Ingo Runkel

We consider the link and three-manifold invariants in De Renzi et al. [Selecta Math. 28 (2022), article no. 42], which are defined in terms of certain non-semisimple ribbon categories \mathcal{C} (locally finite for link invariants, finite for manifold invariants) together with a choice of tensor ideal and modified trace. If the ideal is all of \mathcal{C} , these invariants agree with those defined by Lyubashenko in the 90’s. We show that in that case the invariants depend on the objects labelling the link only through their simple composition factors. In order to detect non-trivial extensions, one needs to pass to proper ideals. We compute examples of link and three-manifold invariants for \mathcal{C} being the category of N pairs of symplectic fermions. Using a quasi-Hopf algebra realisation of \mathcal{C} , we find that the Lyubashenko-invariant of a lens space is equal to the order of its first homology group to the power N , a relation we conjecture to hold for all three-manifolds with finite first homology group. For N \ge 2 , \mathcal{C} allows for tensor ideals \mathcal{I} with a modified trace which are different from all of \mathcal{C} and from the projective ideal. Using the theory of pullback traces and symmetrised cointegrals, we show that the link invariant obtained from \mathcal{I} can distinguish a continuum of indecomposable but reducible objects which all have the same composition series.

The modern way to understand symmetries of a quantum field theory is via its topological defects in various dimensions. In this contribution to the proceedings, we focus on line defects in 2d[ ]QFT and we point out that topological defects naturally embed into a larger class, namely, translation invariant defects. The latter still allow for nonsingular fusion and one obtains a monoidal category of translation invariant defects which contains that of topological defects as a full subcategory. We give a simple perturbative description of translation invariant defects in a perturbed conformal field theory via chiral three-dimensional topological field theory. We show in the example of the Ising CFT and the Lee–Yang CFT that even if no topological defects survive the deformation, some translation invariant defects still do.

The modern way to understand symmetries of a quantum field theory is via its topological defects in various dimensions. In this contribution to the proceedings we focus on line defects in 2d QFT and we point out that topological defects naturally embed into a larger class, namely translation invariant defects. The latter still allow for non-singular fusion and one obtains a monoidal category of translation invariant defects which contains that of topological defects as a full subcategory. We give a simple perturbative description of translation invariant defects in a perturbed conformal field theory via chiral three-dimensional topological field theory. We show in the example of the Ising CFT and the Lee-Yang CFT that even if no topological defects survive the deformation, some translation invariant defects still do.

We investigate non-local conserved charges in perturbed two-dimensional conformal field theories (CFT) from the point of view of the 3d symmetry TFT (SymTFT) of the unperturbed theory. In the SymTFT we state a simple commutation condition which results in a pair of compatible bulk and defect perturbations, such that the perturbed line defects are conserved in the perturbed CFT. In other words, the perturbed defects are rigidly translation invariant, and such defects form a monoidal category which extends the topological symmetries. As examples we study the A-type Virasoro minimal models M ( p , q ) . Our formalism provides one-parameter families of commuting non-local conserved charges for perturbations by a primary bulk field with Kac label (1, 2), (1, 3), or (1, 5), which are the standard integrable perturbations of minimal models. We find solutions to the commutation condition also for other bulk perturbations, such as (1, 7), and we contrast this with the existence of local conserved charges. There has been recent interest in the possibility that in certain cases perturbations by fields such as (1, 7) can be integrable, and our construction provides a new way in which integrability can be found without the need for local conserved charges.

Recently, large degeneracy based on product eigenstates has been found in spin ladders, kagome-like lattices, and motif magnetism, connected to spin liquids, anyonic phases, and quantum scars. We unify these systems by a complete classification of product eigenstates of Heisenberg XXZ Hamiltonians with Dzyaloshinskii-Moriya interaction on general graphs in the form of Kirchhoff rules for spin supercurrents. By this, we construct spin systems with extensive degree of degeneracy linked to exotic condensates which could be studied in atomic gases and quantum spin lattices.

In this paper we continue to investigate the lattice models obtained from 2d CFTs via the construction introduced in [arXiv:2112.01563]. On the side of the 2d CFT we consider the cloaking boundary condition relative to a fixed fusion category F of topological line defects. The resulting lattice model realises the topological symmetry F exactly. We compute the state spaces and Boltzmann weights of these lattice model in the example of unitary Virasoro minimal models. We work directly with amplitudes, rather than with normalised correlators, and we provide a careful treatment of the Weyl anomaly factor in terms of the Liouville action. We numerically evaluate the Ising CFT on the torus with one hole and cloaking boundary condition in two channels, and illustrate in this example that the anomaly factors are essential to obtain matching results for the amplitudes. We show that lattice models obtained from Virasoro minimal models at lowest non-trivial cutoff can be exactly mapped to loop models. This provides a first non-trivial check that our lattice models can contain the 2d CFT they were constructed from in their phase diagram, and we propose a condition on the cloaking boundary condition for F under which we expect this to happen in general.

Mapping class group averages appear in the study of 3D gravity partition functions. In this paper, we work with 3D topological field theories to establish a bulk-boundary correspondence between such averages and correlators of 2D rational CFTs whose chiral mapping class group representations are irreducible and satisfy a finiteness property. We show that Ising-type modular fusion categories satisfy these properties on surfaces with or without field insertions, extending results in [1], and we comment on the absence of invertible global symmetries in the examples we consider.

The skein module for a d-dimensional manifold is a vector space spanned by embedded framed graphs decorated by a category A with suitable extra structure depending on the dimension d, modulo local relations which hold inside d-balls. For a full subcategory S of A, an S-admissible skein module is defined analogously, except that local relations for a given ball may only be applied if outside the ball at least one edge is coloured in S. In this paper we prove that admissible skein modules in any dimension satisfy excision, namely that the skein module of a glued manifold is expressed as a coend over boundary values on the boundary components glued together. We furthermore relate skein modules for different choices of S, apply our result to cylinder categories, and recover the relation to modified traces.