UNIST International Workshop on Calabi-Yau Crystals and BPS Counting 2025
Talks
Talk 1
Torus Knots and Virasoro Minimal Models
Dongmin Gang (Seoul National University)
Thursday, October 30, 2025
108-320, 10:00 - 10:45
In 2013, K. Hikami and A. N. Kirillov discovered an intriguing connection between torus knots T_{(P,Q)} and Virasoro minimal models M(P,Q) by relating the Kashaev invariants of the knots to the characters of the corresponding minimal models. We rediscover this relation by combining the 3D–3D correspondence with the bulk–boundary correspondence. More explicitly, we study 3D N=2 gauge theories associated with torus-knot complements via the Dimofte–Gaiotto–Gukov construction and find that, in the infrared, they either flow to a unitary TQFT (for |P-Q|=1) supporting the chiral algebra of the corresponding unitary minimal model, or to a 3D N=4 rank-0 SCFT (for |P-Q|>1) realizing the non-unitary one after topological twisting.
Talk 2
Gauge Origami and BPS qq-characters
Go Noshita (University of Tokyo)
Thursday, October 30, 2025
108-320, 11:00 - 11:45
Gauge origami is a generalized supersymmetric quiver gauge theory where intersecting D-branes appear. An interesting property of the gauge origami partition functions is the existence of non-perturbative Dyson-Schwinger equations related to the symmetries of adding and removing instantons. The BPS qq-characters are the physical observables characterizing them, and they are related to quantum algebras. In this talk, I will review the constructions of the qq-characters for the gauge origami system on C^4 and then discuss generalizations to toric Calabi-Yau 4-folds. This talk is based on recent works with T. Kimura.
Talk 3
Miura operators as R-matrices from M-brane intersections
Saebyeok Jeong (IBS-CGP, Pohang)
Thursday, October 30, 2025
108-320, 13:30 - 14:15
In this talk, I will discuss how M2-M5 intersections in a twisted M-theory background yield the R-matrices of the quantum toroidal algebra of gl(1). These R-matrices are identified with the Miura operators for the q-deformed W- and Y-algebras. Additionally, I will show how the M2-M5 intersection (or equivalently, the Miura operator) generates the qq-characters of the 5d N=1 gauge theory, offering new insight into the algebraic meaning of the latter.
Talk 4
Nekrasov’s gauge origami via DT4 theory
Woonam Lim (Yonsei University)
Thursday, October 30, 2025
108-320, 14:30 - 15:15
The study of classical instantons on spacetime has led to many interesting developments in mathematics. In a series of papers, Nekrasov introduced the generalized ADHM equations, whose solutions are instantons on the “origami spacetime.” In this talk, I will explain how to interpret gauge origami via DT4 theory. The main result shows that Nekrasov’s origami partition function, defined by local contributions, coincides with a global definition via Oh–Thomas classes in DT4 theory. This global definition is crucial for deriving the Dyson–Schwinger equation, which was one of Nekrasov’s main motivations for introducing gauge origami theory. I will also briefly discuss a conjectural sheaf-theoretic description of gauge origami. This is joint work in progress with N. Arbesfeld and M. Kool.
Colloquium
A Dual Tale of Physics and Geometry
Sebastian Franco (City College, City University of New York)
Thursday, October 30, 2025
108-320, 16:00 - 17:00
Physics and mathematics have long shared a deep and fruitful dialogue, where progress in one field often fuels and reshapes the other. This colloquium will offer a glimpse into the vibrant modern conversation between the two disciplines, centered around Quantum Field Theory (QFT) and its geometric avatars. QFT lies at the heart of our understanding of nature, from the dynamics of fundamental particles to the structure of spacetime itself. Yet, despite its success, our grasp of QFT continues to evolve in surprising ways. I will describe how embedding QFTs within String Theory provides a powerful framework for uncovering new structures, translating intricate physical phenomena into geometric language. One of the most fascinating phenomena that arise in QFTs is Duality: the magical equivalence between theories that seem entirely different. I will give you a flavor of how, through the geometric lens of String Theory, notions such as algebraic geometry and mirror symmetry intertwine with quantum field theory dualities, offering a unified and intuitive picture of the underlying physics. Remarkably, this geometric perspective, often opens the path for new discoveries.
