117371335
Random Kähler Metrics
Abstract The purpose of this article is to propose a new method to define and calculate path integrals over metrics on a Kahler manifold. The main idea is to use finite dimensional spaces of Bergman metrics, as an approximation to the full space of Kahler metrics. We use the theory of large deviations to decide when a sequence of probability measures on the spaces of Bergman metrics tends to a limit measure on the space of all Kahler metrics. Several examples are considered.
119574064
Multi-loop zeta function regularization and spectral cutoff in curved spacetime
Abstract We emphasize the close relationship between zeta function methods and arbitrary spectral cutoff regularizations in curved spacetime. This yields, on the one hand, a physically sound and mathematically rigorous justification of the standard zeta function regularization at one loop and, on the other hand, a natural generalization of this method to higher loops. In particular, to any Feynman diagram is associated a generalized meromorphic zeta function. For the one-loop vacuum diagram, it is directly related to the usual spectral zeta function. To any loop order, the renormalized amplitudes can be read off from the pole structure of the generalized zeta functions. We focus on scalar field theories and illustrate the general formalism by explicit calculations at one-loop and two-loop orders, including a two-loop evaluation of the conformal anomaly.
1817006
Curves of marginal stability, and weak- and strong-coupling BPS spectra in N = 2 supersymmetric QCD
We explicitly determine the global structure of the SL(2, Z) bundle over the Coulomb branch of the moduli space of asymptotically free N = 2 supersymmetric Yang-Mills theories with gauge group SU(2) when massless hypermultiplets are present. For each relevant number of flavours, we show that there is a curve of marginal stability of the Coulomb branch, diffeomorphic to a circle, across which the BPS spectrum is discontinuous. We determine rigorously and completely the BPS spectra inside and outside the curve. In all cases, the spectrum inside the curve consists of only those BPS states that are responsible for the singularities of the low energy effective action (in addition to the massless abelian gauge multiplet which is always present). The predicted decay patterns across the curve of marginal stability are perfectly consistent with all quantum numbers carried by the BPS states. As a byproduct, we also show that the electric and magnetic quantum numbers of the massless states at the singularities proposed by Seiberg and Witten are the only possible ones.
15472365
Quantum parameter space and double scaling limits in N=1 super Yang-Mills theory
We study the physics of $\mathcal{N}=1$ super Yang-Mills theory with the gauge group $\mathrm{U}(N)$ and one adjoint Higgs field, by using the recently derived exact effective superpotentials. Interesting phenomena occur for some special values of the Higgs potential couplings. We find critical points with massless glueballs and/or massless monopoles, confinement without a mass gap, and tensionless domain walls. We describe the transitions between regimes with different patterns of gauge symmetry breaking, or, in the matrix model language, between solutions with a different number of cuts. The standard large N expansion is singular near the critical points, with domain wall tensions scaling as a fractional power of N. We argue that the critical points are four-dimensional analogues of the Kazakov critical points that are commonly found in low dimensional matrix integrals. We define a double scaling limit that yields the exact tension of BPS two-branes in the resulting $\mathcal{N}=1,$ four-dimensional noncritical string theory. D-brane states can be deformed continuously into closed string solitonic states, and vice versa, along paths that go over regions where the string coupling is strong.
119318242
Partial Gauge Fixing and Equivariant Cohomology
Given a gauge theory with gauge group G, it is sometimes useful to find an equivalent formulation in terms of a nontrivial gauge subgroup H⊂G. This amounts to fixing the gauge partially from G down to H. We study this problem systematically, both from the algebraic and from the path integral points of view. We find that the usual BRST cohomology must be replaced by an equivariant version and that the ghost Lagrangian must always include quartic ghost terms, even at tree level. Both the Cartan and Weil models for equivariant cohomology play a role and find natural interpretations within the physics framework. Applications include the construction of D-brane models of emergent space, the 't Hooft Abelian projection scenario in quantum chromodynamics and the formulation of the low-energy effective theories of grand unified models. © 2014 American Physical Society.
