arXiv subjects

Ingo Runkel

Publications and source records attributed to Ingo Runkel.

At least 37 records · Page 2Linked to original sources

arXiv2022

Lattice models from CFT on surfaces with holes I: Torus partition function via two lattice cells

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.

cond-mat.stat-mech↗

arXiv2022

The symplectic fermion ribbon quasi-Hopf algebra and the SL(2,Z)-action on its centre

We introduce a family of factorisable ribbon quasi-Hopf algebras $Q(N)$ for $N$ a positive integer: as an algebra, $Q(N)$ is the semidirect product of $\mathbb{C}\mathbb{Z}_2$ with the direct sum of a Grassmann and a Clifford algebra in $2N$ generators. We show that $Rep Q(N)$ is ribbon equivalent to the symplectic fermion category $SF(N)$ that was computed by the third author from conformal blocks of the corresponding logarithmic conformal field theory. The latter category in turn is conjecturally ribbon equivalent to representations of $V_{ev}$, the even part of the symplectic fermion vertex operator super algebra. Using the formalism developed in our previous paper we compute the projective $SL(2,\mathbb{Z})$-action on the centre of $Q(N)$ as obtained from Lyubashenko's general theory of mapping class group actions for factorisable finite ribbon categories. This allows us to test a conjectural non-semisimple version of the modular Verlinde formula: we verify that the $SL(2,\mathbb{Z})$-action computed from $Q(N)$ agrees projectively with that on pseudo trace functions of $V_{ev}$.

math.QA↗

arXiv2021

3-Dimensional TQFTs From Non-Semisimple Modular Categories

We use modified traces to renormalize Lyubashenko's closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological invariants which we upgrade to 2+1-TQFTs under the additional assumption of factorizability. The resulting functors provide monoidal extensions of Lyubashenko's mapping class group representations, as discussed in arXiv:2010.14852. This general framework encompasses important examples of non-semisimple modular categories coming from the representation theory of quasi-Hopf algebras, which were left out of previous non-semisimple TQFT constructions.

math.GT↗

arXiv2021

Topological field theory on r-spin surfaces and the Arf invariant

We give a combinatorial model for r-spin surfaces with parametrised boundary based on Novak (2015). The r-spin structure is encoded in terms of $\mathbb{Z}_r$-valued indices assigned to the edges of a polygonal decomposition. This combinatorial model is designed for our state sum construction of two-dimensional topological field theories on r-spin surfaces. We show that an example of such a topological field theory computes the Arf-invariant of an r-spin surface as introduced in Geiges, Gonzalo (2012) and Randal-Williams (2014). This implies in particular that the r-spin Arf-invariant is constant on orbits of the mapping class group, providing an alternative proof of that fact.

math.QA↗

arXiv2021

Reshetikhin-Turaev TQFTs close under generalised orbifolds

We specialise the construction of orbifold graph TQFTs introduced in Carqueville et al., arXiv:2101.02482 to Reshetikhin-Turaev defect TQFTs. We explain that the modular fusion category ${\mathcal{C}}_{\mathcal{A}}$ constructed in Mulevičius-Runkel, arXiv:2002.00663 from an orbifold datum $\mathcal{A}$ in a given modular fusion category $\mathcal{C}$ is a special case of the Wilson line ribbon categories introduced as part of the general theory of orbifold graph TQFTs. Using this, we prove that the Reshetikhin-Turaev TQFT obtained from ${\mathcal{C}}_{\mathcal{A}}$ is equivalent to the orbifold of the TQFT for $\mathcal{C}$ with respect to the orbifold datum $\mathcal{A}$.

math.QA↗

arXiv2021

Mapping class group representations and Morita classes of algebras

A modular fusion category C allows one to define projective representations of the mapping class groups of closed surfaces of any genus. We show that if all these representations are irreducible, then C has a unique Morita-class of simple non-degenerate algebras, namely that of the tensor unit. This improves on a result by Andersen and Fjelstad, albeit under stronger assumptions. One motivation to look at this problem comes from questions in three-dimensional quantum gravity.

math.QA↗

arXiv2021

Domain walls between 3d phases of Reshetikhin-Turaev TQFTs

We study surface defects in three-dimensional topological quantum field theories which separate different theories of Reshetikhin-Turaev type. Based on the new notion of a Frobenius algebra over two commutative Frobenius algebras, we present an explicit and computable construction of such defects. It specialises to the construction in Carqueville et al., arXiv:1710.10214 if all 3-strata are labelled by the same topological field theory. We compare the results to the model-independent analysis in Fuchs et al., arXiv:1203.4568 and find agreement.

