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This paper develops an approach to categorical deformation quantization via factorization homology. We show that a quantization of the local coefficients for factorization homology is equivalent to consistent quantizations of its value on manifolds. To formulate our results we introduce the concepts of shifted almost Poisson and BD categories. Our main example is the character stack of flat principal bundles for a reductive algebraic group $G$, where we show that applying the general framework to the Drinfeld category reproduces deformations previously introduced by Li-Bland and \v{S}evera. As a direct consequence, we can conclude a precise relation between their quantization and those introduced by Alekseev, Grosse, and Schomerus. To arrive at our results we compute factorization homology with values in a ribbon category enriched over complete $\mathbb{C}[[\hbar]]$-modules. More generally, we define enriched skein categories which compute factorization homology for ribbon categories enriched over a general closed symmetric monoidal category $\mathcal{V}$., Comment: 84 pages. First version, comments and suggestions welcome

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., Comment: 16 pages, 2 figures

Quantum computing could impact various industries, with the automotive industry with many computational challenges, from optimizing supply chains and manufacturing to vehicle engineering, being particularly promising. This chapter investigates state-of-the-art quantum algorithms to enhance efficiency, accuracy, and scalability across the automotive value chain. We explore recent advances in quantum optimization, machine learning, and numerical and chemistry simulations, highlighting their potential and limitations. We identify and discuss key challenges in near-term and fault-tolerant algorithms and their practical use in industrial applications. While quantum algorithms show potential in many application domains, current noisy intermediate-scale quantum hardware limits scale and, thus, business benefits. In the long term, fault-tolerant systems promise theoretical speedups; however, they also require further progress in hardware and software (e.\,g., related to error correction and data loading). We expect that with this progress, significant practical benefits will emerge eventually.

We prove that every unitary invertible quantum field theory satisfies a generalization of the famous spin statistics theorem. To formulate this extension, we define a `higher spin' action of the stable orthogonal group $O$ on appropriate spacetime manifolds, which extends both the reflection involution and spin flip. On the algebraic side, we define a `higher statistics' action of $O$ on the universal target for invertible field theories, $I\mathbb{Z}$, which extends both complex conjugation and fermion parity $(-1)^F$. We prove that every unitary invertible quantum field theory intertwines these actions., Comment: 6 pages, no figures

Tensor network algorithms can efficiently simulate complex quantum many-body systems by utilizing knowledge of their structure and entanglement. These methodologies have been adapted recently for solving the Navier-Stokes equations, which describe a spectrum of fluid phenomena, from the aerodynamics of vehicles to weather patterns. Within this quantum-inspired paradigm, velocity is encoded as matrix product states (MPS), effectively harnessing the analogy between interscale correlations of fluid dynamics and entanglement in quantum many-body physics. This particular tensor structure is also called quantics tensor train (QTT). By utilizing NVIDIA's cuQuantum library to perform parallel tensor computations on GPUs, our adaptation speeds up simulations by up to 12.1 times. This allows us to study the algorithm in terms of its applicability, scalability, and performance. By simulating two qualitatively different but commonly encountered 2D flow problems at high Reynolds numbers up to $1\times10^7$ using a fourth-order time stepping scheme, we find that the algorithm has a potential advantage over direct numerical simulations in the turbulent regime as the requirements for grid resolution increase drastically. In addition, we derive the scaling $\chi=\mathcal{O}(\text{poly}(1/\epsilon))$ for the maximum bond dimension $\chi$ of MPS representing turbulent flow fields, with an error $\epsilon$, based on the spectral distribution of turbulent kinetic energy. Our findings motivate further exploration of related quantum algorithms and other tensor network methods.

