Showing papers on "Operator algebra published in 2020"

Construction and classification of holomorphic vertex operator algebras

Abstract: We develop an orbifold theory for finite, cyclic groups acting on holomorphic vertex operator algebras. Then we show that Schellekens' classification of $V_1$-structures of meromorphic conformal field theories of central charge 24 is a theorem on vertex operator algebras. Finally we use these results to construct some new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds.

Vertex Algebras for S-duality

Abstract: We define new deformable families of vertex operator algebras $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ associated to a large set of S-duality operations in four-dimensional supersymmetric gauge theory. They are defined as algebras of protected operators for two-dimensional supersymmetric junctions which interpolate between a Dirichlet boundary condition and its S-duality image. The $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ vertex operator algebras are equipped with two $$\mathfrak {g}$$ affine vertex subalgebras whose levels are related by the S-duality operation. They compose accordingly under a natural convolution operation and can be used to define an action of the S-duality operations on a certain space of vertex operator algebras equipped with a $$\mathfrak {g}$$ affine vertex subalgebra. We give a self-contained definition of the S-duality action on that space of vertex operator algebras. The space of conformal blocks (in the derived sense, i.e. chiral homology) for $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ is expected to play an important role in a broad generalization of the quantum Geometric Langlands program. Namely, we expect the S-duality action on vertex operator algebras to extend to an action on the corresponding spaces of conformal blocks. This action should coincide with and generalize the usual quantum Geometric Langlands correspondence. The strategy we use to define the $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ vertex operator algebras is of broader applicability and leads to many new results and conjectures about deformable families of vertex operator algebras.

Higher Index Theory

Abstract: Index theory studies the solutions to differential equations on geometric spaces, their relation to the underlying geometry and topology, and applications to physics. If the space of solutions is infinite dimensional, it becomes necessary to generalise the classical Fredholm index using tools from the K-theory of operator algebras. This leads to higher index theory, a rapidly developing subject with connections to noncommutative geometry, large-scale geometry, manifold topology and geometry, and operator algebras. Aimed at geometers, topologists and operator algebraists, this book takes a friendly and concrete approach to this exciting theory, focusing on the main conjectures in the area and their applications outside of it. A well-balanced combination of detailed introductory material (with exercises), cutting-edge developments and references to the wider literature make this a valuable guide to this active area for graduate students and experts alike.

VOAs and Rank-Two Instanton SCFTs

Abstract: We analyze the $$\mathcal {N}=2$$ superconformal field theories that arise when a pair of D3-branes probe an F-theory singularity from the perspective of the associated vertex operator algebra. We identify these vertex operator algebras for all cases; we find that they have a completely uniform description, parameterized by the dual Coxeter number of the corresponding global symmetry group. We further present free field realizations for these algebras in the style of recent work by three of the authors. These realizations transparently reflect the algebraic structure of the Higgs branches of these theories. We find fourth-order linear modular differential equations for the vacuum characters/Schur indices of these theories, which are again uniform across the full family of theories and parameterized by the dual Coxeter number. We comment briefly on expectations for the still higher-rank cases.

Boundary Chiral Algebras and Holomorphic Twists

Abstract: We study the holomorphic twist of 3d ${\cal N}=2$ gauge theories in the presence of boundaries, and the algebraic structure of bulk and boundary local operators. In the holomorphic twist, both bulk and boundary local operators form chiral algebras (\emph{a.k.a.} vertex operator algebras). The bulk algebra is commutative, endowed with a shifted Poisson bracket and a "higher" stress tensor; while the boundary algebra is a module for the bulk, may not be commutative, and may or may not have a stress tensor. We explicitly construct bulk and boundary algebras for free theories and Landau-Ginzburg models. We construct boundary algebras for gauge theories with matter and/or Chern-Simons couplings, leaving a full description of bulk algebras to future work. We briefly discuss the presence of higher A-infinity like structures.