hep-th↗

arXiv2021

Mapping Class Group Representations From Non-Semisimple TQFTs

In [arXiv:1912.02063], we constructed 3-dimensional Topological Quantum Field Theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modular category $\mathcal{C}$. This allows us to prove that the projective representations induced from the non-semisimple TQFTs of [arXiv:1912.02063] are equivalent to those obtained by Lyubashenko via generators and relations in [arXiv:hep-th/9405167]. Finally, we show that, when $\mathcal{C}$ is the category of finite-dimensional representations of the small quantum group of $\mathfrak{sl}_2$, the action of all Dehn twists for surfaces without marked points has infinite order.

math.GT↗

arXiv2021

Monadic cointegrals and applications to quasi-Hopf algebras

For $\mathcal{C}$ a finite tensor category we consider four versions of the central monad, $A_1, \dots, A_4$ on $\mathcal{C}$. Two of them are Hopf monads, and for $\mathcal{C}$ pivotal, so are the remaining two. In that case all $A_i$ are isomorphic as Hopf monads. We define a monadic cointegral for $A_i$ to be an $A_i$-module morphism $\mathbf{1} \to A_i(D)$, where $D$ is the distinguished invertible object of $\mathcal{C}$. We relate monadic cointegrals to the categorical cointegral introduced by Shimizu (2019), and, in case $\mathcal{C}$ is braided, to an integral for the braided Hopf algebra $\mathcal{L} = \int^X X^\vee \otimes X$ in $\mathcal{C}$ studied by Lyubashenko (1995). Our main motivation stems from the application to finite dimensional quasi-Hopf algebras $H$. For the category of finite-dimensional $H$-modules, we relate the four monadic cointegrals (two of which require $H$ to be pivotal) to four existing notions of cointegrals for quasi-Hopf algebras: the usual left/right cointegrals of Hausser and Nill (1994), as well as so-called $γ$-symmetrised cointegrals in the pivotal case, for $γ$ the modulus of $H$. For (not necessarily semisimple) modular tensor categories $\mathcal{C}$, Lyubashenko gave actions of surface mapping class groups on certain Hom-spaces of $\mathcal{C}$, in particular of $SL(2,\mathbb{Z})$ on $\mathcal{C}(\mathcal{L},\mathbf{1})$. In the case of a factorisable ribbon quasi-Hopf algebra, we give a simple expression for the action of $S$ and $T$ which uses the monadic cointegral.

math.QA↗

arXiv2021

Orbifold graph TQFTs

A generalised orbifold of a defect TQFT $\mathcal{Z}$ is another TQFT $\mathcal{Z}_{\mathcal{A}}$ obtained by performing a state sum construction internal to $\mathcal{Z}$. As an input it needs a so-called orbifold datum $\mathcal{A}$ which is used to label stratifications coming from duals of triangulations and is subject to conditions encoding the invariance under Pachner moves. In this paper we extend the construction of generalised orbifolds of $3$-dimensional TQFTs to include line defects. The result is a TQFT acting on 3-bordisms with embedded ribbon graphs labelled by a ribbon category $\mathcal{W}_{\mathcal{A}}$ that we canonically associate to $\mathcal{Z}$ and $\mathcal{A}$. We also show that for special orbifold data, the internal state sum construction can be performed on more general skeletons than those dual to triangulations. This makes computations with $\mathcal{Z}_{\mathcal{A}}$ easier to handle in specific examples.

math.QA↗

arXiv2020

Fibonacci-type orbifold data in Ising modular categories

An orbifold datum is a collection $\mathbb{A}$ of algebraic data in a modular fusion category $\mathcal{C}$. It allows one to define a new modular fusion category $\mathcal{C}_{\mathbb{A}}$ in a construction that is a generalisation of taking the Drinfeld centre of a fusion category. Under certain simplifying assumptions we characterise orbifold data $\mathbb{A}$ in terms of scalars satisfying polynomial equations and give an explicit expression which computes the number of isomorphism classes of simple objects in $\mathcal{C}_{\mathbb{A}}$. In Ising-type modular categories we find new examples of orbifold data which - in an appropriate sense - exhibit Fibonacci fusion rules. The corresponding orbifold modular categories have 11 simple objects, and for a certain choice of parameters one obtains the modular category for $sl(2)$ at level 10. This construction inverts the extension of the latter category by the $E_6$ commutative algebra.

math.QA↗

arXiv2020

Orbifolds of Reshetikhin-Turaev TQFTs

We construct three classes of generalised orbifolds of Reshetikhin-Turaev theory for a modular tensor category $\mathcal{C}$, using the language of defect TQFT from [arXiv:1705.06085]: (i) spherical fusion categories give orbifolds for the "trivial" defect TQFT associated to vect, (ii) $G$-crossed extensions of $\mathcal{C}$ give group orbifolds for any finite group $G$, and (iii) we construct orbifolds from commutative $Δ$-separable symmetric Frobenius algebras in $\mathcal{C}$. We also explain how the Turaev-Viro state sum construction fits into our framework by proving that it is isomorphic to the orbifold of case (i). Moreover, we treat the cases (ii) and (iii) in the more general setting of ribbon tensor categories. For case (ii) we show how Morita equivalence leads to isomorphic orbifolds, and we discuss Tambara-Yamagami categories as particular examples.

