Monday, June 30

8:00–9:00 am

MPIM

Bruno Nachtergaele, UC Davis

Title: Ground states of quantum lattice systems: Quantum Lattice Systems: observables, dynamics, ground states, GNS representation, ground state gap, examples

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Michael Hopkins, Harvard

10:30–10:45 am

break

10:45–11:45 am

MPIM

Pieter Naajkens, Cardiff

Title: Introduction to superselection sector theory: Motivation and introduction of basic setting

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Alexei Kitaev, Caltech

Title: Local definitions of gapped Hamiltonians and topological and invertible states

Tuesday, July 1

8:00–9:00 am

MPIM

Bruno Nachtergaele, UC Davis

Title: Ground states of quantum lattice systems: Quasilocality: almost local observables and interactions, Lieb-Robinson bounds, quasi-adiabatic evolution, stability I

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Michael Hopkins, Harvard

10:30–10:45 am

break

10:45–11:45 am

MPIM

Pieter Naajkens, Cardiff

Title: Introduction to superselection sector theory: Building the braided (fusion) category of superselection sectors I

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Alexei Kitaev, Caltech

Wednesday, July 2

8:00–9:00 am

MPIM

Bruno Nachtergaele, UC Davis

Title: Ground states of quantum lattice systems: Quantum Entanglement in many-body systems: short-range entangled states, topological entanglement, stability II

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Michael Hopkins, Harvard

10:30–10:45 am

break

10:45–11:45 am

MPIM

Pieter Naajkens, Cardiff

Title: Introduction to superselection sector theory: Building the braided (fusion) category of superselection sectors II

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Alexei Kitaev, Caltech

Title: Local definitions of gapped Hamiltonians and topological and invertible states

Thursday, July 3

8:00–9:00 am

MPIM

Bruno Nachtergaele, UC Davis

Title: Ground states of quantum lattice systems: Quantum Phase Diagrams: order parameters, topological invariants, examples

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Michael Hopkins, Harvard

10:30–10:45 am

break

10:45–11:45 am

MPIM

Pieter Naajkens, Cardiff

Title: Introduction to superselection sector theory: Classification of phases and long-range entanglement

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Alexei Kitaev, Caltech

Title: Local definitions of gapped Hamiltonians and topological and invertible states

No talks Friday July 4

Monday July 7

8:00–9:00 am

MPIM

Jackson van Dyke, UT Austin

Title: Moduli spaces of projective 3d TQFTs

Abstract: A gapped quantum system is well-approximated at low energy by a projective topological field theory. Therefore questions concerning the classification, symmetries, and anomalies of gapped quantum systems can be reinterpreted via the homotopy theory of the moduli space of such theories. I will describe a moduli space of 3-dimensional TQFTs, and the sense in which its homotopy theory informs us about the low energy behavior of gapped systems in 2+1 dimensions. This moduli space depends on the fixed target category: Explicitly, it is built from the classifying spaces of higher groups of automorphisms of ribbon categories. The emphasis will be on target categories which have convenient algebraic features, yet are analytically robust enough to allow for boundary/relative theories defined in terms of unitary representations on topological vector spaces.

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Lukasz Fidkowski, University of Washington

Title: Non-invertible bosonic chiral symmetry on the lattice

Abstract: We construct a Hamiltonian lattice realization of the non-invertible chiral symmetry that mimics an axial rotation at a rational angle in a U(1) gauge theory with bosonic charged matter.  We provide a heuristic argument that this setup allows a symmetric Hamiltonian which flows, at low energies, to a known field theory with this symmetry.

10:30–10:45 am

break

10:45–11:45 am

MPIM

Matthias Ludewig, University of Greifswald

Title: Generalized Kitaev Pairings and Higher Berry curvature in coarse geometry

Abstract: In Appendix C of his “Anyons” paper, Kitaev introduced the notion of a “generalized Chern number” for a 2-dimensional system by diving the system in three ordered parts and measuring a signed rotational flux. This construction has since been used by several authors to measure topological non-triviality of a physical system. In recent work with Guo Chuan Thiang, we observe that the recipe provided by Kitaev can be interpreted in coarse geometry as the pairing of a K-theory class with a coarse cohomology class. A corresponding index theorem then provides a proof that the set of values of this “Kitaev pairing” is always quantized, as already argued by Kitaev. In our work, we generalize Kitaev’s definition and the corresponding quantization result to arbitrary dimensions. By replacing a single Hamiltonian with a whole family of Hamiltonians (parametrized by a space X), we recover and extend the construction of “Higher Berry curvatures” by Kapustin and Spodyneiko. Given a coarse cohomology class, we obtain a characteristic class on the parameter space X, which is integral whenever integrated against a cycle in X that lies in the image of the homological Chern character (so, in particular, spheres in X).

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Theo Johnson-Freyd, Perimeter Institute

Tuesday, July 8

8:00–9:00 am

MPIM

David Reutter, University of Hamburg

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Agnes Beaudry, UC Boulder

10:30–10:45 am

break

10:45–11:45 am

MPIM

João Faria Martins, University of Leeds

Title: A categorification of Quinn’s finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from discrete higher gauge theory

Abstract: Quinn’s Finite Total Homotopy TQFT is a topological quantum field theory defined for any dimension n of space, depending on the choice of a homotopy finite space B. For instance, B can be the classifying space of a finite group or a finite 2-group.
In this talk, I will report on recent joint work with Tim Porter on once-extended versions of Quinn’s Finite Total Homotopy TQFT, taking values in the symmetric monoidal bicategory of groupoids, linear profunctors, and natural transformations between linear profunctors. The construction works in all dimensions, yielding (0,1,2)-, (1,2,3)-, and (2,3,4)-extended TQFTs, given a homotopy finite space B. I will  shown how to compute these once-extended TQFTs when B is the classifying space of a homotopy 2-type, represented by a crossed module of groups.
Reference: Faria Martins J, Porter T: “A categorification of Quinn’s finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids.” arXiv:2301.02491 [math.CT]

