The Batalin–Vilkovisky Formalism in Noncommutative Effective Field Theory

In the context of spin foam models for quantum gravity, group field theories are a useful tool allowing on the one hand a non-perturbative formulation of the partition function and on the other hand admitting an interpretation as generalized matrix models. Focusing on 2d group field theories, we review their explicit relation to matrix models and show their link to a class of non-commutative field theories invariant under a quantum-deformed 3d Poincaré symmetry. This provides a simple relation between matrix models and non-commutative geometry. Moreover, we review the derivation of effective 2d group field theories with non-trivial propagators from Boulatov's group field theory for 3d quantum gravity. Besides the fact that this gives a simple and direct derivation of non-commutative field theories for the matter dynamics coupled to (3d) quantum gravity, these effective field theories can be expressed as multi-matrix models with a non-trivial coupling between matrices of different sizes. It should be interesting to analyze this new class of theories, both from the point of view of matrix models as integrable systems and for the study of non-commutative field theories.

In this chapter, we give a survey of Costello’s approach to quantum field theory, that gives a neat combination of the Batalin-Vilkovisky formalism with Wilson’s effective field theory. We start by describing the energetic viewpoint of effective field theory, and explaining the meaning of effective propagators. We then describe the renormalization group equation, its Feymann graph expansion, and Costello’s main quantization theorem, that describes very explicitly the quantization process for local field theories. We then proceed with the description of the effective quantum Batalin-Vilkovisky formalism, that explains what is an effective field quantization of a classical gauge theory, and gives a nice description of the obstructions to quantization. We then explain the relation between observables and the partition function in this effective formalism and describe Costello’s classification of renormalizable scalar field theories on $\mathbb{R}^{4}$ . We finish by stating the application of the given theory to pure Yang-Mills theory.

In this review we present some of the fundamental mathematical structures\nwhich permit to define noncommutative gauge field theories. In particular, we\nemphasize the theory of noncommutative connections, with the notions of\ncurvatures and gauge transformations. Two different approaches to\nnoncommutative geometry are covered: the one based on derivations and the one\nbased on spectral triples. Examples of noncommutative gauge field theories are\ngiven to illustrate the constructions and to display some of the common\nfeatures.\n

formula omitted] symmetry breaking, scalar mesons and the nucleon spin problem in an effective chiral field theory

We show that the fundamental time-reversal invariant (TRI) insulator exists in $4+1$ dimensions, where the effective-field theory is described by the $(4+1)$-dimensional Chern-Simons theory and the topological properties of the electronic structure are classified by the second Chern number. These topological properties are the natural generalizations of the time reversal-breaking quantum Hall insulator in $2+1$ dimensions. The TRI quantum spin Hall insulator in $2+1$ dimensions and the topological insulator in $3+1$ dimensions can be obtained as descendants from the fundamental TRI insulator in $4+1$ dimensions through a dimensional reduction procedure. The effective topological field theory and the ${Z}_{2}$ topological classification for the TRI insulators in $2+1$ and $3+1$ dimensions are naturally obtained from this procedure. All physically measurable topological response functions of the TRI insulators are completely described by the effective topological field theory. Our effective topological field theory predicts a number of measurable phenomena, the most striking of which is the topological magnetoelectric effect, where an electric field generates a topological contribution to the magnetization in the same direction, with a universal constant of proportionality quantized in odd multiples of the fine-structure constant $\ensuremath{\alpha}={e}^{2}∕\ensuremath{\hbar}c$. Finally, we present a general classification of all topological insulators in various dimensions and describe them in terms of a unified topological Chern-Simons field theory in phase space.

Chapter 1 features lectures that review the formalism of renormalization in quantum field theories with special regard to effective quantum field theories. While renormalization theory is part of every advanced course on quantum field theory, for effective theories some more advanced topics become particularly important. These topics include the renormalization of composite operators, operator mixing under scale evolution, and the resummation of large logarithms of scale ratios. The lectures from this course thus set the basis for any systematic study of the techniques and applications of effective field theories and offer an introduction for the reader to the content within this book.

On string field theory and effective actions

This review article presents the progress made over the last decade, since the introduction of effective field theories (EFTs) into post-Newtonian (PN) gravity. These have been put forward in the context of gravitational waves (GWs) from the compact binary inspiral. The mature development of this interdisciplinary field has resulted in significant advances of wide interest to physics at several levels serving various purposes. The field has firmly demonstrated, that seemingly disparate physical domains, such as quantum field theory and classical gravity, are related, and that the EFT framework is a universal one, where it has been proven to supply a robust methodology to boost progress in the development of PN theory. In this review emphasis was put on an accessible pedagogic presentation of the field theoretic aspects of the subject, with the view, that these are in fact common across the whole of theoretical physics, rather than in their original narrow quantum context. The review is aimed at a broad audience, from general readers new to the field, to specialists and experts in related subjects.The review begins with an overview of the introduction of EFTs into classical gravity and their development. Then, the basic ideas, which form the conceptual foundation of EFTs, are provided, and the strategy of a multi-stage EFT framework, which is utilized for the PN binary inspiral problem, is outlined. The main body of the review is then dedicated to presenting in detail the study of each of the effective theories at each of the intermediate scales in the problem, up to the actual GW observables. First, the EFT for a single compact object is considered, from which one proceeds to the EFT of a compact binary system, viewed as a composite particle with internal binding interactions. Finally, one arrives at the effective theory of the time-dependent multipole moments of the radiating system. The review is concluded with the multiple prospects of building on the progress in the field, and using further modern field theory insights and tools, to specifically address the study of GWs, as well as to broadly expand our fundamental understanding of gauge and gravity theories across the classical and quantum regimes.