In this short note, we classify linear categorified open topological field theories in dimension two by pivotal Grothendieck-Verdier categories. 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., Comment: 14 pages, some diagrams

We present a coherent definition of dagger $(\infty,n)$-category in terms of equivariance data trivialized on parts of the category. Our main example is the bordism higher category $\mathbf{Bord}_{n}^X$. This allows us to define a reflection-positive topological quantum field theory to be a higher dagger functor from $\mathbf{Bord}_{n}^X$ to some target higher dagger category $\mathcal{C}$. Our definitions have a tunable parameter: a group $G$ acting on the $(\infty,1)$-category $\mathbf{Cat}_{(\infty,n)}$ of $(\infty,n)$-categories. Different choices for $G$ accommodate different flavours of higher dagger structure; the universal choice is $G = \operatorname{Aut}(\mathbf{Cat}_{(\infty,n)}) = (\mathbb{Z}/2\mathbb{Z})^n$, which implements dagger involutions on all levels of morphisms. The Stratified Cobordism Hypothesis suggests that there should be a map $\mathrm{PL}(n) \to \operatorname{Aut}(\mathbf{AdjCat}_{(\infty,n)})$, where $\mathrm{PL}(n)$ is the group of piecewise-linear automorphisms of $\mathbb{R}^n$ and $\mathbf{AdjCat}_{(\infty,n)}$ the $(\infty,1)$-category of $(\infty,n)$-categories with all adjoints; we conjecture more strongly that $\operatorname{Aut}(\mathbf{AdjCat}_{(\infty,n)}) \cong \mathrm{PL}(n)$. Based on this conjecture we propose a notion of dagger $(\infty,n)$-category with unitary duality or $\mathrm{PL}(n)$-dagger category. We outline how to construct a $\mathrm{PL}(n)$-dagger structure on the fully-extended bordism $(\infty,n)$-category $\mathbf{Bord}_n^X$ for any stable tangential structure $X$; our outline restricts to a rigorous construction of a coherent dagger structure on the unextended bordism $(\infty,1)$-category $\mathbf{Bord}_{n,n-1}^X$. The article is a report on the results of a workshop held in Summer 2023, and is intended as a sketch of the big picture and an invitation for more thorough development., Comment: 18 pages; v2 corrected definition of dagger $(\infty,n)$-category with unitary duality

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 $Z(\mathcal{C})$., Comment: v1: 33 pages, 5 figures

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 kernel of the mapping class group acts 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 group., Comment: 20 pages, 2 figures

We develop a general theory of 3-dimensional ``orbifold completion'', to describe (generalised) orbifolds of topological quantum field theories as well as all their defects. Given a semistrict 3-category $\mathcal{T}$ with adjoints for all 1- and 2-morphisms (more precisely, a Gray category with duals), we construct the 3-category $\mathcal{T}_{\textrm{orb}}$ as a Morita category of certain $E_1$-algebras in $\mathcal{T}$ which encode triangulation invariance. We prove that in $\mathcal{T}_{\textrm{orb}}$ again all 1- and 2-morphisms have adjoints, that it contains $\mathcal{T}$ as a full subcategory, and we argue, but do not prove, that it satisfies a universal property which implies $(\mathcal{T}_{\textrm{orb}})_{\textrm{orb}} \cong \mathcal{T}_{\textrm{orb}}$. This is a categorification of the work in [CR]. Orbifold completion by design allows us to lift the orbifold construction from closed TQFT to the much richer world of defect TQFTs. We illustrate this by constructing a universal 3-dimensional state sum model with all defects from first principles, and we explain how recent work on defects between Witt equivalent Reshetikhin--Turaev theories naturally appears as a special case of orbifold completion., Comment: 58 pages, many diagrams and figures, v2: references, clarifications and improvements added

The simulation of many industrially relevant physical processes can be executed up to exponentially faster using quantum algorithms. However, this speedup can only be leveraged if the data input and output of the simulation can be implemented efficiently. While we show that recent advancements for optimal state preparation can effectively solve the problem of data input at a moderate cost of ancillary qubits in many cases, the output problem can provably not be solved efficiently in general. By acknowledging that many simulation problems arise only as a subproblem of a larger optimization problem in many practical applications however, we identify and define a class of practically relevant problems that does not suffer from the output problem: Quantum Simulation-based Optimization (QuSO). QuSO represents optimization problems whose objective function and/or constraints depend on summary statistic information on the result of a simulation, i.e., information that can be efficiently extracted from a quantum state vector. In this article, we focus on the LinQuSO subclass of QuSO, which is characterized by the linearity of the simulation problem, i.e., the simulation problem can be formulated as a system of linear equations. By cleverly combining the quantum singular value transformation (QSVT) with the quantum approximate optimization algorithm (QAOA), we prove that a large subgroup of LinQuSO problems can be solved with up to exponential quantum speedups with regards to their simulation component. Finally, we present two practically relevant use cases that fall within this subgroup of QuSO problems., Comment: 24 pages, 12 figure, completely refactored and formalized version with key new isights

