Decoding the geometry of conformal field theories

To certain geometries, string theory associates conformal field theories. We discuss techniques to perform the reverse procedure: To recover geometrical data from abstractly defined conformal field theories. This is done by introducing appropriate notions of limits of conformal field theories and their degenerations, and by applying techniques from noncommutative geometry. This note is a summary of our work hep-th/0308143 , aimed to be less technical than the original paper, along with some new calculations confirming our interpretation of the rescaled limiting zero mode of the Virasoro field.Comment: 12 pages, contribution to the Proceedings of the 7th International Workshop "Lie Theory and Its Applications in Physics", Varna, Bulgari

Fusion of Critical Defect Lines in the 2D Ising Model

Two defect lines separated by a distance delta look from much larger distances like a single defect. In the critical theory, when all scales are large compared to the cutoff scale, this fusion of defect lines is universal. We calculate the universal fusion rule in the critical 2D Ising model and show that it is given by the Verlinde algebra of primary fields, combined with group multiplication in O(1,1)/Z_2. Fusion is in general singular and requires the subtraction of a divergent Casimir energy.Comment: 17 page

The coset D-branes of SU(n)

Using a nested coset construction a collection of D-branes that appear to generate all the K-theory charges of string theory on SU(n) are constructed and their charges are determined.Comment: 12 page

A worldsheet extension of O(d,d;Z)

We study superconformal interfaces between N=(1,1) supersymmetric sigma models on tori, which preserve a u(1)^{2d} current algebra. Their fusion is non-singular and, using parallel transport on CFT deformation space, it can be reduced to fusion of defect lines in a single torus model. We show that the latter is described by a semi-group extension of O(d,d;Q), and that (on the level of Ramond charges) fusion of interfaces agrees with composition of associated geometric integral transformations. This generalizes the well-known fact that T-duality can be geometrically represented by Fourier-Mukai transformations. Interestingly, we find that the topological interfaces between torus models form the same semi-group upon fusion. We argue that this semi-group of orbifold equivalences can be regarded as the \alpha' deformation of the continuous O(d,d) symmetry of classical supergravity.Comment: 71 pages, 1 figure, minor additions and correction

Junctions of surface operators and categorification of quantum groups

We show how networks of Wilson lines realize quantum groups U_q(sl_m), for arbitrary m, in 3d SU(N) Chern-Simons theory. Lifting this construction to foams of surface operators in 4d theory we find that rich structure of junctions is encoded in combinatorics of planar diagrams. For a particular choice of surface operators we reproduce known mathematical constructions of categorical representations and categorified quantum groups

Defects and D-Brane Monodromies

In this paper D-brane monodromies are studied from a world-sheet point of view. More precisely, defect lines are used to describe the parallel transport of D-branes along deformations of the underlying bulk conformal field theories. This method is used to derive B-brane monodromies in Kahler moduli spaces of non-linear sigma models on projective hypersurfaces. The corresponding defects are constructed at Landau-Ginzburg points in these moduli spaces where matrix factorisation techniques can be used. Transporting them to the large volume phase by means of the gauged linear sigma model we find that their action on B-branes at large volume can be described by certain Fourier-Mukai transformations which are known from target space geometric considerations to represent the corresponding monodromies.Comment: 49 pages, 5 figure

Attractor Flows from Defect Lines

Deforming a two dimensional conformal field theory on one side of a trivial defect line gives rise to a defect separating the original theory from its deformation. The Casimir force between these defects and other defect lines or boundaries is used to construct flows on bulk moduli spaces of CFTs. It turns out, that these flows are constant reparametrizations of gradient flows of the g-functions of the chosen defect or boundary condition. The special flows associated to supersymmetric boundary conditions in N=(2,2) superconformal field theories agree with the attractor flows studied in the context of black holes in N=2 supergravity.Comment: 28 page

Limits and Degenerations of Unitary Conformal Field Theories

In the present paper, degeneration phenomena in conformal field theories are studied. For this purpose, a notion of convergent sequences of CFTs is introduced. Properties of the resulting limit structure are used to associate geometric degenerations to degenerating sequences of CFTs, which, as familiar from large volume limits of non-linear sigma models, can be regarded as commutative degenerations of the corresponding ``quantum geometries''. As an application, the large level limit of the A-series of unitary Virasoro minimal models is investigated in detail. In particular, its geometric interpretation is determined.Comment: 57 pages, no figures; minor clarifications and reference added; final version to appear in Commun. Math. Phy

Effects of an 8-week meditation program on mood and anxiety in patients with memory loss.

Abstract Background: This study assesses changes in mood and anxiety in a cohort of subjects with memory loss who participated in an 8-week Kirtan Kriya meditation program. Perceived spirituality also was assessed. Previous reports from this cohort showed changes in cognitive function and cerebral blood flow (CBF). The purpose of this analysis was to assess outcome measures of mood and affect, and also spirituality, and to determine whether or not results correlated with changes in CBF. Methods: Fifteen (15) subjects (mean age 62±7 years) with memory problems were enrolled in an 8-week meditation program. Before and after the 8-week meditation, subjects were given a battery of neuropsychologic tests as well as measures of mood, anxiety, and spirituality. In addition, they underwent single photon emission computed tomography scans before and after the program. A region-of-interest template obtained counts in several brain structures that could also be compared to the results from the affect and spirituality measures. Results: The meditation training program resulted in notable improvement trends in mood, anxiety, tension, and fatigue, with some parameters reaching statistical significance. All major trends correlated with changes in CBF. There were nonsignificant trends in spirituality scores that did not correlate with changes in CBF. Conclusions: An 8-week, 12 minute a day meditation program in patients with memory loss was associated with positive changes in mood, anxiety, and other neuropsychologic parameters, and these changes correlated with changes in CBF. A larger-scale study is needed to confirm these findings and better elucidate mechanisms of change

B-type defects in Landau-Ginzburg models

We consider Landau-Ginzburg models with possibly different superpotentials glued together along one-dimensional defect lines. Defects preserving B-type supersymmetry can be represented by matrix factorisations of the difference of the superpotentials. The composition of these defects and their action on B-type boundary conditions is described in this framework. The cases of Landau-Ginzburg models with superpotential W=X^d and W=X^d+Z^2 are analysed in detail, and the results are compared to the CFT treatment of defects in N=2 superconformal minimal models to which these Landau-Ginzburg models flow in the IR.Comment: 50 pages, 2 figure