Two-dimensional Quantum Field Theory Research Articles

Dilaton shifts, probability measures, and decomposition

Abstract In this paper we discuss dilaton shifts (Euler counterterms) arising in decomposition of two-dimensional quantum field theories with higher-form symmetries. Relative shifts between universes are fixed by locality and take a universal form, reflecting underlying (noninvertible, quantum) symmetries. The first part of this paper constructs a general formula for such dilaton shifts, and discusses related computations. In the second part of this paper, we comment on the relation between decomposition and ensembles.

Noninvertible symmetries in 2D from type IIB string theory

We propose a top-down approach to noninvertible symmetries in two-dimensional quantum field theories and their three-dimensional (3D) associated symmetry topological field theories. We focus on the gauge theory engineered on D1-branes probing a particular Calabi-Yau 4-fold singularity. We show how to derive the symmetry topological field theory, a 3D Dijkgraaf-Witten theory, from the IIB supergravity under dimensional reduction. We also identify branes behind the noninvertible topological lines by dimensionally reducing their world volume actions. The action of noninvertible lines on charged local operators is then realized as the Hanany-Witten transition. Published by the American Physical Society 2024

Gauging non-invertible symmetries: topological interfaces and generalized orbifold groupoid in 2d QFT

Gauging is a powerful operation on symmetries in quantum field theory (QFT), as it connects distinct theories and also reveals hidden structures in a given theory. We initiate a systematic investigation of gauging discrete generalized symmetries in two-dimensional QFT. Such symmetries are described by topological defect lines (TDLs) which obey fusion rules that are non-invertible in general. Despite this seemingly exotic feature, all well-known properties in gauging invertible symmetries carry over to this general setting, which greatly enhances both the scope and the power of gauging. This is established by formulating generalized gauging in terms of topological interfaces between QFTs, which explains the physical picture for the mathematical concept of algebra objects and associated module categories over fusion categories that encapsulate the algebraic properties of generalized symmetries and their gaugings. This perspective also provides simple physical derivations of well-known mathematical theorems in category theory from basic axiomatic properties of QFT in the presence of such interfaces. We discuss a bootstrap-type analysis to classify such topological interfaces and thus the possible generalized gaugings and demonstrate the procedure in concrete examples of fusion categories. Moreover we present a number of examples to illustrate generalized gauging and its properties in concrete conformal field theories (CFTs). In particular, we identify the generalized orbifold groupoid that captures the structure of fusion between topological interfaces (equivalently sequential gaugings) as well as a plethora of new self-dualities in CFTs under generalized gaugings.

Spontaneous symmetry breaking on surface defects

Coleman’s theorem states that continuous internal symmetries cannot be spontaneously broken in two-dimensional quantum field theories (QFTs). In this work we consider surface (i.e. two-dimensional) defects in d-dimensional conformal field theories (CFTs) invariant under a continuous internal symmetry group G. We study under which conditions it is possible for a surface defect to break spontaneously a continuous internal symmetry. We find that spontaneous symmetry breaking (SSB) is impossible under reasonable assumptions on the defect Renormalization Group (RG) flow. Counterexamples are possible only for exotic RG flows, that do not terminate at a fixed-point. We discuss an example of this kind. We also illustrate our no-go result with an effective field theory analysis of generic defect RG flows. We find a generic weakly coupled defect universality class (with no SSB), where correlation functions decay logarithmically. Our analysis generalizes the recent discovery by Metlitski of the extraordinary-log boundary universality class in the O(N) model.

Finite volume form factors in integrable theories

We develop a new method to calculate finite size corrections for form factors in two-dimensional integrable quantum field theories. We extract these corrections from the excited state expectation value of bilocal operators in the limit when the operators are far apart. We elaborate the finite size effects explicitly up to the 3rd Lüscher order and conjecture the structure of the general form. We also fully recover the explicitly known massive fermion finite volume form factors.

Null energy constraints on two-dimensional RG flows

We study applications of spectral positivity and the averaged null energy condition (ANEC) to renormalization group (RG) flows in two-dimensional quantum field theory. We find a succinct new proof of the Zamolodchikov c-theorem, and derive further independent constraints along the flow. In particular, we identify a natural C-function that is a completely monotonic function of scale, meaning its derivatives satisfy the alternating inequalities (–1)nC(n)(μ2) ≥ 0. The completely monotonic C-function is identical to the Zamolodchikov C-function at the endpoints, but differs along the RG flow. In addition, we apply Lorentzian techniques that we developed recently to study anomalies and RG flows in four dimensions, and show that the Zamolodchikov c-theorem can be restated as a Lorentzian sum rule relating the change in the central charge to the average null energy. This establishes that the ANEC implies the c-theorem in two dimensions, and provides a second, simpler example of the Lorentzian sum rule.

