On (orientifold of) type IIA on a compact Calabi‐Yau

We study the gauged sigma model and its mirror Landau‐Ginsburg model corresponding to type IIA on the Fermat degree‐24 hypersurface in WCP ⁴[1,1,2,8,12] (whose blow‐up gives the smooth CY₃(3,243)) away from the orbifold singularities, and its orientifold by a freely‐acting antiholomorphic involution. We derive the Picard‐Fuchs equation obeyed by the period integral as defined in [1, 2], of the parent 𝒩 = 2 type IIA theory of [3]. We obtain the Meijer's basis of solutions to the equation in the large and small complex structure limits (on the mirror Landau‐Ginsburg side) of the abovementioned Calabi‐Yau, and make some remarks about the monodromy properties associated based on [4], at the same and another MATHEMATICAlly interesting point. Based on a recently shown 𝒩 = 1 four‐dimensional triality [6] between Heterotic on the self‐mirror Calabi‐Yau CY₃(11,11), M theory on and F‐theory on an elliptically fibered CY₄ with the base given by CP ¹ × Enriques surface, we first give a heuristic argument that there can be no superpotential generated in the orientifold of of CY₃(3,243), and then explicitly verify the same using mirror symmetry formulation of [2] for the abovementioned hypersurface away from its orbifold singularities. We then discuss briefly the sigma model and the mirror Landau‐Ginsburg model corresponding to the resolved Calabi‐Yau as well.     

Exact Results for Topological Strings on Resolved <Emphasis Type="Italic">Y</Emphasis><Superscript><Emphasis Type="Italic"> p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Superscript> Singularities

We obtain exact results in α′ for open and closed A-model topological string amplitudes on a large class of toric Calabi-Yau threefolds by using their correspondence with five dimensional gauge theories. The toric Calabi-Yaus that we analyze are obtained as minimal resolution of cones over Y ^p,q manifolds and give rise via M-theory compactification to SU(p) gauge theories on . As an application we present a detailed study of the local case and compute open and closed genus zero Gromov-Witten invariants of the orbifold. We also display the modular structure of the topological wave function and give predictions for higher genus amplitudes. The mirror curve in this case is the spectral curve of the relativistic A ₁ Toda chain. Our results also indicate the existence of a wider class of relativistic integrable systems associated to generic Y ^p,q geometries.     

Remodeling the B-Model

We propose a complete, new formalism to compute unambiguously B-model open and closed amplitudes in local Calabi–Yau geometries, including the mirrors of toric manifolds. The formalism is based on the recursive solution of matrix models recently proposed by Eynard and Orantin. The resulting amplitudes are non-perturbative in both the closed and the open moduli. The formalism can then be used to study stringy phase transitions in the open/closed moduli space. At large radius, this formalism may be seen as a mirror formalism to the topological vertex, but it is also valid in other phases in the moduli space. We develop the formalism in general and provide an extensive number of checks, including a test at the orbifold point of A _p fibrations, where the amplitudes compute the ’t Hooft expansion of vevs of Wilson loops in Chern-Simons theory on lens spaces. We also use our formalism to predict the disk amplitude for the orbifold .     

Conformal field theories and modular invariant partition functions for parafermion field theories

Parafermion conformal field theories with D_2N discrete symmetry are examined in detail. The structure of field space of parafermion field theories is studied with the help of a projection operator G. Characters of the representations of the twist sector of parafermion algebra and projected characters are given. A new class of modular invariant partition functions, therefore conformal field theories, for parafermion theories are found. We argue that the principal theories correspond to the generic critical SOS models of Andrew, Baxter and Forrest.     

<Emphasis Type="Italic">W</Emphasis><Subscript>4</Subscript> toda example as hidden Liouville CFT

We construct correlators in the W ₄ Toda 2d conformal field theory for a particular class of representations and demonstrate a relation to a W ₂ (Virasoro) theory with different central charge. The relevance of the classical limits of the constructed 3-point functions and braiding matrices to problems in 4d conformal theories is discussed.     

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.     

The operator algebra of orbifold models

We analyze the chiral properties of (orbifold) conformal field theories which are obtained from a given conformal field theory by modding out by a finite symmetry group. For a class of orbifolds, we derive the fusion rules by studying the modular transformation properties of the one-loop characters. The results are illustrated with explicit calculations of toroidal andc=1 models.     

