XII Workshop on Geometric Correspondences of Gauge Theories

Jul 4, 2022, 9:00 AM → Jul 7, 2022, 6:00 PM Europe/Rome

Giambiagi lecture hall (ICTP)

The focus of the workshop is on recent developments in correspondences among supersymmetric gauge theories in various dimensions, topological field theories, 2D conformal field theories, integrable systems and the corresponding moduli spaces.

The Workshop is organized in the context of the Institute for Geometry and Physics.

Organizers: G. Bonelli, A.Tanzini.

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When: Jul 4, 2022 09:00 ROME
Register in advance for this meeting:
https://zoom.us/meeting/register/tJcrceGvpj8vH9GFKu6nGzp8Nlu2hS9D0uLL

Meeting ID and passcode - it will be the same for all 4 days

ID 936 6878 4555
passcode 081999

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The workshop is financially supported by IGAP, INFN and SISSA.

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The registration to the workshop is closed.

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Link to the 2011 edition website.

Link to the 2012 edition website.

Link to the 2013 edition website

Link to the 2015 edition website.

Link to the 2016 edition website

Link to the 2017 edition website

Link to the 2018 edition website

Link to the 2019 edition website

Link to the 2020 edition website

Link to the 2021 edition website

Monday, July 4

10:00 AM → 11:00 AM TBA: Pasquetti

11:00 AM → 11:30 AM Coffee Break

11:30 AM → 12:30 PM Connecting 5d Higgs Branches via Fayet-Iliopoulos Deformations

I will describe how the geometry of the Higgs branch of 5d SCFTs is transformed under general movement along the extended Coulomb branch. By working directly with the magnetic quiver, I will demonstrate a correspondence between Fayet-Iliopoulos deformations in 3d and 5d mass deformations. This relation provides a new perspective on the interconnectedness of 8 supercharge SCFTs, that can be utilized to establish a local version of mirror symmetry, when the Higgs branch has multiple cones and the mirror map is not globally well-defined.

Speaker: Marieke Van Beest

12:30 PM → 2:30 PM Lunch

2:30 PM → 3:30 PM Intersecting Defects: the Supergroup Side and Geometric Transitions

We consider the BPS intersecting defects that arise upon Higgsing a parent 5d SUSY gauge theory in the Ω-background. They are described by pairs of 3d N=2 SUSY gauge theories interacting through 1d matter at the intersection. We explore the relations between instanton and generalized vortex calculus, pointing out a duality between intersecting defects subject to the generic Ω-background and a deformation of supergroup gauge theories, the exact supergroup point being achieved in the self-dual or unrefined limit. Embedding the setup into refined topological strings, in the simplest case when the parent 5d theory is Abelian we are able to identify the dual of the intersecting defects as the supergroup version of refined Chern–Simons theory via a generalized open/closed duality or geometric transition. We also discuss the BPS/CFT side of the correspondence, finding an interesting large rank duality with equivariant super-instanton counting. This motivates to study diverse Higgsings of a bigger 5d SUSY supergroup gauge theory, which allows us to connect the partition functions of different defect theories via analytic continuation in some of the parameters.

Speaker: Fabrizio Nieri (Trinity College, Dublin)

3:30 PM → 4:00 PM Tea Break

4:00 PM → 5:00 PM On the 6d origin of non-invertible symmetries in 4d

Sometimes it is useful to think of symmetries of quantum fields in terms of quasi-topological defects. One of the advantages of such reformulation is the remark that there are more general notions of symmetries. Recently, several examples of models in 4d with non-invertibile duality and triality symmetry defects have been constructed. In this seminar I will discuss how to exploit 6d SCFTs to reproduce some of these known examples, as well as to generate infinitely many new examples of models with non-invertibile “M-ality”symmetry defects.

Speaker: Michele Del Zotto (Uppsala University)

Tuesday, July 5

10:00 AM → 11:00 AM Perturbative connection formulas for Heun equations

The connection problem for Heun equation seeks to relate bases of Frobenius solutions associated to different Fuchsian singularities. In a recent paper 2201.04491 [hep-th], Bonelli, Iossa, Lichtig and Tanzini have proposed a conjecture relating the relevant connection coefficients to classical conformal blocks of the Virasoro algebra. In practical terms, their conjecture allows to compute the connection coefficients in the form of a perturbative expansion in a suitable parameter. I will explain how to derive the corresponding perturbative formulas directly from Heun’s equation.

