Narrow Results

The Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction for noncommutative U(N) instantons is revisited. The origin of the instanton number is elucidated, including the case of the U(1) instantons. The group actions on the instantons are argued.

We use th Interacting Instanton Liquid Model to investigate the non-perturbative interaction associated to spontaneous chiral symmetry breaking. In particular, we study how hadrons' structure changes as a function of the quark mass. To support the validity of this model in the chiral regime, the quark mass dependencies of several properties are shown to agree with chiral perturbation theory, including the density of eigenmodes of the Dirac operator and the masses of the pion and nucleon. We identify a natural mass scale m* = 80 MeV which sets the boundary of the mass regime where the non-perturbative QCD dynamics is determined by chiral effects.

In order to understand the information loss problem, non-perturbative effects will do significant roles. Instantons are in general helpful for this purpose. There are various and rich thin-shell instantons and tunneling channels that eventually result a trivial geometry without a singularity nor an event horizon. We further discuss that there are some subtle examples in asymptotic de Sitter cases that need further investigations

Paradigm shift in gauge topology, from instantons to their constituents — instantondyons — has recently lead to very significant advances. Unlike instantons, these objects have three different set of charges, therefore allowing to explain and to tie together several important phenomena: confinement, chiral symmetry breaking and magnetic screening.

We discuss the modular anomaly equation satisfied by the the prepotential of 4-dimensional 𝒩 = 2^* theories and show that its validity is related to S-duality. The recursion relations that follow from the modular anomaly equation allow one to write the prepotential in terms of (quasi)-modular forms, thus resumming the instanton contributions. These results can be checked against the microscopic multi-instanton calculus in the case of classical algebras, but are valid also for the exceptional E_6,7,8,F₄ and G₂ algebras, where direct computations are not available.

Quantum tunneling of vortices had been found to be an important novel phenomena for description of low temperature creep in high temperature superconductors (HTSCs). We speculate that quantum tunneling may be also exhibited in mesoscopic superconductors due to vortices trapped by the Bean-Livingston barrier. The London approximation and method of images is used to estimate the shape of the potential well in superconducting HTSC quantum dot. To calculate the escape rate we use the instanton technique. We model the vortex by a quantum particle tunneling from a two-dimensional ground state under magnetic field applied in the transverse direction. The resulting decay rates obtained by the instanton approach and conventional WKB are compared revealing complete coincidence with each other.

We conjecture that the confinement–deconfinement phase transition in QCD at large number of colors N and N_f ≪ N at T ≠ 0 and µ ≠ 0 is triggered by the drastic change in θ behavior. The conjecture is motivated by the holographic model of QCD where confinement–deconfinement phase transition indeed happens precisely at T = T_c where θ dependence experiences a sudden change in behavior. The conjecture is also supported by quantum field theory arguments when the instanton calculations (which trigger the θ dependence) are under complete theoretical control for T > T_c, suddenly break down immediately below T < T_c with sharp changes in the θ dependence. Finally, the conjecture is supported by a number of numerical lattice results. We employ this conjecture to study confinement–deconfinement phase transition of hot and dense QCD in large N limit by analyzing the θ dependence. We estimate the critical values for T_c and µ_c where the phase transition happens by approaching the critical values from the hot and/or dense regions where the instanton calculations are under complete theoretical control. We also describe some defects of various codimensions within a holographic model of QCD by focusing on their role around the phase transition point.

We study four dimensional supersymmetric gauge theory in the Ω-background with the two dimensional super-Poincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional theory. The ε-parameter is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential on Σ which becomes the Yang-Yang function of the integrable system. We present the thermodynamic-Bethe-Ansatz like formulae for these functions and the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the many-body systems, such as the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions, for which we present a complete characterization of the L²-spectrum. We very briefly discuss the quantization of Hitchin system.

The progress on calorons (finite temperature instantons) is sketched. In particular there is some interest for confining temperatures, where the holonomy (the asymptotic value of the Polyakov loop) is non-trivial. In the last section I give more recent results by others.

We report on recent advances in understanding the non-local symmetries of quantum field theory, notably gauge theory. The symmetries relate topologically distinct sectors of the field space. We study these symmetries in some detail in the context of the BPS/CFT correspondence.