Gerald V. Dunne

Self-dual Chern-Simons theories

Abelian Nonrelativistic Model.- Nonabelian Nonrelativistic Model.- Abelian Relativistic Model.- Nonabelian Relativistic Model.- Quantum Aspects.

Aspects Of Chern-Simons Theory

Lectures at the 1998 Les Houches Summer School: Topological Aspects of Low Dimensional Systems. These lectures contain an introduction to various aspects of Chern-Simons gauge theory: (i) basics of planar field theory, (ii) canonical quantization of Chern-Simons theory, (iii) Chern-Simons vortices, and (iv) radiatively induced Chern-Simons terms.

Heisenberg-Euler effective Lagrangians: Basics and extensions

I present a pedagogical review of Heisenberg-Euler effective Lagrangians, beginning with the original work of Heisenberg and Euler, and Weisskopf, for the one loop effective action of quantum electrodynamics in a constant electromagnetic background field, and then summarizing some of the important applications and generalizations to inhomogeneous background fields, nonabelian backgrounds, and higher loop effective Lagrangians.

Resurgence and trans-series in Quantum Field Theory: the \mathbb{C}{{\mathbb{P}}^{N-1 }} model

A bstractThis 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-Écalle resummation. The ideas from QFT use continuity on

Chern-Simons Theory in the Schrodinger Representation

{{\mathbb{R}}^1}\times \mathbb{S}_L^1

Dynamically assisted Schwinger mechanism.

, i.e., the absence of any phase transition as N → ∞ 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 configurations with actions 1/N (kink-instantons), 2/N (bions and bi-kinks), in units where the 2d instanton action is normalized to one. Perturbation theory possesses the IR-renormalon ambiguity that arises due to non-Borel summability of the large-orders perturbation series (of Gevrey-1 type), for which a microscopic cancellation mechanism was unknown. This divergence must be present because the corresponding expansion is on a singular Stokes ray in the complexified coupling constant plane, and the sum exhibits the Stokes phenomenon crossing the ray. We show that there is also a non-perturbative ambiguity inherent to certain neutral topological molecules (neutral bions and bion-anti-bions) in the semiclassical expansion. We find a set of “confluence equations” that encode the exact cancellation of the two different type of ambiguities. There exists a resurgent behavior in the semi-classical trans-series analysis of the QFT, whereby subleading orders of exponential terms mix in a systematic way, canceling all ambiguities. We show that a new notion of “graded resurgence triangle” is necessary to capture the path integral approach to resurgence, and that graded resurgence underlies a potentially rigorous definition of general QFTs. The mass gap and the Θ angle dependence of vacuum energy are calculated from first principles, and are in accord with large-N and lattice results.

Quasi-exactly solvable systems and orthogonal polynomials

Abstract We quantize the (2 + 1)-dimensional Chern-Simons theory in the functional Schrodinger representation. The realization of gauge transformations on states involves a 1-cocycle. We determine this cocycle; we show how solving the Gauss law constraint in the non-Abelian theory requires quantizing the parameter that normalizes the action; we trivialize the 1-cocycle with a spatially non-local cochain related to a 2-dimensional fermion determinant and we find the physical states that satisfy the Gauss law constraint. The quantum holonomy of physical states involves a contribution that is missed when the constraint is solved before quantization. We compute this quantity for the Abelian theory in Minkowski space, where it exhibits an interesting group theoretic structure. (In a note added in proof the corresponding non-Abelian computation is presented.) Also we consider coupling to external sources and offer yet another derivation of the anomalous statistics and spin of the charge and flux carrying particles —a calculation which is especially simple in the functional Schrodinger representation.

Complex periodic potentials with real band spectra

We study electron-positron pair creation from the Dirac vacuum induced by a strong and slowly varying electric field (Schwinger effect) which is superimposed by a weak and rapidly changing electromagnetic field (dynamical pair creation). In the subcritical regime where both mechanisms separately are strongly suppressed, their combined impact yields a pair creation rate which is dramatically enhanced. Intuitively speaking, the strong electric field lowers the threshold for dynamical particle creation--or, alternatively, the fast electromagnetic field generates additional seeds for the Schwinger mechanism. These findings could be relevant for planned ultrahigh intensity lasers.

Resurgence theory, ghost-instantons, and analytic continuation of path integrals

This paper shows that there is a correspondence between quasi‐exactly solvable models in quantum mechanics and sets of orthogonal polynomials {Pn}. The quantum‐mechanical wave function is the generating function for the Pn(E), which are polynomials in the energy E. The condition of quasi‐exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value J. The zeros of the critical polynomial PJ(E) are the quasi‐exact energy eigenvalues of the system.

Large-order perturbation theory for a non-Hermitian PT-symmetric Hamiltonian

Abstract This paper demonstrates that complex P T-symmetric periodic potentials possess real band spectra. However, there are significant qualitative differences in the band structure for these potentials when compared with conventional real periodic potentials. For example, while the potentials V(x) = i sin2N+1 (x) (N = 0, 1, 2,…) have infinitely many gaps, at the band edges there are periodic wave functions but no antiperiodic wave functions. Numerical analysis and higher-order WKB techniques are used to establish these results.