Networks in Physics and Geometry: Spectral, Exponential, and Beyond

Jul 8, 2025, 9:00 AM → Jul 9, 2025, 5:00 PM Europe/Paris

2nd Floor, Seminar Room 2L8 (Laboratoire de Mathématiques d'Orsay)

Spectral and Exponential Networks play a fundamental role in mathematical physics, particularly in the study of moduli spaces and wall-crossing phenomena. This two-days workshop aims to bring together experts working on spectral and exponential networks and their connections to BPS states, mirror symmetry, knot theory, resurgence, and Teichmüller theory, and seeks to advance our understanding of these networks and foster new insights and collaborations.

Organisateurs: Sibasish Banerjee (sbanerjee@ihes.fr), Veronica Fantini, Raphael Senghaas (rsenghaas@mathi.uni-heidelberg.de), Valdo Tatitscheff (vtatitscheff@mathi.uni-heidelberg.de).

Zoom link: https://eu02web.zoom-x.de/j/5835011530?pwd=Nhwk5fKASmJUHfvcThvbZOaXSuFvOm.1 
Meeting ID: 583 501 1530
Passcode: 000000

Tue, July 8

9:30 AM → 10:00 AM Welcome coffee

10:00 AM → 11:00 AM Exponential networks and the life of partitions

Following a quick summary of exponential networks, I will describe in some detail the explicit correspondence, obtained in collaboration with Banerjee, Romo and Senghaas, between torus fixed points of the Hilbert scheme of points in the plane and anomaly-free finite webs attached to the quadratically framed pair of pants.

Speaker: Prof. Johannes Walcher (Heidelberg University)

11:00 AM → 11:30 AM Break

11:30 AM → 12:30 PM Some aspects of exponential networks (Online)

I will describe exponential networks and its uses in defining (some version of) Donaldson-Thomas invariants and BPS quivers. If time allows I will present some other interpretation of these invariants as Euler characteristic of certain families of special Lagrangians.

Speaker: Prof. Mauricio Romo (SIMIS/Fudan University)

12:30 PM → 2:00 PM Lunch break

2:00 PM → 3:00 PM (Partial) abelianization and higher Teichmüller spaces

Higher Teichmüller spaces are connected components in the space of representations from a surface group into a higher rank Lie group. The first examples of these are Hitchin components for split real Lie groups. I will give an overview of the known examples of higher Teichmüller spaces, via the notion of Theta-positivity introduced by Guichard-Wienhard to generalize Fock-Goncharov and Lusztig total positivity. I will then present how spectral networks offer a convenient set of tools to study those higher Teichmüller spaces, in particular Fock-Goncharov-like coordinate systems on them.

Speaker: Dr Clarence Kineider (MPI Leipzig)

3:00 PM → 3:30 PM Break

3:30 PM → 4:30 PM Spectral networks as screening contours (Online)

I will describe a role of spectral networks in 2-dimensional conformal field theory: they can be used as "screening contours" in a new construction of Virasoro conformal blocks from branched double covers. The key new point is that, when three exponentiated screening contours end on a branch point of the cover, they cancel an unwanted singularity of the conformal block there. The talk is intended to be self-contained: in particular I will explain what Virasoro conformal blocks are. This is a report of joint work with Qianyu Hao.

Speaker: Prof. Andrew Neitzke (Yale University)

Wed, July 9

9:30 AM → 10:00 AM Coffee

10:00 AM → 11:00 AM Spectral networks and Betti Lagrangians (Online)

In this talk I will discuss the interface of spectral network theory and real contact symplectic topology. I begin with an introduction to weave theory, which allows one to construct exact Lagrangians from totally degenerate spectral networks (otherwise known as N-graphs). I then broaden the classical notion of a meromorphic spectral curve to the wider setting of Betti Lagrangians and formulate a corresponding concept of generic spectral networks.

We give lots of explicit examples—BNR spectral curves, ADE singularities, and N-triangle configurations. Finally, I outline how spectral networks admit both pseudoholomorphic and categorical characterisations: the former through Floer theory, the latter through the partially wrapped Fukaya category theory, leading to a concrete understanding of the Gaiotto–Moore–Neitzke non-abelianisation functor. I will close by sketching potential applications.

Speaker: Dr Nick Nho

11:00 AM → 11:30 AM Break

11:30 AM → 12:30 PM Thimbles and resummation to networks

I will discuss how parametric resurgence recovers many
of the basic properties of networks and some examples that arise
from q-hypergeometric functions, which play an important role in
Chern-Simons theory and open topological strings. I will explain
how to construct these networks from some basic building blocks
coming from the Faddeev quantum dilogarithm and the geometry
of thimbles in an appropriate homology theory. This is based on joint
work with Andersen-Fantini-Kontsevich.

Speaker: Dr Cambell Wheeler (IHES)

12:30 PM → 2:00 PM Lunch break

2:00 PM → 3:00 PM Alternating diagrams and non-commutative cluster Lagrangians

Given a bipartite graph G on a possibly punctured surface S, there is a (non-commutative) cluster Poisson variety X(G,S). It depends only on the equivalence class of G under certain elementary transformations. A threefold M which bounds the surface S with filled punctures gives rise to a Lagrangian in the generic symplectic fibers of X(G,S). I will explain that it carries a natural non-commutative cluster symplectic structure.
The construction requires a 3d generalization of bipartite surface graphs. The talk reflects joint work with Maxim Kontsevich.

Speaker: Prof. Alexander Goncharov (Yale University)

3:00 PM → 3:30 PM Break

3:30 PM → 4:30 PM Discussion session