Showing papers on "Operator algebra published in 2001"

Noncommutative Manifolds, the Instanton Algebra and Isospectral Deformations

Abstract: We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝn. They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features.

Tachyon potentials, star products and universality

Abstract: We develop an efficient recursive method to evaluate the tachyon potential using the relevant universal subalgebra of the open string star algebra. This method, using off-shell versions of Virasoro Ward identities, avoids explicit computation of conformal transformations of operators and does not require a choice of background. We illustrate the procedure with a pedagogic computation of the level six tachyon potential in an arbitrary gauge, and the evaluation of a few simple star products. We give a background independent construction of the so-called identity of the star algebra, and show how it fits into family of string fields generating a commutative subalgebra.

Locally compact quantum groups in the universal setting

Abstract: In this paper we associate to every reduced C*-algebraic quantum group (A, Δ) (as defined in [11]) a universal C*-algebraic quantum group (Au, Δu). We fine tune a proof of Kirchberg to show that every *-representation of a modified L1-space is generated by a unitary corepresentation. By taking the universal enveloping C*-algebra of a dense sub *-algebra of A we arrive at the C*-algebra Au. We show that this C*-algebra Au carries a quantum group structure which is a rich as its reduced companion. We also establish a bijective correspondence between quantum group morphisms and certain co-actions.

Komaba lectures on noncommutative solitons and D-branes

Abstract: These lectures provide an introduction to noncommutative geometry and its origins in quantum mechanics and to the construction of solitons in noncommutative field theory. These ideas are applied to the construction of D-branes as solitons of the tachyon field in noncommutative open string theory. A brief discussion is given of the K-theory classification of D-brane charge in terms of the K-theory of operator algebras. Based on lectures presented at the Komaba 2000 workshop, Nov. 14-16 2000.

Exact solvability of superintegrable systems

Abstract: It is shown that all four superintegrable quantum systems on the Euclidean plane possess the same underlying hidden algebra sl(3). The gauge-rotated Hamiltonians, as well as their integrals of motion, once rewritten in appropriate coordinates, preserve a flag of polynomials. This flag corresponds to highest-weight finite-dimensional representations of the sl(3)-algebra, realized by first-order differential operators.

Topological structure of the space of composition operators onH

Abstract: Components and isolated points of the topological space of composition operators onH∞ in the uniform operator topology are characterized. Compact differences of two composition operators are also characterized. With the aid of these results, we show that a component inC(H ∞ ) is not in general the set of all composition operators that differ from the given one by a compact operator.

The many faces of Ocneanu cells

Abstract: We define generalised chiral vertex operators covariant under the Ocneanu ``double triangle algebra'' {\cal A}, a novel quantum symmetry intrinsic to a given rational 2-d conformal field theory. This provides a chiral approach, which, unlike the conventional one, makes explicit various algebraic structures encountered previously in the study of these theories and of the associated critical lattice models, and thus allows their unified treatment. The triangular Ocneanu cells, the 3j-symbols of the weak Hopf algebra {\cal A}, reappear in several guises. With {\cal A} and its dual algebra {hat A} one associates a pair of graphs, G and {\tilde G}. While G are known to encode complete sets of conformal boundary states, the Ocneanu graphs {\tilde G} classify twisted torus partition functions. The fusion algebra of the twist operators provides the data determining {\hat A}. The study of bulk field correlators in the presence of twists reveals that the Ocneanu graph quantum symmetry gives also an information on the field operator algebra.

On injectivity and nuclearity for operator spaces

Abstract: An injective operator space $V$ which is dual as a Banach space has the form $eR(1-e)$, where $R$ is an injective von Neumann algebra and where $e$ is a projection in $R$. This is used to show that an operator space $V$ is nuclear if and only if it is locally reflexive and $V^{\ast\ast}$ is injective. It is also shown that any exact operator space is locally reflexive.

