Anatomy of one-loop effective action in non-commutative scalar field theories

The one-loop effective action of non-commutative scalar field theory with cubic self-interaction is studied. Utilizing the worldline formulation, both the planar and non-planar parts of the effective action are computed explicitly. We find complete agreement of the result with the Seiberg–Witten limit of a string worldsheet computation and with the standard Feynman diagrammatics. We prove that, in the low-energy and large non-commutativity limit, the non-planar part of the effective action is simplified enormously and is resummable into a quadratic action of scalar open Wilson line operators. Received: 22 July 2001 / Revised version: 19 October 2001 / Published online: 7 December 2001     

Consistent construction of perturbation theoryon non-commutative spaces

We examine the effect of non-local deformations on the applicability of interaction point time ordered perturbation theory (IPTOPT) based on the free Hamiltonian of local theories. The usual argument for the case of quantum field theory on a non-commutative space (based on the fact that the introduction of star products in bilinear terms does not alter the action) is not applicable to IPTOPT due to several discrepancies compared to the naive path integral approach when non-commutativity involves time. These discrepancies are explained in detail. Besides scalar models, gauge fields are also studied. For both cases, we discuss the free Hamiltonian with respect to non-local deformations. Received: 14 May 2004, Revised: 16 August 2005, Published online: 26 October 2005     

When is a field theory a generalized free field?

We show within a scalar relativistic quantum field theory that if either some even truncatedn-point-function vanishes or some multiple commutator of the field operators is a c-number then the field is necessarily a generalized free field.     

Gravity in non-commutative geometry

We study general relativity in the framework of non-commutative differential geometry. As a prerequisite we develop the basic notions of non-commutative Riemannian geometry, including analogues of Riemannian metric, curvature and scalar curvature. This enables us to introduce a generalized Einstein-Hilbert action for non-commutative Riemannian spaces. As an example we study a space-time which is the product of a four dimensional manifold by a two-point space, using the tools of non-commutative Riemannian geometry, and derive its generalized Einstein-Hilbert action. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.Dedicated to H. ArakiSupported in part by the Swiss National Foundation (SNF)     

Closed star products and cyclic cohomology

We define the notion of a closed star product. A (generalized) star product (deformation of the associative product of functions on a symplectic manifold W) is closed iff integration over W is a trace on the deformed algebra. We show that for these products the cyclic cohomology replaces the Hochschild cohomology in usual star products. We then define the character of a closed star product as the cohomology class (in the cyclic bicomplex) of a well-defined cocycle, and show that, in the case of pseudodifferential operators (standard ordering on the cotangent bundle to a compact Riemannian manifold), the character is defined and given by the Todd class, while in general it fails to satisfy the integrality condition.     

Gauge theory of the star product

The choice of a star product realization for non-commutative field theory can be regarded as a gauge choice in the space of all equivalent star products. With the goal of having a gauge invariant treatment, we develop tools, such as integration measures and covariant derivatives on this space. The covariant derivative can be expressed in terms of connections in the usual way giving rise to new degrees of freedom for non-commutative theories.     

A Geometric Approach to Boundaries and Surface Defects in Dijkgraaf–Witten Theories

Dijkgraaf–Witten theories are extended three-dimensional topological field theories of Turaev–Viro type. They can be constructed geometrically from categories of bundles via linearization. Boundaries and surface defects or interfaces in quantum field theories are of interest in various applications and provide structural insight. We perform a geometric study of boundary conditions and surface defects in Dijkgraaf–Witten theories. A crucial tool is the linearization of categories of relative bundles. We present the categories of generalized Wilson lines produced by such a linearization procedure. We establish that they agree with the Wilson line categories that are predicted by the general formalism for boundary conditions and surface defects in three-dimensional topological field theories that has been developed in Fuchs et al. (Commun Math Phys 321:543–575, 2013)     

Construction of a finite energy momentum tensor

A non-perturbative approach, postulating the existence of a family of Zimmermann normal products, certain linear relations among field operators, and the Wilson short distance expansion, is used to construct a finite energy momentum tensor. The dependence of the tensor on the field operators is made explicit by a suitable limit procedure. The calculations are performed in a scalar A⁴ model as an example. The results obtained are generalizations of the perturbation theory treatment of products of operators.     

The Thirring interaction in the two-dimensional axial–current–pseudoscalar derivative coupling model

We reexamine the two-dimensional model of massive fermions interacting with a massless pseudoscalar field via axial-current derivative coupling. The hidden Thirring interaction in the axial-derivative coupling model is exhibited compactly by performing a canonical field transformation on the Bose field algebra and the model is mapped into the Thirring model with an additional vector–current–scalar derivative interaction (Schroer–Thirring model). The Fermi field operator is rewritten in terms of the Mandelstam soliton operator coupled to a free massless scalar field. The charge sectors of the axial-derivative model are mapped into the charge sectors of the massive Thirring model. The complete bosonized version of the model is presented. The bosonized composite operators of the quantum Hamiltonian are obtained as the leading operators in the Wilson short distance expansions.     

Higgs and Wilson lines in field and string theory

This is a short review on basics of the use of the Wilson line to break gauge symmetry in theories with compact extra dimensions. We show how the computation of the one-loop effective field theory leads to a finite result. We then explain the realization of this breaking and the effective potential computation in an open string theory framework with D-branes. To cite this article: K. Benakli, C. R. Physique 8 (2007).     

