Instantons, quivers and noncommutative Donaldson-Thomas theory

We construct noncommutative Donaldson-Thomas invariants associated with abelian orbifold singularities by analyzing the instanton contributions to a six-dimensional topological gauge theory. The noncommutative deformation of this gauge theory localizes on noncommutative instantons which can be classified in terms of three-dimensional Young diagrams with a colouring of boxes according to the orbifold group. We construct a moduli space for these gauge field configurations which allows us to compute its virtual numbers via the counting of representations of a quiver with relations. The quiver encodes the instanton dynamics of the noncommutative gauge theory, and is associated to the geometry of the singularity via the generalized McKay correspondence. The index of BPS states which compute the noncommutative Donaldson-Thomas invariants is realized via topological quantum mechanics based on the quiver data. We illustrate these constructions with several explicit examples, involving also higher rank Coulomb branch invariants and geometries with compact divisors, and connect our approach with other ones in the literature.     

Topological charge on the lattice: The O(3) case

We discuss the topological classification of the field configurations on the lattice. The regulator may act as a source of topological defects of the order of the lattice spacing we introduce a family of operators Q_L (block topological charge) that discriminates against instantons of sizes less than or equal to L. We prove that the correct charge implies first taking the limit a → 0 with L fixed and then L → 0. Numerical simulations à la Monte Carlo for O(3) in two dimensions are discussed. The observed density of small instantons agrees very well with theoretical calculations in the continuum theory.     

Interacting instantons and quantum mechanical metastability

The decay by tunneling of a quantum mechanical metastable state at finite temperature is described in terms of interacting instantons. The picture of instantons with arbitrary interaction strength is developed using complex classical paths. It is shown that there exists a temperature scale at which multi-instanton configurations “freeze”. Far below this scale the partition function of the theory is that of a gas of weakly interacting particles, while far above it the contribution of only one particle dominates. It is argued that this qualitative behavior is common to all quantum mechanical systems possessing a dilute gas regime at low temperatures.     

Supersymmetric Yang-Mills,octonionic instantons and triholomorphic curves

In four-dimensional gauge theory there exists a well-known correspondence between instantons and holomorphic curves, and a similar correspondence exists between certain octonionic instantons and triholomorphic curves. We prove that this latter correspondence stems from the dynamics of various dimensional reductions of ten-dimensional supersymmetric Yang-Mills theory. More precisely we show that the dimensional reduction of the (5+1)-dimensional supersymmetric sigma model with hyper-Kähler (but otherwise arbitrary) target X to a four-dimensional hyper-Kähler manifold M is a topological sigma model localising on the space of triholomorphic maps M -+ X (or hyperinstantons). When X is the moduli space M_k of instantons on a four-dimensional hyper-Kdhler manifold K, this theory has an interpretation in terms of supersymmetric gauge theory. In this case, the topological sigma model can be understood as an adiabatic limit of the dimensional reduction of ten-dimensional supersymmetric Yang-Mills on the eight-dimensional manifold M × K of holonomy Sp(1) × Sp(1) ⊂ Spin(7), which is a cohomological theory localising on the moduli space of octonionic instantons.     

Instantons and confinement in the SU(2) lattice gauge theory

We study the properties of gauge field configurations obtained by freezing the quantum fluctuations of the gauge fields on the lattice. The freezing process is subdivided into three regions: instanton-anti-instanton vacuum, the annihilation of (anti)instantons, and classical solutions. We obtain the distribution of instanton sizes, ϱ, and the distribution of interinstanton distances, R. The average value of the ratio R/ϱ is 2.3. The contribution of the multi-instanton configurations to the value of the string tension is 5%.     

Instantonic branes, Atiyah–Hirzebruch spectral sequence, and duality of SYM

We study the equivalence among orientifold three-planes in the context of the Atiyah–Hirzebruch spectral sequence. This equivalence refers to the fact that two different cohomology classes in the same cohomology group, which classify the orientifolds, are lifted to the same trivial class in K-theory. The physical interpretation of this mathematical algorithm is given by the role of D-brane instantons. By following some recent ideas, we extend the sequence to include a classification of NS–NS fluxes. We find that such equivalences, in the low energy limit of the dynamics on the worldvolume of type IIB D3-branes on top of the orientifolds, are interpreted as the duality in four-dimensional SYM theories.     

