Showing papers on "Operator algebra published in 2015"

Bulk Locality and Quantum Error Correction in AdS/CFT

Abstract: We point out a connection between the emergence of bulk locality in AdS/CFT and the theory of quantum error correction. Bulk notions such as Bogoliubov transformations, location in the radial direction, and the holographic entropy bound all have natural CFT interpretations in the language of quantum error correction. We also show that the question of whether bulk operator reconstruction works only in the causal wedge or all the way to the extremal surface is related to the question of whether or not the quantum error correcting code realized by AdS/CFT is also a “quantum secret sharing scheme”, and suggest a tensor network calculation that may settle the issue. Interestingly, the version of quantum error correction which is best suited to our analysis is the somewhat nonstandard “operator algebra quantum error correction” of Beny, Kempf, and Kribs. Our proposal gives a precise formulation of the idea of “subregion-subregion” duality in AdS/CFT, and clarifies the limits of its validity.

Braided Tensor Categories and Extensions of Vertex Operator Algebras

Abstract: Let V be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of V and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of V-modules are equivalent.

Rationality of W-algebras: principal nilpotent cases

Abstract: We prove the rationality of all the minimal series principal W -algebras discovered by Frenkel, Kac and Wakimoto, thereby giving a new family of rational and C2-conite vertex operator algebras. A key ingredient in our proof is the study of Zhu’s algebra of simple W -algebras via the quantized Drinfeld-Sokolov reduction. We show that the functor of taking Zhu’s algebra commutes with the reduction functor. Using this general fact we determine the maximal spectrums of the associated graded of Zhu’s algebras of vertex operator algebras associated with admissible representations of ane Kac-Moody algebras as well.

Bounding the Set of Finite Dimensional Quantum Correlations

Abstract: We describe a simple method to derive high performance semidefinite programing relaxations for optimizations over complex and real operator algebras in finite dimensional Hilbert spaces. The method is very flexible, easy to program, and allows the user to assess the behavior of finite dimensional quantum systems in a number of interesting setups. We use this method to bound the strength of quantum nonlocality in Bell scenarios where the dimension of the parties is bounded from above. We derive new results in quantum communication complexity and prove the soundness of the prepare-and-measure dimension witnesses introduced in Gallego et al., Phys. Rev. Lett. 105, 230501 (2010). Finally, we propose a new dimension witness that can distinguish between classical, real, and complex two-level systems.

The Choquet boundary of an operator system

Abstract: We show that every operator system (and hence every unital operator algebra) has sufficiently many boundary representations to generate the C*-envelope.

A new realization of quantum geometry

Abstract: We construct in this article a new realization of quantum geometry, which is obtained by quantizing the recently-introduced flux formulation of loop quantum gravity. In this framework, the vacuum is peaked on flat connections, and states are built upon it by creating local curvature excitations. The inner product induces a discrete topology on the gauge group, which turns out to be an essential ingredient for the construction of a continuum limit Hilbert space. This leads to a representation of the full holonomy-flux algebra of loop quantum gravity which is unitarily-inequivalent to the one based on the Ashtekar-Isham-Lewandowski vacuum. It therefore provides a new notion of quantum geometry. We discuss how the spectra of geometric operators, including holonomy and area operators, are affected by this new quantization. In particular, we find that the area operator is bounded, and that there are two different ways in which the Barbero-Immirzi parameter can be taken into account. The methods introduced in this work open up new possibilities for investigating further realizations of quantum geometry based on different vacua.

Anyons and matrix product operator algebras

Abstract: Quantum tensor network states and more particularly projected entangled-pair states provide a natural framework for representing ground states of gapped, topologically ordered systems. The defining feature of these representations is that topological order is a consequence of the symmetry of the underlying tensors in terms of matrix product operators. In this paper, we present a systematic study of those matrix product operators, and show how this relates entanglement properties of projected entangled-pair states to the formalism of fusion tensor categories. From the matrix product operators we construct a C*-algebra and find that topological sectors can be identified with the central idempotents of this algebra. This allows us to construct projected entangled-pair states containing an arbitrary number of anyons. Properties such as topological spin, the S matrix, fusion and braiding relations can readily be extracted from the idempotents. As the matrix product operator symmetries are acting purely on the virtual level of the tensor network, the ensuing Wilson loops are not fattened when perturbing the system, and this opens up the possibility of simulating topological theories away from renormalization group fixed points. We illustrate the general formalism for the special cases of discrete gauge theories and string-net models.

Spherically symmetric canonical quantum gravity

Abstract: Canonical quantization of spherically symmetric space-times is carried out, using real-valued densitized triads and extrinsic curvature components, with specific factor-ordering choices ensuring in an anomaly free quantum constraint algebra. Comparison with previous work [Nucl. Phys. B399, 211 (1993)] reveals that the resulting physical Hilbert space has the same form, although the basic canonical variables are different in the two approaches. As an extension, holonomy modifications from loop quantum gravity are shown to deform the Dirac space-time algebra, while going beyond ``effective'' calculations.

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.

