Massive integrable soliton theories

T-duality in massive integrable field theories: the homogeneous and complex sine-Gordon models

Three-dimensional massive scalar field theory and the derivative expansion of the renormalization group

We find a general class of pp-wave solutions of type IIB string theory such that the light cone gauge worldsheet lagrangian is that of an interacting massive field theory. When the light cone Lagrangian has (2,2) supersymmetry we can find backgrounds that lead to arbitrary superpotentials on the worldsheet. We consider situations with both flat and curved transverse spaces. We describe in some detail the background giving rise to the N=2 sine Gordon theory on the worldsheet. Massive mirror symmetry relates it to the deformed $CP^1$ model (or sausage model) which seems to elude a purely supergravity target space interpretation.

Some mathematical questions relating to coset conformal field theories (CFT) are considered in the framework of algebraic quantum field theory as developed previously by us. We consider the issue of fix point resolution in the diagonal cosets of type A. We show how to decompose certain reducible representations into irreducibles, and prove that the coset CFT gives rise to a unitary modular category and therefore may be used to construct 3-manifold invariants. We prove that if the coset inclusion satisfies certain conditions which can be checked in examples, the Kac–Wakimoto Hypothesis (KWH) is equivalent to the Kac–Wakimoto Conjecture (KWC), a result which seems to be hard to prove by purely representation considerations. Examples are also presented.

We offer a procedure for evaluating the forces exerted by solitons of weak-coupling field theories on one another. We illustrate the procedure for the kink and the antikink of the two-dimensional ${\ensuremath{\varphi}}^{4}$ theory. To do this, we construct analytically a static solution of the theory which can be interpreted as a kink and an antikink held a distance $R$ apart. This leads to a definition of the potential energy $U(R)$ for the pair, which is seen to have all the expected features. A corresponding evaluation is also done for $U(R)$ between a soliton and an antisoliton of the sine-Gordon theory. When this $U(R)$ is inserted into a nonrelativistic two-body problem for the pair, it yields a set of bound states and phase shifts. These are found to agree with exact results known for the sine-Gordon field theory in those regions where $U(R)$ is expected to be significant, i.e., when $R$ is large compared to the soliton size. We take this agreement as support that our procedure for defining $U(R)$ yields the correct description of the dynamics of well-separated soliton pairs. An important feature of $U(R)$ is that it seems to give strong intersoliton forces when the coupling constant is small, as distinct from the forces between the ordinary quanta of the theory. We suggest that this is a general feature of a class of theories, and emphasize the possible relevance of this feature to real strongly interacting hadrons.

At large scales, magnetostatics of superconductors is described by a massive vector field theory: the London model. The magnetic field cannot penetrate into the bulk unless quantum vortices are formed. These are topological excitations characterized by an invariant: the phase winding. The London model dictates that loops of such vortices are not stable because the kinetic energy of superflow and the magnetic energy are smaller, the smaller vortex loops are. We demonstrate that in two-component superconductors, under certain conditions, such as the proximity to pair-density-wave instabilities, the hydromagnetostatics of the superconducting state and topological excitation changes dramatically: the excitations acquire the form of stable vortex loops and knots characterized by the different topological invariant: the Hopf index and hence termed hopfions. This demonstrates that magnetic properties in a superconducting state can be dramatically different from those of a London's massive vector field theory.