math.QA↗

arXiv2020

Constructing modular categories from orbifold data

In Carqueville et al., arXiv:1809.01483, the notion of an orbifold datum $\mathbb{A}$ in a modular fusion category $\mathcal{C}$ was introduced as part of a generalised orbifold construction for Reshetikhin-Turaev TQFTs. In this paper, given a simple orbifold datum $\mathbb{A}$ in $\mathcal{C}$, we introduce a ribbon category $\mathcal{C}_{\mathbb{A}}$ and show that it is again a modular fusion category. The definition of $\mathcal{C}_{\mathbb{A}}$ is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when $\mathbb{A}$ is given by a simple commutative $Δ$-separable Frobenius algebra $A$ in $\mathcal{C}$; (ii) when $\mathbb{A}$ is an orbifold datum in $\mathcal{C} = \operatorname{Vect}$, built from a spherical fusion category $\mathcal{S}$. We show that in case (i), $\mathcal{C}_{\mathbb{A}}$ is ribbon-equivalent to the category of local modules of $A$, and in case (ii), to the Drinfeld centre of $\mathcal{S}$. The category $\mathcal{C}_{\mathbb{A}}$ thus unifies these two constructions into a single algebraic setting.

math.QA↗

arXiv2020

Fermionic CFTs and classifying algebras

We study fermionic conformal field theories on surfaces with spin structure in the presence of boundaries, defects, and interfaces. We obtain the relevant crossing relations, taking particular care with parity signs and signs arising from the change of spin structure in different limits. We define fermionic classifying algebras for boundaries, defects, and interfaces, which allow one to read off the elementary boundary conditions, etc. As examples, we define fermionic extensions of Virasoro minimal models and give explicit solutions for the spectrum and bulk structure constants. We show how the $A$- and $D$-type fermionic Virasoro minimal models are related by a parity-shift operation which we define in general. We study the boundaries, defects, and interfaces in several examples, in particular in the fermionic Ising model, i.e. the free fermion, in the fermionic tri-critical Ising model, i.e. the first unitary $N=1$ superconformal minimal model, and in the supersymmetric Lee-Yang model, of which there are two distinct versions that are related by parity-shift.

hep-th↗

arXiv2019

Modified traces for quasi-Hopf algebras

Let H be a finite-dimensional unimodular pivotal quasi-Hopf algebra over a field k, and let H-mod be the pivotal tensor category of finite-dimensional H-modules. We give a bijection between left (resp. right) modified traces on the tensor ideal H-pmod of projective modules and left (resp. right) cointegrals for H. The non-zero left/right modified traces are non-degenerate, and we show that non-degenerate left/right modified traces can only exist for unimodular H. This generalises results of Beliakova, Blanchet, and Gainutdinov from Hopf algebras to quasi-Hopf algebras. As an example we compute cointegrals and modified traces for the family of symplectic fermion quasi-Hopf algebras.

math.QA↗

arXiv2019

Projective objects and the modified trace in factorisable finite tensor categories

For C a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show: 1) C always contains a simple projective object; 2) if C is in addition ribbon, the internal characters of projective modules span a submodule for the projective SL(2,Z)-action; 3) the action of the Grothendieck ring of C on the span of internal characters of projective objects can be diagonalised; 4) the linearised Grothendieck ring of C is semisimple iff C is semisimple. Results 1-3 remain true in positive characteristic under an extra assumption. Result 1 implies that the tensor ideal of projective objects in C carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of S-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular S-transformation restricted to internal characters of projective objects. We compute the modified trace in the example of symplectic fermion categories, and we illustrate how the Verlinde-like formula for projective objects can be applied there.

math.QA↗

arXiv2019

String-net models for non-spherical pivotal fusion categories

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 non-spherical 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 labelled by a simple object Z of the Drinfeld centre. This space is found to be non-zero iff Z is isomorphic to a non-unit simple object determined by the non-spherical 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.

math.QA↗

arXiv2018

Line and surface defects in Reshetikhin-Turaev TQFT

A modular tensor category $\mathcal{C}$ gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded $\mathcal{C}$-coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. The surface defects are labelled by $Δ$-separable symmetric Frobenius algebras and the line defects by "multi-modules" which are equivariant with respect to a cyclic group action. Our invariant cannot distinguish non-isotopic embeddings of 2-spheres, but we give an example where it distinguishes non-isotopic embeddings of 2-tori.

math.QA↗