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

Title: Lattice models and topological quantum field theories I

Abstract: This series will cover the relationship between gapped Hamiltonian lattice models and topological quantum field theories, with an emphasis on a conjecture of Kitaev.

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

Abstract: (week 1 lectures) In this series of lectures, I will give an introduction to the operator-algebraic (Doplicher-Haag-Roberts) approach to study the superselection sectors of a (2D) gapped quantum spin system. The sectors have a rich mathematical structure of a braided monoidal category. This category describes all the algebraic properties of the ‘anyons’ or ‘charges’ such quantum spin systems can have. The aim of these lectures is to build up this theory from first principles, using simple examples of topologically ordered models to illustrate the main ideas. If time permits, I will elaborate on how this fits into the larger programme of the classification of gapped phases of matter, and long-range entangled states in particular. No prior knowledge of operator algebras or tensor categories is assumed.

SLIDES (pdf)

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 I

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

Title: Lattice models and topological quantum field theories II

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

SLIDES (pdf)

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 II

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

Title: Lattice models and topological quantum field theories III

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

SLIDES (pdf)

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 III

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

Title: Lattice models and topological quantum field theories IV

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

SLIDES (pdf)

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 IV

No talks Friday July 4

Monday July 7

8:00–9:00 am

MPIM

Jackson van Dyke, TU Munich

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

Constantin Teleman, UC Berkeley

Title: Quantizing homotopy types

Abstract: Kontsevich (90’s) proposed a topological quantization of (sigma-models into) finite homotopy types to top dimensions (d, d+1). Its enhancement to a `fully extended’ TQFT was described later (Freed, Hopkins, Lurie and the speaker) in the target category of iterated algebras. Independently, Chas and Sullivan constructed a (partially defined) 2-dimensional TQFT (d=1) with target compact oriented manifolds.
I will briefly review the features of the finite homotopy theory and its boundary conditions, with particular interest in Dirichlet conditions; their analogue in Chas-Sullivan theory (older work by Blumberg, Cohen and the speaker). Finally, I propose a generalization combining these to a higher-dimensional Chas-Sullivan theory.

Slides (link)

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

Title: Some thoughts about the Kapustin–Kitaev cobordism conjecture

Abstract: In 2013, Kitaev explained that, under some reasonable locality hypotheses, gapped invertible phases of bosonic lattice models in different dimensions are naturally organized into an \Omega-spectrum. The following year, Kapustin conjectured that this spectrum is dual to a Thom spectrum, specifically (smooth) oriented bordism MSO, and that for fermionic lattice models one sees instead the dual to spin bordism. In 2016, Freed and Hopkins proved Kapustin’s conjecture for invertible phases of continuous unitary QFTs valued in an at-the-time conjectural universal target category. Freed and Hopkins put bordism categories into the statement of the problem, by working from the beginning with continuous QFTs. Kapustin’s conjecture for lattice models remains open.
David Reutter and I, in ongoing work in progress, have investigating Kapustin’s conjecture from the perspective of deeper category theory. We have built the universal target category for phases satisfying a finite semisimplicity hypothesis, and we are working on relaxing finite semisimplicity. We can show that any spectrum of invertible finite-semisimple phases will indeed be dual to a Thom spectrum for some topological group G acting on the spectrum of spheres. For example, if one looks just at those bosonic phases which can be topologically condensed from the vacuum, G is almost the (oriented) piecewise linear group, whose Thom spectrum is the bordism spectrum MSPL is the (oriented) *piecewise* smooth manifolds; the difference between MSPL and MSO is only visible in dimensions 7 and above. I say almost because in fact our G is what you would get if you tried to build MSPL, but could only make finitary measurements, which surely is explained by our restriction to condensable semisimple TQFTs. We conjecture that MSPL, rather than MSO, classifies invertible gapped phases of bosonic lattice models.
The general relation between MSPL and topological phases is explained by a certain “surgery exact sequence” for topological phases that mirrors the surgery sequence for MSPL. By studying this sequence, we can also answer the question of which invertible phases admit a gapped boundary condition. In particular that only (the trivial phase and) the Arf–Kervaire invariants admit finite-semisimple gapped boundary conditions.

SLIDES (pdf)

Tuesday, July 8

8:00–9:00 am

MPIM

David Reutter, University of Hamburg

Title: On the categorical spectrum of topological quantum field theories

Abstract: As originally suggested by Kitaev, invertible topological quantum field theories of varying dimensions should assemble into a spectrum/generalized homology theory. A candidate for such a spectrum of invertible TQFTs was proposed by Freed and Hopkins, with the defining property that (isomorphism classes of) n-dimensional invertible TQFTs are completely determined by their partition functions on closed n-manifolds. More generally, not-necessarily-invertible TQFTs should assemble into a ‘categorical spectrum’, an analogue of a spectrum with non-invertible cells at each level. In this talk, I will explain that there exists a unique such categorical spectrum satisfying a list of reasonable assumptions on the collection of (compact/very finite & discrete) TQFTs; one of these assumptions being that its invertibles agree with Freed and Hopkins’ suggestion. I will explain these assumptions, sketch how this categorical spectrum looks like in low-dimensions, outline its construction, and how it may be used to learn about gapped boundaries of anomaly theories in high dimensions. This is based on work in progress with Theo Johnson-Freyd.