<Emphasis Type="Italic">PT</Emphasis> Symmetry,Conformal Symmetry,and the Metrication of Electromagnetism

We present some interesting connections between PT symmetry and conformal symmetry. We use them to develop a metricated theory of electromagnetism in which the electromagnetic field is present in the geometric connection. However, unlike Weyl who first advanced this possibility, we do not take the connection to be real but to instead be PT symmetric, with it being \(iA_{\mu }\) rather than \(A_{\mu }\) itself that then appears in the connection. With this modification the standard minimal coupling of electromagnetism to fermions is obtained. Through the use of torsion we obtain a metricated theory of electromagnetism that treats its electric and magnetic sectors symmetrically, with a conformal invariant theory of gravity being found to emerge. An extension to the non-Abelian case is provided.     

C=1 conformal field theories on Riemann surfaces

We study the theory ofc=1 torus and ₂-orbifold models on general Riemann surfaces. The operator content and occurrence of multi-critical points in this class of theories is discussed. The partition functions and correlation functions of vertex operators and twist fields are calculated using the theory of double covered Riemann surfaces. It is shown that orbifold partition functions are sensitive to the Torelli group. We give an algebraic construction of the operator formulation of these nonchiral theories on higher genus surfaces. Modular transformations are naturally incorporated as canonical transformations in the Hilbert space.     

Type IIB 2-form fields and gauge coupling constant of 4D ${\cal N}=2$ super QCD

We study the relation between the Type IIB (NSNS and RR) 2-form fields and the (complex) gauge coupling constant of the 4D SU(N _c ) super Yang-Mills theory with N _f fundamental matter particles. We start from the analysis of the D2-brane world-volume theory with heavy N _c quarks on the N _f D6 supergravity background. After a sequence of T- and S-dualities, we obtain the (generalized) 2-forms in the configuration with N _c D5-branes wrapping on a vanishing two-cycle under the influence of the background. These 2-forms show the same behavior as the gauge coupling constant of the 4D super QCD. The background reduces to the orbifold in the twelve-dimensional space-time formally realized by introducing the two parameters as additional space coordinates. The 10D gravity dual is suggested as the 2D flip in this twelve-dimensional space-time. In the case of N _f = 2N _c , this gravity dual becomes AdS₅ x S⁵/Z₂ with a D3-charge which depends on the constant generalized NSNS 2-form. This is the result expected from the M-theory QCD configuration. Based on the known exact result, we also discuss this configuration after including non-perturbative effects.Received: 2 May 2003, Revised: 21 July 2003, Published online: 19 September 2003     

Remarks on Wilson lines,modular invariance and possible string relics in Calabi-Yau compactifications

When one mods out a (2,2) conformal field theory by the action of a discrete group, it is possible to include Wilson lines to break the gauge symmetry. We simplify and generalize an earlier analysis by Witten of the constraints that modular invariance places on the allowed symmetry breaking patterns. The analysis does not depend on the details of the original conformal field theory. We then consider the fractionally charged states in such theories, first discussed by Wen and Witten. We note that these are rather generic, and consider the possibilities for their detection. We also note that, while in general they are expected to be massive (∼M_Planck), in models based on free fields, such as orbifold compactifications, there are likely to be massless (very light) fractionally charged states.     

Vector-spinor space and field equations

Generalizing the work of Einstein and Mayer, it is assumed that at each point of space-time there exists a vector-spinor space with N_v vector dimensions and N_s spinor dimensions, where N_v=2k and N_s=2 ^k, k3. This space is decomposed into a tangent space with4 vector and4 spinor dimensions and an internal space with N_v–4 vector and N_s–4 spinor dimension. A variational principle leads to field equations for geometric quantities which can be identified with physical fields such as the electromagnetic field, Yang-Mills gauge fields, and wave functions of bosons and fermions.     