Braiding matrices,modular transformations and topological field theories in 2+1 dimensions

Relations between 3D topological field theories and rational conformal field theories are discussed. In the former framework, we can define the generalized Verlinde operators. Using these operators, we find modular transformations for conformal blocks of one point functions and two point functions on the torus. The result is generalized to higher genus. The correctness of our formulae is illustrated by some examples. We also emphasize the importance of the fusion algebra.Addresses after October 1, 1989: Institute of Theoretical Physics, Academia Sinica, Beijing, P. R. China     

An Extension of Distribution Theory and of the Paley-Wiener-Schwartz Theorem Related to Quantum Gauge Theory

We show that a considerable part of the theory of (ultra)distributions and hyperfunctions can be extended to more singular generalized functions, starting from an angular localizability notion introduced previously. Such an extension is needed to treat quantum gauge field theories with indefinite metric in a generic covariant gauge. Prime attention is paid to the generalized functions defined on the Gelfand-Shilov spaces which gives the widest framework for construction of gauge-like models. We associate a similar test function space with every open and every closed cone, show that these spaces are nuclear and obtain the required formulas for their tensor products. The main results include the generalization of the Paley–Wiener–Schwartz theorem to the case of arbitrary singularity and the derivation of the relevant theorem on holomorphic approximation. Received: 29 December 1995 / Accepted: 13 September 1996     

Higher-Dimensional BF Theories¶in the Batalin–Vilkovisky Formalism:¶The BV Action and Generalized Wilson Loops

This paper analyzes in detail the Batalin–Vilkovisky quantization procedure for BF theories on an n-dimensional manifold and describes a suitable superformalism to deal with the master equation and the search of observables. In particular, generalized Wilson loops for BF theories with additional polynomial B-interactions are discussed in any dimensions. The paper also contains the explicit proofs to the theorems stated in [16]. Received: 25 October 2000 / Accepted: 30 March 2001     

Local ν-Euler Derivations and Deligne's¶Characteristic Class of Fedosov Star Products¶and Star Products of Special Type

In this paper we explicitly construct local ν-Euler derivations , where the ξ_α are local, conformally symplectic vector fields and the are formal series of locally defined differential operators, for Fedosov star products on a symplectic manifold (M,ω) by means of which we are able to compute Deligne's characteristic class of these star products. We show that this class is given by , where is a formal series of closed two-forms on M the cohomology class of which coincides with the one introduced by Fedosov to classify his star products. Moreover, we consider star products that have additional algebraic structures and compute the effect of these structures on the corresponding characteristic classes of these star products. Specifying the constituents of Fedosov's construction we obtain star products with these special properties. Finally, we investigate equivalence transformations between such special star products and prove existence of equivalence transformations being compatible with the considered algebraic structures. Dedicated to the memory of Moshé Flato Received: 28 June 1999 / Accepted: 11 April 2002?Published online: 11 September 2002     

On free massless (pseudo)scalar quantum field theory in 1+1-dimensional space-time

We construct a consistent quantum field theory of a free massless (pseudo)scalar field in 1+1-dimensional space-times free of infrared divergences. We show that in such a quantum field theory (i) a continuous symmetry of (pseudo)scalar field translations is spontaneously broken, (ii) Goldstone bosons appear as quanta of a free massless (pseudo)scalar field and (iii) there is a non-vanishing spontaneous {magnetization}. In spite of the existence of a spontaneous {magnetization} the main inequality between vacuum expectation values of certain operators which have been used for the derivation of the Mermin–Wagner–Hohenberg theorem (C. Itzykson and J.-M. Drouffe, {Statistical field theory}, Vol. I, 1989, pp. 219–224) is fulfilled. Received: 19 December 2001 / Revised version: 31 March 2002 / Published online: 14 June 2002     

Complexified Gravity in Noncommutative Spaces

The presence of a constant background antisymmetric tensor for open strings or D-branes forces the space-time coordinates to be noncommutative. This effect is equivalent to replacing ordinary products in the effective theory by the deformed star product. An immediate consequence of this is that all fields get complexified. The only possible noncommutative Yang–Mills theory is the one with U(N) gauge symmetry. By applying this idea to gravity one discovers that the metric becomes complex. We show in this article that this procedure is completely consistent and one can obtain complexified gravity by gauging the symmetry U(1,D−1) instead of the usual SO(1,D−1). The final theory depends on a Hermitian tensor containing both the symmetric metric and antisymmetric tensor. In contrast to other theories of nonsymmetric gravity the action is both unique and gauge invariant. The results are then generalized to noncommutative spaces. Received: 1 June 2000 / Accepted: 27 November 2000     

Generating functional and large N limit of nonlocal 2D generalized Yang–Mills theories (nlgYM_2's)

Using the path integral method, we calculate the partition function and the generating functional (of the field strengths) on nonlocal generalized 2D Yang–Mills theories (nlgYM's), which are nonlocal in the auxiliary field. This has been considered before by Saaidi and Khorrami. Our calculations are done for general surfaces. We find a general expression for the free energy of in nlgYM theories at the strong coupling phase (SCP) regime () for large groups. In the specific model, we show that the theory has a third order phase transition. Received: 24 June 2000 / Published online: 23 January 2001     

Natural Star Products on Symplectic Manifolds and Quantum Moment Maps

We define a natural class of star products: those which are given by a series of bidifferential operators which at order k in the deformation parameter have at most k derivatives in each argument. This class includes all the standard constructions of star products. We show that any such star product on a symplectic manifold defines a unique symplectic connection. We parametrise such star products, study their invariance properties and give necessary and sufficient conditions for them to have a quantum moment map. We show that Kravchenko's sufficient condition for a moment map for a Fedosov star product is also necessary.     

Zimmermann normal products and the Wilson expansion

The principal part and asymptotic form of the Wilson short distance expansion is derived in the framework of Bogoliubov-Parasiuk-Hepp-Zimmermann renormalized perturbation theory. Use is made of the Zimmermann normal product techniques with all work being performed in the scalar A⁴ model. The expansion of the product of four field operators is given as an explicit example.