Construction Of Field Theory Research Articles

A new kind of quantum field theory of (n – 1)-dimensional defects in 2n dimensions

I describe a project to open a new territory of quantum field theory where the fields live not on a space–time manifold but on certain complete metric spaces of (n − 1)-dimensional objects (defects) in a 2n-dimensional space-time M. These metric spaces are ‘quasi Riemann surfaces’; they are formally analogous to Riemann surfaces. Every construction of a 2d conformal field theory (CFT) is to give an analogous construction of a CFT on the quasi Riemann surfaces, and thereby a cft on M. The global symmetry group of the 2d CFT becomes a local gauge symmetry. Ordinary local quantum fields in space–time are constructed by restricting to small objects. The project is based on writing the free n-form in 2n dimensions as the 2d Gaussian model on the quasi Riemann surfaces.

Three blocks solvable lattice models and Birman–Murakami–Wenzl algebra

Birman–Murakami–Wenzl (BMW) algebra was introduced in connection with knot theory. We treat here interaction round the face solvable (IRF) lattice models. We assume that the face transfer matrix obeys a cubic polynomial equation, which is called the three block case. We prove that the three block theories all obey the BMW algebra. We exemplify this result by treating in detail the SU(2)2×2 fused models, and showing explicitly the BMW structure. We use the connection between the construction of solvable lattice models and conformal field theory. This result is important to the solution of IRF lattice models and the development of new models, as well as to knot theory.

Dynamical and observational constraints on the warm little inflaton scenario

We explore the dynamics and observational predictions of the Warm Little Inflaton scenario, presently the simplest realization of warm inflation within a concrete quantum field theory construction. We consider three distinct types of scalar potentials for the inflaton, namely chaotic inflation with a quartic monomial potential, a Higgs-like symmetry breaking potential and a non-renormalizable plateau-like potential. In each case, we determine the parametric regimes in which the dynamical evolution is consistent for 50-60 e-folds of inflation, taking into account thermal corrections to the scalar potential and requiring, in particular, that the two fermions coupled directly to the inflaton remain relativistic and close to thermal equilibrium throughout the slow-roll regime and that the temperature is always below the underlying gauge symmetry breaking scale. We then compute the properties of the primordial spectrum of scalar curvature perturbations and the tensor-to-scalar ratio in the allowed parametric regions and compare them with Planck data, showing that this scenario is theoretically and observationally successful for a broad range of parameter values.

Compactification of 6D minimal SCFTs on Riemann surfaces

We study compactifications on Riemann surfaces with punctures of N=(1,0) 6d SCFTs with a one dimensional tensor branch and no continuous global symmetries. The effective description of such models on the tensor branch is in terms of pure gauge theories with decoupled tensor. For generic Riemann surfaces, the resulting theories in four dimensions are expected to have N=1 supersymmetry. We compute the anomalies expected from the resulting 4d theories by integrating the anomaly polynomial of the 6d theory on the Riemann surface. For the cases with 6d gauge models with gauge groups SU(3) and SO(8) we further propose a field theory construction for the resulting 4d theories. For the 6d SU(3) theory, we argue that the theories in four dimensions are quivers with SU(3) gauge nodes and free chiral fields. The theories one obtains from the 6d SO(8) gauge theory are quivers with SU(4) gauge groups and chiral fields with R charge a half. In the last case the theories constructed for general Riemann surfaces involve gauging of symmetries appearing at strong coupling. The conformal manifolds of the models are constructed from gauge couplings and baryonic superpotentials. We support our conjectures by matching the dimensions of the conformal manifolds with complex structure moduli of the Riemann surfaces, matching anomalies between six and four dimensions, and checking the dualities related to different pair of pants decompositions of the surfaces. As a simple application of the results we conjecture that SU(3) gauge theory with nine flavors in four dimensions has a duality group acting on the seven dimensional conformal manifold which is the mapping class group of sphere with ten marked points.

Freezing-in dark matter through a heavy invisible Z′

We demonstrate that in a class of the $U(1)'$ extension of the Standard Model (SM), under which all the Standard Model matter fields are uncharged and the additional neutral gauge boson $Z'$ couples to a set of heavy nonstandard fermions, dark matter (DM) production mediated by $Z'$ can proceed through the generation of generalized Chern-Simons (GCS) couplings. The origin of the GCS terms is intimately connected to the cancellation of gauge anomalies. We show that the DM production cross section triggered by GCS couplings is sufficient even for an intermediate scale $Z'$ . A large range of DM and Z masses is then allowed for reasonably high reheating temperature ($T_\rh \gtrsim 10^{10}~\text{GeV}$). This type of scenario opens up a new paradigm for unified models. We also study the UV completion of such effective field theory constructions, augmenting it by a heavy fermionic spectrum. The latter, when integrated out, generates the GCS-like terms and provides a new portal to the dark sector. The presence of a number of derivative couplings in the GCS-like operators induces a high temperature dependence to the DM production rate. The mechanism has novel consequences and leads to a new reheating dependence of the relic abundance.

