resurgence in Gauge and String Theories

Instituto Superior Técnico, Lisbon, Portugal. July 18-22, 2016.

- Michael Berry, Bristol University
## Two Physical Applications of Divergent Series: The Aharonov-Bohm Wave, and Exquisite Sensitivity of Optical Polarization

- Approximate connections between the Aharonov-Bohm wave (AB) scattered by a magnetic flux line and the Fresnel-Sommerfeld edge-diffracted wave have been known for decades. The connection can be made precise by representing AB as a series whose large parameter is distance from the flux; the Erf-based approximation is uniform in angle and flux. The series diverges, and can be resummed to give additional accuracy, even within one wavelength of the flux.
- Divergent series and nonhermitian degeneracies combine in the optics of polarized light traversing a slowly-twisted anisotropic absorbing material, leading to a prediction that the emergent polarization is extraordinarily sensitive to the incident polarization - a dramatic violation of adiabatic intuition, explained by the Stokes phenomenon.

- Jorge Russo, University of Barcelona
## Supersymmetric Gauge Theories and Resurgence

We discuss supersymmetric observables in four-dimensional ${\mathcal N}=2$ gauge theories. Some of them are given in terms of closed, analytic formulas, which exhibit the explicit dependence on couplings, masses and on the rank $N_c$ of the gauge group. We comment on some features of the $1/N_c$ expansion and of the weak coupling expansion. We also discuss some salient aspects of SQCD with two massive flavors, and of ${\mathcal N}=2$ gauge theory on ellipsoids.

- Inês Aniceto, Jagiellonian University
## Applications of Resurgence in AdS/CFT

The perturbative study of physical observables often leads to asymptotic expansions, and resurgence has played a major role in understanding the nonperturbative phenomena associated with this asymptotic behaviour. In the context of AdS/CFT correspondence, perturbation theory in different regimes of the coupling and rank of the gauge group can lead to both convergent or asymptotic expansions. In this talk, we review the asymptotic regime of two observables in the context of AdS/CFT, focusing on the case of $\mathcal{N}=4$ SYM. The first of these is the cusp anomalous dimension, whose asymptotic expansion at large coupling can be determined via the so-called BES equation, obtained from the integrable properties of the gauge theory. The transseries solution is shown to be resurgent from large-order checks, and a resummation of this transseries allows us to perform a strong/weak coupling interpolation and connect to the well know results at weak coupling.

The second example covered in this talk is the study of the nonperturbative effects of black brane quasinormal modes appearing on the gravity side of the AdS/CFT description of $\mathcal{N}=4$ SYM plasma. The corresponding asymptotic hydrodynamic gradient expansion in the case of boost-invariant flow can be obtained from an AdS/CFT perspective, and shown to have resurgent properties. From the study of toy examples of hydrodynamic theories mimicking the existence of these quasinormal modes, one can write a transseries solution, and use resurgence to study its properties in different hydrodynamic regimes.

- Michael Borinsky, Humboldt University Berlin
## Asymptotic Calculus for Combinatorial Dyson-Schwinger Equations

Most perturbative expansions in QFT are asymptotic series. This divergence is believed to be dominated by the factorial growth of the number of Feynman diagrams. Asymptotic expansions of this type have many interesting properties. They form a subring of the ring of formal power series, which is also closed under composition and inversion. On this space a derivative can be defined which fulfills Leibniz and chain rules. This

*asymptotic calculus*can be used to obtain asymptotic expansions of expressions which are only given implicitly, for instance by functional or differential equations.- Tatsuhiro Misumi, Akita & Keio University
## Non-BPS Exact Solutions and their Relations to Bions

We discuss non-BPS exact solutions in $\mathbb{C}\mathbb{P}^N$ and Grassmann sigma models on $\mathbb{R}^1 \times \mathbb{S}^1$ with twisted boundary conditions. We focus on the relation of the non-BPS solutions to the ansatz of multi-instanton (bion) configurations and discuss their significance in the context of the resurgence theory. We find that the transition between seemingly distinct configurations of multi-instantons occur as moduli changes in the non-BPS solutions, and the simplest non-BPS exact solution corresponds to multi-bion configurations with fully-compressed double fractional instantons in the middle. It indicates that the non-BPS solutions make small but nonzero contribution to the resurgent trans-series as special cases of the multi-bion configurations. We observe a generic pattern of transitions between distinct multi-bion configurations (flipping partners), leading to the three essential properties of the non-BPS exact solution: (i) opposite sign for terms corresponding to the left and right infinities, (ii) symmetric location of fractional instantons, and (iii) the transition between distinct bion configurations. By studying the balance of forces, we show that the relative phases between the instanton constituents play decisive roles in stability and instability of the muli-instanton configurations.

