arXiv subjects

Lukas Woike

Publications and source records attributed to Lukas Woike.

At least 19 recordsLinked to original sources

arXiv2026

Modular Functors with Singularities from Vertex Operator Algebras Beyond Rigidity and Finiteness

For a vertex operator algebra $V$ and a suitable category of its modules, we propose a construction for spaces of conformal blocks organized into an open-closed modular functor with singularities. This is inspired by the idea of implementing directly from the start the principle of holomorphic factorization. More precisely, using the strategy of modular extension introduced by Costello and developed further in our previous work, we build for each surface $\Sigma$ with at least one boundary component per path component and specified boundary labels attached to marked intervals or boundary circles a representation $\Omega_V(\Sigma;-)$ of the mapping class group of $\Sigma$. The construction can be described explicitly on generating Dehn twists. This approach is a priori independent from other constructions based on algebraic geometry or topological techniques involving e.g. surgery, but we include an overview over the available comparisons. In the special case in which the module category of $V$ is a not necessarily semisimple modular category $\mathcal{A}$, the spaces $\Omega_V(\Sigma)$ are equivalent to the string-net spaces for $\mathcal{A}$ and hence to the modular functor for the Drinfeld center $Z(\mathcal{A})\simeq \bar{\mathcal{A}}\boxtimes\mathcal{A}$. However, the construction of $\Omega_V$ in this paper has the advantage of being available beyond rationality, rigidity, self-contragredience and finiteness. Moreover, we prove that $\Omega_V$ satisfies excision, is finite-dimensional in the $C_2$-cofinite case and produces representations of surface braid groups generalizing the ones of Brochier-Jordan. We prove for the triplet $\mathcal{W}_{2,3}$ with non-exact fusion product that the boundary conditions introduced by Gaberdiel-Runkel-Wood produce correlation functions, provided that one uses the notion of a modular functor with singularities that we develop.

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arXiv2025

Pivotal Module Categories, Factorization Homology and Modular Invariant Modified Traces

The algebraic notion of a pivotal module category was developed by Schaumann and Shimizu and is central to the description of boundary conditions in conformal field theory according to a proposal by Fuchs and Schweigert. In this paper, we present a large class of examples of pivotal module categories of topological origin: For a unimodular finite ribbon category $\mathcal{A}$, we prove that the factorization homology $\int_\Sigma \mathcal{A}$ of a compact oriented surface $\Sigma$ with $n$ marked boundary intervals, at least one per connected component, comes with the structure of a pivotal module category over $\mathcal{A}^{\boxtimes n}$. This endows the internal skein algebras of Ben-Zvi-Brochier-Jordan, in particular the elliptic double, with a symmetric Frobenius structure. As application, we obtain, for each choice of $\mathcal{A}$, a family of full open conformal field theories, each of which comes with correlation functions for all surfaces with marked boundary intervals that are explicitly computable using factorization homology. As a further application, we explain how modified traces can be 'integrated' over surfaces: We show that the modified trace for $\mathcal{A}$ extends in a canonical way to the factorization homology of $\Sigma$. The resulting traces have the remarkable property of being modular invariant, i.e. fixed by the mapping class group action.

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arXiv2025

Reflection Equivariance and the Heisenberg Picture for Spaces of Conformal Blocks

Monoidal product, braiding, balancing and weak duality are pieces of algebraic information that are well-known to have their origin in oriented genus zero surfaces and their mapping classes. More precisely, each of them correspond to operations of the cyclic framed $E_2$-operad. We extend this correspondence to include another algebraic piece of data, namely the modified trace, by showing that it amounts to a homotopy fixed point structure with respect to the homotopy involution that reverses the orientation of surfaces and dualizes the state spaces. We call such a homotopy fixed point structure reflection equivariance. As an application, we describe the effect of orientation reversal on spaces of conformal blocks and skein modules in the non-semisimple setting, throughout relying on their factorization homology description. This has important consequences: For a modular functor that is reflection equivariant relative to a rigid duality, i) the circle category is modular, and the resulting mapping class group representations are automatically the ones built by Lyubashenko, and ii) the modules over the internal skein algebras are generated by one simple representation, carrying a unique projective mapping class group representation making the action equivariant. While i) is a new topological characterization of not necessarily semisimple modular categories, ii) generalizes the implicit description of spaces of conformal blocks purely through the representation theory of moduli algebras given by Alekseev-Grosse-Schomerus from rational conformal field theories admitting a Hopf algebra description to finite rigid logarithmic conformal field theories. This also generalizes several results of Faitg from ribbon factorizable Hopf algebras to arbitrary modular categories.