Local field theory construction of very special conformal symmetry

Cohen and Glashow argued that very special conformal field theories of a particular kind (i.e. with HOM(2) or SIM(2) invariance) cannot be constructed within the framework of local field theories. We, however, show examples of local construction by using non-linear realization. We further construct linear realization from the topological twist at the cost of unitarity. To demonstrate the ubiquity of our idea, we also present corresponding holographic models.

Vector Effective Field Theories from Soft Limits.

We present a bottom-up construction of vector effective field theories using the infrared structure of scattering amplitudes. Our results employ two distinct probes of soft kinematics: multiple soft limits and single soft limits after dimensional reduction applicable in four and general dimensions, respectively. Both approaches uniquely specify the Born-Infeld (BI) model as the only theory of vectors completely fixed by certain infrared conditions which generalize the Adler zero for pions. These soft properties imply new recursion relations for on-shell scattering amplitudes in BI theory and suggest the existence of a wider class of vector effective field theories.

Constructive tensorial group field theory II: the model

In this paper, we continue our program of non-pertubative constructions of tensorial group field theories (TGFT). We prove analyticity and Borel summability in a suitable domain of the coupling constant of the simplest super-renormalizable TGFT which contains some ultraviolet divergencies, namely the color-symmetric quartic melonic rank-four model with Abelian gauge invariance, nicknamed . We use a multiscale loop vertex expansion. It is an extension of the loop vertex expansion (the basic constructive technique for non-local theories) which is required for theories that involve non-trivial renormalization.

Constructive tensorial group field theory I: The - model

The loop vertex expansion (LVE) is a constructive technique using canonical combinatorial tools. It works well for quantum field theories without renormalization, which is the case of the field theory studied in this paper. Tensorial group field theories (TGFTs) are a new class of field theories proposed to quantize gravity. This paper is devoted to a very simple TGFT for rank three tensors with U(1) group and quartic interactions, hence nicknamed -. It has no ultraviolet divergence, and we show, with the LVE, that it is Borel summable in its coupling constant.

Very special conformal field theories and their holographic duals

Cohen and Glashow introduced the notion of very special relativity as viable space-time symmetry of elementary particle physics. As a natural generalization of their idea, we study the subgroup of the conformal group, dubbed very special conformal symmetry, which is an extension of the very special relativity. We classify all of them and construct field theory examples as well as holographic realization of the very special conformal field theories.

A Cohomological Perspective on Algebraic Quantum Field Theory

Algebraic quantum field theory is considered from the perspective of the Hochschild cohomology bicomplex. This is a framework for studying deformations and symmetries. Deformation is a possible approach to the fundamental challenge of constructing interacting QFT models. Symmetry is the primary tool for understanding the structure and properties of a QFT model.This perspective leads to a generalization of the algebraic quantum field theory framework, as well as a more general definition of symmetry. This means that some models may have symmetries that were not previously recognized or exploited.To first order, a deformation of a QFT model is described by a Hochschild cohomology class. A deformation could, for example, correspond to adding an interaction term to a Lagrangian. The cohomology class for such an interaction is computed here. However, the result is more general and does not require the undeformed model to be constructed from a Lagrangian. This computation leads to a more concrete version of the construction of perturbative algebraic quantum field theory.

Self-dual phase space for ( 3+1 )-dimensional lattice Yang-Mills theory

I propose a self-dual deformation of the classical phase space of lattice Yang--Mills theory, in which both the electric and magnetic fluxes take value in the gauge Lie group. A local construction of the deformed phase space requires the machinery of "quasi-Hamiltonian spaces" by Alekseev et al., which is here reviewed. The results is a full-fledged finite-dimensional and gauge-invariant phase space, whose self-duality properties are largely enhanced in (3+1) spacetime dimensions. This enhancement is due to a correspondence with the moduli space of an auxiliary non-commutative flat connection living on a Riemann surface defined from the lattice itself, which in turn equips the duality between electric and magnetic fluxes with a neat geometrical interpretation in terms of a Heegaard splitting of the space manifold. Finally, I discuss the consequences of the proposed deformation on the quantization of the phase space, its quantum gravitational interpretation, as well as its relevance for the construction of (3+1) dimensional topological field theories with defects.

Conformal field theory construction for non-Abelian hierarchy wave functions

The fractional quantum Hall effect is the paradigmatic example of topologically ordered phases. One of its most fascinating aspects is the large variety of different topological orders that may be realized, in particular nonabelian ones. Here we analyze a class of nonabelian fractional quantum Hall model states which are generalizations of the abelian Haldane-Halperin hierarchy. We derive their topological properties and show that the quasiparticles obey nonabelian fusion rules of type su(q)_k. For a subset of these states we are able to derive the conformal field theory description that makes the topological properties - in particular braiding - of the state manifest. The model states we study provide explicit wave functions for a large variety of interesting topological orders, which may be relevant for certain fractional quantum Hall states observed in the first excited Landau level.