We give a complete classification of topological field theories with reflection structure and spin-statistics in one and two spacetime dimensions. Our answers can be naturally expressed in terms of an internal fermionic symmetry group $G$ which is different from the spacetime structure group. Fermionic groups encode symmetries of systems with fermions and time reversing symmetries. We show that 1-dimensional topological field theories with reflection structure and spin-statistics are classified by finite dimensional hermitian representations of $G$. In spacetime dimension two we give a classification in terms strongly $G$-graded stellar Frobenius algebras. Our proofs are based on the cobordism hypothesis. Along the way, we develop some useful tools for the computation of homotopy fixed points of 2-group actions on bicategories., Comment: 74+56 pages, 6 figures

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., Comment: 21 pages, diagrams partly in color; v2: minor changes, Cor. 3.1 strengthened; v3: 23 pages, minor changes in response to reports

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., Comment: 27 pages, 3 figures; v3: some changes throughout, mostly additional explanations in response to reports

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 $\mathbb{S}^1 \times \mathbb{D}^2$., Comment: v1: 12 pages; v2: minor changes

For a finite group $D$, we study categorical factorisation homology on oriented surfaces equipped with principal $D$-bundles, which `integrates' a (linear) balanced braided category $\mathcal{A}$ with $D$-action over those surfaces. For surfaces with at least one boundary component, we identify the value of factorisation homology with the category of modules over an explicit algebra in $\mathcal{A}$, extending the work of Ben-Zvi, Brochier and Jordan to surfaces with $D$-bundles. Furthermore, we show that the value of factorisation homology on annuli, boundary conditions, and point defects can be described in terms of equivariant representation theory. Our main example comes from an action of Dynkin diagram automorphisms on representation categories of quantum groups. We show that in this case factorisation homology gives rise to a quantisation of the moduli space of flat twisted bundles., Comment: 39 pages, 14 figures, comments welcome; references for twisted bundles updated

We characterize cyclic algebras over the associative and the framed little 2-disks operad in any symmetric monoidal bicategory. The cyclicity is appropriately treated in a coherent way, i.e. up to coherent isomorphism. When the symmetric monoidal bicategory is specified to be a certain symmetric monoidal bicategory of linear categories subject to finiteness conditions, we prove that cyclic associative and cyclic framed little 2-disks algebras, respectively, are equivalent to pivotal Grothendieck-Verdier categories and ribbon Grothendieck-Verdier categories, a type of category that was introduced by Boyarchenko-Drinfeld based on Barr's notion of a $\star$-autonomous category. We use these results and Costello's modular envelope construction to obtain two applications to quantum topology: I) We extract a consistent system of handlebody group representations from any ribbon Grothendieck-Verdier category inside a certain symmetric monoidal bicategory of linear categories and show that this generalizes the handlebody part of Lyubashenko's mapping class group representations. II) We establish a Grothendieck-Verdier duality for the category extracted from a modular functor by evaluation on the circle (without any assumption on semisimplicity), thereby generalizing results of Tillmann and Bakalov-Kirillov., Comment: 66 pages, lots of figures and diagrams; v3: minor changes in terminology to be consistent with follow-up papers (ribbon GV instead of balanced braided GV, auxiliary notion of self-dual algebra), comment on invertibility of maps of modular algebras included, minor changes in the presentation in several places; v4: some changes following comments by the referee, to appear in Quart. J. Math

We study bundle gerbes on manifolds $M$ that carry an action of a connected Lie group $G$. We show that these data give rise to a smooth 2-group extension of $G$ by the smooth 2-group of hermitean line bundles on $M$. This 2-group extension classifies equivariant structures on the bundle gerbe, and its non-triviality poses an obstruction to the existence of equivariant structures. We present a new global approach to the parallel transport of a bundle gerbe with connection, and use it to give an alternative construction of this smooth 2-group extension in terms of a homotopy-coherent version of the associated bundle construction. We apply our results to give new descriptions of nonassociative magnetic translations in quantum mechanics and the Faddeev-Mickelsson-Shatashvili anomaly in quantum field theory. We also propose a definition of smooth string 2-group models within our geometric framework. Starting from a basic gerbe on a compact simply-connected Lie group $G$, we prove that the smooth 2-group extensions of $G$ arising from our construction provide new models for the string group of $G$., Comment: 79 pages, 2 figures; v2: minor corrections, comments and references added; Final version to be published in Communications in Mathematical Physics

