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.
18373561
Biharmonic conformal field theories
Abstract We construct in this paper a new class of conformal field theories (CFT's) in which the fields are characterized by equations of the biharmonic type. All these theories are interacting and admit an explicit Lagrangian formulation. For that reason, the biharmonic CFT's are useful in the study of interacting CFT's and can have also applications in string theories. Besides, it is possible to interprete the biharmonic CFT's as theories of conformal gravity in two dimensions, which represent an alternative to Liouville field theory.
116932271
Applications of a generalization of the nonlinear sigma model with O(d) group of symmetry to the dynamics of a constrained chain
Subject of this work are the applications of a field theoretical model, called here generalized nonlinear sigma model or simply GNLSM,to the dynamics of a chain subjected to constraints. Chains with similar properties and constraints have been discussed in a seminal paper of Edwards and Goodyear using an approach based on the Langevin equation. The GNLSM has been proposed in a previous publication in order to describe the dynamics of a two dimensional chain. In this paper the model is extended to d dimensions and a bending energy term is added to its action. As an application, two observables are computed in the case of a very stiff chain. The first observable is the dynamical form factor of a ring shaped chain. The second observable is a straightforward generalization to dynamics of the static form factor. This observable is relevant in order to estimate the average distance between two arbitrary points of the chain. Finally, a variant of the GNLNM is presented, in which the topological conditions which constrain the motion of two linked chains are imposed with the help of the Gauss linking invariant.
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.
121244510
On the stochastic quantization of the Chern-Simons theory
Abstract We present in this paper a refined treatment of the stochastic quantization of Chern-Simons field theories. We show in particular that the naive Langevin equation does not lead to a well-defined equilibrium limit. Our main issue is the introduction of a nontrivial kernel in the Langevin equation which yields the correct equilibrium limit. We restrict ourselves to the free field theory case.
117767120
Operator formalism for b-c systems with lambda = 1 on general algebraic curves
In this letter we develope an operator formalism for the $b-c$ systems with conformal weight $\lambda=1$ defined on a general closed and orientable Riemann surface. The advantage of our approach is that the Riemann surface is represented as an affine algebraic curve. In this way it is possible to perform explicit calculations in string theory at any perturbative order. Besides the obvious applications in string theories and conformal field theories, (the $b-c$ systems at $\lambda=1$ are intimately related to the free scalar field theory), the operator formalism presented here sheds some light also on the quantization of field theories on Riemann surfaces. In fact, we are able to construct explicitly the vacuum state of the $b-c$ systems and to define creation and annihilation operators. All the amplitudes are rigorously computed using simple normal ordering prescriptions as in the flat case.
17626338
Spectral asymmetry and supersymmetry
Fractional charges, and in particular the spectral asymmetry eta of certain Dirac operators, can appear in the central charge of supersymmetric field theories. This yields unexpected analyticity constraints on eta from which classic results can be recovered in an elegant way. The method could also be applied in the context of string theory.
15521431
On the geometry of super Yang-Mills theories: phases and irreducible polynomials
We study the algebraic and geometric structures that underly the space of vacua of = 1 super Yang-Mills theories at the non-perturbative level. Chiral operators are shown to satisfy polynomial equations over appropriate rings, and the phase structure of the theory can be elegantly described by the factorization of these polynomials into irreducible pieces. In particular, this idea yields a powerful method to analyse the possible smooth interpolations between different classical limits in the gauge theory. As an application in U(N) theories, we provide a simple and completely general proof of the fact that confining and Higgs vacua are in the same phase when fundamental flavors are present, by finding an irreducible polynomial equation satisfied by the glueball operator. We also derive the full phase diagram for the theory with one adjoint when N ≤ 7 using computational algebraic geometry programs.
18522220
The BPS spectra and superconformal points in massive N = 2 supersymmetric QCD
We present a detailed study of the analytic structure, BPS spectra and superconformal points of the N = 2 susy SU(2) gauge theories with Nf = 1, 2, 3 massive quark hypermultiplets. We compute the curves of marginal stability with the help of the explicit solutions for the low-energy effective actions in terms of standard elliptic functions. We show that only a few of these curves are relevant. As a generic example, the case of Nf = 2 with two equal bare masses is studied in depth. We determine the precise existence domains for each BPS state, and show how they are compatible with the RG flows. At the superconformal point, where two singularities coincide, we prove that (for Nf = 2) the massless spectrum consists of four distinct BPS states and is S-invariant. This is due to the monodromy around the superconformal point being S, providing strong evidence for exact S-duality of the SCFT For all Nf, we compute the slopes ω of the β-functions at the fixed point couplings and show that they are related to the anomalous dimensions α of u = 〈trƒ2〉 by ω = 2(α − 1).
119183237
Gauge theories, D-branes and holography
Abstract Based on a generalization of the string theoretic concept of D-brane probe, we propose a new approach to large N gauge theories which makes the holographic properties manifest. For any gauge theory, we define from first principles an effective action for a fixed number of “probe” D-branes in the presence of N “background” D-branes on which the gauge theory lives. This effective action is shown to encode all the information about the large N gauge theory. The analysis of the planar diagram expansion which computes the effective action yields a simple and generic mechanism explaining the emergence of holographic space dimensions: the probe D-branes move in a higher dimensional dual holographic space–time. The construction yields a new perspective on the notion of bulk space–time locality and draws unexpected links with some aspects of the 't Hooft Abelian projection ideas. It also provides a new non-perturbative approximation scheme, able to capture both the weak and strong coupling regimes. We sketchily illustrate the basic ideas on a few examples, including the pure four dimensional Yang–Mills theory.
15289132
On exact superpotentials in confining vacua
We consider the N=1 super Yang-Mills theory with gauge group U(Nc) or SU(Nc) and one adjoint Higgs field with an arbitrary polynomial superpotential. We provide a purely field theoretic derivation of the exact effective superpotential W(S) for the glueball superfield S in the confining vacua. We show that the result matches with the Dijkgraaf-Vafa matrix model proposal. The proof brings to light a deep relationship between non-renormalization theorems first discussed by Intriligator, Leigh and Seiberg, and the fact that W(S) is given by a sum over planar diagrams.