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Emil Prodan, Yeshiva University

Title: Mapping the landscape of frustration-free models

Abstract: Frustration-free models are of great interest because they are amenable to specialized techniques and their understanding is more complete among the general quantum spin models. In this talk, I will establish an almost bijective relation between frustration-free families of projections and a subclass of hereditary subalgebras defined by an intrinsic property. This relation sets further synergies between frustration-free models and open projections in double duals, and subsets of pure states spaces. These connections enable a better understanding of the class of frustration-free models. For example, the open projections in the double dual derived from frustration-free models is dense in the norm-topology in the space of generic open projections, thus assuring us that, for many purposes, we can choose to work with frustration-free models without losing generality. Furthermore, the Cuntz semigroup, originally designed to classify the positive elements of C*-algebra, has been proven to also classify the open projections. Given the mentioned connections, we now have a new device to investigate the ground states of quantum spin models.

Wednesday, July 9

8:00–9:00 am

MPIM

Alexander Schenkel, University of Nottingham

Title: C*-categorical prefactorization algebras for superselection sectors and topological order

Abstract: I will present a geometric framework to encode the algebraic structures on the category of superselection sectors of an algebraic quantum field theory on the n-dimensional lattice Z^n. I will show that, under certain assumptions which are implied by Haag duality, the monoidal C*-categories of localized superselection sectors carry the structure of a locally constant prefactorization algebra over the category of cone-shaped subsets of Z^n. Employing techniques from higher algebra, one extracts from this datum an underlying locally constant prefactorization algebra defined on open disks in the cylinder R^1 x S^{n-1}. While the sphere S^{n-1} arises geometrically as the angular coordinates of cones, the origin of the line R^1 is analytic and rooted in Haag duality. The usual braided (for n=2) or symmetric (for n>2) monoidal C*-categories of superselection sectors are recovered by removing a point of the sphere and using the equivalence between E_n-algebras and locally constant prefactorization algebras defined on open disks in R^n. The non-trivial homotopy groups of spheres induce additional algebraic structures on these E_n-monoidal C*-categories, which in the simplest case of Z^2 is given by a braided monoidal self-equivalence arising geometrically as a kind of ‘holonomy’ around the circle S^1.
This talk is based on joint work with Marco Benini, Victor Carmona and Pieter Naaijkens.

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Constantin Teleman, UC Berkeley

10:30–10:45 am

break

10:45–11:45 am

MPIM

Nils Carqueville, University of Vienna

Title: Gauging categorical symmetries

Abstract: Orbifold data are categorical symmetries that can be gauged in oriented defect topological quantum field theories. We review the general construction and apply it to 2-group symmetries of 3-dimensional TQFTs; upon further specialisation this leads to equivariantisation of G-crossed braided fusion categories. We also describe a proposal, via higher dagger categories, to gauging categorical symmetries in the context of other tangential structures. This is based on separate projects with Benjamin Haake and Tim Lüders.

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Nikita Sopenko, IAS

Thursday, July 10

8:00–9:00 am

MPIM

Ilka Brunner, Ludwig-Maximilians University of Munich

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

David Penneys, Ohio State

10:30–10:45 am

break

10:45–11:45 am

MPIM

Christoph Schweigert, University of Hamburg

Title: Tensor network states: a topological field theory perspective.

Abstract: Projected entangled pair states (PEPS) and matrix product operators (MPO) are standard tools in quantum information theory and quantum many-body physics. We explain how to understand them in terms of Turaev-Viro models on manifolds with boundary. We then sketch how a recently developed categorical Morita theory for spherical module categories can be used to find generalizations of the standard PEPS tensors.

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Greg Moore, Rutgers

Friday, July 11

8:00–9:00 am

CMSA

Markus Pflaum, UC Boulder

Title: A tour d’horizon through homotopical aspects of C*-algebraic quantum spin systems

Abstract: In the talk I report on joint work with Beaudry, Hermele, Moreno, Qi and Spiegel, where a homotopy theoretic framework for studying state spaces of quantum lattice spin systems has been introduced using the language of C*-algebraic quantum mechanics. First some old and new results about the state space of the quasi-local algebra of a quantum lattice spin system when endowed with either the natural metric topology or the weak* topology will be presented. Switching to the algebraic topological side, the homotopy groups of the unitary group of a UHF algebra will then be determined and it will be indicated that the pure state space of any UHF algebra in the weak* topology is weakly contractible. In addition, I will show at the example of non-commutative tori that also in the case of a not commutative C*-algebra, the homotopy type of the state space endowed with the weak* topology can be non-trivial and is neither deformation nor Morita invariant. Finally, I indicate how such tools together with methods from higher homotopy theory such as E_infinity spaces may lead to a framework for constructing Kitaev’s loop-spectrum of bosonic invertible gapped phases of matter.

9:00–9:30 am

Breakfast break

9:30–10:30 am

MPIM

Speed Talks

Hataishi, Leung, Oyama, Stewart, Virelizier, Yu, Wasserman

10:30–10:45 am

break

10:45–11:45 am

CMSA

Speed Talks

Lukasz Fidkowski, University of Washington

Ben Gripaios, University of Cambridge

Eric Roon, Michigan State University

Bowen Shi, University of Illinois Urbana-Champaign

Qiyu Zhang, Yang Institute of Theoretical Physics/Stony Brook

Roman Geiko, UCLA