Thermal States in Conformal QFT. I

We analyze the set of locally normal KMS states w.r.t. the translation group for a local conformal net A{{\mathcal A}} of von Neumann algebras on \mathbb R{\mathbb R} . In this first part, we focus on the completely rational net A{{\mathcal A}} . Our main result here states that, if A{{\mathcal{A}}} is completely rational, there exists exactly one locally normal KMS state j{\varphi} . Moreover, j{\varphi} is canonically constructed by a geometric procedure. A crucial r?le is played by the analysis of the “thermal completion net” associated with a locally normal KMS state. A similar uniqueness result holds for KMS states of two-dimensional local conformal nets w.r.t. the time-translation one-parameter group.     

On the Representation Theory of Virasoro Nets

We discuss various aspects of the representation theory of the local nets of von Neumann algebras on the circle associated with positive energy representations of the Virasoro algebra (Virasoro nets). In particular we classify the local extensions of the c=1 Virasoro net for which the restriction of the vacuum representation to the Virasoro subnet is a direct sum of irreducible subrepresentations with finite statistical dimension (local extensions of compact type). Moreover we prove that if the central charge c is in a certain subset of (1, ), including [2, ), and h(c–1)/24, the irreducible representation with lowest weight h of the corresponding Virasoro net has infinite statistical dimension. As a consequence we show that if the central charge c is in the above set and satisfies c25 then the corresponding Virasoro net has no proper local extensions of compact type.Supported in part by the Italian MIUR and GNAMPA-INDAM.     

On the Uniqueness of Diffeomorphism Symmetry in Conformal Field Theory

A Möbius covariant net of von Neumann algebras on S¹ is diffeomorphism covariant if its Möbius symmetry extends to diffeomorphism symmetry. We prove that in case the net is either a Virasoro net or any at least 4-regular net such an extension is unique: the local algebras together with the Möbius symmetry (equivalently: the local algebras together with the vacuum vector) completely determine it. We draw the two following conclusions for such theories. (1) The value of the central charge c is an invariant and hence the Virasoro nets for different values of c are not isomorphic as Möbius covariant nets. (2) A vacuum preserving internal symmetry always commutes with the diffeomorphism symmetries. We further use our result to give a large class of new examples of nets (even strongly additive ones), which are not diffeomorphism covariant; i.e. which do not admit an extension of the symmetry to Diff⁺(S¹).Supported in part by the Italian MIUR and GNAMPA-INDAM.     

Classification of Two-Dimensional Local Conformal Nets with <Emphasis Type="Italic">c</Emphasis> < 1 and 2-Cohomology Vanishing for Tensor Categories

We classify two-dimensional local conformal nets with parity symmetry and central charge less than 1, up to isomorphism. The maximal ones are in a bijective correspondence with the pairs of A-D-E Dynkin diagrams with the difference of their Coxeter numbers equal to 1. In our previous classification of one-dimensional local conformal nets, Dynkin diagrams D _{2n +1} and E ₇ do not appear, but now they do appear in this classification of two-dimensional local conformal nets. Such nets are also characterized as two-dimensional local conformal nets with -index equal to 1 and central charge less than 1. Our main tool, in addition to our previous classification results for one-dimensional nets, is 2-cohomology vanishing for certain tensor categories related to the Virasoro tensor categories with central charge less than 1.Supported in part by JSPS.Supported in part by GNAMPA and MIUR.     

Boundary Quantum Field Theory on the Interior of the Lorentz Hyperboloid

We construct local, boost covariant boundary QFT nets of von Neumann algebras on the interior of the Lorentz hyperboloid \({\mathfrak{H}_R}\), x ² ? t ² > R ², x > 0, in the two-dimensional Minkowski spacetime. Our first construction is canonical, starting with a local conformal net on \({\mathbb{R}}\), and is analogous to our previous construction of local boundary CFT nets on the Minkowski half-space. This net is in a thermal state at Hawking temperature. Then, inspired by a recent construction by E. Witten and one of us, we consider a unitary semigroup that we use to build up infinitely many nets. Surprisingly, the one-particle semigroup is again isomorphic to the semigroup of symmetric inner functions of the disk. In particular, by considering the U(1)-current net, we can associate with any given symmetric inner function a local, boundary QFT net on \({\mathfrak{H}_R}\). By considering different states, we shall also have nets in a ground state, rather than in a KMS state.     

KMS States and Conformal Measures

From a non-constant holomorphic map on a connected Riemann surface we construct an étale second countable locally compact Hausdorff groupoid whose associated groupoid C*-algebra admits a one-parameter group of automorphisms with the property that its KMS states correspond to conformal measures in the sense of Sullivan. In this way certain quadratic polynomials give rise to quantum statistical models with a phase transition arising from spontaneous symmetry breaking.     