Topological Holography: The Example of The D2-D4 Brane System

Abstract: We propose a toy model for holographic duality. The model is constructed by embedding a stack of $N$ D2-branes and $K$ D4-branes (with one dimensional intersection) in a 6D topological string theory. The world-volume theory on the D2-branes (resp. D4-branes) is 2D BF theory (resp. 4D Chern-Simons theory) with $\mathrm{GL}_N$ (resp. $\mathrm{GL}_K$) gauge group. We propose that in the large $N$ limit the BF theory on $\mathbb{R}^2$ is dual to the closed string theory on $\mathbb R^2 \times \mathbb R_+ \times S^3$ with the Chern-Simons defect on $\mathbb R \times \mathbb R_+ \times S^2$. As a check for the duality we compute the operator algebra in the BF theory, along the D2-D4 intersection -- the algebra is the Yangian of $\mathfrak{gl}_K$. We then compute the same algebra, in the guise of a scattering algebra, using Witten diagrams in the Chern-Simons theory. Our computations of the algebras are exact (valid at all loops). Finally, we propose a physical string theory construction of this duality using a D3-D5 brane configuration in type IIB -- using supersymmetric twist and $\Omega$-deformation.

On extensions of glmn⏜$$ \mathfrak{gl}\widehat{\left(\left.m\right|n\right)} $$ Kac-Moody algebras and Calabi-Yau singularities

Abstract: We discuss a class of vertex operator algebras $$ {\mathcal{W}}_{\left.m\right|n\kern0.33em \times \kern0.33em \infty } $$ generated by a super- matrix of fields for each integral spin 1, 2, 3, . . . . The algebras admit a large family of truncations that are in correspondence with holomorphic functions on the Calabi-Yau singularity given by solutions to xy = zmwn. We propose a free-field realization of such truncations generalizing the Miura transformation for $$ {\mathcal{W}}_N $$ algebras. Relations in the ring of holomorphic functions lead to bosonization-like relations between different free-field realizations. The discussion provides a concrete example of a non-trivial interplay between vertex operator algebras, algebraic geometry and gauge theory.

Approximate recovery and relative entropy I. general von Neumann subalgebras

Abstract: We prove the existence of a universal recovery channel that approximately recovers states on a v. Neumann subalgebra when the change in relative entropy, with respect to a fixed reference state, is small. Our result is a generalization of previous results that applied to type-I v. Neumann algebras by Junge at al. [arXiv:1509.07127]. We broadly follow their proof strategy but consider here arbitrary v. Neumann algebras, where qualitatively new issues arise. Our results hinge on the construction of certain analytic vectors and computations/estimations of their Araki-Masuda $L_p$ norms. We comment on applications to the quantum null energy condition.

On the tensor structure of modules for compact orbifold vertex operator algebras

Abstract: Suppose $$V^G$$ is the fixed-point vertex operator subalgebra of a compact group G acting on a simple abelian intertwining algebra V. We show that if all irreducible $$V^G$$ -modules contained in V live in some braided tensor category of $$V^G$$ -modules, then they generate a tensor subcategory equivalent to the category $${{\,\mathrm{Rep}\,}}G$$ of finite-dimensional representations of G, with associativity and braiding isomorphisms modified by the abelian 3-cocycle defining the abelian intertwining algebra structure on V. Additionally, we show that if the fusion rules for the irreducible $$V^G$$ -modules contained in V agree with the dimensions of spaces of intertwiners among G-modules, then the irreducibles contained in V already generate a braided tensor category of $$V^G$$ -modules. These results do not require rigidity on any tensor category of $$V^G$$ -modules and thus apply to many examples where braided tensor category structure is known to exist but rigidity is not known; for example they apply when $$V^G$$ is $$C_2$$ -cofinite but not necessarily rational. When $$V^G$$ is both $$C_2$$ -cofinite and rational and V is a vertex operator algebra, we use the equivalence between $$\mathrm {Rep}\,G$$ and the corresponding subcategory of $$V^G$$ -modules to show that V is also rational. As another application, we show that a certain category of modules for the Virasoro algebra at central charge 1 admits a braided tensor category structure equivalent to $${{\,\mathrm{Rep}\,}}SU(2)$$ , up to modification by an abelian 3-cocycle.

