# Schedule for: 17w5167 - Field Theories and Higher Structures in Mathematics and Physics

Arriving in Oaxaca, Mexico on Sunday, June 4 and departing Friday June 9, 2017

Sunday, June 4 | |
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14:00 - 23:59 | Check-in begins (Front desk at your assigned hotel) |

19:30 - 22:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

20:30 - 21:30 | Informal gathering (Hotel Hacienda Los Laureles) |

Monday, June 5 | |
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07:30 - 08:45 | Breakfast (Restaurant at your assigned hotel) |

08:45 - 09:00 | Introduction and Welcome (Conference Room San Felipe) |

09:00 - 09:50 |
Alberto Cattaneo: An introduction to the BV-BFV formalism ↓ The BV-BFV formalism unifies the BV formalism (which deals with the problem of fixing the gauge of field theories on closed manifolds) with the BFV formalism (which yields a cohomological resolution of the reduced phase space of a classical field theory). I will explain how this formalism arises and how it can be quantized. (Conference Room San Felipe) |

09:50 - 10:20 | Coffee Break (Conference Room San Felipe) |

10:20 - 11:10 |
Stephan Stolz: From factorization algebras to functorial field theories ↓ There are various, quite different mathematical approaches to quantum field theories, among them functorial field theories in the sense of Atiyah and Segal and the factorization algebras of quantum observables of Costello and Gwilliam.
In the talk I will describe a construction that produces a twisted functorial field theory from a factorization algebra, thus relating these two approaches. This is joint work with Bill Dwyer and Peter Teichner. (Conference Room San Felipe) |

11:30 - 12:00 |
Marco Benini: Algebraic quantum field theory meets homotopical algebra ↓ An algebraic quantum field theory (AQFT) presents a QFT on Lorentzian manifolds as an assignment of algebras to spacetimes, subject to physical axioms (e.g. Einstein causality). Such algebras are interpreted as quantizations of the function algebras on the moduli space of a classical field theory. In many cases, e.g. the stack of a gauge theory, moduli spaces encode "higher structures". As a consequence, functions on such spaces form "higher algebras", which can be analyzed by homotopical algebra (à la Quillen). Therefore, to investigate the quantization of such moduli spaces, one needs to infuse AQFT with homotopical algebra, resulting in "homotopical AQFT", i.e. the assignment of "higher algebras" to spacetimes. After motivating our approach with a concrete application of homotopical algebra to the Cauchy problem of the Yang-Mills stack, I will provide a "working definition" of homotopical AQFT, emphasize its role in relation to gauge theories and present two toy examples arising via homotopy Kan extensions.
Based on [arXiv: 1503.08839, 1610.06071, 1704.01378]. (Conference Room San Felipe) |

12:30 - 14:00 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

15:40 - 16:30 |
Vladimir Dotsenko: Noncommutative cohomological field theories ↓ I shall introduce a noncommutative version of the notion of a CohFT, where
the role of Deligne-Mumford compactifications of moduli spaces is played by toric varieries
of Loday's realisations of associahedra. The corresponding noncommutative analogues of
Gerstenhaber and Batalin-Vilkovisky algebras will also be discussed. This is a joint
work with S.Shadrin and B.Vallette. (Conference Room San Felipe) |

16:30 - 17:00 | Coffee Break (Conference Room San Felipe) |

17:00 - 17:30 |
Florian Naef: Linearization of the Goldman-Turaev BV algebra using Kashiwara-Vergne theory ↓ Using local intersections, Goldman and Turaev defined a BV operator on the exterior algebra of homotopy classes of loops on a surface. On a genus zero surface with three boundary components the linearization problem of this structure is equivalent to the Kashiwara-Vergne problem in Lie theory. Motivated by this result a generalization of the Kashiwara-Vergne problem in higher genera is proposed and solutions are constructed in analogy with elliptic associators.
This is joint work with A. Alekseev, N. Kawazumi and Y. Kuno. (Conference Room San Felipe) |

18:00 - 20:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Tuesday, June 6 | |
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07:30 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 09:50 |
Pavel Mnev: Cellular BV-BFV-BF theory ↓ We will present an example of a topological field theory living on cobordisms endowed
with CW decomposition (this example corresponds to the so-called BF theory in its abelian and
non-abelian variants), which satisfies the Batalin-Vilkovisky master equation, satisfies (a version of)
Segal's gluing axiom w.r.t. concatenation of cobordisms and is compatible with
cellularaggregations. In non-abelian case, the action functional of the theory is constructed out of
local unimodular L-infinity algebras on cells; the partition function carries the information about the
Reidemeister torsion, together with certain information pertaining to formal geometry of the moduli
space of local systems. This theory provides an example of the BV-BFV programme for
quantization of field theories on manifolds with boundary in cohomological formalism. This is a joint
work with Alberto S. Cattaneo and Nicolai Reshetikhin. (Conference Room San Felipe) |