Mirror Extensions of Local Nets

In this paper we prove a general theorem on the extensions of local nets which was inspired by recent examples of exotic extensions for Virasoro nets with central charge less than one and earlier work on cosets and conformal inclusions. When applying the theorem to conformal inclusions and diagonal inclusions, we obtain infinite series of new examples of completely rational nets. Supported in part by NSF.     

Conformal blocks in Virasoro and W theories: Duality and the Calogero–Sutherland model

We study the properties of the conformal blocks of the conformal field theories with Virasoro or W-extended symmetry. When the conformal blocks contain only second-order degenerate fields, the conformal blocks obey second order differential equations and they can be interpreted as ground-state wave functions of a trigonometric Calogero–Sutherland Hamiltonian with non-trivial braiding properties. A generalized duality property relates the two types of second order degenerate fields. By studying this duality we found that the excited states of the Calogero–Sutherland Hamiltonian are characterized by two partitions, or in the case of _WAk−1${\mathrm{WA}}_{k-1}$ theories by k   partitions. By extending the conformal field theories under consideration by a u(1)$u(1)$ field, we find that we can put in correspondence the states in the Hilbert state of the extended CFT with the excited non-polynomial eigenstates of the Calogero–Sutherland Hamiltonian. When the action of the Calogero–Sutherland integrals of motion is translated on the Hilbert space, they become identical to the integrals of motion recently discovered by Alba, Fateev, Litvinov and Tarnopolsky in Liouville theory in the context of the AGT conjecture. Upon bosonization, these integrals of motion can be expressed as a sum of two, or in general k, bosonic Calogero–Sutherland Hamiltonian coupled by an interaction term with a triangular structure. For special values of the coupling constant, the conformal blocks can be expressed in terms of Jack polynomials with pairing properties, and they give electron wave functions for special Fractional Quantum Hall states.     

A newN= 6 superconformal algebra

In this paper we construct a newN = 6 superconformal algebra which extends the Virasoro algebra by theSO ₆ current algebra, by 6 odd primary fields of conformal weight 3/2 and by 10 odd primary fields of conformal weight 1/2. The commutation relations of this algebra, which we will refer to asCK ₆, are represented by short distance operator product expansions (OPE). We constructCK ₆, as a subalgebra of theSO(6) superconformal algebra K₆, thus giving it a natural representation as first order differential operators on the circle withN = 6 extended symmetry. We show thatCK ₆ has no nontrivial central extensions. Partially supported by NSC grant 85-2121-M-006-019 of the ROC. Partially supported by NSF grant DMS-9622870.     

Spectral flow invariants and twisted cyclic theory for the Haar state on

In [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], we presented a K-theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely SU_q(2) and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes from the generator of the modular group of the Haar state. In contrast to the Cuntz algebras studied in [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], the computations are considerably more complex and interesting, because there are non-trivial ‘eta’ contributions to this index.     

Operator algebras and conformal field theory

We define and study two-dimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite type III₁ factor. The conformal net determined by the algebras of local observables is proven to satisfy Haag duality. The representation of the Moebius group (and presumably of the entire Virasoro algebra) on the vacuum sector of a conformal field theory is uniquely determined by the Tomita-Takesaki modular operators associated with its vacuum state and its conformal net. We then develop the theory of Moebius covariant representations of a conformal net, using methods of Doplicher, Haag and Roberts. We apply our results to the representation theory of loop groups. Our analysis is motivated by the desire to find a background-independent formulation of conformal field theories.     

Quasi-particle structure around a single vortex in high-Tc superconductors

For analyzing the checker-board like modulation of the local density of states (LDOS) around a vortex observed in the slightly overdoped Bi₂Sr₂CaCu₂O_x, we examined the effect of pseudogap state of high-T_c superconductors to the LDOS around the vortex. We first derived the Bogoliubov-de Gennes equation for d-wave superconductivity (d-SC) in the presence of d-spin density wave (d-SDW). Using the Fourier–Bessel expansion, we solved this equation for a single vortex state, numerically. We found that the peak of the bound states around E = 0 becomes small and modulation of the LDOS is observed for larger d-SDW order parameter.     

The W N Minimal Model Classification

We first rigourously establish, for any N ≥ 2, that the toroidal modular invariant partition functions for the (not necessarily unitary) W _N (p, q) minimal models biject onto a well-defined subset of those of the SU(N) × SU(N) Wess-Zumino-Witten theories at level (p − N, q − N). This permits considerable simplifications to the proof of the Cappelli-Itzykson-Zuber classification of Virasoro minimal models. More important, we obtain from this the complete classification of all modular invariants for the W ₃(p, q) minimal models. All should be realised by rational conformal field theories. Previously, only those for the unitary models, i.e. W ₃(p, p + 1), were classified. For all N our correspondence yields for free an extensive list of W _N (p, q) modular invariants. The W ₃ modular invariants, like the Virasoro minimal models, all factorise into SU(3) modular invariants, but this fails in general for larger N. We also classify the SU(3) × SU(3) modular invariants, and find there a new infinite series of exceptionals.     