Product Algebras for Galerkin Discretisations of Boundary Integral Operators and their Applications

Abstract: Operator products occur naturally in a range of regularised boundary integral equation formulations. However, while a Galerkin discretisation only depends on the domain space and the test (or dual) space of the operator, products require a notion of the range. In the boundary element software package Bempp, we have implemented a complete operator algebra that depends on knowledge of the domain, range, and test space. The aim was to develop a way of working with Galerkin operators in boundary element software that is as close to working with the strong form on paper as possible, while hiding the complexities of Galerkin discretisations. In this article, we demonstrate the implementation of this operator algebra and show, using various Laplace and Helmholtz example problems, how it significantly simplifies the definition and solution of a wide range of typical boundary integral equation problems.

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

Abstract: Practically and intrinsically, inclusions of operator algebras are of fundamental interest. The subject of this paper is intermediate operator algebras of inclusions. There are two previously known theorems which naturally and completely describe all intermediate operator algebras: the Galois Correspondence Theorem and the Tensor Splitting Theorem. Here we establish the third, new complete description theorem which gives a canonical bijective correspondence between intermediate operator algebras and intermediate extensions of dynamical systems. One can also regard this theorem as a crossed product splitting theorem, analogous to the Tensor Splitting Theorem. We then give concrete applications, particularly to maximal amenability problem and a new realization result of intermediate operator algebra lattice.

Topological Gaps in Quasi-Periodic Spin Chains: A Numerical and K-Theoretic Analysis

Abstract: Topological phases supported by quasi-periodic spin-chain models and their bulk-boundary principles are investigated by numerical and K-theoretic methods. We show that, for both the un-correlated and correlated phases, the operator algebras that generate the Hamiltonians are non-commutative tori, hence the quasi-periodic chains display physics akin to the quantum Hall effect in two and higher dimensions. The robust topological edge modes are found to be strongly shaped by the interaction and, generically, they have hybrid edge-localized and chain-delocalized structures. Our findings lay the foundations for topological spin pumping using the phason of a quasi-periodic pattern as an adiabatic parameter, where selectively chosen quantized bits of magnetization can be transferred from one edge of the chain to the other.

Noncommutative Furstenberg boundary

Abstract: We introduce and study the notions of boundary actions and of the Furstenberg boundary of a discrete quantum group. As for classical groups, properties of boundary actions turn out to encode significant properties of the operator algebras associated with the discrete quantum group in question; for example we prove that if the action on the Furstenberg boundary is faithful, the quantum group C*-algebra admits at most one KMS-state for the scaling automorphism group. To obtain these results we develop a version of Hamana's theory of injective envelopes for quantum group actions, and establish several facts on relative amenability for quantum subgroups. We then show that the Gromov boundary actions of free orthogonal quantum groups, as studied by Vaes and Vergnioux, are also boundary actions in our sense; we obtain this by proving that these actions admit unique stationary states. Moreover, we prove these actions are faithful, hence conclude a new unique KMS-state property in the general case, and a new proof of unique trace property when restricted to the unimodular case. We prove equivalence of simplicity of the crossed products of all boundary actions of a given discrete quantum group, and use it to obtain a new simplicity result for the crossed product of the Gromov boundary actions of free orthogonal quantum groups.

On orbifold constructions associated with the Leech lattice vertex operator algebra

Abstract: In this paper, we study orbifold constructions associated with the Leech lattice vertex operator algebra. As an application, we prove that the structure of a strongly regular holomorphic vertex operator algebra of central charge 24 is uniquely determined by its weight one Lie algebra if the Lie algebra has the type A3,43A1,2, A4,52, D4,12A2,6, A6,7, A7,4A1,13, D5,8A1,2 or D6,5A1,12 by using the reverse orbifold construction. Our result also provides alternative constructions of these vertex operator algebras (except for the case A6,7) from the Leech lattice vertex operator algebra.

