Showing papers on "Operator algebra published in 2021"
TL;DR: In this article, a log-modular quantum group at even order roots of unity was constructed, both as finite-dimensional ribbon quasi-Hopf algebras and as finite ribbon tensor categories, via a de-equivariantization procedure.
Abstract: We construct log-modular quantum groups at even order roots of unity, both as finite-dimensional ribbon quasi-Hopf algebras and as finite ribbon tensor categories, via a de-equivariantization procedure. The existence of such quantum groups had been predicted by certain conformal field theory considerations, but constructions had not appeared until recently. We show that our quantum groups can be identified with those of Creutzig-Gainutdinov-Runkel in type $$A_1$$
, and Gainutdinov-Lentner-Ohrmann in arbitrary Dynkin type. We discuss conjectural relations with vertex operator algebras at (1, p)-central charge. For example, we explain how one can (conjecturally) employ known linear equivalences between the triplet vertex algebra and quantum $$\mathfrak {sl}_2$$
, in conjunction with a natural $${{\,\mathrm{PSL}\,}}_2$$
-action on quantum $$\mathfrak {sl}_2$$
provided by our de-equivariantization construction, in order to deduce linear equivalences between “extended” quantum groups, the singlet vertex operator algebra, and the (1, p)-Virasoro logarithmic minimal model. We assume some restrictions on the order of our root of unity outside of type $$A_1$$
, which we intend to eliminate in a subsequent paper.
39 citations
TL;DR: This article focuses on optimization of polynomials in noncommuting variables, while taking into account sparsity in the input data, and a converging hierarchy of semidefinite relaxations for eigenvalue and trace optimization is provided.
Abstract: This article focuses on optimization of polynomials in noncommuting variables, while taking into account sparsity in the input data. A converging hierarchy of semidefinite relaxations for eigenvalue and trace optimization is provided. This hierarchy is a noncommutative analogue of results due to Lasserre (SIAM J Optim 17(3):822–843, 2006) and Waki et al. (SIAM J Optim 17(1):218–242, 2006). The Gelfand–Naimark–Segal construction is applied to extract optimizers if flatness and irreducibility conditions are satisfied. Among the main techniques used are amalgamation results from operator algebra. The theoretical results are utilized to compute lower bounds on minimal eigenvalue of noncommutative polynomials from the literature.
33 citations
TL;DR: In this article, a braided tensor category structure on the category of C 1 -cofinite modules for the (universal or simple) Virasoro vertex operator algebras of arbitrary central charge is presented.
31 citations
TL;DR: In this paper, the authors studied four-dimensional gauge theories with arbitrary simple gauge group with global center symmetry and discrete chiral symmetry and showed that the mixed $0-form/$1$-form 't Hooft anomaly results in a central extension of the global-symmetry operator algebra.
Abstract: We study four-dimensional gauge theories with arbitrary simple gauge group with $1$-form global center symmetry and $0$-form parity or discrete chiral symmetry. We canonically quantize on $\mathbb{T}^3$, in a fixed background field gauging the $1$-form symmetry. We show that the mixed $0$-form/$1$-form 't Hooft anomaly results in a central extension of the global-symmetry operator algebra. We determine this algebra in each case and show that the anomaly implies degeneracies in the spectrum of the Hamiltonian at any finite-size torus. We discuss the consistency of these constraints with both older and recent semiclassical calculations in $SU(N)$ theories, with or without adjoint fermions, as well as with their conjectured infrared phases.
24 citations
TL;DR: In this article, it was shown that every strongly rational, holomorphic vertex algebra V of central charge 24 with V 1 ≠ { 0 } can be obtained by an orbifold construction from the Leech lattice vertex operator algebra V Λ, which suffices to restrict the possible Lie algebras that can occur as weight-one space of V to the 71 of Schellekens.
20 citations
TL;DR: In this article, the authors adapt the classical notion of building models by games to the setting of continuous model theory, and show that the set of continuous functions on the pseudoarc is an enforceable model of the theory of unital, projectionless, abelian -algebras.