Polarization free generators and the S matrix

Abstract: Polarization-free generators, i.e. “interacting” Heisenberg operators which are localized in wedge-shaped regions of Minkowski space and generate single particle states from the vacuum, are a novel tool in the analysis and synthesis of two-dimensional integrable quantum field theories. In the present article, the status of these generators is analyzed in a general setting. It is shown that such operators exist in any theory and in any number of spacetime dimensions. But in more than two dimensions they have rather delicate domain properties in the presence of interaction. If, for example, they are defined and temperate on a translation-invariant, dense domain, then the underlying theory yields only trivial scattering. In two-dimensional theories, these domain properties are consistent with non-trivial interaction, but they exclude particle production. Thus the range of applications of polarization-free generators seems to be limited to the realm of two-dimensional theories.

Komaba Lectures on Noncommutative Solitons and D-Branes

Abstract: These lectures provide an introduction to noncommutative geometry and its origins in quantum mechanics and to the construction of solitons in noncommutative field theory. These ideas are applied to the construction of D-branes as solitons of the tachyon field in noncommutative open string theory. A brief discussion is given of the K-theory classification of D-brane charge in terms of the K-theory of operator algebras. Based on lectures presented at the Komaba 2000 workshop, Nov. 14-16 2000.

The operator algebra and twisted KZ equations of WZW orbifolds

Abstract: We obtain the operator algebra of each twisted sector of all WZW orbifolds, including the general twisted current algebra and the algebra of the twisted currents with the twisted affine primary fields. Surprisingly, the twisted right and left mover current algebras are not a priori copies of each other. Using the operator algebra we also derive world-sheet differential equations for the twisted affine primary fields of all WZW orbifolds. Finally we include ground state properties to obtain the twisted Knizhnik-Zamolodchikov equations of the WZW permutation orbifolds and the inner-automorphic WZW orbifolds.

Some characterizations of the automorphisms of B(H) and C(X)

Abstract: We present some nonlinear characterizations of the automorphisms of the operator algebra B(H) and the function algebra C(X) by means of their spectrum preserving properties

Higher-dimensional algebra and Planck scale physics

Abstract: This is a nontechnical introduction to recent work on quantum gravity using ideas from higher-dimensional algebra. We argue that reconciling general relativity with the Standard Model requires a `background-free quantum theory with local degrees of freedom propagating causally'. We describe the insights provided by work on topological quantum field theories such as quantum gravity in 3-dimensional spacetime. These are background-free quantum theories lacking local degrees of freedom, so they only display some of the features we seek. However, they suggest a deep link between the concepts of `space' and `state', and similarly those of `spacetime' and `process', which we argue is to be expected in any background-free quantum theory. We sketch how higher-dimensional algebra provides the mathematical tools to make this link precise. Finally, we comment on attempts to formulate a theory of quantum gravity in 4-dimensional spacetime using `spin networks' and `spin foams'.

Dynamical quantum groups at roots of 1

Abstract: Given a dynamical twist for a finite-dimensional Hopf algebra, we construct two weak Hopf algebras, using methods of P. Xu and of P. Etingof and A. Varchenko, and we show that they are dual to each other. We generalize the theory of dynamical quantum groups to the case when the quantum parameter q is a root of unity. These objects turn out to be self-dual—which is a fundamentally new property, not satisfied by the usual Drinfeld-Jimbo quantum groups.

Weak modules and logarithmic intertwining operators for vertex operator algebras

Abstract: We consider a class of weak modules for vertex operator algebras that we call logarithmic modules. We also construct nontrivial examples of intertwining operators between certain logarithmic modules for the Virasoro vertex operator algebra. At the end we speculate about some possible logarithmic intertwiners at the level $c=0$.