-Algebras, False Theta Functions and Quantum Modular Forms, I

Abstract: In this paper, we study certain partial and false theta functions in connection to vertex operator algebras and conformal eld theory. We prove a variety of results concerning the asymptotics of modied characters of irreducible modules of certain W-algebras of singlet type, which allows us in particular to determine their (analytic) quantum dimensions. Our results are fully consistent with the previous conjectures on fusion rings for these vertex algebras. More importantly, we prove quantum modularity ( a la Zagier) of the numerator part of irreducible characters of singlet algebra modules, thus demonstrating that quantum modular forms naturally appear in many \suciently nice" irrational vertex algebras. It is interesting that quantum modularity persists on the whole set of rationals as in the original Zagier’s example coming from Kontsevich’s \strange series". In the last part, slightly independent of all this, we also discuss Nahm-type q-hypergometric series in connection to tails of colored Jones polynomials of certain torus knots and characters of modules for the (1;p)-singlet algebra.

The Moonshine Module for Conway’s Group

Abstract: We exhibit an action of Conway’s group – the automorphism group of the Leech lattice – on a distinguished super vertex operator algebra, and we prove that the associated graded trace functions are normalized principal moduli, all having vanishing constant terms in their Fourier expansion. Thus we construct the natural analogue of the Frenkel–Lepowsky–Meurman moonshine module for Conway’s group. The super vertex operator algebra we consider admits a natural characterization, in direct analogy with that conjectured to hold for the moonshine module vertex operator algebra. It also admits a unique canonically twisted module, and the action of the Conway group naturally extends. We prove a special case of generalized moonshine for the Conway group, by showing that the graded trace functions arising from its action on the canonically twisted module are constant in the case of Leech lattice automorphisms with fixed points, and are principal moduli for genus-zero groups otherwise.

Algebraic Effects, Linearity, and Quantum Programming Languages

Abstract: We develop a new framework of algebraic theories with linear parameters, and use it to analyze the equational reasoning principles of quantum computing and quantum programming languages. We use the framework as follows: we present a new elementary algebraic theory of quantum computation, built from unitary gates and measurement; we provide a completeness theorem or the elementary algebraic theory by relating it with a model from operator algebra; we extract an equational theory for a quantum programming language from the algebraic theory; we compare quantum computation with other local notions of computation by investigating variations on the algebraic theory.

Rokhlin Dimension: Obstructions and Permanence Properties

Abstract: This paper is a further study of finite Rokhlin dimension for actions of finite groups and the integers on C ∗ -algebras, intro- duced by the first author, Winter, and Zacharias. We extend the definition of finite Rokhlin dimension to the nonunital case. This def- inition behaves well with respect to extensions, and is sufficient to establish permanence of finite nuclear dimension and Z-absorption. We establish K-theoretic obstructions to the existence of actions of finite groups with finite Rokhlin dimension (in the commuting tower version). In particular, we show that there are no actions of any non- trivial finite group on the Jiang-Su algebra or on the Cuntz algebra O∞ with finite Rokhlin dimension in this sense. 2010 Mathematics Subject Classification: 46L55 1 The study of group actions on C ∗ -algebras, and their associated crossed prod- ucts, has always been a central research theme in operator algebras. One would like to identify properties of group actions which on the one hand occur com- monly and naturally enough to be of interest, and on the other hand are strong enough to be used to derive interesting properties of the action or of the crossed product. Examples of important properties for a group action meeting these criteria are the various forms of the Rokhlin property, which arose early on in the theory. See, for instance, (Izu01) and references therein for actions of Z and (Izu04a, Izu04b, Phi09, OP12) for the finite group case. The Rokhlin property for the single automorphism case is quite prevalent, and generic in some cases, forming a dense Gset in the automorphism group (see (HWZ15)). However, it

Conformal field theory, tensor categories and operator algebras

Abstract: This is a set of lecture notes on the operator algebraic approach to 2-dimensional conformal field theory. Representation theoretic aspects and connections to vertex operator algebras are emphasized. No knowledge on operator algebras or quantum field theory is assumed.

Mock modular Mathieu moonshine modules

Abstract: We construct super vertex operator algebras which lead to modules for moonshine relations connecting the four smaller sporadic simple Mathieu groups with distinguished mock modular forms. Starting with an orbifold of a free fermion theory, any subgroup of $$\textit{Co}_0$$ that fixes a 3-dimensional subspace of its unique non-trivial 24-dimensional representation commutes with a certain $${\mathcal {N}}=4$$ superconformal algebra. Similarly, any subgroup of $$\textit{Co}_0$$ that fixes a 2-dimensional subspace of the 24-dimensional representation commutes with a certain $${\mathcal {N}}=2$$ superconformal algebra. Through the decomposition of the corresponding twined partition functions into characters of the $${\mathcal {N}}=4$$ (resp. $${\mathcal {N}}=2$$ ) superconformal algebra, we arrive at mock modular forms which coincide with the graded characters of an infinite-dimensional $${\mathbb {Z}}$$ -graded module for the corresponding group. The Mathieu groups are singled out amongst various other possibilities by the moonshine property: requiring the corresponding weak Jacobi forms to have certain asymptotic behaviour near cusps. Our constructions constitute the first examples of explicitly realized modules underlying moonshine phenomena relating mock modular forms to sporadic simple groups. Modules for other groups, including the sporadic groups of McLaughlin and Higman–Sims, are also discussed.