Abstract: We adapt the classical notion of building models by games to the setting of continuous model theory. As an application, we study to what extent canonical operator algebras are enforceable models. For example, we show that the hyperfinite II1 factor is an enforceable II1 factor if and only if the Connes Embedding Problem has a positive solution. We also show that the set of continuous functions on the pseudoarc is an enforceable model of the theory of unital, projectionless, abelian -algebras and use this to show that it is the prime model of its theory.
19 citations
TL;DR: In this article, a dilation-theoretic characterization of the Choquet order on the space of measures on a compact convex set using ideas from the theory of operator algebras is established.
18 citations
TL;DR: The study of operator algebras of operators on Hilbert spaces is one of the most active areas within functional analysis as discussed by the authors, and a modern overview can be traced back to the 50's, and has seen renewed attention in the last decade through the infusion of new techniques.
15 citations
TL;DR: In this article, the authors studied correlation functions involving generalized ANEC operators in free and holographic Conformal Field Theories and derived the algebra of these light-ray operators.
Abstract: We study correlation functions involving generalized ANEC operators of the form $$ \int {dx}^{-}{\left({x}^{-}\right)}^{n+2}{T}_{--}\left(\overrightarrow{x}\right) $$
in four dimensions. We compute two, three, and four-point functions involving external scalar states in both free and holographic Conformal Field Theories. From this information, we extract the algebra of these light-ray operators. We find a global subalgebra spanned by n = {−2, −1, 0, 1, 2} which annihilate the conformally invariant vacuum and transform among themselves under the action of the collinear conformal group that preserves the light-ray. Operators outside this range give rise to an infinite central term, in agreement with previous suggestions in the literature. In free theories, even some of the operators inside the global subalgebra fail to commute when placed at spacelike separation on the same null-plane. This lack of commutativity is not integrable, presenting an obstruction to the construction of a well defined light-ray algebra at coincident $$ \overrightarrow{x} $$
coordinates. For holographic CFTs the behavior worsens and operators with n ≠ −2 fail to commute at spacelike separation. We reproduce this result in the bulk of AdS where we present new exact shockwave solutions dual to the insertions of these (exponentiated) operators on the boundary.
15 citations
TL;DR: In particular, the authors showed that the category of all such twisted V-modules is a braided G-crossed (super) category, and that the G-equivariantization of this braided tensor equivalent to the original V-algebra is a BRAIDED tensor equivalence.
Abstract: A two-dimensional chiral conformal field theory can be viewed mathematically as the representation theory of its chiral algebra, a vertex operator algebra. Vertex operator algebras are especially well suited for studying logarithmic conformal field theory (in which correlation functions have logarithmic singularities arising from non-semisimple modules for the chiral algebra) because of the logarithmic tensor category theory of Huang, Lepowsky, and Zhang. In this paper, we study not-necessarily-semisimple or rigid braided tensor categories $${\mathcal {C}}$$
of modules for the fixed-point vertex operator subalgebra $$V^G$$
of a vertex operator (super)algebra V with finite automorphism group G. The main results are that every $$V^G$$
-module in $${\mathcal {C}}$$
with a unital and associative V-action is a direct sum of g-twisted V-modules for possibly several $$g\in G$$
, that the category of all such twisted V-modules is a braided G-crossed (super)category, and that the G-equivariantization of this braided G-crossed (super)category is braided tensor equivalent to the original category $${\mathcal {C}}$$
of $$V^G$$
-modules. This generalizes results of Kirillov and Muger proved using rigidity and semisimplicity. We also apply the main results to the orbifold rationality problem: whether $$V^G$$
is strongly rational if V is strongly rational. We show that $$V^G$$
is indeed strongly rational if V is strongly rational, G is any finite automorphism group, and $$V^G$$
is $$C_2$$
-cofinite.