Actions for noncommutative (NC) gauge field theories can be expanded perturbatively in powers of the noncommutativity parameter theta using the Seiberg–Witten map between ordinary classical fields and their NC counterparts. The leading order term represents classical (theta =0) action while higher-order terms give us theta -dependent NC corrections that ought to capture some aspects of quantum gravity. Building on previous work of Aschieri and Castellani on NC Chern–Simons (CS) gauge and gravity theories, showing that non-trivial theta -dependence exists only for spacetime dimensions Dge 5, we investigate a correlated effect of these extra spatial dimensions and noncommutativity on four-dimensional physics, up to first-order in theta . Assuming that one spatial dimension is compactified into a circle, we apply the Kaluza–Klein reduction procedure on the NC D=5 CS theory for the conformal gauge group SO(4, 2), to obtain an effective, theta -dependent four-dimensional theory of gravity that has Einstein–Hilbert gravity with negative cosmological constant as its commutative limit. We derive field equations for this modified theory of gravity and study the effect of NC interactions on some classical geometries, such as the AdS-Schwarzschild black hole. We find that this NC background spacetime gives rise to chiral gravitational anomaly due to the nonvanishing theta -dependent Pontryagin density.

We present a gauge and Lorentz invariant effective field theory model for the interaction of a charged scalar matter field with a magnetic monopole source, described by an external magnetic current. The quantum fluctuations of the monopole field are described effectively by a strongly coupled “dual” Ud(1) gauge field, which is independent of the electromagnetic Uem(1) gauge field. The effective interactions of the charged matter with the monopole source are described by a gauge invariant mixed Chern-Simons-like (Pontryagin-density) term between the two U(1) gauge fields. The latter interaction coupling is left free, and a lattice study of the system is performed with the aim of determining the phase structure of this effective theory. Our study shows that, in the spontaneously broken-symmetry phase, the monopole source triggers, via the mixed Chern-Simons term, which is nontrivial in its presence, the generation of a dynamical singular configuration (magnetic-monopolelike) for the respective gauge fields. The scalar field also behaves in the broken phase in a way similar to that of the scalar sector of the ’t Hooft-Polyakov monopole. Moreover, we show that the modest size of the lattices involved does not have significant effects on the main conclusions of our analysis. Published by the American Physical Society 2024

Non commutative geometry and super Yang–Mills theory

An effective field theory is a quantum field theory that is taken to describe the physical world in a limited range of length scales. While a fabled final theory of physics is supposed to describe the world down to arbitrarily small length scales, effective field theories provide a coarse-grained description of the behaviour of quantum fields and particles above some finite length scale. This way of thinking about quantum field theory models arose in the 1970s along with a framework called the renormalisation group. From this, and related techniques, a new methodology has arisen for constructing models of the behaviour of quantum particles at a particular physical scale. Effective field theories have been connected to a number of epistemological and metaphysical debates in the philosophy of science. Some have seen the effective field theory concept as hostile to scientific realism, while others have attempted to use it to articulate and defend a form of scientific realism. Some have seen the relationship between effective field theories at different scales as exemplifying a form of emergence in physics, while others have argued that it falls under the purview of standard accounts of intertheoretic reduction. While little consensus has been reached on these questions it is, at least, agreed that the effective field theory ideology represents an important conceptual shift in the way that physicists think about scale which philosophers of science need to take into account.

This paper gives a construction, using heat kernels, of differential forms on the moduli space of metrised ribbon graphs, or equivalently on the moduli space of Riemann surfaces with boundary. The construction depends on a manifold with a bundle of Frobenius algebras, satisfying various conditions. These forms satisfy gluing conditions which mean they form an open topological conformal field theory, i.e. a kind of open string theory. If the integral of these forms converged, it would yield the purely quantum part of the partition function of a Chern-Simons type gauge theory. Yang-Mills theory on a four manifold arises as one of these Chern-Simons type gauge theories.

Basic properties of gauge theories in the framework of the Faddeev-Popov (FP) method, Batalin-Vilkovisky (BV) formalism, and functional renormalization group (FRG) approach are considered. The FP and BV quantizations are characterized by the Becchi-Rouet-Stora-Tyutin (BRST) symmetry, while the BRST symmetry is broken in the FRG approach. It is shown that the FP method, the BV formalism, and the FRG approach can be provided with the Slavnov-Taylor identity, the Ward identity, and the modified Slavnov-Taylor identity, respectively. It is proven that using the background field method the background gauge invariance of the effective action within the FP and FRG quantization procedures can be achieved in nonlinear gauges. The gauge-dependence problem within the FP, BV, and FRG quantizations is studied. Arguments allowing us to state the existence of principal problems of the FRG in the case of gauge theories are given.

We present a formalism to explicitly construct non-Abelian gauge theories on noncommutative spaces (induced via a star product with aconstant Poisson tensor) from a consistency relation. This results in an expansion of the gauge parameter, the noncommutative gauge potential and fields in the fundamental representation, in powers of a parameter of the noncommutativity. This allows the explicit construction of actions for these gauge theories.