Multiplier tests and subhomogeneity of multiplier algebras

Abstract: Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of $n \times n$ matrices analogous to the classical Pick matrix. We study for which reproducing kernel Hilbert spaces it suffices to consider matrices of bounded size $n$. We connect this problem to the notion of subhomogeneity of non-selfadjoint operator algebras. Our main results show that multiplier algebras of many Hilbert spaces of analytic functions, such as the Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence one has to test Pick matrices of arbitrarily large matrix size $n$. To treat the Drury-Arveson space, we show that multiplier algebras of certain weighted Dirichlet spaces on the disc embed completely isometrically into the multiplier algebra of the Drury-Arveson space.

Lorentzian Dynamics and Factorization Beyond Rationality

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 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.

Feynman diagrams and $\Omega$-deformed M-theory

Abstract: We derive the simplest commutation relations of operator algebras associated to M2 branes and an M5 brane in the $\Omega$-deformed M-theory, which is a natural set-up for Twisted holography. Feynman diagram 1-loop computations in the twisted-holographic dual side reproduce the same algebraic relations.

Adapting Logic to Physics: The Quantum-Like Eigenlogic Program

Abstract: Considering links between logic and physics is important because of the fast development of quantum information technologies in our everyday life. This paper discusses a new method in logic inspired from quantum theory using operators, named Eigenlogic. It expresses logical propositions using linear algebra. Logical functions are represented by operators and logical truth tables correspond to the eigenvalue structure. It extends the possibilities of classical logic by changing the semantics from the Boolean binary alphabet {0,1} using projection operators to the binary alphabet {+1, −1} employing reversible involution operators. Also, many-valued logical operators are synthesized, for whatever alphabet, using operator methods based on Lagrange interpolation and on the Cayley-Hamilton theorem. Considering a superposition of logical input states one gets a fuzzy logic representation where the fuzzy membership function is the quantum probability given by the Born rule. Historical parallels from Boole, Post, Poincare and Combinatory Logic are presented in relation to probability theory, non-commutative quaternion algebra and Turing machines. An extension to first order logic is proposed inspired by Grover's algorithm. Eigenlogic is essentially a logic of operators and its truth-table logical semantics is provided by the eigenvalue structure which is shown to be related to the universality of logical quantum gates, a fundamental role being played by non-commutativity and entanglement.

A remark on matrix product operator algebras, anyons and subfactors

Abstract: We show that the mathematical structures in a recent work of Bultinck–Mariena–Williamson–Sahinoglu–Haegemana–Verstraete are the same as those of flat symmetric bi-unitary connections and the tube algebra in subfactor theory. More specifically, a system of flat symmetric bi-unitary connections arising from a subfactor with finite index and finite depth satisfies all their requirements for tensors and the tube algebra for such a subfactor, and the anyon algebra for such tensors are isomorphic up to the normalization constants. Furthermore, the matrix product operator algebras arising from tensors corresponding to possibly non-flat symmetric bi-unitary connections are isomorphic to those arising from flat symmetric bi-unitary connections for subfactors.

The holographic map as a conditional expectation

Abstract: We study the holographic map in AdS/CFT, as modeled by a quantum error correcting code with exact complementary recovery. We show that the map is determined by local conditional expectations acting on the operator algebras of the boundary/physical Hilbert space. Several existing results in the literature follow easily from this perspective. The Black Hole area law, and more generally the Ryu-Takayanagi area operator, arises from a central sum of entropies on the relative commutant. These entropies are determined in a state independent way by the conditional expectation. The conditional expectation can also be found via a minimization procedure, similar to the minimization involved in the RT formula. For a local net of algebras associated to connected boundary regions, we show the complementary recovery condition is equivalent to the existence of a standard net of inclusions -- an abstraction of the mathematical structure governing QFT superselection sectors given by Longo and Rehren. For a code consisting of algebras associated to two disjoint regions of the boundary theory we impose an extra condition, dubbed dual-additivity, that gives rise to phase transitions between different entanglement wedges. Dual-additive codes naturally give rise to a new split code subspace, and an entropy bound controls which subspace and associated algebra is reconstructable. We also discuss known shortcomings of exact complementary recovery as a model of holography. For example, these codes are not able to accommodate holographic violations of additive for overlapping regions. We comment on how approximate codes can fix these issues.

