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.
14626342
Exact amplitudes in four-dimensional non-critical string theories
Abstract The large N expansion of N =2 supersymmetric Yang–Mills theory with gauge group SU( N ) has recently been shown to break down at singularities on the moduli space. We conjecture that by taking N →∞ and approaching the singularities in a correlated way, all the observables of the theory have a finite universal limit yielding amplitudes in string theories dual to field theories describing the light degrees of freedom. We explicitly calculate the amplitudes corresponding to the Seiberg–Witten period integrals for an A n −1 series of multicritical points as well as for other critical points exhibiting a scaling reminiscent of the c =1 matrix model. Our results extend the matrix model approach to non-critical strings in less than one dimension to non-critical strings in four dimensions.
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.
119513982
Super Yang-Mills, matrix models and geometric transitions
Super Yang-Mills, modeles de matrices et transitions geometriques. J'expose deux applications de la relation entre les theories de jauge supersymetriques N = 1 a quatre dimensions, les modeles de matrices jauges a zero dimension, et les transitions geometriques en theorie des cordes. La premiere traite du spectre des parois ou branes BPS. Il est demontre qu'un etat de type D-brane, dont la tension en couplage faible est proportionnelle a N ∼ 1/g s , et un etat de type soliton de corde fermee, dont la tension en couplage faible est proportionnelle a N 2 ∼ 1/g 2 s , peuvent etre continument deformes l'un en l'autre. Ceci est un aspect de la theorie plus generale des espaces quantiques de parametres N = 1. La deuxieme est une nouvelle methode, purement geometrique, pour calculer exactement la somme sur les diagrammes planaires en zero dimension. C'est un exemple de dualite entre cordes ouvertes et cordes fermees.
119106648
Emergent D4-brane background from D-particles
Abstract We show that the solution of a pre-geometric strongly coupled quantum mechanical model describing K D-particles in the presence of N D4-branes in type IIA string theory, at fixed K and large N , yields an effective action describing the motion of the K D-particles in a classical ten-dimensional curved spacetime. By comparing the effective action with the non-abelian D-brane action in an arbitrary supergravity background, we identify the metric, dilaton and Ramond–Ramond fields and find a precise match with the near-horizon D4-brane geometry.
15116130
Galois symmetries in super Yang-Mills theories
Classifying the phases of gauge theories is hindered by the lack of local order parameters. In particular, the standard Wilson's and 't Hooft's non-local order parameters are known to be insufficient to explain the existence of the plethora of phases that are found in supersymmetric gauge theories. Motivated by these observations, we reanalyze the concept of gauge symmetry breaking using Galois theory. Unlike the ordinary classical notion of unbroken gauge group, the Galois symmetry makes sense in the full quantum theory and must be a phase invariant. The algebraic structure underlying the space of vacua of supersymmetric gauge theories, that we have developed recently, is precisely designed to allow a rigorous mathematical implementation of these ideas.
16103872
On the Schwinger model on Riemann surfaces
Abstract In this paper the Schwinger model or two-dimensional quantum electrodynamics is exactly solved on a Riemann surface providing the explicit expression of the partition function and of the generating functional of the amplitudes between the fermionic currents. This offers one of the few examples in which it is possible to integrate in an explicit way a gauge field theory interacting with matter on a Riemann surface.
122396324
Polymer topology and Chern-Simons field theory
Abstract In this paper the statistical mechanical problem of two topologically linked polymers is related to a topological Ginzburg-Landau model coupled to two Chern-Simons field theories. The Chern-Simons fields play the role of propagators of the topological interactions. The effects of the topological constraints is investigated using the method of effective potential in one-loop approximation.
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.
119596204
Operator formalism on general algebraic curves
Abstract The usual Laurent expansion of the analytic tensors on the complex plane is generalized to any closed and orientable Riemann surface represented as an affine algebraic curve. As an application, the operator formalism for the b − c systems is developed. The physical states are expressed by means of creation and annihilation operators as in the complex plane and the correlation functions are evaluated starting from simple normal ordering rules. The Hilbert space of the theory exhibits an interesting internal structure, being splitted into n ( n is the number of branches of the curve) independent Hilbert spaces. In this way we are able to realize new kinds of conformal field theories at genus zero with symmetry group Vir n ⊗ G , Vir being the Virasoro group and G denoting a discrete and nonabelian crystallographic group. Exploiting the operator formalism a large collection of explicit formulas of string theory is derived. Finally, we develop as an important byproduct new methods in order to handle differential equations related to monodromy, like the Riemann monodromy problem.
118089538
Gauge Field Theories on Riemann Surfaces
In this paper the free gauge field theories on a Riemann surface of any genus are quantized in the covariant gauge. The propagators of the gauge fields are explicitly derived and their properties are analysed in details. As an application, the correlation functions of a Yang-Mills field theory with gauge group $SU(N)$ are computed at the lowest order.