Topology and Flux of T-Dual Manifolds with Circle Actions

We present an explicit formula for the topology and H-flux of the T-dual of a general type II, compactification, significantly generalizing earlier results. Our results apply to T-dualities with respect to any circle action on spacetime X. As before, T-duality exchanges type IIA and type IIB string theories. A new consequence is that the T-dual spacetime is a singular space when the fixed point set ${X^\mathbb{T}}$ is non-empty; the singularities correspond to Kaluza-Klein monopoles. We propose that the Ramond-Ramond charges of type II string theories on the singular dual are classified by twisted equivariant cohomology groups. We also discuss the K-theory approach.     

Macroscopic Quantum Coherence in an Antiferromagnetic Spin Chain with Biaxial Anisotropy

Instanton configurations of (1+1)-dimensions in an antiferromagnetic biaxial-anisotropy-spin-chain are obtained explicitly in the strong anisotropy limit, which interpolate between degenerate equilibrium orientations of the Néel vector along easy axis and are seen to be responsible for quantum tunneling. Macroscopic quantum coherence of the domain walls is demonstrated in terms of the instantons.     

Instantons and the 1/N expansion

We show that a non-interacting gas of instantons, gives a contribution which vanishes, in QCD, exponentially withN(the number of colours). If we include some of the interactions between instantons, then the resulting dilute gas of instantons—which is now in the form that is used in practice to investigate the structure of hadrons—gives a contribution that is essentialyconstant withN. Thus it is entirely consistent for both theN → ∞ limit to be useful and for the dilute gas of instantons to be relevant for the properties of hadrons.     

Moduli Spaces of Instantons on the Taub-NUT Space

We present ADHM-Nahm data for instantons on the Taub-NUT space and encode these data in terms of Bow Diagrams. We study the moduli spaces of the instantons and present these spaces as finite hyperkähler quotients. As an example, we find an explicit expression for the metric on the moduli space of one SU(2) instanton.We motivate our construction by identifying a corresponding string theory brane configuration. By following string theory dualities we are led to supersymmetric gauge theories with impurities.     

Structure of higher order corrections to the Lipatov asymptotic form

High orders of perturbation theory can be calculated by the Lipatov method, whereby they are determined by saddle-point configurations, or instantons, of the corresponding functional integrals. For most field theories, the Lipatov asymptotic form has the functional form ca ^NΓ(N+b) (N is the order of perturbation theory) and the relative corrections to it are series in powers of 1/N. It is shown that this series diverges factorially and its high-order coefficients can be calculated using a procedure similar to the Lipatov one: the Kth expansion coefficient has the form const[ln(S ₁/S ₀)]^?K Γ(K+(r ₁? r ₀)/2), where S ₀ and S ₁ are the values of the action for the first and second instantons of this particular field theory, and r ₀ and r ₁ are the corresponding number of zeroth-order modes; the instantons satisfy the same equation as in the Lipatov method and are assumed to be renumbered in order of their increasing action. This result is universal and is valid in any field theory for which the Lipatov asymptotic form is as specified above.     

How instantons solve the U(1) problem

The gauge theory for strong interactions, QCD, has an apparent U(1) symmetry that is not realized in the real world. The violation of the U(1) symmetry can be attributed to a well-known anomaly in the regularization of the theory, which in field configurations called “instantons” can be seen to give rise to interactions that explicitly break the symmetry. A simple polynomial effective Lagrangian describes these effects qualitatively very well. In particular it is seen that no unwanted Goldstone bosons appear and the eta particle owes a large fraction of its mass to instantons. There is no need for field configurations with fractional winding numbers and it is explained how a spurious U(1) symmetry that remains in QCD even after introducing instantons, does not affect these results.     

Wormholes in higher dimensions with Gauss-Bonnet terms

A class of wormhole solutions permitted in a theory with Gauss-Bonnet terms in the gravitational action in higher dimensions have been studied. The case of de-Sitter type instantons, with a compact inner space, are of particular interest here. Some of the configurations, when continued analytically to the Lorentzian metric lead to the standard inflationary universe. Some multiple-sphere configurations of the type studied by Myers have also been noted. The Euclidean action for the solutions has been calculated and the relevance of the solutions in the quantum creation of the universe has been considered.     