09:50 - 10:20 | Coffee Break (Conference Room San Felipe) |

10:20 - 11:10 |
David Jordan: Braided tensor categories and the cobordism hypothesis ↓ In work with David Ben-Zvi and Adrien Brochier, we introduced a (would-be) 4-D
topological field theory which relates to N=4 d=4 SYM in the same way that the Reshetikhin-Turaev
3-D theory relates to Chern-Simons theory. On surfaces it assigns certain explicit categories
quantizing quasi-coherent sheaves on the character variety of the surface (along the Atiyah-Bott/
Goldman/Fock-Rosly Poisson bracket), and these in turn relate to many well-known constructions
in quantum algebra.
The parenthetical "would be" above means that, while the theory had an a priori definition on
*surfaces* via factorization homology -- due to work of Ayala-Francis, Lurie, and Scheimbauer,
these techniques do not apply to 3- and 4-manifolds. In this talk I'll explain work with Adrien Brochier and Noah Snyder, which constructs the 3-manifold invariants following the prescription of
the cobordism hypothesis. This is in the spirit of Douglas-Schommer-Pries-Snyder's work on finite
tensor categories -- but in the infinite setting -- and also echoes early ideas of Lurie and Walker.
The resulting 3-manifold invariants quantize Lagrangians in the character variety of the boundary.
They are not at all well-understood or computed explicitly in general, but they appear
phenomenologically to relate to many emerging structures, such as quantum A-polyonomials,
DAHA-Jones polynomials, and Khovanov-Rozansky knot homologies. (Conference Room San Felipe) |

11:30 - 12:00 |
Ernesto Lupercio: Quantum Toric Geometry, Complex Systems, and Mirror Symmetry ↓ In this talk I will survey our investigations regarding quantum toric varieties (Katzarkov, Lupercio, Meersseman, Verjovsky), its relation to sandpiles, tropical geometry and complex systems (Guzman, Kalinin, Lupercio, Prieto, Shkolnikov) and Mirror Symmetry (Katzarkov, Kerr, Lupercio, Meerssemann). (Conference Room San Felipe) |

12:10 - 12:25 | Group photo (Hotel Hacienda Los Laureles) |

12:30 - 14:00 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

16:00 - 16:40 | Coffee Break (Conference Room San Felipe) |

16:40 - 17:30 |
Maxim Zabzine: Virasoro constraints and localization ↓ During last years numerous results were obtained for the exact partition functions and
other supersymmetric observables for supersymmetric gauge theories in diverse dimensions.
Typically the result can be expressed through the matrix model which satisfy the different versions
of Virasoro constraints (or its deformations). I will review the subject and provide some examples
for 3D and 4D gauge theories. (Conference Room San Felipe) |

18:00 - 20:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

20:00 - 22:00 | Gong Show (Conference Room San Felipe) |

Wednesday, June 7 | |
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07:30 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 13:30 | Excursion (Oaxaca) |

13:30 - 15:00 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

15:20 - 16:10 |
Dmitry Tamarkin: Axiomatic microlocal category. ↓ I am going to present a construction of an infinity stable category associated to a
closed symplectic manifold whose symplectic form has integer periods. The category looks like
the Fukaya category of M with coefficients in a certain local system. One first define an infinity
category C_{rR} associated to the product of two symplectic balls B_r times B_R whose objects
are (roughly) graphs of symplectomorphic embeddings B_r to B_R and homs are positive
isotopies (it is defined via listing axioms which characterize it). We have a composition
C_{r_1r_2} times C_{r_2r_3} to C_{r_1r_3} so that we have an infinity 2-category C whose 0-
objects are balls and the category of morphisms between B_r and B_R is C_{rR} One has a
functor F_M} from C to the infinity 2 category of infinity categories, where F_M(B_r) is the
category of symplectic embeddings B_r—>M. One also has another functor P between the same
infinity categories and one defines the micro local category on M as hom(P,F_M). (Conference Room San Felipe) |

16:10 - 16:40 | Coffee Break (Conference Room San Felipe) |

16:40 - 17:30 |
Hiro Lee Tanaka: Morse theory and the stack of broken lines ↓ I will talk about ongoing progress in understanding Morse theory as a deformation
problem encoded by (a sheaf living on) the stack of broken lines. This is joint work with Jacob
Lurie. (Conference Room San Felipe) |

18:00 - 18:50 |
Jonathan Weitsman: Fermionization of Gauge Theories ↓ We discuss fermionization of quantum gauge theories in 3 and 4 dimensions, with momentum cut offs. The corresponding Fermionic quantum field theories are nonlocal and have convergent perturbation expansions. We discuss some conjecturesthat arise from studying the structure of these theories. (Conference Room San Felipe) |