We prove a decomposition formula for the dimensional reduction of an extended topological field theory that arises as an orbifold of an equivariant topological field theory. Our decomposition formula can be expressed in terms of a categorification of the integral with respect to groupoid cardinality. The application of our result to topological field theories of Dijkgraaf-Witten type proves a recent conjecture of Qiu-Wang., Comment: 12 pages; very small changes, to appear in Bull. London Math. Soc

We develop a general framework for the description of anomalies using extended functorial field theories extending previous work by Freed and Monnier. In this framework, anomalies are described by invertible field theories in one dimension higher and anomalous field theories live on their boundaries. We provide precise mathematical definitions for all concepts involved using the language of symmetric monoidal bicategories. In particular, field theories with anomalies will be described by symmetric monoidal transformations. The use of higher categorical concepts is necessary to capture the Hamiltonian picture of anomalies. The relation to the path integral and the Hamiltonian description of anomalies will be explained in detail. Furthermore, we discuss anomaly inflow in detail. We apply the general framework to the parity anomaly in fermionic systems coupled to background gauge and gravitational fields on odd-dimensional spacetimes. We use the extension of the Atiyah-Patodi-Singer index theorem to manifolds with corners due to Loya and Melrose to explicitly construct an extended invertible field theory encoding the anomaly. This allows us to compute explicitly the 2-cocycle of the projective representation of the gauge symmetry on the quantum state space. As a second application, we study discrete symmetries of Dijkgraaf-Witten theories and their gauging. Non-abelian group cohomology is used to describe discrete symmetries and we derive concrete conditions for such a symmetry to admit 't Hooft anomalies in terms of the Lyndon-Hochschild-Serre spectral sequence. We give an explicit realization of a discrete gauge theory with 't Hooft anomaly as a state on the boundary of a higher-dimensional Dijkgraaf-Witten theory., Comment: 187 pages, PhD Thesis based on arXiv:1709.03860, arXiv:1802.10455 and arXiv:1811.05446

We describe discrete symmetries of two-dimensional Yang-Mills theory with gauge group $G$ associated to 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., Comment: 51 pages, 1 table, lots of figures; v2: references added; Final version to be published in Letters in Mathematical Physics

These are notes from an informal mini-course on factorization homology, infinity-categories, and topological field theories. The target audience was imagined to be graduate students who are not homotopy theorists., Comment: Preprint of a book of the same title to be published 2021 by SpringerBriefs in Mathematical Physics, reproduced with permission of Springer International Publishing

Hurwitz spaces are homotopy quotients of the braid group action on the moduli space of principal bundles over a punctured plane. By considering a certain model for this homotopy quotient we build an aspherical topological operad that we call the little bundles operad. As our main result, we describe this operad as a groupoid-valued operad in terms of generators and relations and prove that the categorical little bundles algebras are precisely Turaev's crossed categories. Moreover, we prove that the evaluation on the circle of a homotopical two-dimensional equivariant topolological field theory yields a little bundles algebra up to coherent homotopy., Comment: 33 pages, 6 figures, lots of diagrams; small changes; final version to appear in Algebr. Geom. Topol

We study discrete symmetries of Dijkgraaf-Witten theories and their gauging in the framework of (extended) functorial quantum field theory. Non-abelian group cohomology is used to describe discrete symmetries and we derive concrete conditions for such a symmetry to admit 't Hooft anomalies in terms of the Lyndon-Hochschild-Serre spectral sequence. We give an explicit realization of a discrete gauge theory with 't Hooft anomaly as a state on the boundary of a higher-dimensional Dijkgraaf-Witten theory. This allows us to calculate the 2-cocycle twisting the projective representation of physical symmetries via transgression. We present a general discussion of the bulk-boundary correspondence at the level of partition functions and state spaces, which we make explicit for discrete gauge theories., Comment: 46 pages, 1 figure; v2: minor corrections and clarifying comments added, references updated; Final version to appear in Communications in Mathematical Physics