Anomalous finite-temperature transport has recently been observed in numerical studies of various integrable models in one dimension; these models share the feature of being invariant under a continuous non-Abelian global symmetry. This work offers a comprehensive group-theoretic account of this elusive phenomenon. For an integrable quantum model with local interactions, invariant under a global non-Abelian simple Lie group G, we find that finite-temperature transport of Noether charges associated with symmetry G in thermal states that are invariant under G is universally superdiffusive and characterized by the dynamical exponent z=3/2. This conclusion holds regardless of the Lie algebra symmetry, local degrees of freedom (on-site representations), Lorentz invariance, or particular realization of microscopic interactions: We accordingly dub it “superuniversal.” The anomalous transport behavior is attributed to long-lived giant quasiparticles dressed by thermal fluctuations. We provide an algebraic viewpoint on the corresponding dressing transformation and elucidate formal connections to fusion identities amongst the quantum-group characters. We identify giant quasiparticles with nonlinear soliton modes of classical field theories that describe low-energy excitations above ferromagnetic vacua. Our analysis of these field theories also provides a complete classification of the low-energy (i.e., Goldstone-mode) spectra of quantum isotropic ferromagnetic chains.Received 24 October 2020Revised 4 May 2021Accepted 19 May 2021DOI:https://doi.org/10.1103/PhysRevX.11.031023Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasNonequilibrium statistical mechanicsQuantum statistical mechanicsQuantum transportTechniquesIntegrable systemsStatistical PhysicsGeneral Physics

In this chapter, we shall look at various forms of the ASD condition on the curvature of a metric in four dimensions. The conditions are naturally expressed in terms of ‘Lax pairs’ of vector fields, either on space-time itself or on some bundle over it. By interpreting a vector field as a generator of a diffeomorphism group, we can represent some of the conditions as special cases of the ASDYM equations, but with infinite-dimensional gauge groups. In §13.1-4 we derive various forms of the ASD condition and its subcases on the curvature of a metric. In §13.5 we describe the extension of the twistor correspondence by which ASD conformal structures are encoded in a curved twistor space P, and the way in which various supplementary conditions on the metric (vacuum, Einstein, Kāhler) give rise to additional structures on P. In §13.6 we discuss the reductions by one symmetry to the equations of an Einstein-Weyl space and to the ‘SU(∞) Toda field equations’. In §13.7 we shall explain how many standard integrable equations arise as reductions of the ASD conditions on a metric, and how the curved twistor construction relates to the Penrose-Ward transform.

Aspects of the cohomology of the infinite-dimensional Heisenberg group as represented on the free boson field over a given Hilbert space are treated. The 1-cohomology is shown to be trivial in certain spaces of generalized vectors. From this derives a canonical quantization mapping from classical (unquantized) forms to generalized operators on the boson field. An example, applied here to scalar relativistic fields, is the quantization of a given classical interaction Lagrangian or Hamiltonian, i.e., the establishment and characterization of corresponding boson field operators. For example, if ϕ \phi denotes the free massless scalar field in d-dimensional Minkowski space ( d ≥ 4 d \geq 4 , even) and if q is an even integer greater than or equal to 4, then ∫ M 0 : ϕ ( X ) q : d X {\smallint _{{{\mathbf {M}}_0}}}:\phi {(X)^q}:dX exists as a nonvanishing, Poincaré invariant, hermitian, selfadjointly extendable operator, where : ϕ ( X ) q \phi {(X)^q} : denotes the Wick power. Applications are also made to the rigorous establishment of basic symbolic operators in heuristic quantum field theory, including certain massive field theories; to a class of pseudo-interacting fields obtained by substituting the free field into desingularized expressions for the total Hamiltonian in the conformally invariant case d = q = 4 d = q = 4 and to corresponding scattering theory.

In two dimensions, conformai invariance provides a natural way to classify the universality classes for critical phenomena [1]. It is of growing interest to extend the Conformai Field Theories (CFT) away from criticality to reach a complete understanding of the scaling region around the fixed points. Zamolodchikov [2] found that particular deformations of the CFT preserve the integrability of the models and the corresponding massive field theories can be described by a factorizable S-matrix. The 5-matrix of the three-state Potts model [3] and the S-matrix of the Ising model in a magnetic field [2] are the first examples of this approach. The hidden E 8 symmetry of Ising model is responsible for the rich structure of the S-matrix in the latter case. Recently, other models have been discussed: the 5-matrix of non-unitary Yang-Lee edge singularity [4] and the S-matrix of the thermal perturbation of the tricritical Ising model [5]. P. Christe will discuss the latter model, expecially its hidden E 7 symmetry, in his talk [6].