Classification of holomorphic framed vertex operator algebras of central charge 24

Abstract: This article is a continuation of our work on the classification of holomorphic framed vertex operator algebras of central charge 24. We show that a holomorphic framed VOA of central charge 24 is uniquely determined by the Lie algebra structure of its weight one subspace. As a consequence, we completely classify all holomorphic framed vertex operator algebras of central charge 24 and show that there exist exactly 56 such vertex operator algebras, up to isomorphism.

On the Extended W-Algebra of Type 2 at Positive Rational Level

Abstract: The extended W-algebra of type sl2 at positive rational level, denoted by Mp+,p− , is a vertex operator algebra (VOA) that was originally proposed in [16]. This VOA is an extension of the minimal model VOA and plays the role of symmetry algebra for certain logarithmic conformal field theories. We give a construction of Mp+,p− in terms of screening operators and use this construction to prove that Mp+,p− satisfies Zhu's c2-cofiniteness condition, calculate the structure of the zero mode algebra (also known as Zhu's algebra), and classify all simple Mp+,p− -modules.

Quantum dimensions and fusion rules for parafermion vertex operator algebras

Abstract: The quantum dimensions and the fusion rules for the parafermion vertex operator algebra associated to the irreducible highest weight module for the affine Kac-Moody algebra A (1) 1 of level k are determined.

Quantum Probability Theory and the Foundations of Quantum Mechanics

Abstract: By and large, people are better at coining expressions than at filling them with interesting, concrete contents. Thus, it may not be very surprising that there are many professional probabilists who may have heard the expression but do not appear to be aware of the need to develop “quantum probability theory” into a thriving, rich, useful field featured at meetings and conferences on probability theory. Although our aim, in this essay, is not to contribute new results on quantum probability theory, we hope to be able to let the reader feel the enormous potential and richness of this field. What we intend to do, in the following, is to contribute some novel points of view to the “foundations of quantum mechanics”, using mathematical tools from “quantum probability theory” (such as the theory of operator algebras).

Vertex operator algebras associated to the Virasoro algebra over an arbitrary field

Abstract: The vertex operator algebras and modules associated to the highest weight modules for the Virasoro algebra over an arbitrary field F with chF 6 2 are studied. The irreducible modules of vertex operator algebra L( 1 ,0)F are classified. The rationality of L( 1 ,0)F is established if chF 6 7.

3d mirror symmetry as a canonical transformation

Abstract: We generalize the free Fermi-gas formulation of certain 3d $$ \mathcal{N}=3 $$ super-symmetric Chern-Simons-matter theories by allowing Fayet-Iliopoulos couplings as well as mass terms for bifundamental matter fields. The resulting partition functions are given by simple modifications of the argument of the Airy function found previously. With these extra parameters it is easy to see that mirror-symmetry corresponds to linear canonical transformations on the phase space (or operator algebra) of the 1-dimensional fermions.

General Mazur–Ulam Type Theorems and Some Applications

Abstract: Recently we have presented several structural results on certain isometries of spaces of positive definite matrices and on those of unitary groups. The aim of this paper is to put those previous results into a common perspective and extend them to the context of operator algebras, namely, to that of von Neumann factors.

$\mathcal{W}$-algebras with non-admissible levels and the Deligne exceptional series

Abstract: Structure of certain simple $\mathcal{W}$-algebras assocated with the Deligne exceptional Lie algebras and non-admissible levels are described as the {\it simple current extensions} of certain vertex operator algebras. As an application, the $C_2$-cofiniteness and $\mathbb{Z}_2$-rationality of the algebras are proved.

Operators on Hilbert Space

Abstract: This book's principle goals are (i) to present the spectral theorem as a statement on the existence of a unique continuous and measurable functional calculus, (ii) to present a proof without digressing into a course on the Gelfand theory of commutative Banach algebras, (iii) to introduce the reader to the basic facts concerning the various von Neumann-Schatten ideals, the compact operators, the trace-class operators and all bounded operators, and finally, (iv) to serve as a primer on the theory of bounded linear operators on separable Hilbert space.

Real positivity and approximate identities in Banach algebras

Abstract: Blecher and Read have recently introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain $C^*$-algebraic results and theories to more general algebras. In the present paper we generalize some part of this, and some other facts, to larger classes of Banach algebras.

II_1 factors with non-isomorphic ultrapowers

Abstract: We prove that there exist uncountably many separable II$_1$ factors whose ultrapowers (with respect to arbitrary ultrafilters) are non-isomorphic. In fact, we prove that the families of non-isomorphic II$_1$ factors originally introduced by McDuff \cite{MD69a,MD69b} are such examples. This entails the existence of a continuum of non-elementarily equivalent II$_1$ factors, thus settling a well-known open problem in the continuous model theory of operator algebras.