Construction of the Unitary Free Fermion Segal CFT

In this article, we provide a detailed construction and analysis of the mathematical conformal field theory of the free fermion, defined in the sense of Graeme Segal. We verify directly that the operators assigned to disks with two disks removed correspond to vertex operators, and use this to deduce analytic properties of the vertex operators. One of the main tools used in the construction is the Cauchy transform for Riemann surfaces, for which we establish several properties analogous to those of the classical Cauchy transform in the complex plane.

Minimally extended SILH

Higgs boson compositeness is a phenomenologically viable scenario addressing the hierarchy problem. In minimal models, the Higgs boson is the only degree of freedom of the strong sector below the strong interaction scale. We present here the simplest extension of such a framework with an additional composite spin-zero singlet. To this end, we adopt an effective field theory approach and develop a set of rules to estimate the size of the various operator coefficients, relating them to the parameters of the strong sector and its structural features. As a result, we obtain the patterns of new interactions affecting both the new singlet and the Higgs boson's physics. We identify the characteristics of the singlet field which cause its effects on Higgs physics to dominate over the ones inherited from the composite nature of the Higgs boson. Our effective field theory construction is supported by comparisons with explicit UV models.

From supersymmetric quantum mechanics to scalar field theories

In this work we address the reconstruction problem, investigating the construction of field theories from supersymmetric quantum mechanics. The procedure is reviewed, starting from reflectionless potentials that admit one and two bound states. We show that, although the field theory reconstructed from the potential that support a single bound state is unique, it may break unicity in the case of two bound states. We illustrate this with an example, which leads us with two distinct field theories.

Solutions in Constructive Field Theory

Constructive field theory aims to rigorously construct concrete, nontrivial solutions to Lagrangians used in particle physics. I examine the relationship of solutions in constructive field theory to both axiomatic and Lagrangian quantum field theory (QFT). I argue that Lagrangian QFT provides conditions for what counts as a successful constructive solution and other information that guides constructive field theorists to solutions. Solutions matter because they describe the behavior of QFT systems and thus what QFT says the world is like. Constructive field theory clarifies existing disputes about which parts of QFT are philosophically relevant and how rigor relates to these disputes.

A general state-sum construction of 2-dimensional topological quantum field theories with defects

We derive a general state sum construction for 2D topological quantum field theories (TQFTs) with source defects on oriented curves, extending the state-sum construction from special symmetric Frobenius algebra for 2D TQFTs without defects (cf. Lauda and Pfeiffer [State-sum construction of two-dimensional open-closed topological quantum field theories, J. Knot Theory Ramifications 16 (2007) 1121–1163, doi: 10.1142/S0218216507005725]). From the extended Pachner moves (Crane and Yetter [Moves on filtered PL manifolds and stratified PL spaces, arXiv:1404.3142 ]), we derive equations that we subsequently translate into string diagrams so that we can easily observe their properties. As in Dougherty, Park and Yetter [On 2-dimensional Dijkgraaf–Witten theory with defects, to appear in J. Knots Theory Ramifications], we require that triangulations be flaglike, meaning that each simplex of the triangulation is either disjoint from the defect curve, or intersects it in a closed face, and that the extended Pachner moves preserve flaglikeness.

Height fluctuations in interacting dimers

We consider a non-integrable model for interacting dimers on the two-dimensional square lattice. Configurations are perfect matchings of $\mathbb Z^2$, i.e. subsets of edges such that each vertex is covered exactly once ("close-packing" condition). Dimer configurations are in bijection with discrete height functions, defined on faces $\boldsymbol{\xi}$ of $\mathbb Z^2$. The non-interacting model is "integrable" and solvable via Kasteleyn theory; it is known that all the moments of the height difference $h_{\boldsymbol{\xi}}-h_{\boldsymbol{\eta}}$ converge to those of the massless Gaussian Free Field (GFF), asymptotically as $|{\boldsymbol{\xi}}-{\boldsymbol{\eta}}|\to \infty$. We prove that the same holds for small non-zero interactions, as was conjectured in the theoretical physics literature. Remarkably, dimer-dimer correlation functions are instead not universal and decay with a critical exponent that depends on the interaction strength. Our proof is based on an exact representation of the model in terms of lattice interacting fermions, which are studied by constructive field theory methods. In the fermionic language, the height difference $h_{\boldsymbol{\xi}}-h_{\boldsymbol{\eta}}$ takes the form of a non-local operator, consisting of a sum of monomials along an {\it arbitrary} path connecting $\boldsymbol{\xi}$ and $\boldsymbol{\eta}$. As in the non-interacting case, this path-independence plays a crucial role in the proof.

Radiative corrections in kaon decays

This account provides a brief overview of various non-perturbative methods that have been developed in order to address the issue of radiative corrections to kaon decays: chiral lagrangians, non-relativistic effective field theory and dispersive constructions. Only a small number of applications is mentioned, mainly for illustrative purposes.