Fermionization of fusion category symmetries in 1+1 dimensions

We discuss the fermionization of fusion category symmetries in two-dimensional topological quantum field theories (TQFTs). When the symmetry of a bosonic TQFT is described by the representation category Rep(H) of a semisimple weak Hopf algebra H, the fermionized TQFT has a superfusion category symmetry SRep(H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}u), which is the supercategory of super representations of a weak Hopf superalgebra H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}u. The weak Hopf superalgebra H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}u depends not only on H but also on a choice of a non-anomalous ℤ2 subgroup of Rep(H) that is used for the fermionization. We derive a general formula for H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}u by explicitly constructing fermionic TQFTs with SRep(H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}u) symmetry. We also construct lattice Hamiltonians of fermionic gapped phases when the symmetry is non-anomalous. As concrete examples, we compute the fermionization of finite group symmetries, the symmetries of finite gauge theories, and duality symmetries. We find that the fermionization of duality symmetries depends crucially on F-symbols of the original fusion categories. The computation of the above concrete examples suggests that our fermionization formula of fusion category symmetries can also be applied to non-topological QFTs.

Topological gauging and double current deformations

We study solvable deformations of two-dimensional quantum field theories driven by a bilinear operator constructed from a pair of conserved U(1) currents Ja. We propose a quantum formulation of these deformations, based on the gauging of the corresponding symmetries in a path integral. This formalism leads to an exact dressing of the S-matrix of the system, similarly as what happens in the case of a textrm{T}overline{textrm{T}} deformation. For conformal theories the deformations under study are expected to be exactly marginal. Still, a peculiar situation might arise when the conserved currents Ja are not well-defined local operators in the original theory. A simple example of this kind of system is provided by rotation currents in a theory of multiple free, massless, non-compact bosons. We verify that, somewhat unexpectedly, such a theory is indeed still conformal after deformation and that it coincides with a TsT transformation of the original system. We then extend our formalism to the case in which the conserved currents are non-Abelian and point out its connection with Deformed T-dual Models and homogeneous Yang-Baxter deformations. In this case as well the deformation is based on a gauging of the symmetries involved and it turns out to be non-trivial only if the symmetry group admits a non-trivial central extension. Finally we apply what we learned by relating the textrm{T}overline{textrm{T}} deformation to the central extension of the two-dimensional Poincaré algebra.

Toverline{T} deformed scattering happens within matrices

We show the Toverline{T} deformation of two-dimensional quantum field theories is equivalent to replacing the spacetime dependence of the fields with dependence on the indices of infinitely large matrices. We show how this correspondence explains the CDD phase dressing of the S-matrix and the general formula for the deformation of arbitrary correlation functions. We also describe how the Moyal deformation of self-dual gravity is a Toverline{T} deformation of the theory described by the Chalmers-Siegel action, where the Toverline{T} deformation is defined on the two-dimensional plane of interactions.

Asymptotics in an asymptotic CFT

In this work we illustrate the resurgent structure of the λ-deformation; a two-dimensional integrable quantum field theory that has an RG flow with an SU(N)k Wess-Zumino-Witten conformal fixed point in the UV. To do so we use modern matched asymptotic techniques applied to the thermodynamic Bethe ansatz formulation to compute the free energy to 38 perturbative orders in an expansion of large applied chemical potential. We find numerical evidence for factorial asymptotic behaviour with both alternating and non-alternating character which we match to an analytic expression. A curiosity of the system is that the leading non-alternating factorial growth vanishing when k divides N. The ambiguities associated to Borel resummation of this series are suggestive of non-perturbative contributions. This is verified with an analytic study of the TBA system demonstrating a cancellation between perturbative and non-perturbative ambiguities.

The Hermitian axiom on two-dimensional topological quantum field theories

We examine Atiyah’s Hermitian axiom for two-dimensional complex topological quantum field theories. Building on the correspondence between 2D topological quantum field theories (TQFTs) and Frobenius algebras, we find the algebraic objects corresponding to Hermitian and unitary TQFTs, respectively, and prove structure theorems about them. We then clarify a few older results on unitary TQFTs using our structure theorems.

Root-TT[over ¯] Deformations in Two-Dimensional Quantum Field Theories.

In this Letter, we introduce a one-parameter deformation of two-dimensional quantum field theories generated by a nonanalytic operator that we call Root-TT[over ¯]. For a conformal field theory, the operator coincides with the square root of the TT[over ¯] operator. More generally, the operator is defined so that classically it is marginal and generates a flow that commutes with the TT[over ¯] flow. Intriguingly, the Root-TT[over ¯] flow is closely related to the ModMax theory recently constructed by Bandos, Lechner, Sorokin, and Townsend.

Fixed points of (0,2) Landau-Ginzburg renormalization group flows and the chiral algebra

We discuss renormalization group flows in two-dimensional quantum field theories with (0,2) supersymmetry. We focus on theories with UV described by a Landau-Ginzburg Lagrangian and use the chiral algebra to constrain the IR dynamics. We present examples where the structure of the chiral algebra is incompatible with unitarity of the IR superconformal theory and discuss the implications of this result for programs of classifying (0,2) SCFTs as endpoints of flows from simple Lagrangian theories.