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arXiv2025

The Construction of Correlators in Finite Rigid Logarithmic Conformal Field Theory

For logarithmic conformal field theories whose monodromy data is given by a not necessarily semisimple modular category, we solve the problem of constructing and classifying the consistent systems of correlators. The correlator construction given in this article applies to the open-closed sector and generalizes the well-known one for rational conformal field theories given by Fuchs-Runkel-Schweigert roughly twenty years ago and solves conjectures of Fuchs, Gannon, Schaumann and Schweigert. The strategy is, even in the rational special case, entirely different. The correlators are constructed using the extension procedures that can be devised by means of the modular microcosm principle. It is shown that, as in the rational case, the correlators admit a holographic description, with the main difference that the holographic principle is phrased in terms of factorization homology. The latter description is used to prove that the coefficients obtained by the evaluation of the torus partition function at projective objects are non-negative integers. Moreover, we show that the derived algebra of local operators associated to a consistent system of correlators carries a Batalin-Vilkovisky structure. We prove that it is equivalent to the Batalin-Vilkovisky structure on the Hochschild cohomology of the pivotal module category of boundary conditions, for the notion of pivotality due to Schaumann and Shimizu. This proves several expectations formulated by Kapustin-Rozansky and Fuchs-Schweigert for general conformal field theories.

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arXiv2025

Modular functors from conformal blocks of rational vertex operator algebras

For a vertex operator algebra $V$, one may naturally define spaces of conformal blocks following a construction of Frenkel-Ben-Zvi generalized by Damiolini-Gibney-Tarasca. If $V$ is strongly rational, these spaces of conformal blocks form vector bundles over a suitable moduli space of algebraic curves. In this article, we establish, under the same assumptions, the widely expected topological result that the spaces of conformal blocks produce a modular functor, i.e. a modular algebra over an extension of the surface operad. This entails that the category $\mathcal{C}_V$ of admissible $V$-modules inherits from the topology of genus zero surfaces a ribbon Grothendieck-Verdier structure that leads even to the structure of a modular fusion category whose structure comes directly from the spaces of conformal blocks of $V$. As a direct consequence, we prove that the modular functor from conformal blocks extends to a three-dimensional topological field theory and comes with a description in terms of factorization homology.

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arXiv2024

Admissible Skein Modules and Ansular Functors: A Comparison

Given a finite ribbon category, which is a particular case of a cyclic algebra over the operad of genus zero surfaces, there are two possibilities for an extension defined on all three-dimensional handlebodies: On the one hand, one can use the admissible skein module construction of Costantino-Geer-Patureau-Mirand. On other hand, by a construction of the authors using Costello's modular envelope, one can build a so-called ansular functor, a handlebody version of the notion of a modular functor. Unlike the admissible skein modules with their construction through the Reshetikhin-Turaev graphical calculus, the ansular functor is defined purely through a universal property. In this note, we prove the widely held expectation that these constructions are related by giving an isomorphism between them, with the somewhat surprising subtlety that we need to include consistently on one of the sides an additional boundary component labeled by the distinguished invertible object of Etingof-Nikshych-Ostrik. In other words, the constructions agree on handlebodies up to a `background charge' that becomes trivial in the unimodular case. Our comparison result includes the handlebody group action as well as the skein algebra action.