Attention to the two-dimensional Liouville Field Theory (LFT) is drawn basically for two reasons. First it was recognized [1] as an effective field theory of the 2d. quantum gravity. In particular it is very relevant in the string theory [1–4]. Second, it is an example of non-rational conformai field theory (CFT) which is very likely exactly solvable (e.g., the classical equations of motion are integrable).The interest to LFT was renewed recently with the development of the matrix model approach to 2d gravity [5, 6]. It was shown that LFT is able to reproduce some of the predictions of the matrix model approach, in particular the scaling behavior [7–9], the genus one partition functions [10] and some of the integrated correlation functions [11–15]. It is very plausible therefore that LFT describes the same 2d quantum gravity as the matrix models do (at least in the “weak coupling region” C L > 25).

We argue that the notion of identical particles is no longer well defined in quantum systems governed by non-commutative deformations of space-time symmetries. Such models are characterized by four-momentum space given by a non-abelian Lie group. Our analysis is based on the observation that, for states containing more than one particle, only the total momentum of the system is a well defined quantum number. Such total momentum is obtained from the non-abelian composition of the particles individual momenta which are no longer uniquely defined. The main upshot of our analysis is that all previous attempts to construct Fock spaces for these models rested on wrong assumptions and indeed have been unsuccessful. We also show how the natural braiding of momentum quantum numbers which characterizes the exchange of factors in the tensor product of states is covariant under relativistic transformations thus solving a long standing problem in the field.

K-theoretical boundary rings in [formula omitted] coset models

We analyze various perspectives on the elliptic genus of non-compact supersymmetric coset conformal field theories with central charge larger than three. We calculate the holomorphic part of the elliptic genus via a free field description of the model, and show that it agrees with algebraic expectations. The holomorphic part of the elliptic genus is directly related to an Appell-Lerch sum and behaves anomalously under modular transformation properties. We analyze the origin of the anomaly by calculating the elliptic genus through a path integral in a coset conformal field theory. The path integral codes both the holomorphic part of the elliptic genus, and a non-holomorphic remainder that finds its origin in the continuous spectrum of the non-compact model. The remainder term can be shown to agree with a function that mathematicians introduced to parameterize the difference between mock theta functions and Jacobi forms. The holomorphic part of the elliptic genus thus has a path integral completion which renders it non-holomorphic and modular.

We study conformal field theory with the symmetry algebra $ \mathcal{A}\left( {2,\ p} \right)={{{\widehat{\mathfrak{gl}}{(n)_2}}} \left/ {{\widehat{\mathfrak{gl}}{{{\left( {n-p} \right)}}_2}}} \right.} $ . In order to support the conjecture that this algebra acts on the moduli space of instantons on ℂ2/ℤ p , we calculate the characters of its representations and check their coincidence with the generating functions of the fixed points of the moduli space of instantons. We show that the algebra $ \mathcal{A}\left( {2,\ p} \right) $ can be realized in two ways. The first realization is connected with the cross-product of p Virasoro and p Heisenberg algebras: $ {{\mathcal{H}}^p} $ × Vir p . The second realization is connected with: $ {{\mathcal{H}}^p} \times \widehat{\mathfrak{sl}}{(p)_2}\times \left( {\widehat{\mathfrak{sl}}{(2)_p}\times {{{\widehat{\mathfrak{sl}}{(2)_{n-p }}}} \left/ {{\widehat{\mathfrak{sl}}{(2)_n}}} \right.}} \right) $ . The equivalence of these two realizations provides the non-trivial identity for the characters of $ \mathcal{A}\left( {2,\ p} \right) $ . The moduli space of instantons on ℂ2/ℤ p admits two different compactifications. This leads to two different bases for the representations of $ \mathcal{A}\left( {2,\ p} \right) $ . We use this fact to explain the existence of two forms of the instanton pure partition functions.