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Agnes Beaudry, UC Boulder

Title: An algebraic theory of planon-only fracton orders

Abstract: In this talk, I will describe an algebraic theory for planon-only abelian fracton orders. These are three-dimensional gapped phases with the property that fractional excitations are abelian particles restricted to move in parallel planes. The fusion and statistics data can be identified with a finitely generated module over a Laurent polynomial ring together with a U(1)-valued quadratic form. These systems thus lend themselves to an elegant algebraic theory which we expect will lead to easily computable phase invariants and a classification. As a starting point, we establish a necessary condition for physical realizability, the excitation-detector principle, which I will explain. We conjecture that this criterion is also sufficient for realizability. I will also discuss preliminary classification results.
This talk is based on joint with Michael Hermele, Wilbur Shirley and Evan Wickenden.

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  show 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.

SLIDES (pdf)

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

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

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

Title: Reflection positivity and invertible phases of 2d quantum many-body systems

Abstract: Reflection positivity is a property that is usually taken as an assumption in the classification of topological phases of matter via continuous quantum field theories. For general quantum many-body systems, this property does not hold. This raises the question of whether it somehow emerges in the effective theory from the microscopic description, thereby justifying the field-theoretic approach.
In this talk, I will discuss reflection positivity in the context of invertible phases of two-dimensional lattice systems. I will explain why every such phase admits a reflection-positive representative, and why inverse phases are represented by complex conjugate states. I will also introduce an index that distinguishes these phases and is conjecturally related to the chiral central charge.

Thursday, July 10

8:00–9:00 am

MPIM

Ilka Brunner, Ludwig-Maximilians University of Munich

Title: Defects as functors between phases of Abelian gauged linear sigma models

Abstract: Defects act naturally on boundary conditions, providing functors between D-brane categories. In the context of gauged linear sigma models, one can use defects to transport branes from one phase to another. In this talk, I will show how to construct such defects explicitly.

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

David Penneys, Ohio State

Title: Holography for bulk-boundary local topological order

Abstract: In previous joint work [arXiv:2307.12552] with C. Jones, Naaijkins and Wallick, we introduced local topological order (LTO) axioms for quantum spin systems which allowed us to define a physical boundary manifested by a net of boundary algebras in one dimension lower. This gives a formal setting for topological holography, where the braided tensor category of DHR bimodules of the physical boundary algebra captures the bulk topological order.
In joint work with C. Jones and Naaijkens, we extend the LTO axioms to quantum spin systems equipped with a topological boundary, again producing a physical boundary algebra for the bulk-boundary system, whose category of (topological) boundary DHR bimodules recovers the topological boundary order. We perform this analysis in explicit detail for Levin-Wen and Walker-Wang bulk-boundary systems.
Along the way, we introduce a 2D braided categorical net of algebras built from a unitary braided fusion category (UBFC), which arise as boundary algebras of Walker-Wang models. We consider the canonical state on this braided categorical net corresponding to the standard topological boundary for the Walker-Wang model. Interestingly, in this state, the cone von Neumann algebras are type I with finite dimensional centers, in contrast with the type II and III cone von Neumann algebras from the Levin-Wen models studied in [arXiv:2307.12552]. Their superselection sectors recover the underlying unitary category of our UBFC, and we conjecture the superselection category also captures the fusion and braiding.

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

Title: p-form puzzles

Abstract: It is commonly stated that level k BF theory for a p-form (and a form of complementary dimension) is equivalent to a homotopy sigma model with target space K(A,p) where A is a cyclic group of order k.  Some aspects of this standard statement are puzzling me. I’ll explain what they are. (Perhaps someone in the audience can resolve my puzzles.) Then I’ll revisit the (again standard) electromagnetic duality of p-form electrodynamics. The conclusion will be that a slightly modified version of Ray-Singer torsion is the partition function of an invertible topological field theory.

Friday, July 11
Note: On Friday, there will be separate schedules for Bonn and CMSA.

To view the Bonn schedule, please visit the program page at: https://www.mpim-bonn.mpg.de/qft25

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

9:30–11:00 am

CMSA

Speed Talks

Ben Gripaios, University of Cambridge
Title: Locality and smoothness of QFTs

Carolyn Zhang, Harvard University
Title: SymTFT approach for (non-)invertible symmetries of mixed states

Roman Geiko, UCLA
Title: Omega-spectrum of stabilizer invertible phases

11:00–11:15 am

break

11:15–12:45 pm

CMSA

Speed Talks continued

Eric Roon, Michigan State University
Title: Finitely Correlated States Driven by Topological Dynamics

Dmitri Pavlov, Texas Tech University
Title: The classification of two-dimensional extended conformal field theories

Bowen Shi, University of Illinois Urbana-Champaign
Title: Mathematical Puzzles from the Entanglement Bootstrap: On Immersions and regular homotopy
SLIDES (pdf)