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Higher Categorical Tools for Quantum Phases of Matter

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Monday, March 18, 20248:30 AM RegistrationRegistration8:30 AM - 9:00 AM9:00 AM Opening RemarksOpening Remarks9:00 AM - 9:15 AMRoom: PI/4-405 - Bob Room9:15 AM Analogies between QFT and lattice systems - Anton Kapustin (California Institute of Technology)Analogies between QFT and lattice systems
- Anton Kapustin (California Institute of Technology)

9:15 AM - 10:15 AMRoom: PI/4-405 - Bob Room I discuss some analogies between the Haag-Kastler approach to QFT and quantum statistical mechanics of lattice systems. As an illustrative example, I consider the interpretation of the Hall conductance of gapped 2d lattice systems as an obstruction to gauging a global symmetry of a gapped state. I argue that in order to define a proper analog of the net of algebras of observables one needs to study a category of subsets of the lattice equipped with a natural Grothendieck topology.10:15 AM BreakBreak10:15 AM - 10:45 AMRoom: PI/1-124 - Lower Bistro10:45 AM Models of anyons with symmetry: a bulk-boundary correspondence - Fiona Burnell (University of Minnesota)Models of anyons with symmetry: a bulk-boundary correspondence- Fiona Burnell (University of Minnesota)

10:45 AM - 11:45 AMRoom: PI/4-405 - Bob Room I will describe models with on-site symmetry that permutes anyons with non-trivial mutual statistics, and show that the action of this symmetry on the boundary can effectively be that of a non-invertible symmetry such as Kramers-Wannier duality. I will sketch some implications of this for anomalies in non-invertible symmetries. Finally, I will introduce a construction (based on idempotent completion) that allows us to realize all possible anyon permuting symmetries of a given topological order in an on-site way.12:00 PM LunchLunch12:00 PM - 1:00 PMRoom: PI/2-251 - Upper Bistro1:00 PM Free DiscussionFree Discussion1:00 PM - 2:00 PM2:00 PM Twisted Tools for (Untwisted) Quantum Field Theory - Justin Kulp (Stony Brook University)Twisted Tools for (Untwisted) Quantum Field Theory- Justin Kulp (Stony Brook University)

2:00 PM - 3:00 PMRoom: PI/4-405 - Bob Room3:00 PM BreakBreak3:00 PM - 3:30 PMRoom: PI/1-124 - Lower Bistro3:30 PM Gong ShowGong Show3:30 PM - 5:00 PMRoom: PI/4-405 - Bob Room -
Tuesday, March 19, 20249:15 AM Quantum double models and Dijkgraaf-Witten theory with defects - Catherine Meusburger (FAU Erlangen-Nürnberg)Quantum double models and Dijkgraaf-Witten theory with defects
- Catherine Meusburger (FAU Erlangen-Nürnberg)

9:15 AM - 10:15 AMRoom: PI/4-405 - Bob Room We use 3d defect TQFTs and state sum models with defects to give a gauge theoretical formulation of Kitaev's quantum double model (for a finite group) and (untwisted) Dijkgraaf-Witten TQFT with defects. This leads to a simple description in terms of embedding quivers, groupoids and their representations. Defect Dijkgraaf-Witten TQFTs is then formulated in terms of spans of groupoids and their representations. This is work in progress with João Faría-Martins, University of Leeds.10:15 AM BreakBreak10:15 AM - 10:45 AMRoom: PI/1-124 - Lower Bistro10:45 AM Topological sectors in quantum lattice models - Clement Delcamp (Institut des Hautes Études Scientifiques (IHES))Topological sectors in quantum lattice models- Clement Delcamp (Institut des Hautes Études Scientifiques (IHES))

10:45 AM - 11:45 AMRoom: PI/4-405 - Bob Room11:45 AM Group PhotoGroup Photo11:45 AM - 12:00 PMRoom: PI/4-405 - Bob Room12:00 PM LunchLunch12:00 PM - 1:00 PMRoom: PI/2-251 - Upper Bistro1:00 PM Free DiscussionFree Discussion1:00 PM - 2:00 PM2:00 PM Douglas-Reutter 4d TQFT as a generalised orbifold - Vincentas Mulevičius (Vilnius University)Douglas-Reutter 4d TQFT as a generalised orbifold- Vincentas Mulevičius (Vilnius University)

