Quantum systems with multiple degenerate classical harmonic minima exhibit new non-perturbative phenomena which are not present for the double-well and periodic potentials. The simplest characteristic example of this family is the triple-well potential. Despite the fact that instantons are exact semiclassical solutions with finite and minimal action, they do not contribute to the energy spectrum at leading order in the semiclassical analysis. This is because the instanton fluctuation prefactor vanishes, which can be interpreted as the action becoming infinite quantum mechanically. Instead, the non-perturbative physics is governed by different types of {\it bion} configurations. A generalization to supersymmetric and quasi-exactly soluble models is also discussed. An interesting pattern of interference between topological and neutral bions, depending on the hidden topological angle, the discrete theta angle and the perturbative level number, leads to an intricate pattern of divergent/convergent expansions for low lying states, and provides criteria for the exact solvability of some of the states. We confirm these semiclassical bion predictions using the Bender-Wu Mathematica package to study the structure of the associated perturbative expansions. It also turns out that all the systems we study have a curious exact one-to-one relationship between the perturbative coefficients of the three wells, which we check using the BenderWu package.
Bions are multiple fractional instanton configurations with zero instanton charge playing important roles in quantum field theories on a compactified space with a twisted boundary condition. We classify fractional instantons and bions in the $SU(N)$ principal chiral model on ${\mathbb R}^2 \times S^1$ with twisted boundary conditions. We find that fractional instantons are global vortices wrapping around $S^1$ with their $U(1)$ moduli twisted along $S^1$, that carry $1/N$ instanton (baryon) numbers for the ${\mathbb Z}_N$ symmetric twisted boundary condition and irrational instanton numbers fo…
Recently, multiple fractional instanton configurations with zero instanton charge, called bions, have been revealed to play important roles in quantum field theories on compactified spacetime. In two dimensions, fractional instantons and bions have been extensively studied in the ${\mathbb C}P^{N-1}$ model and the Grassmann sigma model on ${\mathbb R}^1 \times S^1$ with the ${\mathbb Z}_N$ symmetric twisted boundary condition. Fractional instantons in these models are domain walls with a localized $U(1)$ modulus twisted half along their world volume. In this paper, we classify fractional insta…
We classify bions in the Grassmann $Gr_{N_{\rm F},N_{\rm C}}$ sigma model (including the ${\mathbb C}P^{N_{\rm F}-1}$ model) on ${\mathbb R}^{1}\times S^{1}$ with twisted boundary conditions. We formulate these models as $U(N_{\rm C})$ gauge theories with $N_{\rm F}$ flavors in the fundamental representations. These theories can be promoted to supersymmetric gauge theories and further can be embedded into D-brane configurations in type II superstring theories. We focus on specific configurations composed of multiple fractional instantons, termed neutral bions and charged bions, which are ident…
We study the dynamics of four dimensional gauge theories with adjoint fermions for all gauge groups, both in perturbation theory and non-perturbatively, by using circle compactification with periodic boundary conditions for the fermions. There are new gauge phenomena. We show that, to all orders in perturbation theory, many gauge groups are Higgsed by the gauge holonomy around the circle to a product of both abelian and nonabelian gauge group factors. Non-perturbatively there are monopole-instantons with fermion zero modes and two types of monopole-anti-monopole molecules, called bions. One ty…
We investigate non-BPS exact solutions in ${\mathbb C}P^{N-1}$ sigma models on ${\mathbb R}^1 \times S^{1}$ with twisted boundary conditions, by using the Din-Zakrzewski projection method. 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-co…
Classical configurations in the CP^{N-1} model on R^1 x S^1 is studied with twisted boundary conditions. Starting from fractional instantons with the Z_N twisted boundary conditions, we review briefly the relevance of our study to resurgence phenomenon in field theory. We consider primarily configurations composed of multiple fractional instantons, called "neutral bions", which are identified as "perturbative infrared renormalons". We construct an explicit ansatz corresponding to topologically trivial configurations containing one fractional instanton (nu=1/N) and one fractional anti-instanton…
We study classical configurations in the ${\mathbb C}P^{N-1}$ model on ${\mathbb R}^{1}\times S^{1}$ with twisted boundary conditions. We focus on specific configurations composed of multiple fractionalized-instantons, termed "neutral bions", which are identified as "perturbative infrared renormalons" by Ünsal and his collaborators. For ${\mathbb Z}_N$ twisted boundary conditions, we consider an explicit ansatz corresponding to topologically trivial configurations containing one fractionalized instanton ($ν=1/N$) and one fractionalized anti-instanton ($ν=-1/N$) at large separations, and exhibi…