Instanton effects in supersymmetric gauge theories

We investigate how in supersymmetric gauge theories non-perturbative effects are able to generate non-trivial vacuum properties otherwise forbidden by perturbative non-renormalization theorems. This conclusion can be reliably drawn since the constancy of certain Green functions — due to supersymmetry (SUSY) — allows one to connect vacuum-dominated large distances with short-distance behaviour which is reliably computed by instanton methods. In all the cases we discuss (without matter, with massive or massless matter in real representations and, finally, with matter in complex representations) instanton calculations imply the occurrence of a variety of condensates. For the pure SUSY gauge theory, a gluino condensate induces the spontaneous breaking of Z_2N. For massive super-quantum chromodynamics (SQCD) we find a peculiar mass dependence of matter condensates whose origin is traced to mass singularities of non-zero mode instanton contributions. These contributions force the massless limit of SQCD to differ from the strictly massless case, in which the spontaneous breaking of chiral symmetries is induced. Inconsistency with an anomaly equation forces either infinite matter condensates or spontaneous SUSY breaking in the massless cases. For non-constant Green functions, instantons are shown to provide new calculable short-distance singularities of an obvious non-perturbative nature.     

Instanton solutions from Abelian sinh-Gordon and Tzitzeica vortices

We study the Abelian Higgs vortex solutions to the sinh-Gordon equation and the elliptic Tzitzeica equation. Starting from these particular vortices, we construct solutions to the Taubes equation with higher vortex number, on surfaces with conical singularities.We then, analyse more general properties of vortices on such singular surfaces and propose a method to obtain vortices on conifolds from vortices on surfaces of revolution. We apply our method to construct explicit vortex solutions on the Poincaré disk with a conical singularity in the centre, to which we refer as the “hyperbolic cone”.We uplift the Abelian sinh-Gordon and Tzitzeica vortex solutions to four dimensions and construct cylindrically symmetric, self-dual Yang–Mills instantons on a non-self-dual (nor anti-self-dual) 4-dimensional Kähler manifold with non-vanishing scalar curvature. The instantons we construct in this way cannot be obtained via a twistorial approach.     

Bose-Einstein condensation beyond perturbation theory: Goldstone singularities and instanton solution

Temperature Green functions are applied to the analysis of Bose-condensation of weaklyinteracting gas. The character of Goldstone singularities of correlation functions isestablished to all orders in perturbation theory. These singularities are regularized bythe system volume. An anomalous volume dependence of the correlation functions isrevealed. Quantum-field perturbation series are studied in the framework of the instantonapproach. It is shown that there are no time-dependent instantons and that thetime-independent instanton solutions exhibit factorial growth in large orders of thequantum-field perturbation expansion.     

Weak charge quantization as an instanton of the interacting sigma model

Coulomb blockade in a quantum dot attached to a diffusive conductor is considered in the framework of the nonlinear sigma model. It is shown that the weak charge quantization on the dot is associated with instanton configurations of the Q field in the conductor. The instantons have a finite action and are replica nonsymmetric. It is argued that such instantons may play a role in the transition regime to the interacting insulator.     

The effective potential,vacuum tunneling and the 1/N expansion

The problem of vacuum decay in quantum field theory, when the instability is the result of radiative corrections, is discussed. The large-N expansion of the O(N) quartic model in four (Euclideanized) dimensions is analysed in detail and it is shown that, although the effective potential has no lower bound, tunnelling solutions of the usual type (instantons) do not exist. This is shown to happen because that expansion of the action which begins with the effective potential is inappropriate for the kind of field configurations in question. The relevance of this result to the related problem in the Salam-Weinberg model is also discussed.     

A statistical approach to quantum mechanics

A Monte Carlo method is used to evaluate the Euclidean version of Feynman's sum over particle histories. Following Feynman's treatment, individual paths are defined on a discrete (imaginary) time lattice with periodic boundary conditions. On each lattice site, a continuous position variable x_i specifies the spacial location of the particle. Using a modified Metropolis algorithm, the low-lying energy eigenvalues, |ψ₀(x)|², the propagator, and the effective potential for the anharmonic oscillator are computed, in good agreement with theory. For a deep double-well potential, instantons were found in our computer simulations appearing as multi-kink configurations on the lattice.