Quantum spin dynamics (QSD): VII. Symplectic structures and continuum lattice formulations of gauge field theories

Abstract: Interesting nonlinear functions on the phase spaces of classical field theories can never be quantized immediately because the basic fields of the theory become operator-valued distributions. Therefore, one is usually forced to find a smeared substitute for such a function which corresponds to a regularization. The smeared functions define a new symplectic manifold of their own which is easy to quantize. Finally, one must remove the regulator and establish that the final operator, if it exists, has the correct classical limit. In this paper we begin the investigation of these steps for diffeomorphism-invariant quantum field theories of connections. We introduce a (generalized) projective family of symplectic manifolds, coordinatized by the smeared fields, which is labelled by a pair consisting of a graph and another graph dual to it. We show that a subset of the corresponding projective limit can be identified with the symplectic manifold that one started from. Then we illustrate the programme outlined above by applying it to the Gauss constraint. This paper also complements, as a side result, earlier work by Ashtekar, Corichi and Zapata who observed that certain operators are non-commuting on certain states, although the Poisson brackets between the corresponding classical functions vanish. These authors showed that this is not a contradiction provided that one refrains from a phase space quantization but rather applies a quantization based on the Lie algebra of vector fields on the configuration space of the theory. Here we show that one can provide a phase space quantization, that is, one can find other functions on the classical phase space which give rise to the same operators but whose Poisson algebra precisely mirrors the quantum commutator algebra. The framework developed here is the classical cornerstone on which the semiclassical analysis in a new series of papers called `gauge theory coherent states' is based.

Resolution of stringy singularities by non-commutative algebras

Abstract: In this paper we propose a unified approach to (topological) string theory on certain singular spaces in their large volume limit. The approach exploits the non-commutative structure of D-branes, so the space is described by an algebraic geometry of non-commutative rings. The paper is devoted to the study of examples of these algebras. In our study there is an auxiliary commutative algebraic geometry of the center of the (local) algebras which plays an important role as the target space geometry where closed strings propagate. The singularities that are resolved will be the singularities of this auxiliary geometry. The singularities are resolved by the non-commutative algebra if the local non-commutative rings are regular. This definition guarantees that D-branes have a well defined K-theory class. Homological functors also play an important role. They describe the intersection theory of D-branes and lead to a formal definition of local quivers at singularities, which can be computed explicitly for many types of singularities. These results can be interpreted in terms of the derived category of coherent sheaves over the non-commutative rings, giving a non-commutative version of recent work by M. Douglas. We also describe global features like the Betti numbers of compact singular Calabi-Yau threefolds via global holomorphic sections of cyclic homology classes.

On q analog of McKay correspondence and ADE classification of affine sl(2) conformal field theories

Abstract: The goal of this paper is to classify ``finite subgroups in U_q sl(2)'' where $q=e^{\pi\i/l}$ is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of U_q sl(2); we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related with extensions of the chiral (vertex operator) algebra corresponding to sl^(2) at level k=l-2. We show that ``finite subgroups in U_q sl(2)'' are classified by Dynkin diagrams of types A_n, D_{2n}, E_6, E_8 with Coxeter number equal to $l$, give a description of this correspondence similar to the classical McKay correspondence, and discuss relation with modular invariants in (sl(2))_k conformal field theory.

Notes for a Quantum Index Theorem

Abstract: We view DHR superselection sectors with finite statistics as Quantum Field Theory analogs of elliptic operators where KMS functionals play the role of the trace composed with the heat kernel regularization. We extend our local holomorphic dimension formula and prove an analogue of the index theorem in the Quantum Field Theory context. The analytic index is the Jones index, more precisely the minimal dimension, and, on a 4-dimensional spacetime, the DHR theorem gives the integrality of the index. We introduce the notion of holomorphic dimension; the geometric dimension is then defined as the part of the holomorphic dimension which is symmetric under charge conjugation. We apply the AHKT theory of chemical potential and we extend it to the low dimensional case, by using conformal field theory. Concerning Quantum Field Theory on a curved spacetime, the geometry of the manifold enters in the expression for the dimension. If a quantum black hole is described by a spacetime with bifurcate Killing horizon and sectors are localizable on the horizon, the variation of logarithm of the geometric dimension is proportional to the incremental free energy, due to the addition of the charge, and to the inverse temperature, hence to the inverse of the surface gravity in the Hartle–Hawking KMS state. For this analysis we consider a conformal net obtained by restricting the field to the horizon (“holography”). Compared with our previous work on Rindler spacetime, this result differs inasmuch as it concerns true black hole spacetimes, like the Schwarzschild–Kruskal manifold, and pertains to the entropy of the black hole itself, rather than of the outside system. An outlook concerns a possible relation with supersymmetry and noncommutative geometry.

