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.     

Green functions in a super self-dual Yang-Mills background

In euclidean supersymmetric theories of chiral superfields and vector superfields coupled to a super-self-dual Yang-Mills background, we define Green functions for the Laplace-type differential operators which are obtained from the quadratic part of the action. These Green functions are expressed in terms of the Green function on the space of right chiral superfields, and an explicit expression for the right chiral Green function in the fundamental representation of an SU(n) gauge group is presented using the supersymmetric version of the ADHM formalism. The superfield kernels associated with the Laplace-type operators are used to obtain the one-loop quantum corrections to the super-self-dual Yang-Mills action, and also to provide a superfield version of the super-index theorems for the components of chiral superfields in a self-dual background.     

Spectrum of lattice gauge theories with fermions from a 1/d expansion at strong coupling

We study the strong coupling limit of U(N) or SU(N) gauge theories with fermions on a lattice. The integration over the gauge and fermion degrees of freedom is performed by analytic methods, leading to a partition function in terms of localized meson and baryon fields. A method for deriving a systematic expansion in the inverse of the space-time dimension of the corresponding Green functions is developed. It is applied to the study of spontaneous breakdown of chiral symmetry, which occurs for any U(N) or SU(N) theory with fermions in the fundamental representation. Meson and baryon spectra are then computed, and found to be in close agreement with those obtained by numerical methods at finite coupling. The pion decay constant is estimated.     

Non-perturbative aspects in supersymmetric gauge theories

We review our present understanding of those non-perturbative features of supersymmetric gauge theories that are believed to determine the properties of their ground states (vacua). A wide variety of theories is discussed in detail: pure Yang-Mills, theories with massive or massless real matter fields, theories with chiral matter, both for SU(N) and for the case of a general compact gauge group (as for instance E₈). Depending on some general features of the theory under consideration, various (perhaps) unexpected phenomena are shown to occur. Among these the breakdown of the (perturbatively established) non-renormalization theorem, the occurrence of runaway vacua in certain limits, the spontaneous dynamical breaking of supersymmetry itself in some chiral theories. Throughout the report we restrict ourselves to the confining picture instanton method, occasionally complementing it with the information coming from chiral and supersymmetric Ward-Takahashi identities. We compare our results with the ones suggested earlier by effective Lagrangian methods and, only briefly, with those obtained by other groups in the Higgs picture.     

Chiral U(1) symmetry and weak neutral currents at high energy

If chiral U(1) symmetry is a gauge symmetry, CP is automatically conserved despite the instanton effects, and the weak neutral currents have a definite structure. A realistic SU(2) _L ?U(1)?U(1) _R model contains an axion which is consistent with present data. Furthermore the neutrino interactions to lowest order are identical to the Weinberg-Salam model. Implications for the chiral U(1) currents are discussed.     

Exact analysis of Coulomb gauge vacuum degeneracies in (2 + 1) dimensions

The solutions of the two-dimensional euclidean σ-model provide an infinite number of pure gauge field configurations satisfying the Coulomb gauge condition, in (2 + 1) dimensions. For vacuum gauge fields associated with finite action instanton solutions of the σ-model, we find that the winding number n configuration leads to n negative eigenvalues for the ghost operator, up to a finite calculable degeneracy.     

Stochastic quantization of supersymmetric field theory

The Parisi-Wu stochastic quantization method is applied to supersymmetric field theory. The Langevin equation, which reproduces the Green functions of euclidean field theory, is written in terms of superfields. Supersymmetric U(1) theory under gauge fixing and the large N reduction in chiral SU(N) theory are discussed. Regularization based on the stochastic method is studied also.     

Non-abelian gauge theories in a background field

Green functions of on-mass-shell transverse gauge fields in a background gauge are proved to be obtained from Green functions in the perturbative vacuum by a mere translation.     

Various Full Green Functions in QED

We are interested in deriving various full Green functions through general Ward–Takahashi identities (WTIs) for quantized field theories. With the help of a postulate of gauge group parameter, the general local gauge transformation laws preserving the gauge-invariance of the generating functional itself of QED model have been established successfully. By using path-integral technique, the various WTIs with resulting anomaly terms are derived under the gauge transformations. The arising of Jacobian factor from the integration measure gives a viable possibility to express full Green function. As a consequence, the complete expressions of the full vector, the full axial-vector, the full tensor vertex functions and so on are presented respectively by solving the complete set of the WTIs in the momentum space without considering the constraint imposing any Ansatz. In addition, anomaly function also provides an effective means to judge the divergence of variant coupling currents on fields.     

Instanton zero modes and β-functions in gauged supergravity

We prove that one loop infinities in N?3 gauged O(N) supergravities are produced only by zero modes both on de Sitter and gauge instanton backgrounds. This strongly suggests (in analogy with the super Yang-Mills case) that instantons may be used for establishing exact β-functions in supergravities. We also derive the general formula for the number of gauge instanton zero modes for arbitrary spin and find that the “magnetic moment” part of the contribution in the gauge β-function is completely due to zero modes.     

Uniqueness of the BPST instanton

The uniqueness of the BPST instanton solution is established by solving the field equations subject to prescribed boundary conditions, but without imposing a priori restrictions of self-duality on the gauge fields.     