Observations on the moduli space of superconformal field theories

Some aspects of the moduli space of superconformal field theories are discussed. It is helpful to consider the conformal field theory as a background for propagation of strings and to exploit the space-time interpretation. Using this point of view we show that the metric on the moduli space of N = 4 superconformal field theory with c = 6 is locally that of O(20,4)/O(20) × O(4). We also discover some properties of the moduli space of N = 2 superconformal field theories with c = 9. Particular examples of these conformal field theories are sigma models on four- and six-dimensional Calabi-Yau spaces. Therefore, we can use this technique to learn about the moduli space of these spaces. For c = 6 we recover the known moduli space of K₃. Our analysis of the c = 9 system leads to a new coupling in four dimensional supergravity. As a by-product, we prove that gauge couplings cannot depend on the moduli of N = 1 space-time supersymmetric compactifications.     

AdS 5 Solutions of Type IIB Supergravity and Generalized Complex Geometry

We use the formalism of generalized geometry to study the generic supersymmetric AdS ₅ solutions of type IIB supergravity that are dual to ${\mathcal{N}=1}We use the formalism of generalized geometry to study the generic supersymmetric AdS ₅ solutions of type IIB supergravity that are dual to N=1{\mathcal{N}=1} superconformal field theories (SCFTs) in d = 4. Such solutions have an associated six-dimensional generalized complex cone geometry that is an extension of Calabi-Yau cone geometry. We identify generalized vector fields dual to the dilatation and R-symmetry of the dual SCFT and show that they are generalized holomorphic on the cone. We carry out a generalized reduction of the cone to a transverse four-dimensional space and show that this is also a generalized complex geometry, which is an extension of K?hler-Einstein geometry. Remarkably, provided the five-form flux is non-vanishing, the cone is symplectic. The symplectic structure can be used to obtain Duistermaat-Heckman type integrals for the central charge of the dual SCFT and the conformal dimensions of operators dual to BPS wrapped D3-branes. We illustrate these results using the Pilch-Warner solution.     

Linear bosonic quantum field theories arising from causal variational principles

We describe discrete symmetries of two-dimensional Yang–Mills theory with gauge group G associated with outer automorphisms of G, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted G-bundles and calculate it exactly. We argue that its weak-coupling limit computes the symplectic volume of the moduli space of flat twisted G-bundles on a surface. Using the defect network approach to generalised orbifolds, we gauge the discrete symmetry and construct the corresponding orbifold theory, which is again two-dimensional Yang–Mills theory but with gauge group given by an extension of G by outer automorphisms. With the help of the orbifold completion of the topological defect bicategory of two-dimensional Yang–Mills theory, we describe the reverse orbifold using a Wilson line defect for the discrete gauge symmetry. We present our results using two complementary approaches: in the lattice regularisation of the path integral, and in the functorial approach to area-dependent quantum field theories with defects via regularised Frobenius algebras.

     

Rational Conformal Field Theories and Complex Multiplication

We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on Calabi-Yau manifolds. We perform a detailed study of RCFTs corresponding to the T² target and identify the Cardy branes with geometric branes. The T²s leading to RCFTs admit complex multiplication which characterizes Cardy branes as specific D0-branes. We propose a condition for the conformal sigma model to be RCFT for arbitrary Calabi-Yau n-folds, which agrees with the known cases. Together with recent conjectures by mathematicians it appears that rational conformal theories are not dense in the space of all conformal theories, and sometimes appear to be finite in number for Calabi-Yau n-folds for n>2. RCFTs on K3 may be dense. We speculate about the meaning of these special points in the moduli spaces of Calabi-Yau n-folds in connection with freezing geometric moduli.     

Axiomatic Conformal Theory in Dimensions &gt;2 and AdS/CT Correspondence

We formulate axioms of conformal theory (CT) in dimensions >2 modifying Segal’s axioms for two-dimensional CFT. (In the definition of higher-dimensional CFT, one includes also a condition of existence of energy-momentum tensor.) We use these axioms to derive the AdS/CT correspondence for local theories on AdS. We introduce a notion of weakly local quantum field theory and construct a bijective correspondence between conformal theories on the sphere S^d and weakly local quantum field theories on \({H^{d+1}}\) that are invariant with respect to isometries. (Here \({H^{d+1}}\) denotes hyperbolic space = Euclidean AdS space.) We give an expression of AdS correlation functions in terms of CT correlation functions. The conformal theory has conserved energy-momentum tensor iff the AdS theory has graviton in its spectrum.