XII Workshop on Geometric Correspondences of Gauge Theories

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Monday, July 4, 202210:00 AM PasquettiPasquetti10:00 AM - 11:00 AMRoom: Giambiagi lecture hall11:00 AM Coffee BreakCoffee Break11:00 AM - 11:30 AM11:30 AM Connecting 5d Higgs Branches via Fayet-Iliopoulos Deformations - Marieke Van BeestConnecting 5d Higgs Branches via Fayet-Iliopoulos Deformations
- Marieke Van Beest

11:30 AM - 12:30 PMRoom: Giambiagi lecture hall 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.12:30 PM LunchLunch12:30 PM - 2:30 PM2:30 PM Intersecting Defects: the Supergroup Side and Geometric Transitions - Fabrizio Nieri (Trinity College, Dublin)Intersecting Defects: the Supergroup Side and Geometric Transitions- Fabrizio Nieri (Trinity College, Dublin)

2:30 PM - 3:30 PMRoom: Giambiagi lecture hall 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.3:30 PM Tea BreakTea Break3:30 PM - 4:00 PM4:00 PM On the 6d origin of non-invertible symmetries in 4d - Michele Del Zotto (Uppsala University)On the 6d origin of non-invertible symmetries in 4d- Michele Del Zotto (Uppsala University)

4:00 PM - 5:00 PMRoom: Giambiagi lecture hall 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. -
Tuesday, July 5, 202210:00 AM Perturbative connection formulas for Heun equations - Oleg Lissovyy (Universite’ de Tours)Perturbative connection formulas for Heun equations
- Oleg Lissovyy (Universite’ de Tours)

10:00 AM - 11:00 AMRoom: Giambiagi lecture hall 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.11:00 AM Coffee BreakCoffee Break11:00 AM - 11:30 AM11:30 AM Isomonodromic deformations: Confluence, Reduction and Quantisation - Marta Mazzocco (Birmingham University)Isomonodromic deformations: Confluence, Reduction and Quantisation- Marta Mazzocco (Birmingham University)

11:30 AM - 12:30 PMRoom: Giambiagi lecture hall 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.12:30 PM LunchLunch12:30 PM - 2:30 PM2:30 PM Curve counting on surfaces and topological strings - Andrea Brini (Sheffield University)Curve counting on surfaces and topological strings- Andrea Brini (Sheffield University)

2:30 PM - 3:30 PMRoom: Giambiagi lecture hall 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.3:30 PM Tea BreakTea Break3:30 PM - 4:00 PM4:00 PM Equivariant indices and geometry - Maxim Zabzine (Uppsala University)Equivariant indices and geometry- Maxim Zabzine (Uppsala University)

4:00 PM - 5:00 PMRoom: Giambiagi lecture hall 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. -
Wednesday, July 6, 202210:00 AM VafaVafa10:00 AM - 11:00 AMRoom: Giambiagi lecture hall11:00 AM Coffee BreakCoffee Break11:00 AM - 11:30 AM11:30 AM OPE coefficient in Argyres Douglas theories. - Agnese Bissi (Uppsala University)OPE coefficient in Argyres Douglas theories.
- Agnese Bissi (Uppsala University)

11:30 AM - 12:30 PMRoom: Giambiagi lecture hall 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.12:30 PM LunchLunch12:30 PM - 2:30 PM2:30 PM Bootstrapping the half-BPS line defect CFT in N=4 SYM at strong coupling - Carlo Meneghelli (Parma University)Bootstrapping the half-BPS line defect CFT in N=4 SYM at strong coupling- Carlo Meneghelli (Parma University)

2:30 PM - 3:30 PMRoom: Giambiagi lecture hall 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.3:30 PM Studies of five-dimensional BPS spectra with applications to enumerative geometry - Pietro Longhi (ETH Zurigo)Studies of five-dimensional BPS spectra with applications to enumerative geometry- Pietro Longhi (ETH Zurigo)

3:30 PM - 4:30 PMRoom: Giambiagi lecture hall 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.4:30 PM Tea BreakTea Break4:30 PM - 5:00 PM5:00 PM Tetrahedron instantons and M-theory lift. - Xinyu Zhang (DESY, Hamburg)Tetrahedron instantons and M-theory lift.- Xinyu Zhang (DESY, Hamburg)

5:00 PM - 6:00 PMRoom: Giambiagi lecture hall : 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. -
Thursday, July 7, 202210:00 AM BPS Dendroscopy on local Calabi-Yau threefolds - Boris Pioline (Sorbonne Universite')BPS Dendroscopy on local Calabi-Yau threefolds
- Boris Pioline (Sorbonne Universite')

10:00 AM - 11:00 AMRoom: Giambiagi lecture hall 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.11:00 AM Coffee BreakCoffee Break11:00 AM - 11:30 AM11:30 AM Cluster structure of (non-autonomous) relativistic Toda chains described by K-matrices - Pavel Gavrylenko (Max Planck Institute)Cluster structure of (non-autonomous) relativistic Toda chains described by K-matrices- Pavel Gavrylenko (Max Planck Institute)

11:30 AM - 12:30 PMRoom: Giambiagi lecture hall 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.12:30 PM LunchLunch12:30 PM - 2:30 PM2:30 PM Mutations, cluster reductions and Painleve equations - Andrei MarshakovMutations, cluster reductions and Painleve equations- Andrei Marshakov

2:30 PM - 3:30 PMRoom: Giambiagi lecture hall 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.3:30 PM Tea BreakTea Break3:30 PM - 4:00 PM4:00 PM Local systems, reductions and quivers - Mikhail Bershtein (Skolkovo Institute)Local systems, reductions and quivers- Mikhail Bershtein (Skolkovo Institute)

4:00 PM - 5:00 PMRoom: Giambiagi lecture hall 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