14 citations
TL;DR: In this paper, the authors investigated the emergence of topological defect lines in the conformal Regge limit of two-dimensional conformal field theory, and derived a formula relating the infinite boost limit, which holographically encodes the opacity of bulk scattering, to the action of topologically defect lines on local operators.
Abstract: We investigate the emergence of topological defect lines in the conformal Regge limit of two-dimensional conformal field theory. We explain how a local operator can be factorized into a holomorphic and an anti-holomorphic defect operator connected through a topological defect line, and discuss implications on analyticity and Lorentzian dynamics including aspects of chaos. We derive a formula relating the infinite boost limit, which holographically encodes the “opacity” of bulk scattering, to the action of topological defect lines on local operators. Leveraging the unitary bound on the opacity and the positivity of fusion coefficients, we show that the spectral radii of a large class of topological defect lines are given by their loop expectation values. Factorization also gives a formula relating the local and defect operator algebras and fusion categorical data. We then review factorization in rational conformal field theory from a defect perspective, and examine irrational theories. On the orbifold branch of the c = 1 free boson theory, we find a unified description for the topological defect lines through which the twist fields are factorized; at irrational points, the twist fields factorize through “non-compact” topological defect lines which exhibit continuous defect operator spectra. Along the way, we initiate the development of a formalism to characterize non-compact topological defect lines.
TL;DR: In this article, the authors identify vertex operator algebras (VOAs) of a class of Argyres-Douglas (AD) matters with two types of non-abelian flavor symmetries.
Abstract: We identify vertex operator algebras (VOAs) of a class of Argyres-Douglas (AD) matters with two types of non-abelian flavor symmetries. They are the W algebras defined using nilpotent orbit with partition [qm, 1s]. Gauging above AD matters, we can find VOAs for more general $$ \mathcal{N} $$
= 2 SCFTs engineered from 6d (2, 0) theories. For example, the VOA for general (AN − 1, Ak − 1) theory is found as the coset of a collection of above W algebras. Various new interesting properties of 2d VOAs such as level-rank duality, conformal embedding, collapsing levels, coset constructions for known VOAs can be derived from 4d theory.
Posted Content•
TL;DR: In this paper, a sub-extensive number of exact eigenstates for a large family of density-density interaction terms are found, embedded in a continuum of strongly correlated excited states.
Abstract: While the chiral linear Luttinger liquid is integrable via bosonization, its non-linear counterpart does not admit for an analytic solution. In this work, we find a sub-extensive number of exact eigenstates for a large family of density-density interaction terms. These states are embedded in a continuum of strongly-correlated excited states. The real-space entanglement entropy of some exact states scales logarithmically with system size, while that of others has volume-law scaling. We introduce momentum-space entanglement as an unambiguous differentiator between these exact states and the remaining excited states. With regard to momentum space, the exact states behave as bona fide quantum many body scars: they exhibit identically zero momentum-space entanglement, while typical eigenstates behave thermally. We corroborate this finding by a level statistics analysis. Furthermore, we detail the general formalism for systematically finding all interaction terms and associated exact states, and present a number of infinite exact state sequences extending to arbitrarily high energies. Unlike many previous examples of quantum many body scars, the exact states uncovered here do not lie at equidistant energies, and do not follow from a special operator algebra. Instead, they are uniquely enabled by the interplay of Fermi statistics and chirality.
TL;DR: In this paper, the large-N limit of random matrix models can be realized using operator algebras, and the notion of the Brown measure is introduced to play the role of the eigenvalue distribution for operators in an operator algebra.
Abstract: This article begins with a brief review of random matrix theory, followed by a discussion of how the large-N limit of random matrix models can be realized using operator algebras. I then explain the notion of “Brown measure,” which play the role of the eigenvalue distribution for operators in an operator algebra.
Posted Content•
TL;DR: In this article, a notion of completeness in QFT arising from the analysis of basic properties of the set of operator algebras attached to regions is presented. But this completeness is not applicable to all QFT theories, as shown by the fact that for non-complete theories, the existence of generalized symmetries is unavoidable.
