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We construct a one-parameter family of lattice models starting from a two-dimensional rational conformal field theory on a torus with a regular lattice of holes, each of which is equipped with a conformal boundary condition. The lattice model is obtained by cutting the surface into triangles with clipped-off edges using open channel factorisation. The parameter is given by the hole radius. At finite radius, high energy states are suppressed and the model is effectively finite. In the zero-radius limit, it recovers the CFT amplitude exactly. In the touching hole limit, one obtains a topological field theory. If one chooses a special conformal boundary condition which we call ‘cloaking boundary condition’, then for each value of the radius the fusion category of topological line defects of the CFT is contained in the lattice model. The fact that the full topological symmetry of the initial CFT is realised exactly is a key feature of our lattice models. We provide an explicit recursive procedure to evaluate the interaction vertex on arbitrary states. As an example, we study the lattice model obtained from the Ising CFT on a torus with one hole, decomposed into two lattice cells. We numerically compare the truncated lattice model to the CFT expression obtained from expanding the boundary state in terms of the hole radius and we find good agreement at intermediate values of the radius. [ABSTRACT FROM AUTHOR]

We build two-dimensional quantum field theories on spin surfaces starting from theories on oriented surfaces with networks of topological defect lines and junctions. The construction uses a combinatorial description of the spin structure in terms of a triangulation equipped with extra data. The amplitude for the spin surfaces is defined to be the amplitude for the underlying oriented surface equipped with a defect network dual to the triangulation. Independence of the triangulation and of the other choices follows if the line defect and junctions are obtained from a Δ-separable Frobenius algebra with involutive Nakayama automorphism in the monoidal category of topological defects. For rational conformal field theory, we can give a more explicit description of the defect category, and we work out two examples related to free fermions in detail: the Ising model and the so(n) WZW model at level 1. [ABSTRACT FROM AUTHOR]

A string-net model associates a vector space to a surface in terms of graphs decorated by objects and morphisms of a pivotal fusion category modulo local relations. String-net models are usually considered for spherical fusion categories, and in this case, the vector spaces agree with the state spaces of the corresponding Turaev–Viro topological quantum field theory. In the present work, some effects of dropping the sphericality condition are investigated. In one example of nonspherical pivotal fusion categories, the string-net space counts the number of r -spin structures on a surface and carries an isomorphic representation of the mapping class group. Another example concerns the string-net space of a sphere with one marked point labeled by a simple object Z of the Drinfeld center. This space is found to be nonzero iff Z is isomorphic to a nonunit simple object determined by the nonspherical pivotal structure. The last example mirrors the effect of deforming the stress tensor of a two-dimensional conformal field theory, such as in the topological twist of a supersymmetric theory. [ABSTRACT FROM AUTHOR]

We establish an action of the representations of N = 2-superconformal symmetry on the category of matrix factorisations of the potentials xd and xd - yd, for d odd. More precisely we prove a tensor equivalence between (a) the category of Neveu-Schwarz-type representations of the N = 2 minimal super vertex operator algebra at central charge 3-6/d, and (b) a full subcategory of graded matrix factorisations of the potential xd - yd. The subcategory in (b) is given by permutation-type matrix factorisations with consecutive index sets. The physical motivation for this result is the Landau-Ginzburg/conformal field theory correspondence, where it amounts to the equivalence of a subset of defects on both sides of the correspondence. Our work builds on results by Brunner and Roggenkamp [BR], where an isomorphism of fusion rules was established. [ABSTRACT FROM AUTHOR]

In this paper we relate a problem in representation theory -- the study of Yetter-Drinfeld modules over certain braided Hopf algebras -- to a problem in two-dimensional quantum field theory, namely, the identification of integrable perturba-tions of a conformal field theory. A prescription that parallels Lusztig's construction allows one to read off the quantum group governing the integrable symmetry. As an example, we illustrate how the quantum group for the loop algebra of sl(2) appears in the integrable structure of the perturbed uncompactified and compactified free boson. [ABSTRACT FROM AUTHOR]

The chiral conformal field theory of free super-bosons is generated by weight one currents whose mode algebra is the affinisation of an abelian Lie super-algebra h with non-degenerate super-symmetric pairing. The mode algebras of a single free boson and of a single pair of symplectic fermions arise for even|odd dimension 1|0 and 0|2 of h, respectively. In this paper, the representations of the untwisted mode algebra of free super-bosons are equipped with a tensor product, a braiding, and an associator. In the symplectic fermion case, i.e., if h is purely odd, the braided monoidal structure is extended to representations of the Z/2Z-twisted mode algebra. The tensor product is obtained by computing spaces of vertex operators. The braiding and associator are determined by explicit calculations from three- and four-point conformal blocks. [ABSTRACT FROM AUTHOR]

An introduction is presented in which the editor discusses various reports within the issue on topics including continuum approach to logarithmic conformal field theory (CFT), problem of constructing local logarithmic CFTs, and role of logarithmic CFTs with local degrees of freedom.

We establish an action of the representations of N = 2-superconformal symmetry on the category of matrix factorisations of the potentials xd and xd - yd, for d odd. More precisely we prove a tensor equivalence between (a) the category of Neveu-Schwarz-type representations of the N = 2 minimal super vertex operator algebra at central charge 3-6/d, and (b) a full subcategory of graded matrix factorisations of the potential xd - yd. The subcategory in (b) is given by permutation-type matrix factorisations with consecutive index sets. The physical motivation for this result is the Landau-Ginzburg/conformal field theory correspondence, where it amounts to the equivalence of a subset of defects on both sides of the correspondence. Our work builds on results by Brunner and Roggenkamp [BR], where an isomorphism of fusion rules was established. [ABSTRACT FROM AUTHOR]

This is the part II of a two-part work started in [18]. In part I, Cardy algebras were studied, a notion which arises from the classification of genus-0, 1 open-closed rational conformal field theories. In this part, we prove that a Cardy algebra also satisfies the higher genus factorisation and modular-invariance properties formulated in [7] in terms of the notion of a solution to the sewing constraints. We present the proof by showing that the latter notion, which is defined as a monoidal natural transformation, can be expressed in terms of generators and relations, which correspond exactly to the defining data and axioms of a Cardy algebra. [ABSTRACT FROM AUTHOR]