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119
A topological degree counting for some Liouville systems of mean field type
 Comm. Pure Appl. Math
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PROFILE OF BUBBLING SOLUTIONS TO A LIOUVILLE SYSTEM
, 2009
"... In several fields of Physics, Chemistry and Ecology, some models are described by Liouville systems. In this article we first prove a uniqueness result for a Liouville system in R 2. Then we establish an uniform estimate for bubbling solutions of a locally defined Liouville system near an isolated ..."
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Cited by 9 (5 self)
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In several fields of Physics, Chemistry and Ecology, some models are described by Liouville systems. In this article we first prove a uniqueness result for a Liouville system in R 2. Then we establish an uniform estimate for bubbling solutions of a locally defined Liouville system near an isolated blowup point. The uniqueness result, as well as the local uniform estimates are crucial ingredients for obtaining a priori estimate, degree counting formulas and existence results for Liouville systems defined on Riemann surfaces.
Asymptotic Behavior of Blowup Solutions for Elliptic Equations with Exponential Nonlinearity and Singular Data
, 2008
"... We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of BartolucciChenLinTarantello it is proved that the profile of the solutions dif ..."
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Cited by 9 (6 self)
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We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of BartolucciChenLinTarantello it is proved that the profile of the solutions differs from global solutions of a Liouville type equation only by a uniformly bounded term. The present paper improves their result and establishes an expansion of the solutions near the blowup points with a sharp error estimate.
A system of elliptic equations arising in Chern–Simons field theory
 J. Funct. Anal
"... Abstract. We prove the existence of topological vortices in a relativistic selfdual Abelian ChernSimons theory with two Higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over R2. We present two approaches to prove existence of solutions on ..."
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Cited by 8 (5 self)
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Abstract. We prove the existence of topological vortices in a relativistic selfdual Abelian ChernSimons theory with two Higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over R2. We present two approaches to prove existence of solutions on bounded domains: via minimization of an indefinite functional and via a fixed point argument. We then show that we may pass to the full R2 limit from the boundeddomain solutions to obtain a topological solution in R2. Contents
Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles. arXiv:1403.1277
"... Abstract: Resurgence theory implies that the nonperturbative (NP) and perturbative (P) data in a QFT are quantitatively related, and that detailed information about nonperturbative saddle point field configurations of path integrals can be extracted from perturbation theory. Traditionally, only s ..."
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Cited by 7 (1 self)
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Abstract: Resurgence theory implies that the nonperturbative (NP) and perturbative (P) data in a QFT are quantitatively related, and that detailed information about nonperturbative saddle point field configurations of path integrals can be extracted from perturbation theory. Traditionally, only stable NP saddle points are considered in QFT, and homotopy group considerations are used to classify them. However, in many QFTs the relevant homotopy groups are trivial, and even when they are nontrivial they leave many NP saddle points undetected. Resurgence provides a refined classification of NPsaddles, going beyond conventional topological considerations. To demonstrate some of these ideas, we study the SU(N) principal chiral model (PCM), a two dimensional asymptotically free matrix field theory which has no instantons, because the relevant homotopy group is trivial. Adiabatic continuity is used to reach a weakly coupled regime where NP effects are calculable. We then use resurgence theory to uncover the existence and role of novel ‘fracton ’ saddle points, which turn out to be the fractionalized constituents of previously observed unstable ‘uniton’ saddle points. The fractons play a crucial role in the physics of the PCM, and are responsible for the dynamically generated mass gap of the theory. Moreover, we show that the fractonantifracton events are the weak coupling realization of ’t Hooft’s renormalons, and argue that the renormalon ambiguities are systematically cancelled in the semiclassical expansion. Along the way, we also observe that the semiclassical expansion of the path integral can be geometrized as a sum over Lefschetz thimbles.
Dynamics of N=2 Supersymmetric ChernSimons Theories
, 2000
"... We discuss several aspects of three dimensional N = 2 supersymmetric gauge theories coupled to chiral multiplets. The generation of ChernSimons couplings at lowenergies results in novel behaviour including compact Coulomb branches and interesting patterns of dynamical supersymmetry breaking. We fur ..."
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We discuss several aspects of three dimensional N = 2 supersymmetric gauge theories coupled to chiral multiplets. The generation of ChernSimons couplings at lowenergies results in novel behaviour including compact Coulomb branches and interesting patterns of dynamical supersymmetry breaking. We further show how, given any pair of mirror theories with N = 4 supersymmetry, one may flow to a pair of mirror theories with N = 2 supersymmetry by gauging a suitable combination of the Rsymmetries. The resulting theories again have interesting properties due to Many three dimensional gauge theories with N = 4 supersymmetry exhibit mirror symmetry, a phenomenon in which two theories with different ultraviolet descriptions flow to the same infrared physics. Initially discovered by Intriligator and Seiberg [1], many further pairs of mirror theories have since been constructed using various
On Liouville systems at critical parameters, Part 1: One bubble
 J. Funct. Anal
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ChernSimons Solitons, Toda Theories and the Chiral
 in Proceedings of XXII nd International Conference on Differential Geometric Methods in Physics, Ixtapa (Mexico
, 1992
"... The twodimensional selfdual Chern–Simons equations are equivalent to the conditions for static, zeroenergy solutions of the (2+1)dimensional gauged nonlinear Schrödinger equation with Chern–Simons mattergauge dynamics. In this paper we classify all finite charge SU(N) solutions by first transfo ..."
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The twodimensional selfdual Chern–Simons equations are equivalent to the conditions for static, zeroenergy solutions of the (2+1)dimensional gauged nonlinear Schrödinger equation with Chern–Simons mattergauge dynamics. In this paper we classify all finite charge SU(N) solutions by first transforming the selfdual Chern–Simons equations into the twodimensional chiral model (or harmonic map) equations, and then using the Uhlenbeck–Wood classification of harmonic maps into the unitary groups. This construction also leads to a new relationship between the SU(N) Toda and SU(N) chiral model solutions. 2
Proof of the Julia–Zee theorem
 Comm. Math. Phys
"... It is a well accepted principle that finiteenergy static solutions in the classical relativistic gauge field theory over the (2 + 1)dimensional Minkowski spacetime must be electrically neutral. We call such a statement the Julia–Zee theorem. In this paper, we present a mathematical proof of this f ..."
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It is a well accepted principle that finiteenergy static solutions in the classical relativistic gauge field theory over the (2 + 1)dimensional Minkowski spacetime must be electrically neutral. We call such a statement the Julia–Zee theorem. In this paper, we present a mathematical proof of this fundamental structural property. Key words and phrases: Gauge fields, static electromagnetism, temporal gauge, entire solutions, the ’t Hooft tensor. PACS numbers: 10.15.q, 04.50.h, 03.70.+k, 12.10.g 1