Abstract: We review a notion of completeness in QFT arising from the analysis of basic properties of the set of operator algebras attached to regions. In words, this completeness asserts that the physical observable algebras produced by local degrees of freedom are the maximal ones compatible with causality. We elaborate on equivalent statements to this completeness principle such as the non-existence of generalized symmetries and the uniqueness of the net of algebras. We clarify that for non-complete theories, the existence of generalized symmetries is unavoidable, and further, that they always come in dual pairs with precisely the same ``size''. Moreover, the dual symmetries are always broken together, be it explicitly or effectively. Finally, we comment on several issues raised in recent literature, such as the relationship between completeness and modular invariance, dense sets of charges, and absence of generalized symmetries in the bulk of holographic theories.
TL;DR: In this article, contractive projections, isometries, and real positive maps on algebras of operators on a Hilbert space were studied, and a new Banach-Stone type theorem was proved for real positive projections.
Abstract: We study contractive projections, isometries, and real positive maps on algebras of operators on a Hilbert space. For example we find generalizations and variants of certain classical results on contractive projections on C*-algebras and JB-algebras due to Choi, Effros, Stormer, Friedman and Russo, and others. In fact most of our arguments generalize to contractive `real positive' projections on Jordan operator algebras, that is on a norm-closed space A of operators on a Hilbert space which are closedunder the Jordan product. We also prove many new general results on real positive maps which are foundational to the study of such maps, and of interest in their own right. We also prove a new Banach-Stone type theorem for isometries between operator algebras or Jordan operator algebras. An application of this is given to the characterization of symmetric real positive projections.
TL;DR: In this paper, simple current extensions of tensor products of two vertex operator algebras satisfying certain conditions are studied, and the relationship between the fusion rule for the simple current extension and the fusion rules for a tensor factor is established.
Abstract: We study simple current extensions of tensor products of two vertex operator algebras satisfying certain conditions. We establish the relationship between the fusion rule for the simple current extension and the fusion rule for a tensor factor. In a special case, we construct a chain of simple current extensions. We discuss certain irreducible twisted modules for the simple current extension as well.
TL;DR: The theory of real positivity of operator algebras initiated by the author and Charles Read was discussed in this paper, where it was shown that positivity is often the right replacement in a general algebra A for positivity in C∗-algebra.
Abstract: Most of this article is an expanded version of our talk at the Positivity X conference. It is essentially a survey, but some part, like most of the lengthy Sect. 5, is comprised of new results whose proofs are unpublished elsewhere. We begin by reviewing the theory of real positivity of operator algebras initiated by the author and Charles Read. Then we present several new general results (mostly joint work with Matthew Neal) about real positive maps. The key point is that real positivity is often the right replacement in a general algebra A for positivity in C∗-algebras. We then apply this to studying contractive projections (‘conditional expectations’) and isometries of operator algebras. For example we generalize and find variants of certain classical results on positive projections on C∗-algebras and JB algebras due to Choi, Effros, Stormer, Friedman and Russo, and others. In previous work with Neal we had done the ‘completely contractive’ case; we focus here on describing the real positive contractive case from recent work with Neal. We also prove here several new and complementary results on this topic due to the author, indeed this new work constitutes most of Sect. 5. Finally, in the last section we describe a related part of some recent joint work with Labuschagne on what we consider to be a good noncommutative generalization of the ‘characters’ (i.e. homomorphisms into the scalars) on an algebra. Such characters are a special case of the projections mentioned above, and are shown to be intimately related to conditional expectations. The idea is to try to use these to generalize certain classical function algebra results involving characters.
TL;DR: In this paper, a q-deformation of the Miura transformation is derived for q-formed YL,M,N, which is obtained as a reduction of the quantum toroidal algebra.
