Mathematics and Physics in Heidelberg Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik Mathematisches Institut Karl-Theodor-Brücke Surprise Fakultät für Physik und Astronomie Fakultät für Physik und Astronomie Institut für Theoretische Physik Universität Heidelberg Universität Heidelberg Universität Heidelberg

Flat Connections in Physics and Geometry

A Physical Mathematics Workshop at Heidelberg University


From July 20 to 22, 2017, the Mathematical Physics group in the Mathematisches Institut at Universität Heidelberg will host a three-day workshop on recent topics at the intersection of Physics and Geometry. The workshop is intended to serve as an unofficial satellite workshop of String Math 2017, held in Hamburg July 24–29.

Confirmed speakers

  • Matthew Bullimore (Oxford)
  • Mykola Dedushenko (Pasadena)
  • Ori Ganor (Berkeley)
  • Owen Gwilliam (Bonn)
  • Justin Hilburn (Philadelphia)
  • Kentaro Hori (Tokyo)
  • Si Li (Beijing)
  • Hyenho Lho (Zurich)
  • Mauricio Romo (Princeton)
  • Piotr Sułkowski (Warsaw)
  • Brian Williams (Evanston)
  • Piljin Yi (Seoul)
  • Michele del Zotto (Stony Brook)


Picture of Participants



Thursday, July 20

Time Speaker Title, Abstract
9:30am Ganor Quadratic Reciprocity and Janus Configurations
Compactification of N=4 Super-Yang-Mills theory on a circle with various SL(2,Z) duality twists will be discussed. A technique for counting ground states on a torus will be presented, and simple Wilson loops will be analyzed as well. The partition function on a mapping torus, calculated in two different ways, leads to the Landsberg-Schaar identity for quadratic Gauss sums, and generalizations.
Coffee break
11:00am Sułkowski Knots are quivers?
I will present a surprising relation between knot invariants and quiver representation theory, motivated by various string theory constructions involving BPS states. Consequences of this relation include the proof of the famous Labastida-Marino-Ooguri-Vafa conjecture, explicit (and unknown before) formulas for colored HOMFLY polynomials for various knots, new viewpoint on knot homologies, new dualities between quivers, and many others. The crucial role in this relation is play by A-polynomials -- which represent certain flat connections in Chern-Simons theory -- and their quantization.
Lunch break
1:30pm Bullimore The Twisted Hilbert Space of 3d N = 2 Theories
I will discuss a description of 3d N = 2 gauge theories on R x C with topological twist on a Riemann surface C as a supersymmetric quantum mechanics on R. I will focus on a mathematical description of the Hilbert space of supersymmetric ground states, which in general contains more information than the twisted partition function on S^1 x C. In particular, I will consider some simple abelian examples and demonstrate invariance of the Hilbert space under three-dimensional mirror symmetry.
3:00pm Yi Index, Partition Functions, and Rational Invariants
In recent years, varieties of index-like quantitites have been computed by exact path integral, a.k.a. the localization. For gauge theories, the path integral reduces to contour integrals, albeit with many subtleties. However, it turns out that interpretation of the result, which should be really called the twisted partition function rather than the refined index, requires even more care. After a cursory review of the recent derivations and accompanying subtleties, we consider theories with noncompact Coulomb phases. The rational nature of the twisted partition function is observed and physically explained, which also connects to the rational invariant of the Kontsevich-Soibelman wall-crossing algebra. This gives us a nontrivial tool for extracting the refined and integral index out of such rational twisted partition functions. An application to pure Yang-Mills quantum mechanics solves a decades-old problem of counting D0 bound states with an Orientifold point. Along the way, we also resolve a critical conflict between Kac/Smilga and Staudacher/Pestun, circa 1999~2002, by isolating the notion of H-saddles. The latter also proves to be a universal feature of partition functions in the high "temperature" limit.
4:15pm del Zotto Discrete Integrable Systems, Supersymmetric Quantum Mechanics, and Framed BPS States
It is possible to understand whether a given BPS spectrum is generated by a relevant deformation of a 4D N=2 SCFT or of an asymptotically free theory from the periodicity properties of the corresponding quantum monodromy. With the aim of giving a better understanding of the above conjecture, we revisit the description of framed BPS states of four-dimensional relativistic quantum field theories with eight conserved supercharges in terms of supersymmetric quantum mechanics. We unveil aspects of the deep interrelationship in between the Seiberg-dualities of the latter, the discrete symmetries of the theory in the bulk, and quantum discrete integrable systems.