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arXiv2024

The Cyclic and Modular Microcosm Principle in Quantum Topology

Monoidal categories with additional structure such as a braiding or some form of duality abound in quantum topology. They often appear in tandem with Frobenius algebras inside them. Motivations for this range from the theory of module categories to the construction of correlators in conformal field theory. We generalize the Baez-Dolan microcosm principle to consistently describe all these types of algebras by extending it to cyclic and modular algebras in the sense of Getzler-Kapranov. Our main result links the microcosm principle for cyclic algebras to the one for modular algebras via Costello's modular envelope. The result can be understood as a local-to-global construction or an integration procedure for various flavors of Frobenius algebras that substantially generalizes and unifies the available (and often intrinsically semisimple) methods using for example triangulations or classical skein theory. As the main application of this rather abstract result, we solve the problem of classifying consistent systems of correlators for open conformal field theories and show that the genus zero correlators for logarithmic conformal field theories constructed by Fuchs-Schweigert can be uniquely extended to handlebodies. This establishes a very general correspondence between full genus zero conformal field theory in dimension two and skein theory in dimension three.

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arXiv2024

Categorified Open Topological Field Theories

In this short note, we classify linear categorified open topological field theories in dimension two by pivotal Grothendieck-Verdier categories, a type of monoidal category equipped with a weak, not necessarily rigid duality. In combination with recently developed string-net techniques, this leads to a new description of the spaces of conformal blocks of Drinfeld centers $Z(\mathcal{C})$ of pivotal finite tensor categories $\mathcal{C}$ in terms of the modular envelope of the cyclic associative operad. If $\mathcal{C}$ is unimodular, we prove that the space of conformal blocks inherits the structure of a module over the algebra of class functions of $\mathcal{C}$ for every free boundary component. As a further application, we prove that the sewing along a boundary circle for the modular functor for $Z(\mathcal{C})$ can be decomposed into a sewing procedure along an interval and the application of the partial trace. Finally, we construct mapping class group representations from Grothendieck-Verdier categories that are not necessarily rigid and make precise how these generalize existing constructions.

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arXiv2023

The Lyubashenko Modular Functor for Drinfeld Centers via Non-Semisimple String-Nets

The Levin-Wen string-nets of a spherical fusion category $\mathcal{C}$ describe, by results of Kirillov and Bartlett, the representations of mapping class groups of closed surfaces obtained from the Turaev-Viro construction applied to $\mathcal{C}$. We provide a far-reaching generalization of this statement to arbitrary pivotal finite tensor categories, including non-semisimple or non-spherical ones: We show that the finitely cocompleted string-net modular functor built from the projective objects of a pivotal finite tensor category is equivalent to Lyubashenko's modular functor built from the Drinfeld center $Z(\mathcal{C})$.

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arXiv2023

The Dehn Twist Action for Quantum Representations of Mapping Class Groups

We calculate the Dehn twist action on the spaces of conformal blocks of a not necessarily semisimple modular category. In particular, we give the order of the Dehn twists under the mapping class group representations of closed surfaces. For Dehn twists about non-separating simple closed curves, we prove that this order is the order of the ribbon twist, thereby generalizing a result that De Renzi-Gainutdinov-Geer-Patureau-Mirand-Runkel obtained for the small quantum group. In the separating case, we express the order using the order of the ribbon twist on monoidal powers of the canonical end. As an application, we prove that the Johnson kernels of the mapping class groups act trivially if and only if for the canonical end the ribbon twist and double braiding with itself are trivial. We give a similar result for the visibility of the Torelli groups.

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arXiv2023

Classification of Consistent Systems of Handlebody Group Representations

The classifying spaces of handlebody groups form a modular operad. Algebras over the handlebody operad yield systems of representations of handlebody groups that are compatible with gluing. We prove that algebras over the modular operad of handlebodies with values in an arbitrary symmetric monoidal bicategory $\mathcal{M}$ (we introduce for these the name ansular functor) are equivalent to self-dual balanced braided algebras in $\mathcal{M}$. After specialization to a linear framework, this proves that consistent systems of handlebody group representations on finite-dimensional vector spaces are equivalent to ribbon Grothendieck-Verdier categories in the sense of Boyarchenko-Drinfeld. Additionally, it produces a concrete formula for the vector space assigned to an arbitrary handlebody in terms of a generalization of Lyubashenko's coend. Our main result can be used to obtain an ansular functor from vertex operator algebras subject to mild finiteness conditions. This includes examples of vertex operator algebras whose representation category has a non-exact monoidal product.