119685691
Gravitational actions in two dimensions and the Mabuchi functional
Abstract The Mabuchi energy is an interesting geometric functional on the space of Kahler metrics that plays a crucial role in the study of the geometry of Kahler manifolds. We show that this functional, as well as other related geometric actions, contribute to the effective gravitational action when a massive scalar field is coupled to gravity in two dimensions in a small mass expansion. This yields new theories of two-dimensional quantum gravity generalizing the standard Liouville models.
15099856
Topologically nontrivial sectors of the Maxwell field theory on algebraic curves
Abstract In this paper a large family of nondegenerate metrics is derived on general algebraic curves. In this way, it is possible to treat many differential equations arising in quantum mechanics and field theories on Riemann surfaces as differential equations on the complex sphere. The case of the Maxwell field theories on curves with Z n group of automorphisms is studied in details. These curves are particularly important because they cover the entire moduli space spanned by the Riemann surfaces of genus g ≤ 2. All the classical solutions of the Maxwell equations are explicitly constructed. Also the examples of the scalar fields and of an electron immersed in a constant magnetic field will be briefly investigated.
116973667
Quantum Field Theories on Algebraic Curves
In this talk the main features of the operator formalism for the $b-c$ systems on general algebraic curves developed in refs. [1-2] are reviewed. The first part of the talk is an introduction to the language of algebraic curves. Some explicit techniques for the construction of meromorphic tensors are explained. The second part is dedicated to the discussion of the $b-c$ systems. Some new results concerning the concrete representation of the basic operator algebra of the $b-c$ systems and the calculation of divisors on algebraic curves have also been included.
119179081
On a singular Fredholm-type integral equation arising in N=2 super-Yang–Mills theories
Abstract In this work we study the Nekrasov–Shatashvili limit of the Nekrasov instanton partition function of Yang–Mills field theories with N = 2 supersymmetry and gauge group SU ( N c ) . The theories are coupled with N f flavors of fundamental matter. The equation that determines the density of eigenvalues at the leading order in the saddle-point approximation is exactly solved when N f = 2 N c . The dominating contribution to the instanton free energy is computed. The requirement that this energy is finite imposes quantization conditions on the parameters of the theory that are in agreement with analogous conditions that have been derived in previous works. The instanton energy and thus the instanton contribution to the prepotential of the gauge theory is computed in closed form.
119806289
O(N) Generalized nonlinear sigma model and its applications
In this work the results of an interdisciplinary research between field theory and statistical mechanics will be presented. It is shown that the dynamics of an inextensible chain is described by a slight generalization of the O(d) nonlinear sigma model. It is checked, that in the large time limit the correct equilibrium distribution is reached. Our approach is based on path integrals, but it may be also connected to the usual description of the dynamics of a chain as a diffusion stochastic process. Some applications of our results will be discussed, like for instance the calculation of the average distance between two polymer segments.
119961620
NONSTANDARD CONFORMAL FIELD THEORIES FROM THE b-c SYSTEMS ON GENERAL ALGEBRAIC CURVES
In this paper we develop an operator formalism for the b–c systems with conformalweight λ=1 defined on a general closed and orientable Riemann surface. The advantageof our approach is that the Riemann surface is represented as an affine algebraic curve.In this way it is possible to show that the b–c systems at higher genera are equivalentto nonstandard conformal field theories on the complex sphere. The amplitudes of theseconformal field theories, rigorously computed using simple normal ordering prescriptions,are single-valued on the algebraic curve and coincide with the correlation functions of theoriginal b–c systems. Besides the obvious applications in string theories and conformalfield theories, (the b–c systems at λ=1 are intimately related to the free scalar fieldtheory), the operator formalism presented here also sheds some light on the quantizationof field theories on Riemann surfaces.