Abstract: Recently, Gaiotto and Rapcak proposed a generalization of WN algebra by considering the symmetry at the corner of the brane intersection (corner vertex operator algebra). The algebra, denoted as YL,M,N, is characterized by three non-negative integers L, M, N. It has a manifest triality automorphism which interchanges L, M, N, and can be obtained as a reduction of W1+∞ algebra with a “pit” in the plane partition representation. Later, Prochazka and Rapcak proposed a representation of YL,M,N in terms of L + M + N free bosons by a generalization of Miura transformation, where they use the fractional power differential operators. In this paper, we derive a q-deformation of the Miura transformation. It gives a free field representation for q-deformed YL,M,N, which is obtained as a reduction of the quantum toroidal algebra. We find that the q-deformed version has a “simpler” structure than the original one because of the Miki duality in the quantum toroidal algebra. For instance, one can find a direct correspondence between the operators obtained by the Miura transformation and those of the quantum toroidal algebra. Furthermore, we can show that the both algebras share the same screening operators.
TL;DR: In this article, it was shown that any unitary extension of a completely unitary VOA is completely Unitary, i.e., it is unitary with respect to the Q-system.
Abstract: We relate extensions of completely unitary VOAs and (commutative) Q-systems. As an application, we show that any unitary extension of a completely unitary VOA is completely unitary.
TL;DR: In this paper, Gaiotto and Rapcak proposed a generalized version of the Miura transformation for the quantum toroidal algebra, denoted as the Corner Vertex Algebra (CVA), where the symmetry at the corner of the brane intersection is considered.
Abstract: Recently, Gaiotto and Rapcak proposed a generalization of $W_N$ algebra by considering the symmetry at the corner of the brane intersection (corner vertex operator algebra). The algebra, denoted as $Y_{L,M,N}$, is characterized by three non-negative integers $L, M, N$. It has a manifest triality automorphism which interchanges $L, M, N$, and can be obtained as a reduction of $W_{1+\infty}$ through a "pit" in the plane partition representation. Later, Prochazka and Rapcak proposed a representation of $Y_{L,M,N}$ in terms of $L+M+N$ free bosons through a generalization of Miura transformation, where they use the fractional power differential operators. In this paper, we derive a $q$-deformation of their Miura transformation. It gives the free field representation for $q$-deformed $Y_{L,M,N}$, which is obtained as a reduction of the quantum toroidal algebra. We find that the $q$-deformed version has a "simpler" structure than the original one because of the Miki duality in the quantum toroidal algebra. For instance, one can find a direct correspondence between the operators obtained by the Miura transformation and those of the quantum toroidal algebra. Furthermore, we can show that the screening charges of both the symmetries are identical.
TL;DR: The notion of quantized characters was introduced in this paper as a natural quantization of characters in the context of asymptotic representation theory for quantum groups, and the representation associated with any extreme quantized character generates a von Neumann factor. From the viewpoint of operator algebras, it is natural to ask what is the Murray-von Neumann-Connes type of the resulting factor.
Abstract: The notion of quantized characters was introduced in our previous paper as a natural quantization of characters in the context of asymptotic representation theory for quantum groups. As in the case of ordinary groups, the representation associated with any extreme quantized character generates a von Neumann factor. From the viewpoint of operator algebras (and measurable dynamical systems), it is natural to ask what is the Murray–von Neumann–Connes type of the resulting factor. In this paper, we give a complete solution to this question when the inductive system is of quantum unitary groups .
TL;DR: This study investigated extremal objects from the aspect of design theory by investigating their analogies between codes, lattices, and vertex operator algebras.
Abstract: There are many analogies between codes, lattices, and vertex operator algebras. For example, extremal objects are good examples of combinatorial, spherical, and conformal designs. In this study, we investigated these objects from the aspect of design theory.
TL;DR: The two-character level-1 WZW models corresponding to Lie algebras in the Cvitanovic-Deligne series A1, A2, G2, D4, F4, E6, E7 have been argued to form coset pairs with respect to the meromorphic E8,1 CFT as mentioned in this paper.