Notes on Certain (0,2) Correlation Functions

In this paper we shall describe some correlation function computations in perturbative heterotic strings that, for example, in certain circumstances can lend themselves to a heterotic generalization of quantum cohomology calculations. Ordinary quantum chiral rings reflect worldsheet instanton corrections to correlation functions involving products of elements of Dolbeault cohomology groups on the target space. The heterotic generalization described here involves computing worldsheet instanton corrections to correlation functions defined by products of elements of sheaf cohomology groups. One must not only compactify moduli spaces of rational curves, but also extend a sheaf (determined by the gauge bundle) over the compactification, and linear sigma models provide natural mechanisms for doing both. Euler classes of obstruction bundles generalize to this language in an interesting way.     

Dimensional regularisation and instantons

Dimensional regularisation is applied to the calculation of the quantum corrections to the instanton tunnelling amplitude in an SU(2) gauge theory. The principal feature is the introduction of an n-dimensional field configuration (a “quasi-instanton”), which generalises the O(5) invariance of the instanton and allows a coordinatisation of the function space of fields in its neighbourhood. This enables the functional integral measure to be factorised, with integrations over the translation and dilatation degrees of freedom being extracted. It is shown that a conformally invariant definition of orthogonality must be used in relation to the zero-mode eigenfunctions of the small oscillations expansion, irrespective of regularisation. An O(n + 1) covariant formalism is employed. An unconventional choice of gauge fixing term, which is not a perfect square, is made and is shown to allow the important freedom of calculating in a gauge specified by an arbitrary parameter α.     

Semiclassical functional integrals for self-dual gauge fields

The semiclassical approximation to the functional integral for four-dimensional Euclidean gauge theories is discussed in detail for general stationary points of the action. It is shown how to take the limit from a compact space to flat space, and the zero modes corresponding to global gauge transformations are carefully discussed. The results are specialised to general self-dual multi-instanton gauge fields given by the general construction of Atiyah et al. It is shown how the normalization matrix of the zero modes can be determined and the complete expression for the functional measure is given for the two instanton case. This is shown to factorise for well-separated instantons. Some technical matters are discussed in an appendix and a resume of results for multi-instanton functional determinants is included.     

The “instanton liquid”

We review the theory of interacting topological fluctuations in the ground state of quantum gauge theories, the so called “instanton liquid”. First we outline some known phenomenological facts, both coming from “real” experiments (for QCD) and from the lattice data. Then we describe interaction of instantons and the statistical mechanics of their ensemble for theSU(2) gauge theory. The very essential role of the light quarks is considered using numerical experiments. One of the main conclusions is that instantons induce chiral symmetry breaking in vacuum, but as they are suppressed (e.g. by the nonzero temperature) this symmetry is restored. Phase transition is found to be very strong and of the first order.     

Plane-wave,coordinate-space,and moment techniques in the operator-product expansion: Equivalence,improved methods,and the heavy quark expansion

We compare plane-wave, coordinate-space and moment methods for evaluating operator-product expansion (OPE) coefficients of the light-quark and gluon condensates. Equivalence of these methods for quark condensate contributions is proven to all orders in the quark mass parameterm. The three methods are also shown to yield equivalent gluon condensate contributions to two-current correlation functions, regardless of the gauge chosen for external gluon fields in the coordinate space approach. An improved method for evaluating quarkcondensate OPE coefficients is presented for several (two-current) correlation functions. Gauge-dependent Green functions are also discussed. It is shown that contradictory expressions for the gluon-condensate contribution to the quark propagator occurring from the plane-wave and coordinate-space approaches yield identical relations between the heavy-quark and gluon condensates, as anticipated from the gauge invariance of the heavy-quark expansion.     

Left chiral “zero modes” in a super-self-dual Yang-Mills background

A super-self-duality constraint can be imposed on the curvature two-form in supersymmetric Yang-Mills theories formulated in euclidean superspace. If left covariantly chiral superfields are coupled to such a background, part of this space of superfields is annihilated by a superspace operator constructed from the quadratic part of the action. These are analogous to the zero modes of the Dirac equation which occur in the presence of a multi-instanton. Using the supersymmetric version of the ADHM formalism, this space of left covariantly chiral “zero mode” superfields is constructed in the fundamental representation of an SU(n) gauge group. It is shown to be spanned by a set of k superfields, where k is the instanton number of the bosonic component of the Yang-Mills background.     

The off-shell infrared problem in N = 1 supersymmetric yang-mills theories (II). General massless models

We prove that in a general massless N = I SYM theory off-shell Green functions exist such that Green functions of gauge invariant operators are supersymmetrically covariant. The off-shell infrared problem present in the superfield treatment of these theories is thus shown to remain a gauge artefact. The N = 2, 4 pure SYM theories are covered by this result and thus exist as N = 1 SYM theories.     

The phase structure of Einstein–Cartan theory

In the Einstein–Cartan theory of torsion-free gravity coupling to massless fermions, the four-fermion interaction is induced and its strength is a function of the gravitational and gauge couplings, as well as the Immirzi parameter. We study the dynamics of the four-fermion interaction to determine whether effective bilinear terms of massive fermion fields are generated. Calculating one-particle-irreducible two-point functions of fermion fields, we identify three different phases and two critical points for phase transitions characterized by the strength of four-fermion interaction: (1) chiral symmetric phase for massive fermions in strong coupling regime; (2) chiral symmetric broken phase for massive fermions in intermediate coupling regime; (3) chiral symmetric phase for massless fermions in weak coupling regime. We discuss the scaling-invariant region for an effective theory of massive fermions coupled to torsion-free gravity in the low-energy limit.