Aspects of Nonrelativistic Strings

We review recent developments on nonrelativistic string theory. In flat spacetime, the theory is defined by a two-dimensional relativistic quantum field theory with nonrelativistic global symmetries acting on the worldsheet fields. This theory arises as a self-contained corner of relativistic string theory. It has a string spectrum with a Galilean dispersion relation, and a spacetime S-matrix with nonrelativistic symmetry. This string theory also gives a unitary and ultraviolet complete framework that connects different corners of string theory, including matrix string theory and noncommutative open strings. In recent years, there has been a resurgence of interest in the non-Lorentzian geometries and quantum field theories that arise from nonrelativistic string theory in background fields. In this review, we start with an introduction to the foundations of nonrelativistic string theory in flat spacetime. We then give an overview of recent progress, including the appropriate target-space geometry that nonrelativistic strings couple to. This is known as (torsional) string Newton–Cartan geometry, which is neither Lorentzian nor Riemannian. We also give a review of nonrelativistic open strings and effective field theories living on D-branes. Finally, we discuss applications of nonrelativistic strings to decoupling limits in the context of the AdS/CFT correspondence.

Fermionic quantum field theories as probabilistic cellular automata

A class of fermionic quantum field theories with interactions is shown to be equivalent to probabilistic cellular automata, namely cellular automata with a probability distribution for the initial states. Probabilistic cellular automata on a one-dimensional lattice are equivalent to two-dimensional quantum field theories for fermions. They can be viewed as generalized Ising models on a square lattice and therefore as classical statistical systems. As quantum field theories they are quantum systems. Thus quantum mechanics emerges from classical statistics. As an explicit example for an interacting fermionic quantum field theory we describe a type of discretized Thirring model as a cellular automaton. The updating rule of the automaton is encoded in the step evolution operator that can be expressed in terms of fermionic annihilation and creation operators. The complex structure of quantum mechanics is associated to particle-hole transformations. The naive continuum limit exhibits Lorentz symmetry. We exploit the equivalence to quantum field theory in order to show how quantum concepts as wave functions, density matrix, noncommuting operators for observables and similarity transformations are convenient and useful concepts for the description of probabilistic cellular automata.

Thermalization and hydrodynamics of two-dimensional quantum field theories

We consider 2d QFTs as relevant deformations of CFTs in the thermodynamic limit. Using causality and KPZ universality, we place a lower bound on the timescale characterizing the onset of hydrodynamics. The bound is determined parametrically in terms of the temperature and the scale associated with the relevant deformation. This bound is typically much stronger than \frac{1}{T}1T, the expected quantum equilibration time. Subluminality of sound further allows us to define a thermodynamic CC-function, and constrain the sign of the T\bar TTT‾ term in EFTs.

Inelastic Scattering of a Photon by a Quantum Phase Slip.

Spontaneous decay of a single photon is a notoriously inefficient process in nature irrespective of the frequency range. We report that a quantum phase-slip fluctuation in high-impedance superconducting waveguides can split a single incident microwave photon into a large number of lower-energy photons with a near unit probability. The underlying inelastic photon-photon interaction has no analogs in nonlinear optics. Instead, the measured decay rates are explained without adjustable parameters in the framework of a new model of a quantum impurity in a Luttinger liquid. Our result connects circuit quantum electrodynamics to critical phenomena in two-dimensional boundary quantum field theories, important in the physics of strongly correlated systems. The photon lifetime data represent a rare example of verified and useful quantum many-body simulation.

A general definition of $JT_a$ -- deformed QFTs

We propose a general path-integral definition of two-dimensional quantum field theories deformed by an integrable, irrelevant vector operator constructed from the components of the stress tensor and those of a U(1) current. The deformed theory is obtained by coupling the original QFT to a flat dynamical gauge field and ``half'' a flat dynamical vielbein. The resulting partition function is shown to satisfy a geometric flow equation, which perfectly reproduces the flow equations for the deformed energy levels that were previously derived in the literature. The S-matrix of the deformed QFT differs from the original S-matrix only by an overall phase factor that depends on the charges and momenta of the external particles, thus supporting the conjecture that such QFTs are UV complete, although intrinsically non-local. For the special case of an integrable QFT, we check that this phase factor precisely reproduces the change in the finite-size spectrum via the Thermodynamic Bethe Ansatz equations.

Orbifold groupoids

We review the properties of orbifold operations on two-dimensional quantum field theories, either bosonic or fermionic, and describe the “Orbifold groupoids” which control the composition of orbifold operations. Three-dimensional TQFT’s of Dijkgraaf-Witten type will play an important role in the analysis. We briefly discuss the extension to generalized symmetries and applications to constrain RG flows.

Long strings and symmetric product orbifold from the AdS3Bethe equations

A particularly rich class of integrable systems arises from the AdS/CFT duality. There, the two-dimensional quantum field theory living on the string worldsheet may be understood in terms of a non-relativistic factorized S matrix, and the energy spectrum may be derived by techniques such as the mirror thermodynamic Bethe ansatz or the quantum spectral curve. In the case of AdS3/CFT2 without Ramond-Ramond fluxes, the worldhseet theory is a Wess-Zumino-Witten model with continous and discrete representations which, for the lowest allowed level, is dual to the symmetric product orbifold of a free theory. I will show how continuous representations may arise from integrability, and that at the lowest level the Bethe equations yield the symmetric product orbifold partition function on the nose.