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arXiv2023

A Parallel Section Functor for 2-Vector Bundles

We associate to a 2-vector bundle over an essentially finite groupoid a 2-vector space of parallel sections, or, in representation theoretic terms, of higher invariants, which can be described as homotopy fixed points. Our main result is the extension of this assignment to a symmetric monoidal 2-functor $\operatorname{Par} : \mathbf{2VecBunGrpd} \to \mathbf{2Vect}$. It is defined on the symmetric monoidal bicategory $\mathbf{2VecBunGrpd}$ whose morphisms arise from spans of groupoids in such a way that the functor $\operatorname{Par}$ provides pull-push maps between 2-vector spaces of parallel sections of 2-vector bundles. The direct motivation for our construction comes from the orbifoldization of extended equivariant topological field theories.

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arXiv2023

The differential graded Verlinde Formula and the Deligne Conjecture

A modular category $\mathcal{C}$ gives rise to a differential graded modular functor, i.e. a system of projective mapping class group representations on chain complexes. This differential graded modular functor assigns to the torus the Hochschild complex and, in the dual description, the Hochschild cochain complex of $\mathcal{C}$. On both complexes, the monoidal product of $\mathcal{C}$ induces the structure of an $E_2$-algebra, to which we refer as the differential graded Verlinde algebra. At the same time, the modified trace induces on the tensor ideal of projective objects in $\mathcal{C}$ a Calabi-Yau structure so that the cyclic Deligne Conjecture endows the Hochschild cochain and chain complex of $\mathcal{C}$ with a second $E_2$-structure. Our main result is that the action of a specific element $S$ in the mapping class group of the torus transforms the differential graded Verlinde algebra into this second $E_2$-structure afforded by the Deligne Conjecture. This result is established for both the Hochschild chain and the Hochschild cochain complex of $\mathcal{C}$. In general, these two versions of the result are inequivalent. In the case of Hochschild chains, we obtain a block diagonalization of the Verlinde algebra through the action of the mapping class group element $S$. In the semisimple case, both results reduce to the Verlinde formula. In the non-semisimple case, we recover after restriction to zeroth (co)homology earlier proposals for non-semisimple generalizations of the Verlinde formula.

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arXiv2023

The Diffeomorphism Group of the Solid Closed Torus and Hochschild Homology

We prove that for a self-injective ribbon Grothendieck-Verdier category $\mathcal{C}$ in the sense of Boyarchenko-Drinfeld the cyclic action on the Hochschild complex of $\mathcal{C}$ extends to an action of the diffeomorphism group of the solid closed torus $\mathbb{S}^1 \times \mathbb{D}^2$.

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arXiv2022

A Classification of Modular Functors via Factorization Homology

Modular functors are traditionally defined as systems of projective representations of mapping class groups of surfaces that are compatible with gluing. They can formally be described as modular algebras over central extensions of the modular surface operad, with the values of the algebra lying in a suitable symmetric monoidal $(2,1)$-category $\mathcal{S}$ of linear categories. In this paper, we prove that modular functors in $\mathcal{S}$ are equivalent to self-dual balanced braided algebras $\mathcal{A}$ in $\mathcal{S}$ (a categorification of the notion of a commutative Frobenius algebra) for which a condition formulated in terms of factorization homology with coefficients in $\mathcal{A}$ is satisfied; we call such $\mathcal{A}$ connected. The equivalence in one direction is afforded by genus zero restriction. Our construction of the inverse equivalence is entirely topological and can be thought of as a far reaching generalization of the construction of modular functors from skein theory. In order to verify the connectedness condition in practice, we prove that it can be reduced to a single condition in genus one. Moreover, we show that cofactorizability of $\mathcal{A}$, a condition known to be satisfied for modular categories, is sufficient. Therefore, we recover in particular Lyubashenko's construction of a modular functor from a (not necessarily semisimple) modular category and show that it is determined by its genus zero part. Additionally, we exhibit modular functors that do not come from modular categories and outline applications to the theory of vertex operator algebras.