Speaker: Oleg Lissovyy (Universite’ de Tours)

11:00 AM → 11:30 AM Coffee Break

11:30 AM → 12:30 PM Isomonodromic deformations: Confluence, Reduction and Quantisation

We study the isomonodromic deformations of systems of differential equations with poles of any order on the Riemann sphere as Hamiltonian flows on the product of co-adjoint orbits of the truncated current algebra. Our motivation is to produce confluent versions of the celebrated Knizhnik--Zamolodchikov equations and explain how their quasiclassical solution can be expressed via the isomonodromic $\tau$-function.
In order to achieve this, we study the confluence cascade of $r+ 1$ simple poles to give rise to a singularity of arbitrary Poincar\'e rank $r$ as a Poisson morphism and explicitly compute the isomonodromic Hamiltonians.

Speaker: Marta Mazzocco (Birmingham University)

12:30 PM → 2:30 PM Lunch

2:30 PM → 3:30 PM Curve counting on surfaces and topological strings

I will describe a surprising web of correspondences linking together several a priori distant classes of enumerative invariants associated to a pair (X,D), with X a complex algebraic surface and D a singular anticanonical divisor in it. These include the log Gromov--Witten invariants of the pair, the Gromov--Witten invariants of an associated higher dimensional Calabi--Yau variety, the open Gromov--Witten invariants of certain special Lagrangians in toric Calabi--Yau threefolds, the Donaldson--Thomas theory of a class of symmetric quivers, and certain open and closed Gopakumar--Vafa-type invariants. I will also discuss how these correspondences can be effectively used to provide a complete closed-form solution to the calculation of all these invariants.

Speaker: Andrea Brini (Sheffield University)

3:30 PM → 4:00 PM Tea Break

4:00 PM → 5:00 PM Equivariant indices and geometry

I will discuss the role of equivariant parameters for the calculation of different
quantities (equivariant volumes, disk partition function, indices) for toric non-compact
Kahler manifolds. The talk is based on two papers: 2111.07663 with N.Nekrasov, N.Piazzalunga
(with appendix by M.Vergne) and the work in progress with N.Piazzalunga, L.Cassia.

Speaker: Maxim Zabzine (Uppsala University)

Wednesday, July 6

10:00 AM → 11:00 AM TBA: Vafa

11:00 AM → 11:30 AM Coffee Break

11:30 AM → 12:30 PM OPE coefficient in Argyres Douglas theories.

In this talk I will discuss how to put bounds on the OPE coefficients of some Argyres Douglas theories using conformal bootstrap techniques. I will also discuss how to contrast these bounds with results obtained using localisation.

Speaker: Agnese Bissi (Uppsala University)

12:30 PM → 2:30 PM Lunch

2:30 PM → 3:30 PM Bootstrapping the half-BPS line defect CFT in N=4 SYM at strong coupling

In this talk I will present how the 1d CFT defined by the half-BPS Wilson line in planar N=4 super Yang-Mills can be solved in a perturbative expansion around strong coupling using analytic bootstrap methods. I will also comment on how the results obtained in this way reproduce the available data from integrability-based methods. The talk is based on joint work with Pietro Ferrero.

Speaker: Carlo Meneghelli (Parma University)

3:30 PM → 4:30 PM Studies of five-dimensional BPS spectra with applications to enumerative geometry

The framework of spectral networks was introduced in physics as a way to compute BPS states of 4d N=2 gauge theories. In this talk I will review an extension of this framework, known as exponential networks, which arises in the study of 5d N=1 BPS states. Geometric engineering connects 5d N=1 BPS spectra to enumerative invariants of certain Calabi-Yau threefolds, I will review the computation of old and new results in the setting of local toric threefolds. I will also sketch a new perspective on this framework, which elucidates the geometric meaning of the invariants computed by networks, in terms of elementary data of A-branes in the mirror geometry, recovering an old conjecture of Joyce.

Speaker: Pietro Longhi (ETH Zurigo)

4:30 PM → 5:00 PM Tea Break

5:00 PM → 6:00 PM Tetrahedron instantons and M-theory lift.