We derive the semiclassical contributions from the real and complex bions in the two-dimensional $\mathbb C P^{N-1}$ sigma model on ${\mathbb R} \times S^{1}$ with a twisted boundary condition. The bion configurations are saddle points of the complexified Euclidean action, which can be viewed as bound states of a pair of fractional instantons with opposite topological charges. We first derive the bion solutions by solving the equation of motion in the model with a potential which simulates an interaction induced by fermions in the $\mathbb C P^{N-1}$ quantum mechanics. The bion solutions have …
We discuss the non-perturbative contributions from real and complex saddle point solutions in the $\mathbb{C}P^1$ quantum mechanics with fermionic degrees of freedom, using the Lefschetz thimble formalism beyond the gaussian approximation. We find bion solutions, which correspond to (complexified) instanton-antiinstanton configurations stabilized in the presence of the fermionic degrees of freedom. By computing the one-loop determinants in the bion backgrounds, we obtain the leading order contributions from both the real and complex bion solutions. To incorporate quasi zero modes which become …
Perturbation series in quantum field theory are generally divergent asymptotic series which are also typically not Borel resummable in the sense that the resummed series is ambiguous. The ambiguity is associated with singularities in the Borel plane on the positive real axis. In quantum mechanics there are cases in which the ambiguity that arises in perturbation theory cancels against a similarly ambiguous contribution from instanton--anti-instanton events. In asymptotically free gauge theories this mechanism does not suffice because perturbation theory develops ambiguities associated with sin…
It has been argued that many non-perturbative phenomena in quantum mechanics (QM) and quantum field theory (QFT) are determined by complex field configurations, and that these contributions should be understood in terms of of Picard-Lefschetz theory. In this work we compute the contribution from non-BPS multi-instanton configurations, such as instanton-anti-instanton $[{I}\bar{I}]$ pairs, and argue that these contributions should be interpreted as exact critical points at infinity. The Lefschetz thimbles associated with such critical points have a specific structure arising from the presence o…
We study the non-perturbative dynamics of the two dimensional ${O(N)}$ and Grassmannian sigma models by using compactification with twisted boundary conditions on $\mathbb R \times S^1$, semi-classical techniques and resurgence. While the $O(N)$ model has no instantons for $N>3$, it has (non-instanton) saddles on $\mathbb R^2$, which we call 2d-saddles. On $\mathbb R \times S^1$, the resurgent relation between perturbation theory and non-perturbative physics is encoded in new saddles, which are associated with the affine root system of the ${\frak o}(N) $ algebra. These events may be viewed as…
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. For the configurations with only instantons (anti-instantons), we obtain unambiguous results. For those with both inst…
We present a broad conceptual introduction to some new ideas in non-perturbative QFT. The large-$N$ orbifold-orientifold equivalence connects a natural large-$N$ limit of QCD to QCD with adjoint fermions. QCD(adj) with periodic boundary conditions and double-trace deformation of Yang-Mills theory satisfy large-$N$ volume independence, a type of orbifold equivalence. Certain QFTs that satisfy volume independence at $N=\infty$ exhibit adiabatic continuity at finite-$N$, and also become semi-classically calculable on small $\mathbb R^3 \times S^1$. We discuss the role of monopole-instantons, and …
We investigate the phase structure of the compactified $2$-dimensional nonlinear $SU(3)/U(1)^2$ flag sigma model with respect to two $θ$-terms. Based on the circle compactification with the ${\mathbb Z}_{3}$-twisted boundary condition, which preserves an 't Hooft anomaly of the original uncompactified theory, we perform the semiclassical analysis based on the dilute instanton gas approximation (DIGA). We clarify classical vacua of the theory and derive fractional instanton solutions connecting these vacua. The resulting phase structure based on DIGA exhibits the quantum phase transitions and t…
Real-Time Instanton Dynamics in a Triple-Well to Double-Well Homotopy
Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles University（查尔斯大学数学与物理学院粒子与核物理研究所）；Department of Low-Temperature Physics, Faculty of Mathematics and Physics, Charles University（查尔斯大学数学与物理学院低温物理系）；Department of Physics, Faculty of Applied Sciences, University of West Bohemia（西波希米亚大学应用科学学院物理系）
We show that the semi-classical analysis of generic Euclidean path integrals necessarily requires complexification of the action and measure, and consideration of complex saddle solutions. We demonstrate that complex saddle points have a natural interpretation in terms of the Picard-Lefschetz theory. Motivated in part by the semi-classical expansion of QCD with adjoint matter on ${\mathbb R}^3\times S^1$, we study quantum-mechanical systems with bosonic and fermionic (Grassmann) degrees of freedom with harmonic degenerate minima, as well as (related) purely bosonic systems with harmonic non-de…