19:00 - 21:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Thursday, June 8 | |
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07:30 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 09:50 |
Chris Rogers: Towards an adjunction between the homotopy theories of dg manifolds and Lie ∞-groupoids ↓ Lie ∞-groupoids are simplicial manifolds which satisfy conditions similar to the
horn filling conditions for Kan simplicial sets. Lie ∞-groupoids are to non–negatively
graded dg manifolds, or L∞-algebroids, as Lie groups are to Lie algebras. In particular,
there is an integration procedure based on a smooth analog of Sullivan’s realization
functor from rational homotopy theory that pro- duces a Lie ∞-groupoid from dg–
manifold. There is also a differentiation functor due to Sˇevera, which uses supergeometry
to construct the 1-jet of a simplicial manifold.
In this talk, I will present joint work (arXiv:1609.01394) in progress with Chenchang Zhu
in which we study the relationship between these integration and differentiation
procedures, in analogy with Lie’s Second Theorem. A crucial first step involves
constructing a user–friendly homotopy theory for Lie ∞-groupoids. This is a subtle
problem, due to the fact that the category of manifolds lacks limits. I will describe how
results of Behrend and Getzler can be generalized to develop a homotopy theory for Lie
∞-groups/groupoids that is compatible with the well–known homotopy theory of L∞-
algebras/algebroids. If time permits, I will mention some possible applications to AKSZ
σ-models via Kotov–Strobl’s theory of characteristic classes for (non–trivial) Q-bundles. (Conference Room San Felipe) |

10:00 - 10:20 | Coffee Break (Conference Room San Felipe) |

10:20 - 11:10 |
Raimar Wulkenhaar: Matricial quantum field theory ↓ This subject combines ideas from quantum field theory on noncommutative spaces with
technologies developed for matrix models to rigorously compute all renormalised correlation
functions of these toy models. There is a natural projection of matricial correlation functions to
Schwinger functions of an ordinary Euclidean quantum field theory. The main question we are
working on is whether or not these functions satisfy Osterwalder-Schrader reflection positivity. If so
the model would define a true relativistic (but very simple) quantum field theory in four dimensions.
We have partial results that this could be the case. (Conference Room San Felipe) |

11:30 - 12:00 |
Pavel Safronov: Shifted geometric quantization ↓ Geometric quantization attaches vector spaces to symplectic manifolds equipped with extra data.
In this talk I will discuss how higher, or categorified, geometric quantization looks like. The input
this time will be a shifted symplectic space. As an example, in shift (-1) one discovers BV
quantization. (Conference Room San Felipe) |

12:30 - 14:00 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

15:40 - 16:30 |
Chris Hull: Higher Gauge Theory and Higher Gravity ↓ Generalisations of gauge theory and gravity suggested by supersymmetry and string
theory are explored. Their physical and mathematical structure are discussed and superconformal
symmetry in six dimensions is seen to play an interesting role. (Conference Room San Felipe) |

16:30 - 17:00 | Coffee Break (Conference Room San Felipe) |

17:00 - 17:30 |
Robert Oeckl: Functorial quantization of linear field theory ↓ Working towards the aim of axiomatizing realistic quantum field theories in a TQFT-type framework
we focus on the simplest class of examples: linear field theories and their perturbation theory. In
order to understand quantization it turns out to be useful to introduce an axiomatization of classical
field theory also, on manifolds with boundary. We show how geometric quantization together with
the Feynman path integral then leads to a quantization functor from (augmented) classical field
theories to quantum field theories. We discuss scope, applications, limitations and future directions
of this approach. (Conference Room San Felipe) |

18:00 - 20:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Friday, June 9 | |
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07:30 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

09:30 - 10:00 |
Christian Blohmann: The hamiltonian Lie algebroids of field theories ↓ I will report on progress of a joint project with Alan Weinstein in which we try to understand the initial value constraints of field theories with external symmetries as momenta of a Lie algebroid action on the presymplectic manifold of fields. Having shown earlier how a Lie algebroid symmetry naturally appears in General Relativity, we are currently studying a notion of hamiltonian momentum map for Lie algebroids. I will explain how hamiltonian Lie algebroids generally arise in lagrangian field theories with spacetime diffeomorphism symmetry. (Conference Room San Felipe) |

10:10 - 10:40 |
Ivan Contreras: Poisson sigma models and the symplectic category. ↓ The classical BV-BFV formulation of the Poisson sigma model with boundary produces isotropic
evolution relations, i.e. immersed submanifolds of products of the spaces of boundary fields. In this
talk we will prove that such relations are in fact split Lagrangian, i.e. isotropic with isotropic
complements, and they form a groupoid object in an extended version of the symplectic category. (Conference Room San Felipe) |

10:40 - 11:10 | Coffee Break (Conference Room San Felipe) |

11:10 - 12:00 |
Ezra Getzler: Vanishing of BV cohomology ↓ Consider the BV cohomology associated to a solution of the BV master equation for field
theory on a d-dimensional world sheet. When d=0, Felder and Kazhdan have suggested the
additional axiom that the cohomology for a physically relevant theory vanish below dimension 0.
The natural generalization of this to d>0 is that the cohomology vanish below dimension d.
In earlier work, we have shown that this axiom is violated for the spinning particle, which is a toy
model of a supersymmetric field coupled to supergravity in d=1. Sean Pohorence and I have
shown that in contrast, the axiom holds for the superparticle, which is a toy model of the Green-
Schwartz superstring in d=1. This theory exhibits some interesting features: there is an infinite
tower of ghosts, so it is important to work with the correct completion of the space of local
observables, and also one must work with Cech cochains at every stage of the calculation. (Conference Room San Felipe) |

12:30 - 14:00 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

14:00 - 15:00 | Final discussions and Goodbye (Conference Room San Felipe) |