Abstract: The two-character level-1 WZW models corresponding to Lie algebras in the Cvitanovic-Deligne series A1, A2, G2, D4, F4, E6, E7 have been argued to form coset pairs with respect to the meromorphic E8,1 CFT. Evidence for this has taken the form of holomorphic bilinear relations between the characters. We propose that suitable 4-point functions of primaries in these models also obey bilinear relations that combine them into current correlators for E8,1, and provide strong evidence that these relations hold in each case. Different cases work out due to special identities involving tensor invariants of the algebra or hypergeometric functions. In particular these results verify previous calculations of correlators for exceptional WZW models, which have rather subtle features. We also find evidence that the intermediate vertex operator algebras A0.5 and E7.5, as well as the three-character A4,1 theory, also appear to satisfy the novel coset relation.
TL;DR: In this article, the role of representation theory in terms of tensor categories is emphasized and connections to 2-dimensional conformal field theory are also presented, in particular, anyon condensation, gapped domain walls and matrix product operators.
Abstract: We review recent interactions between mathematical theory of two-dimensional topological order and operator algebras, particularly the Jones theory of subfactors. The role of representation theory in terms of tensor categories is emphasized. Connections to 2-dimensional conformal field theory are also presented. In particular, we discuss anyon condensation, gapped domain walls and matrix product operators in terms of operator algebras.
TL;DR: In this article, it was shown that for hyperrigid tensor algebras of C*-correspondences, each one of these conjectures is equivalent to the Hao-Ng problem.
Abstract: In an earlier work, the authors proposed a non-selfadjoint approach to the Hao-Ng isomorphism problem for the full crossed product, depending on the validity of two conjectures stated in the broader context of crossed products for operator algebras. By work of Harris and Kim, we now know that these conjectures in the generality stated may not always be valid. In this paper we show that in the context of hyperrigid tensor algebras of C*-correspondences, each one of these conjectures is equivalent to the Hao-Ng problem. This is accomplished by studying the representation theory of non-selfadjoint crossed products of C*-correspondence dynamical systems; in particular we show that there is an appropriate dilation theory. A large class of tensor algebras of C*-correspondences, including all regular ones, are shown to be hyperrigid. Using Hamana's injective envelope theory, we extend earlier results from the discrete group case to arbitrary locally compact groups; this includes a resolution of the Hao-Ng isomorphism for the reduced crossed product and all hyperrigid C*-correspondences. A culmination of these results is the resolution of the Hao-Ng isomorphism problem for the full crossed product and all row-finite graph correspondences; this extends a recent result of Bedos, Kaliszewski, Quigg and Spielberg.
TL;DR: In this article, the Jones theory of subfactors has been studied in the context of topological order and operator algebras, and the role of representation theory has been discussed.
Abstract: We review recent interactions between mathematical theory of two-dimensional topological order and operator algebras, particularly the Jones theory of subfactors. The role of representation theory ...
TL;DR: Borders on Hilbert space dimension are established in terms of properties of a tuple of operators that guarantee a matricial range is non-empty and hence additionally guarantee the existence of hybrid codes for a given quantum channel.
Abstract: We introduce and initiate the study of a family of higher rank matricial ranges, taking motivation from hybrid classical and quantum error correction coding theory and its operator algebra framewor...
TL;DR: In this paper, Popa's orthogonality method was applied to a new class of groups, and the authors obtained the first examples of prime AFD factors and tensorially prime simple AF-algebras.
Abstract: Applying Popa's orthogonality method to a new class of groups, we construct amenable group factors which are prime and have no infinite dimensional regular abelian *-subalgebras. By adjusting Farah--Katsura's solution of Dixmier's problem to the von Neumann algebra setting, we obtain the first examples of prime AFD factors and tensorially prime simple AF-algebras. Our results are proved in ZFC, thus in particular answering questions asked by Farah--Hathaway--Katsura--Tikuisis. We also directly determine central sequences of certain crossed products. This concludes the failure of the Kirchberg $\mathcal{O}_\infty$-absorption theorem in the non-separable setting.