- Ovidiu Costin, The Ohio State University
## Borel Plane Resurgence in Hyperasymptotics and Factorial Series

Hyperasymptotics is one of the choice tools in obtaining precise numerical results from divergent series. In recent work with MV Berry, RD Costin and C Howls we found that Borel plane resurgence analysis leads to important improvements: already the corrected second stage reexpansion is more accurate than infinitely many stages of usual hyperasymptotics.

Another classical technique of resummation, factorial series, also benefits substantially from Borel plane analysis. The factorial series limitations, ranging from slow convergence, small domain of validity and inability to describe the Stokes sector are removed when resurgence tools are used.

- Alexander Gorsky, Moscow Institute of Physics and Technology
## Instanton-Torus Knot Duality in SQCD

We present the duality between the instantons in 5d QED and SQCD and torus knots. It is snown that the fermion condensate serves as the generating function for the torus knot polynomials. The integer numbers $(n,m)$ characterizing the torus knot are identified as the instanton and electric charge. This implies the interesting interplay between the perturbative and nonperturbative contributions in terms of knots. A somewhat related phenomena of eigenvalue tunneling in the random topological networks is also mentioned.

- Ricardo Couso-Santamaría, Instituto Superior Técnico, Universidade de Lisboa
## Latest News on Resurgence Applied to Topological Strings

Topological string theory, along with matrix models, is one of the most fruitful settings in which to explore resurgence and obtain valuable nonperturbative information about observables. Lessons learned from working with practical, tangible examples will surely prove useful when approaching more difficult problems like realistic quantum field theories.

In this talk I'll present the latest advances on the topic which involve the exploration of asymptotic expansions of perturbative and nonperturbative free energies, the resummation of transseries and agreement with independent nonperturbative completions, the discovery of new transseries sectors related to NS-brane type effects, and possible consequences for the theory of Gromov–Witten enumerative invariants.

- Tin Sulejmanpasic, North Carolina State University
## Semi-Classics, Complexified Path Integrals and Lefshetz Thimbles

I will discuss semi-classics in quantum field theory, and argue that the notion of semi-classics requires the complexification of paths in the path integral. I will further argue that complex saddles which are at occasions singular and multi-valued may contribute to the path integral.

- Pavel Buividovich, Regensburg University
## Generalized Series Expansions in Asymptotically Free Large-$N$ Quantum Field Theories

We consider lattice perturbation theory for large-$N$ quantum field theories with unitary $U(N)$ matrices as fundamental degrees of freedom, such as principal chiral models and lattice gauge theories. We argue that a proper treatment of the Jacobian of the transformation which maps $U(N)$ group manifold to a space of Hermitian matrices, which is a necessary ingredient in the construction of perturbative expansions, results in a massive bare propagator. Since the number of planar diagrams grows only exponentially with order, this ensures the absence of any factorial divergences in (bare) lattice perturbation theory. The price to pay is that the perturbative expansion is no longer organized as an expansion in powers of coupling, but rather has a look of generalized series involving, in general, both powers and logs of coupling. I demonstrate the convergence of such expansion on the examples of several large-$N$ models, where results are available either from exact solution or numerical studies with infinite $N$ extrapolations.

- Muneto Nitta, Keio University
## Fractional Instantons and Bions

Fractional instantons and their molecules called bions have been turned out to play significant roles in recent development of resurgence theory. I will review fractional instantons and bions and discuss non-Abelian bions and complex bions.

- Yasuyuki Hatsuda, University of Geneva
## Resummation Problems and Nonperturbative Corrections

I will present some remarks on resummation problems in two-parameter expansions. In the first part, I will start with a simple example, known as the Faddeev (or non-compact) quantum dilogarithm, and discuss its (Borel) resummation. I will explain how nonperturbative corrections appear in this case. In the second part, I will report our recent result on an exact version of the Bohr-Sommerfeld quantization conditions for the relativistic Toda lattice in terms of topological invariants for certain local Calabi-Yau threefold. It turned out that these quantization conditions have the very similar structure to the quantum dilogarithm.