2:00 PM - 3:00 PMRoom: PI/4-405 - Bob Room The state-sum invariants of 4d manifolds obtained from spherical fusion 2-categories due to Douglas-Reutter offer an exciting entrypoint to the study of 4d TQFTs. In this talk we will argue that these invariants arise from a TQFT, obtained by filling the trivial 4d TQFT with a defect foam. Such construction is known as a generalised orbifold, the Turaev-Viro-Barrett-Westbury (i.e. 3d state-sum) models are also known to arise in this way from the defects in the trivial 3d TQFT (a result by Carqueville-Runkel-Schaumann). Advantages of this point of view offer e.g. realisations of state-spaces, examples of domain walls and commuting-projector realisations of (3+1)-dimensional topological phases. Based on a joint project with Nils Carqueville and Lukas Müller.3:00 PM BreakBreak3:00 PM - 3:30 PMRoom: PI/1-124 - Lower Bistro3:30 PM Organized DiscussionOrganized Discussion3:30 PM - 5:00 PMRoom: PI/3-394 - Skyroom -
Wednesday, March 20, 20249:15 AM Weak Hopf symmetric tensor networks - Andras Molnar (University of Vienna)Weak Hopf symmetric tensor networks
- Andras Molnar (University of Vienna)

9:15 AM - 10:15 AMRoom: PI/4-405 - Bob Room10:15 AM BreakBreak10:15 AM - 10:45 AMRoom: PI/1-124 - Lower Bistro10:45 AM Hierarchy construction for fractional quantum hall states via condensable algebras - Carolyn Zhang (Harvard University)Hierarchy construction for fractional quantum hall states via condensable algebras- Carolyn Zhang (Harvard University)

10:45 AM - 11:45 AMRoom: PI/4-405 - Bob Room12:00 PM LunchLunch12:00 PM - 1:00 PMRoom: PI/2-251 - Upper Bistro1:00 PM Free DiscussionFree Discussion1:00 PM - 2:00 PM2:00 PM Zesting topological order and symmetry-enriched topological order in (2+1)D - Colleen Delaney (University of California, Berkeley)Zesting topological order and symmetry-enriched topological order in (2+1)D- Colleen Delaney (University of California, Berkeley)

2:00 PM - 3:00 PMRoom: PI/4-405 - Bob Room Zesting is a construction that takes a (2+1)D topological order and produces a new one by changing the fusion rules of its anyons. We'll discuss properties of zesting from a physical and computational point of view and explain how the theory produces some closely related families of topological orders, like Kitaev's 16-fold way and modular isotopes. Time permitting we'll cover a generalization of zesting to symmetry-enriched topological order and comment on connections to fusion 2-categories.3:00 PM BreakBreak3:00 PM - 3:30 PMRoom: PI/1-124 - Lower Bistro3:30 PM Motion Groupoids - Fiona Torzewska (University of Bristol)Motion Groupoids- Fiona Torzewska (University of Bristol)

3:30 PM - 4:30 PMRoom: PI/4-405 - Bob Room The braiding statistics of point particles in 2-dimensional topological phases are given by representations of the braid groups. One approach to the study of generalised particles in topological phases, loop particles in 3-dimensions for example, is to generalise (some of) the several different realisations of the braid group. In this talk I will construct for each manifold M its motion groupoid $Mot_M$, whose object class is the power set of M. I will discuss several different, but equivalent, quotients on motions leading to the motion groupoid. In particular that the quotient used in the construction $Mot_M$ can be formulated entirely in terms of a level preserving isotopy relation on the trajectories of objects under flows -- worldlines (e.g. monotonic `tangles'). I will also give a construction of a mapping class groupoid $MCG_M$ associated to a manifold M with the same object class. For each manifold M I will construct a functor $F \colon Mot_M \to MCG_M$, and prove that this is an isomorphism if $\pi_0$ and $\pi_1$ of the appropriate space of self-homeomorphisms of M is trivial. In particular there is an isomorphism in the physically important case $M=[0,1]^n$ with fixed boundary, for any $n\in\mathbb{N}$. I will discuss several examples throughout.6:00 PM BanquetBanquet6:00 PM - 8:00 PMRoom: PI/2-251 - Upper Bistro -
Thursday, March 21, 20249:15 AM Categorical Aspects of Symmetry in Fermionic Systems - Kantaro Ohmori (University of Tokyo)Categorical Aspects of Symmetry in Fermionic Systems
- Kantaro Ohmori (University of Tokyo)

9:15 AM - 10:15 AMRoom: PI/4-405 - Bob Room In this talk we explore the symmetry in 1+1d fermionic systems from a category theory perspective. We argue that It requires additional structure than a tensor category over sVect, and rather captured by a fusion category equipped with a braided central functor from a specific braided fusion category, dependent on the chiral central charge. The same data defines a 1-morphism in the 4-category of braided tensor categories, which lead us to a 4d-3d-2d picture via cobordism hypothesis. This talk is based on an ongoing work with K. Inamura.10:15 AM BreakBreak10:15 AM - 10:45 AMRoom: PI/1-124 - Lower Bistro10:45 AM Bulk Excitations of Invertible Phases - Wenjie Ji (California Institute of Technology)Bulk Excitations of Invertible Phases- Wenjie Ji (California Institute of Technology)