This work is a step towards a non-perturbative continuum definition of quantum field theory (QFT), beginning with asymptotically free two dimensional non-linear sigma-models, using recent ideas from mathematics and QFT. The ideas from mathematics are resurgence theory, the trans-series framework, and Borel-Ecalle resummation. The ideas from QFT use continuity on R^1 x S^1_L, i.e, the absence of any phase transition as N \to infinity, or rapid-crossovers for finite-N, and the small-L weak coupling limit to render the semi-classical sector well-defined and calculable. We classify semi-classical …
We investigate the lattice ${\mathbb C} P^{N-1}$ sigma model on $S_{s}^{1}$(large) $\times$ $S_τ^{1}$(small) with the ${\mathbb Z}_{N}$ symmetric twisted boundary condition, where a sufficiently large ratio of the circumferences ($L_{s}\gg L_τ$) is taken to approximate ${\mathbb R} \times S^1$. We find that the expectation value of the Polyakov loop, which is an order parameter of the ${\mathbb Z}_N$ symmetry, remains consistent with zero ($|\langle P\rangle|\sim 0$) from small to relatively large inverse coupling $β$ (from large to small $L_τ$). As $β$ increases, the distribution of the Polya…
Non-Abelian Josephson junction is a junction of non-Abelian color superconductors sandwiching an insulator, or non-Abelian domain wall if flexible, whose low-energy dynamics is described by a $U(N)$ principal chiral model with the conventional pion mass. A non-Abelian Josephson vortex is a non-Abelian vortex (color magnetic flux tube) residing inside the junction, that is described as a non-Abelian sine-Gordon soliton. In this paper, we propose Josephson instantons and Josephson monopoles, that is, Yang-Mills instantons and monopoles inside a non-Abelian Josephson junction, respectively, and s…
This is an introductory level review of recent applications of resurgent trans-series and Picard-Lefschetz theory to quantum mechanics and quantum field theory. Resurgence connects local perturbative data with global topological structure. In quantum mechanical systems, this program provides a constructive relation between different saddles. For example, in certain cases it has been shown that all information around the instanton saddle is encoded in perturbation theory around the perturbative saddle. In quantum field theory, such as sigma models compactified on a circle, neutral bions provide…
We obtain $D_k$ ALF gravitational instantons by a gluing construction which captures, in a precise and explicit fashion, their interpretation as non-linear superpositions of the moduli space of centred $SU(2)$ monopoles, equipped with the Atiyah-Hitchin metric, and $k$ copies of the Taub-NUT manifold. The construction proceeds from a finite set of points in euclidean space, reflection symmetric about the origin, and depends on an adiabatic parameter which is incorporated into the geometry as a fifth dimension. Using a formulation in terms of hyperKähler triples on manifolds with boundaries, we…
Resurgent transseries have recently been shown to be a very powerful construction in order to completely describe nonperturbative phenomena in both matrix models and topological or minimal strings. These solutions encode the full nonperturbative content of a given gauge or string theory, where resurgence relates every (generalized) multi-instanton sector to each other via large-order analysis. The Stokes phase is the adequate gauge theory phase where an 't Hooft large N expansion exists and where resurgent transseries are most simply constructed. This paper addresses the nonperturbative study …
For the 11-dimensional supergravity over AdS_4 x S^7/Z_k, beginning with a general 4-from ansatz and the main geometry unchanged, we get a tower of massive and tachyonic pseudoscalars. Indeed, the resultant equations can be assigned to the so-called $ϕ^4$ actions of the non-minimally coupled scalar-tensor theories with a cosmological constant. We focus on a well-known tachyonic and a new massive bulk mode, which are singlet under the internal group and break all supersymmetries, associated with skew-whiffing and Wick-rotating of the background 4-from flux, respectively. The first one is the co…
本文基于IIA型弦论中的非BPS D-瞬子和IIB型弦论中的D-瞬子-反D-瞬子系统，研究了一种新的矩阵理论，我们称之为K-矩阵理论。这个理论正确地结合了D-膜的产生和湮灭过程。该理论的结构用谱三元组来表示，谱三元组是黎曼几何a la Connes的非交换推广，它们代表了高维D-膜世界体积上的几何。值得注意的是，在K-矩阵理论中D-膜的构型自然地被同调群的K-理论版本分类，称为K-同调。此外，我们认为，K-同调正确分类的D-膜配置从几何的角度来看。我们还构造了与K-矩阵理论的构型相对应的边界态，并明确地表明它们代表了高维D-膜。
For a wide variety of quantum potentials, including the textbook `instanton' examples of the periodic cosine and symmetric double-well potentials, the perturbative data coming from fluctuations about the vacuum saddle encodes all non-perturbative data in all higher non-perturbative sectors. Here we unify these examples in geometric terms, arguing that the all-orders quantum action determines the all-orders quantum dual action for quantum spectral problems associated with a classical genus one elliptic curve. Furthermore, for a special class of genus one potentials this relation is particularly…