- Yoshitsugu Takei, Kyoto University
## Exact WKB Analysis for Continuous and Discrete Painlevé Equations

Generalizing the exact WKB analysis for one-dimensional Schrödinger equations established by Voros, Pham, Delabere and others, Aoki, Kawai and I developed the exact WKB analysis for continuous Painlevé equations and clarified, in particular, their Stokes geometry and connection formula. Later Iwaki discussed the wall-crossing formula for Painlevé equations as well. In this talk, after reviewing these previous works, I would like to talk about my recent research jointly done in part with N. Joshi (Sydney) on the exact WKB analysis for discrete Painlevé equations that are obtained from continuous Painlevé equations through the Bäcklund transformation. In the analysis of such discrete Painlevé equations both connection formula and wall-crossing formula for continuous Painlevé equations play the same role and appear as different kinds of connection formula.

- Christopher Howls, University of Southampton
## Postponing the Inevitable: Airey on Airy

We take a resurgence approach to Airey converging factors as a tool for extending numerical accuracy by an extension of hyperasymptotics. The approach explains the structure and behaviour of the terms in the Airey expansion. Using a method derived from the exact remainder term of Berry and Howls (1991) and allowing for the presence of additional singularities, we are also able to illustrate geometrically how to overcome one of the main historical issues with Airey converging factors, namely their failure as Stokes lines are approached. We also indicate how the situation changes when more than one Borel-plane singularity is involved.

- Amirkian Kashanipoor, Ecole Normale Supérieure
## Exact WKB for Difference Equations and Quantum Curves

Well-developed tools exist to study the Stokes behavior of WKB solutions of differential equations, and to determine their monodromy behavior. In this talk, we study to what extent these tools can be extended to difference equations. We apply these ideas to the setting of quantum curves in topological string theory.

- Aleksey Cherman, University of Washington
## Resurgence Out of the Box

I will discuss some applications of resurgence theory to the two-dimensional $O(N)$ non-linear sigma model, an asympotitically-free quantum field theory. In contrast to many applications of resurgence to QFT, our analysis does not rely compactifying the theory, and works directly on $\mathbb{R}^2$, by taking advantage of the large $N$ limit.

- Lutz Klaczynski, Humboldt University Berlin
## Resurgence through Dyson-Schwinger Equations

After a brief review of Dyson-Schwinger equations, I will discuss whether and how these equations may serve as a potential source of resurgence for perturbative quantum field theory. Although my latest pertinent results in Yukawa theory and quantum electrodynamics are negative for a fairly generic transseries ansatz, I contend that there are signs of resurgence that should, at least in principle, hold for a much wider class of resurgent transseries. Finally, I will go on to consider the possible role played by renormalisation and specific transmonomials to encode nonperturbative quantum states.

- Marcel Vonk, University of Amsterdam
## Two-Parameter Transseries for Painlevé I

The Painlevé I equation is one of the most studied non-linear ordinary differential equations. Its solutions have been constructed in many different frameworks, including numerics, perturbative series, resurgent transseries and transasymptotic expansions. Several of these approaches have also been related to each other in the literature, but mostly for a particular one-parameter transseries subfamily of solutions, and often only in restricted domains of the complex plane. Using resurgent transseries as a starting point, we show how this domain can be extended, which allows to study the famous pole fields in the Painlevé I solutions in much more detail. Moreover, our techniques can be extended to the full two-parameter family of Painlevé transseries solutions, and uncover interesting relations to modularity.

- David Sauzin, CNRS Paris & SNS Pisa
## Nonlinear Analysis with Endlessly Continuable Functions

A holomorphic germ at 0 is said to be endlessly continuable if it enjoys a certain property of analytic continuation which guarantees that the possible singularities are locally isolated; the singular locus is not fixed in advance and, in projection on the complex plane, it can have accumulation points. It is natural to call resurgent the formal series whose Borel transforms enjoy this property. We give estimates for the convolution product of an arbitrary number of endlessly continuable functions. This allows us to deal with nonlinear operations for the corresponding resurgent series, e.g. substitution into a convergent power series. In particular, this proves that the exponential or the logarithm of a resurgent series is resurgent.