Friday, July 21

Time Speaker Title, Abstract
9:30am Williams Symmetries of higher dimensional holomorphic field theories
The most well known examples of holomorphic field theories come from chiral CFTs in complex dimension one. Of the symmetries of these theories, a special role is played by both the Kac-Moody and Virasoro vertex algebras. In this talk I will discuss generalizations of these symmetries to higher dimensions in the language of factorization algebras. For example, holomorphic gauge theories in complex dimension d (theories depending on the data of a holomorphic connection) generically have the symmetry of a certain dg Lie algebra studied recently by Faonte, Hennion, and Kapranov. My talk will focus on this higher Kac-Moody algebra and I will relate it to a shadow of a richer OPE algebra. Time permitting, I will discuss a similar picture for the higher Virasoro algebras.
Coffee break
11:00am Gwilliam Chern-Simons theory and Koszul duality
The observables of perturbative Chern-Simons theory naturally form an algebra over the little 3-disks operad, as I will explain. In consequence, line operators form a braided monoidal category. We then use higher abstract nonsense, notably Koszul duality, to recover a quantum group with formal parameter. This work in progress is joint with K. Costello and J. Francis.
Lunch break
1:30pm Lho Stable quotient and holomorphic anomaly equations
I will prove holomorphic anomaly equations for the stable quotient invariant of Local P^2 in precise form predicted by B-model Physics. After that I will also discuss about the holomorphic anomaly equation for [C3/Z3] and formal quintic invariants. This talk is based on joint work with Rahul Pandharipande.
3:00pm Li Quantum master equation and integrable hierarchy
We establish the Batalin-Vilkovisky quantization theory for chiral deformation of free conformal field theories. As an application, we describe a universal approach to intebrable hierarchies associated to topological B-model on Calabi-Yau geometry. The talk is based on arXiv: 1612.01292[math.QA] and a joint work in progress with Weiqiang He and Philsang Yoo.
4:15pm Hilburn S-duality of boundary conditions for 4d N=4 and ring objects in the geometric Satake category
In a recent paper Braverman, Finkelberg, and Nakajima showed how one can associate a ring object in the category of equivariant D-modules on the affine Grassmannian Gr_G to a 3d N=4 gauge theory whose cohomology gives the ring of functions on the Coulomb branch. I will explain a physical context for this result in terms of boundary conditions to 4d N=4 super Yang-Mills and use this to give a mathematical formulation of some ideas of Gaiotto.

Saturday, July 22

Time Speaker Title, Abstract
9:30am Hori 2d Seiberg duality, with boundary
I will talk about 2d Seiberg duality and discuss how it may act on the B-type boundary conditions. The grade restriction rule, obtained from the hemisphere partition function, plays an important role. Based on a joint work in progress with Richard Eager.
Coffee break
11:00am Dedushenko Four-manifold invariants and 2d CFT
Recently, it has been conjectured that there exists a class of 2d N=(0,2) SCFTs T[M_4, G], labeled by a 4-manifold M_4 and a Lie group G, which could provide a new type of smooth structure invariants for M_4. Such theories are given by twisted compactifications of the 6d (2,0) theory on M_4. We study one of the simplest cases when G=U(1). Even though such T[M_4, U(1)] looks trivial (in particular, it is free), from the string theory point of view, it is natural to include certain local defect operators in this theory. They are naturally defined in the UV, while in the IR description they explicitly depend on the Seiberg-Witten (SW) invariants of M_4. T[M_4, U(1)] perspective leads to the equivariant multi-monopole generalization of SW invariants. We study them from several points of view, including the 4d gauge theoretic approach, and identify the structure of the VOA T[M_4, U(1)]. We also propose some further directions for generalizations, including the case of non-abelian G.
Lunch break
1:30pm Romo All-Order Volume Conjecture for Closed 3-Manifolds from Complex Chern-Simons Theory
I will review some general aspects of complex Chern-Simons theory on hyperbolic 3-manifolds, focusing on the case of gauge group G=SL(2,C). After a brief introduction to the Volume Conjecture (VC), for knot complements and, a very recent mathematical proposal, for closed hyperbolic 3-manifolds, I'll show how complex Chern-Simons theory is related with them and how this connection leads to a novel generalization of the most recently proposed VC for closed 3-manifolds.

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