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arXiv2022

The distinguished invertible object as ribbon dualizing object in the Drinfeld center

We prove that the Drinfeld center $Z(\mathcal{C})$ of a pivotal finite tensor category $\mathcal{C}$ comes with the structure of a ribbon Grothendieck-Verdier category in the sense of Boyarchenko-Drinfeld. Phrased operadically, this makes $Z(\mathcal{C})$ into a cyclic algebra over the framed $E_2$-operad. The underlying object of the dualizing object is the distinguished invertible object of $\mathcal{C}$ appearing in the well-known Radford isomorphism of Etingof-Nikshych-Ostrik. Up to equivalence, this is the unique ribbon Grothendieck-Verdier structure on $Z(\mathcal{C})$ extending the canonical balanced braided structure that $Z(\mathcal{C})$ already comes equipped with. The duality functor of this ribbon Grothendieck-Verdier structure coincides with the rigid duality if and only if $\mathcal{C}$ is spherical in the sense of Douglas-Schommer-Pries-Snyder. The main topological consequence of our algebraic result is that $Z(\mathcal{C})$ gives rise to an ansular functor, in fact even a modular functor regardless of whether $\mathcal{C}$ is spherical or not. In order to prove the aforementioned uniqueness statement for the ribbon Grothendieck-Verdier structure, we derive a seven-term exact sequence characterizing the space of ribbon Grothendieck-Verdier structures on a balanced braided category. This sequence features the Picard group of the balanced version of the M\"uger center of the balanced braided category.

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arXiv2022

Homotopy Invariants of Braided Commutative Algebras and the Deligne Conjecture for Finite Tensor Categories

It is easy to find algebras $\mathbb{T}\in\mathcal{C}$ in a finite tensor category $\mathcal{C}$ that naturally come with a lift to a braided commutative algebra $\mathsf{T}\in Z(\mathcal{C})$ in the Drinfeld center of $\mathcal{C}$. In fact, any finite tensor category has at least two such algebras, namely the monoidal unit $I$ and the canonical end $\int_{X\in\mathcal{C}} X\otimes X^\vee$. Using the theory of braided operads, we prove that for any such algebra $\mathbb{T}$ the homotopy invariants, i.e. the derived morphism space from $I$ to $\mathbb{T}$, naturally come with the structure of a differential graded $E_2$-algebra. This way, we obtain a rich source of differential graded $E_2$-algebras in the homological algebra of finite tensor categories. We use this result to prove that Deligne's $E_2$-structure on the Hochschild cochain complex of a finite tensor category is induced by the canonical end, its multiplication and its non-crossing half braiding. With this new and more explicit description of Deligne's $E_2$-structure, we can lift the Farinati-Solotar bracket on the Ext algebra of a finite tensor category to an $E_2$-structure at cochain level. Moreover, we prove that, for a unimodular pivotal finite tensor category, the inclusion of the Ext algebra into the Hochschild cochains is a monomorphism of framed $E_2$-algebras, thereby refining a result of Menichi.

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arXiv2022

The Hochschild Complex of a Finite Tensor Category

Modular functors, i.e. consistent systems of projective representations of mapping class groups of surfaces, have been constructed for non-semisimple modular categories already decades ago. Concepts from homological algebra have not been used in this construction although it is an obvious question how they should enter in the non-semisimple case. In the present paper, we elucidate the interplay between the structures from topological field theory and from homological algebra by constructing a homotopy coherent projective action of the mapping class group $\text{SL}(2,\mathbb{Z})$ of the torus on the Hochschild complex of a modular category. This is a further step towards understanding the Hochschild complex of a modular category as a differential graded conformal block for the torus. Moreover, we describe a differential graded version of the Verlinde algebra.

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