10:45 AM - 11:45 AMRoom: PI/4-405 - Bob Room Recent development in the study of topological defects highlights the importance of understanding the multi-dimensional structure of bulk excitations inside a quantum system. When the bulk ground state is trivial, i.e. a product state, excitations on top of it are decoupled from each other and correspond to lower-dimensional phases and their defects within. In this talk, we will expand the discussion to invertible phases and study the bulk excitations in, for example, SPT phases, majorana chain, p + ip superconductor etc. We find that there is a one-to-one correspondence between bulk excitations inside a nontrivial invertible phase and those in a product state. For SPT phases, this can be shown using the symmetric Quantum Cellular Automaton that maps from the product state to the SPT state. In this talk, we will demonstrate the correspondence for more general invertible phases, those realizable using the Symmetry Topological-Field-Theory construction. Our demonstration is built upon a key property of topological orders: certain gapped boundary conditions of a topological bulk state have only relative distinctions but no absolute ones - they cannot be distinguished by any local experiments near the boundary. This talk is based on a work under preparation with David Stephen and Xie Chen.12:00 PM LunchLunch12:00 PM - 1:00 PMRoom: PI/2-251 - Upper Bistro1:00 PM Free DiscussionFree Discussion1:00 PM - 2:00 PM2:00 PM Quantum homotopy groups - Theo Johnson-Freyd (Perimeter Institute / Dalhousie University)Quantum homotopy groups- Theo Johnson-Freyd (Perimeter Institute / Dalhousie University)

2:00 PM - 3:00 PMRoom: PI/4-405 - Bob Room An *open-closed tqft* is a tqft with a choice of boundary condition. Example: the sigma model for a sufficiently finite space, with its Neumann boundary. Slogan: every open-closed tqft is (sigma model, Neumann boundary) for some “quantum space”. In this talk, I will construct homotopy groups for every such “quantum space” (and recover usual homotopy groups). More precisely, these “groups” are Hopf in some category. Given a “quantum fibre bundle” (a relative open-closed tqft), I will construct a Puppe long exact sequence. Retracts in 3-categories and a higher Beck-Chevalley condition will make appearances. This project is joint work in progress with David Reutter.3:00 PM BreakBreak3:00 PM - 3:30 PMRoom: PI/1-124 - Lower Bistro3:30 PM Organized DiscussionOrganized Discussion3:30 PM - 5:00 PMRoom: PI/3-394 - Skyroom -
Friday, March 22, 20249:15 AM Once extended unitary topological field theories - Luuk Stehouwer (Dalhousie University)Once extended unitary topological field theories
- Luuk Stehouwer (Dalhousie University)

9:15 AM - 10:15 AMRoom: PI/4-405 - Bob Room Motivated by the dagger category of Hilbert spaces, we explore the mathematical axiomatization of unitary topological field theory (TFT) using dagger categories. Recent advancements have elucidated the structure of higher dagger categories, paving the way for a precise definition of extended unitary TFTs. I will present an explicit formulation of once extended TFTs utilizing various versions of dagger bicategories. This framework enables a complete classification of unitary extended two-dimensional TFTs with arbitrary symmetry groups in terms of their unitary 2-representations. This is joint work in progress with Lukas Muller.10:15 AM BreakBreak10:15 AM - 10:45 AMRoom: PI/1-124 - Lower Bistro10:45 AM Bosonization and anomalies of 3d fermionic topological orders - Arun Debray (Purdue University)Bosonization and anomalies of 3d fermionic topological orders- Arun Debray (Purdue University)

10:45 AM - 11:45 AMRoom: PI/4-405 - Bob Room There are two notions of a symmetry of a group G on a 3d topological order (TO): an "algebraic" symmetry, where G acts by automorphisms on the tensor category defining the (TO), and a "field-theoretic" symmetry, where the TFT corresponding to the TO is extended to manifolds with a principal G-bundle. The "field-theoretic" notion is stronger than the "algebraic" one, and the obstruction is sometimes referred to as the anomaly of the TO. The goal of this talk is to discuss a project joint with Weicheng Ye and Matthew Yu on computing these anomalies for fermionic TOs/spin TFTs: we develop a general framework employing Gaiotto-Kapustin's bosonic shadow construction. I will discuss both the mathematical conjectures our framework rests on as well as its use in examples. The Smith long exact sequence appears in our computations.11:45 AM Closing RemarksClosing Remarks11:45 AM - 12:00 PMRoom: PI/4-405 - Bob Room12:00 PM LunchLunch12:00 PM - 1:00 PMRoom: PI/2-251 - Upper Bistro1:00 PM Discussion and CollaborationDiscussion and Collaboration1:00 PM - 4:00 PMRoom: PI/1-118 - Reflecting Lounge4:00 PM Perimeter Institute Friday SocialPerimeter Institute Friday Social4:00 PM - 5:00 PMRoom: PI/1-124 - Lower Bistro