Covering Dimension for Nuclear C*-algebras

Abstract: We introduce the completely positive rank, a notion of covering dimension for nuclear $C^*$-algebras and analyze some of its properties. The completely positive rank behaves nicely with respect to direct sums, quotients, ideals and inductive limits. For abelian $C^*$-algebras it coincides with covering dimension of the spectrum and there are similar results for continuous trace algebras. As it turns out, a $C^*$-algebra is zero-dimensional precisely if it is $AF$. We consider various examples, particularly of one-dimensional $C^*$-algebras, like the irrational rotation algebras, the Bunce-Deddens algebras or Blackadar's simple unital projectionless $C^*$-algebra. Finally, we compare the completely positive rank to other concepts of noncommutative covering dimension, such as stable or real rank.

Embeddings of reduced free products of operator algebras

Abstract: Given reduced amalgamated free products of C$^*$--algebras $(A,\phi)=\freeprodi(A_\iota,\phi_\iota)$ and $(D,\psi)=\freeprodi(D_\iota,\psi_\iota)$, an embedding $A\hookrightarrow D$ is shown to exist assuming there are conditional expectation preserving embeddings $A_\iota\hookrightarrow D_\iota$. This result is extended to show the existence of the reduced amalgamated free product of certain classes of unital completely positive maps. Finally, the reduced amalgamated free product of von Neumann algebras is defined in the general case and analogues of the above mentioned results are proved for von Neumann algebras.

Gauge-Invariant Couplings of Noncommutative Branes to Ramond-Ramond Backgrounds

Abstract: We derive the couplings of noncommutative D-branes to spatially varying Ramond-Ramond fields, extending our earlier results in hep-th/0009101. These couplings are expressed in terms of *n products of operators involving open Wilson lines. Equivalence of the noncommutative to the commutative couplings implies interesting identities as well as an expression for the Seiberg-Witten map that was previously conjectured. We generalise our couplings to include transverse scalars, thereby obtaining a Seiberg-Witten map relating commutative and noncommutative descriptions of these scalars. RR couplings for unstable non-BPS branes are also proposed.

Isometric Dilations of Non-Commuting Finite Rank n-Tuples

Abstract: A contractive n-tuple A = (A1, . . . , An) has a minimal joint isometric dilation S = (S1, . . . , Sn) where the Si's are isometries with pairwise orthogonal ranges. This determines a rep- resentation of the Cuntz-Toeplitz algebra. When A acts on a finite dimensional space, the wot-closed nonself-adjoint algebra S generated by S is completely described in terms of the properties of A. This provides complete unitary invariants for the corresponding representations. In addition, we show that the algebra S is always hyper-reflexive. In the last section, we describe similarity invariants. In partic- ular, an n-tuple B of d × d matrices is similar to an irreducible n-tuple A if and only if a certain finite set of polynomials vanish on B.

On the Hopf algebraic origin of Wick normal ordering

Abstract: A combinatorial formula of Rota and Stein is taken to perform Wick reordering in quantum field theory. Wick's theorem becomes a Hopf algebraic identity called Cliffordization. The combinatorial method relying on Hopf algebras is highly efficient in computations and yields closed algebraic expressions.

Operator Algebras, Topology and Subgroups of Quantum Symmetry - Construction of Subgroups of Quantum Groups -

Abstract: In this article, we will discuss several interactions between the non commutative Galois problems, i.e., inclusions of operator algebras and topological quantum field theory in three dimensions. Those interactions are shown to be concretely solved and are related to both the quantum subgroups of the quantum group sU(2)~ at the deformation parameter q = exp(2rilN).