[Joint work with Shingo Kamimoto]

- Simeon Hellerman, Kavli IPMU, The University of Tokyo
## Vacuum Manifolds from Local Operators

In this talk we consider ${\mathcal N}=2$ superconformal field theories in three dimensions with vacuum manifolds of a single complex dimension. In such theories we show there always exists a special one-complex-parameter family of normalizable states on the sphere with the property that expectation values of products of chiral primaries in these states are precisely products one-point functions. These states are themselves coherent superpositions of scalar chiral primaries, with coefficients that can be expressed in terms of conformal bootstrap data. We show that in a certain scaling limit, expectation values of local operators in these states become expectation values in a super-Poincaré invariant vacuum with nonzero vev for the complex modulus. Using this relationship, we express observables in states on the vacuum manifold in terms of bootstrap data. We also make certain observations about the analytic structure of observables as a function of the coherent state parameter, and use this relationship to draw conclusions about the convergence properties of the $1/J$ series expansion for observables in states with large R-charge $J$.

- Gökçe Basar, University of Maryland
## Resurgence, Exact Quantization and Complex Instantons

The theory of resurgence connects perturbative and non-perturbative physics. In this talk I will demonstrate this connection by focusing on certain one-dimensional quantum mechanical systems with degenerate harmonic minima. I will explain how the resurgent trans-series expansions for the low lying energy eigenvalues, which constitute the semi-classical expansion including non-perturbative terms, follow from the exact quantization condition. In contrast, in the opposite spectral region (with high lying eigenvalues), the relevant expansions are convergent. However, due to the poles in the expansion coefficients, they contain non-perturbative contributions which can be identified with complex instantons. I will demonstrate that in each spectral region there are striking relation between perturbative and non-perturbative expansions even though the nature of these expansions are very different. Furthermore there is a simple geometric interpretation of the perturbative non-perturbative connection. Notably, the spectra of these quantum mechanical examples encode the vacua of certain $N=2$ supersymmetric gauge theories in the Nekrasov-Shatashvili limit.

- Yuya Tanizaki, RIKEN BNL Research Center
## Studying the Silver Blaze Problem based on Picard-Lefschetz Theory

The sign problem is one of the biggest issue in theoretical physics. Because of the oscillatory nature of the path integral, it is very difficult to capture the correct physical behavior even with the help of numerical computations. In this talk, I would like to demonstrate that the idea of applied Picard-Lefschetz theory is quite helpful for the deep understanding of the sign problem. For this purpose, the Silver Blaze problem will be considered: path integral on Lefschetz thimbles is used to explain the trivial physical phenomenon that becomes nontrivial because of the sign problem. We will see that two perturbatively equivalent formalisms give different answers.

- Norisuke Sakai, Keio University
## Resurgence in Sine-Gordon Quantum Mechanics: Exact Agreement between Multi-Instantons and Uniform WKB

We compute multi-instanton amplitudes in the sine-Gordon quantum mechanics (periodic cosine potential) by integrating out quasi-moduli parameters corresponding to separations of instantons and anti-instantons. We propose an extension of Bogomolnyi--Zinn-Justin prescription for multi-instanton configurations and an appropriate subtraction scheme. We obtain the multi-instanton contributions to the energy eigenvalue of the lowest band at the zeroth order of the coupling constant. We show that the imaginary parts of the multi-instanton amplitudes precisely cancel the imaginary parts of the Borel resummation of the perturbation series, and verify that our results completely agree with those based on the uniform-WKB calculations, thus confirming the resurgence structure: divergent perturbation series combined with the nonperturbative multi-instanton contributions conspire to give unambiguous results. We also study the neutral bion contributions in the ${\mathbb C}{\mathbb P}^{N-1}$ model on ${\mathbb R}^1\times {\mathbb S}^{1}$ with a small circumference, taking account of the relative phase moduli between the fractional instanton and anti-instanton. We find that the sign of the interaction potential depends on the relative phase moduli, and that both the real and imaginary parts resulting from quasi-moduli integral of the neutral bion get quantitative corrections compared to the sine-Gordon quantum mechanics.