KMS States, Entropy and the Variational Principle¶in Full C*-Dynamical Systems

To any periodic and full C ^*-dynamical system , an invertible operator s acting on the Banach space of trace functionals of the fixed point algebra is canonically associated. KMS states correspond to positive eigenvectors of s. A Perron–Frobenius type theorem asserts the existence of KMS states at inverse temperatures equals the logarithms of the inner and outer spectral radii of s (extremal KMS states). Examples arising from subshifts in symbolic dynamics, self-similar sets in fractal geometry and noncommutative metric spaces are discussed. Certain subshifts are naturally associated to the system, and criteria for the equality of their topological entropy and inverse temperatures of extremal KMS states are given. Unital completely positive maps implemented by partitions of unity {x _j } of grade 1 are considered, resembling the “canonical endomorphism” of the Cuntz algebras. The relationship between the Voiculescu topological entropy of and the topological entropy of the associated subshift is studied. Examples where the equality holds are discussed among Matsumoto algebras associated to non finite type subshifts. In the general case is bounded by the sum of the entropy of the subshift and a suitable entropic quantity of the homogeneous subalgebra. Both summands are necessary. The measure-theoretic entropy of , in the sense of Connes–Narnhofer–Thirring, is compared to the classical measure-theoretic entropy of the subshift. A noncommutative analogue of the classical variational principle for the entropy is obtained for the “canonical endomorphism” of certain Matsumoto algebras. More generally, a necessary condition is discussed. In the case of Cuntz–Krieger algebras an explicit construction of the state with maximal entropy from the unique KMS state is done. Received: 1 February 2000 / Accepted: 23 February 2000     

Tilted two-fluid Bianchi type I models

In this paper we investigate expanding Bianchi type I models with two tilted fluids with the same linear equation of state, characterized by the equation of state parameter w. Individually the fluids have non-zero energy fluxes w.r.t. the symmetry surfaces, but these cancel each other because of the Codazzi constraint. We prove that when w = 0 the model isotropizes to the future. Using numerical simulations and a linear analysis we also find the asymptotic states of models with w > 0. We find that future isotropization occurs if and only if w ￡ \frac13{w\leq \frac{1}{3}} . The results are compared to similar models investigated previously where the two fluids have different equation of state parameters.     

NMR and NQR studies in high-T c oxides: YBa2Cu3O y (6.0≤y≤6.91)

The NQR and NMR techniques have been utilized to characterize the local oxygen coordination of inequivalent Cu sites and the electronic properties in both the normal and superconducting states of YBa₂Cu₃O_y (6.0<-y≤6.91). The distinct NQR lines associated with the different oxygen-coordinated Cu sites, hence the locally differentiated charged states, have been observed. The degree of charge differentiation at the Cu(2) plane sites was found to be increased with decreasingy from 6.91, which might be related with the decrease ofT _c. An anomalous temperature dependence of Cu nuclear spin-lattice relaxation timeT ₁ has been observed for both the Cu(1) chain and Cu(2) plane sites fory=6.91 and it is discussed in connection with antiferromagnetic spin fluctuations in the normal state.     

Pohlmeyer reduction of superstring sigma model

Motivated by a desire to find a useful 2d Lorentz-invariant reformulation of the AdS₅×S⁵ superstring world-sheet theory in terms of physical degrees of freedom we construct the “Pohlmeyer-reduced” version of the corresponding sigma model. The Pohlmeyer reduction procedure involves several steps. Starting with a coset space string sigma model in the conformal gauge and writing the classical equations in terms of currents one can fix the residual conformal diffeomorphism symmetry and kappa-symmetry and introduce a new set of variables (related locally to currents but non-locally to the original string coordinate fields) so that the Virasoro constraints are automatically satisfied. The resulting equations can be obtained from a Lagrangian of a non-Abelian Toda type: a gauged WZW model with an integrable potential coupled also to a set of 2d fermionic fields. A gauge-fixed form of the Pohlmeyer-reduced theory can be found by integrating out the 2d gauge field of the gauged WZW model. The small-fluctuation spectrum near the trivial vacuum contains 8 bosonic and 8 fermionic degrees of freedom with equal mass. We conjecture that the reduced model has world-sheet supersymmetry and is ultraviolet-finite. We show that in the special case of the AdS₂×S² superstring model the reduced theory is indeed supersymmetric: it is equivalent to the N=2 supersymmetric extension of the sine-Gordon model.     

Proving AGT relations in the large-c limit

In the limit of large central charge c   the 4-point Virasoro conformal block becomes a hypergeometric function. It is represented by a sum of chiral Nekrasov functions, which can also be explicitly evaluated. In this way the known proof of the AGT relation is extended from special to generic set of external states, but in the special limit of c=∞$c=\infty$.