Tensor Categories arising from the Virasoro Algebra

Abstract: We show that there is 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. In the generic case of central charge $c=13-6(t+t^{-1})$, with $t otin \mathbb{Q}$, we prove semisimplicity, rigidity and non-degeneracy and also compute the fusion rules of this tensor category.

Inertia groups and uniqueness of holomorphic vertex operator algebras

Abstract: We continue our program on classiffication of holomorphic vertex operator algebras of central charge 24. In this article, we show that there exists a unique strongly regular holomorphic VOA of central charge 24, up to isomorphism, if its weight one Lie algebra has the type C4,10, D7,3A3,1G2,1, A5,6C2,3A1,2, A3,1C7,2, D5,4C3,2A $$ {A}_{1,1}^2 $$ , or E6,4C2,1A2,1. As a consequence, we have verified that the isomorphism class of a strongly regular holomorphic vertex operator algebra of central charge 24 is determined by its weight one Lie algebra structure if the weight one subspace is nonzero.

Tensor Decomposition, Parafermions, Level-Rank Duality, and Reciprocity Law for Vertex Operator Algebras

Abstract: For the semisimple Lie algebra sln, the basic representation L c sln (1,0) of the affine Lie algebra c sln is a lattice vertex operator algebra. The first main result of the paper is to prove that the commutant vertex operator algebra of L c sln (l,0) in the l- fold tensor product L c sln (1,0) l is isomorphic to the parafermion vertex operator algebra K(sll,n), which is the commutant of the Heisenberg vertex operator algebra Lb (n,0) in Lcl (n,0). The result provides a version of level-rank duality. The second main result of the paper is to prove more general version of the first result that the commutant of L c sln (l1+···+ls,0) in L c sln (l1,0)⊗···⊗L c sln (ls,0) is isomorphic to the commutant of the vertex operator algebra generated by a Levi Lie subalgebra of sll1+···+ls corresponding to the composition (l1,··· ,ls) in the rational vertex operator algebra Lb sll1+···+ls (n,0). This general version also resembles a version of reciprocity law discussed by Howe in the context of reductive Lie groups. In the course of the proof of the main results, certain Howe duality pairs also appear in the context of vertex operator algebras. �) are infinite dimensional. The obvious highest weight vector v + ⊗···⊗v + in Lb(1,0) ⊗l generates an irreducible b-submodule isomorphic to Lb(l,0). Thus Mb(l,0) 6 0. Since Lb(1,0) and Lbg(l,0) are vertex operator algebras, Lbg(1,0) ⊗l has a tensor product vertex operator algebra structure with Lbg(l,0) being a vertex operator subalgebra (with

Cyclic orbifolds of lattice vertex operator algebras having group-like fusions

Abstract: Let L be an even (positive definite) lattice and $$g\in O(L)$$. In this article, we prove that the orbifold vertex operator algebra $$V_{L}^{{\hat{g}}}$$ has group-like fusion if and only if g acts trivially on the discriminant group $${\mathcal {D}}(L)=L^*/L$$ (or equivalently $$(1-g)L^*

Deformation quantizations from vertex operator algebras

Abstract: In this note we address the question whether one can recover from the vertex operator algebra associated with a four-dimensional $$ \mathcal{N} $$ = 2 superconformal field theory the deformation quantization of the Higgs branch of vacua that appears as a protected subsector in the three-dimensional circle-reduced theory. We answer this question positively if the UV R-symmetries do not mix with accidental (topological) symmetries along the renormalization group flow from the four-dimensional theory on a circle to the three-dimensional theory. If they do mix, we still find a deformation quantization but at different values of its period.