: In this talk, I will discuss various aspects of tetrahedron instantons and the M-theory lift. Tetrahedron instantons are realized in type IIA superstring theory by bound states of D0-branes and four stacks of intersecting D6-branes with a suitable constant B-field. The tetrahedron instanton partition function admits an elegant closed-form expression depending only on the Omega-deformation parameters. Remarkably, the tetrahedron instanton partition function is equal to certain specialization of the magnificent four partition function, indicating the creation of intersecting D6-branes when a D8-brane annihilates with an anti-D8-brane. Meanwhile, the M-theory lift of our configuration is expected to be $\mathbb{R} \times X$, where $X$ is a noncompact Calabi-Yau fivefold and can be thought of as a superposition of Kaluza-Klein magnetic monopoles. We detect the property of $X$ using the index of M-theory on $X$, which coincides with the tetrahedron instanton partition function up to an appropriate perturbative factor. The talk is based on joint work with E. Pomoni and W. Yan.

Speaker: Xinyu Zhang (DESY, Hamburg)

Thursday, July 7

10:00 AM → 11:00 AM BPS Dendroscopy on local Calabi-Yau threefolds

The spectrum of BPS states in D=4 supersymmetric field theories and string vacua famously jumps across codimension-one walls in vector multiplet moduli space. The Attractor Flow Tree conjecture postulates that the BPS index $\Omega(\gamma,z)$ for given charge $\gamma$ and moduli $z$ can be reconstructed from the `attractor indices’ $\Omega_*(\gamma_i)$ counting BPS states of charge $\gamma_i$ in their respective attractor chamber, by summing over all possible decompositions $\gamma=\sum_i \gamma_i$ and over decorated rooted flow trees. Physically, flow trees provide a mesoscopic representation of BPS states as nested multi-centered bound states of elementary constituents. I will present a rigorous version of this formula in the context of quivers with potential, which governs the BPS spectrum in type IIA string theory compactified on certain conical Calabi-Yau threefolds, in the vicinity of orbifold-type points in Kahler moduli space. Moving away from such orbifold points requires generalizing the flow tree formula from the Abelian category of quiver representations to the derived category of the same. I will present recent progress in this direction in the simplest case of local $P^2$, and argue that its global BPS spectrum and flow tree structure can be deduced from a scattering diagram with simple initial data.

Speaker: Boris Pioline (Sorbonne Universite')

11:00 AM → 11:30 AM Coffee Break

11:30 AM → 12:30 PM Cluster structure of (non-autonomous) relativistic Toda chains described by K-matrices

Usual relativistic Toda chains have standard cluster algebra description, since they live on double Bruhat cells in the loop groups. This cluster structure gives rise to discrete flows in such systems. These discrete flows can be deautonomized to some non-linear q-difference equations of q-isomonodromic type, closely related to 5d gauge theories. It turns out that exactly the same happens for the relativistic Toda chains described by K-matrices (particular cases are B-type and C-type Toda systems). These systems turn out to be cluster, and even more, they have more discrete flows, which in this case form some affine Weyl groups.
The talk will be based on a part of work in progress with A. Liashyk, A. Marshakov, I. Motorin, M. Semenyakin.

Speaker: Pavel Gavrylenko (Max Planck Institute)

12:30 PM → 2:30 PM Lunch

2:30 PM → 3:30 PM Mutations, cluster reductions and Painleve equations

We consider the cluster integrable systems and their invariances, generated by mutations of quivers, induced by intersection forms on base and dual Goncharov-Kenyon surfaces. This leads to natural extension of the Goncharov-Kenyon class of cluster integrable systems by their Hamiltonian reductions. In particular, it allows to fill the gap in cluster construction of the q-difference Painleve equations, corresponding to the self-dual systems in the above sense, and related with the 5d supersymmetric gauge theories.

Based on joint work with M.Bershtein, P.Gavrylenko and M.Semenyakin.

Speaker: Andrei Marshakov

3:30 PM → 4:00 PM Tea Break

4:00 PM → 5:00 PM Local systems, reductions and quivers

Fock and Goncharov defined cluster structure on the space of decorated local systems on a surface. The corresponding quivers are BPS quivers for supersymmetric 4d theories, the corresponding X-cluster varieties are related to Coulomb branches. But the original construction is defined only for the case of generic monodromies. It appears that varieties and quivers for special monodromies can be obtained via Hamiltonian reduction. The examples include Ruijsenaars model and XXZ spin chain.

Based on joint work in progress with P. Gavrylenko, A. Marshakov, M. Semenyakin

Speaker: Mikhail Bershtein (Skolkovo Institute)