Talk 5
4d Crystals for Toric Calabi-Yau 4-Folds: the Exploration Continues
Sebastian Franco (City College, City University of New York)
Friday, October 31, 2025
108-320, 10:00 - 10:45
I will then introduce a family of 4d crystal melting models describing the BPS spectrum of D-branes on toric Calabi–Yau 4-folds. Their crystalline structures are determined by the associated brane brick models. The crystals provide a discretized version of the underlying toric geometries. I will present new methods to analyze these crystals, explore their interplay with triality, and report recent results on their structure.
Talk 6
Quiver Algebras from Crystals
Jiakang Bao (University of Tokyo)
Friday, October 31, 2025
108-320, 11:00 - 11:45
Quiver Yangians are BPS algebras arised from Type IIA string theory on toric Calabi-Yau threefolds. The crystals in the BPS counting form representations of the quiver Yangians. In this talk, I will discuss some recent progress on constructing the crystals and certain quiver algebras encoding the information of the BPS states for generic quivers (not necessarily toric) satisfying the so-called no-overlap condition and also for theories with fewer supercharges (such as those arised from Calabi-Yau fourfolds). If time permits, I will mention the work in progress with Duncan Laurie on the cases of affine DE-types, where we do not have the no-overlap condition.
Talk 7
Lagrangian classes, Donaldson-Thomas theory, and gauged linear sigma models
Hyeonjun Park (KIAS, Seoul)
Friday, October 31, 2025
108-320, 13:30 - 14:15
In this talk, I will explain the construction of Lagrangian classes for perverse sheaves in cohomological Donaldson-Thomas theory, whose existence was conjectured by Joyce. The two key ingredients are a relative version of the DT perverse sheaves and a hyperbolic version of the dimensional reduction theorem. As a special case, we recover Borisov-Joyce/Oh-Thomas virtual classes in DT4 theory. As applications, I will explain how to construct the following structures from the Lagrangian classes: (1) cohomological field theories for gauged linear sigma models, (2) cohomological Hall algebras for 3-Calabi-Yau categories, (3) relative Donaldson-Thomas invariants for Fano 4-folds with anti-canonical divisors, (4) refined surface counting invariants for Calabi-Yau 4-folds. This is joint work in progress with Adeel Khan, Tasuki Kinjo, and Pavel Safronov.
Talk 8
Aspects of dualities in supersymmetric quantum mechanics and DT4 invariants from quivers
Cyril Closset (University of Birmingham)
Friday, October 31, 2025
108-320, 14:30 - 15:15
I will discuss Seiberg-like dualities of N=2 supersymmetric quantum mechanics (SQM), emphasizing new aspects compared to dualities of 2d N=(0,2) gauge theories. In particular, I will present new mutation dualities for 1d N=2 SQCD with unitary gauge group. I will also comment on the relationship between 1d N=2 SQM quivers and DT4 invariants of local fourfolds.
Talk 9
Calabi Yau Lattice Paths and BPS Counting
Piotr Sulkowski (University of Warsaw)
Friday, October 31, 2025
108-320, 16:00 - 16:45
I will present a class of lattice path models – specifically what we call generalized Schröder paths – that encode colored HOMFLY-PT invariants of torus knots. A basic set of such paths corresponds to the generators of uncolored HOMFLY-PT homology, and invoking the knots-quivers correspondence, the generating functions of such paths can be expressed as quiver generating series. These lattice paths enumerate BPS states associated to knots via brane constructions.
Talk 10
Non-perturbative quantum geometry of topological strings and BPS invariants
Murad Alim (Technical University Munich)
Friday, October 31, 2025
108-320, 17:00 - 17:45
The partition function of topological string theory on any family of Calabi-Yau threefolds is defined perturbatively as an asymptotic series in the topological string coupling and encodes, in a holomorphic limit, higher genus Gromov-Witten as well as Gopakumar-Vafa invariants. I will prove that the partition function of topological strings of any CY in this limit can be written as a product, where each factor is given by the partition function of the resolved conifold with shifted arguments, raised to the power of certain sheaf invariants. I will use this result to study the Borel summation of topological string theory.
Talk 11
A pullback between sheaves on log Calabi-Yau 4-folds
Jeongseok Oh (Seoul National University)
Saturday, November 1, 2025
108-320, 10:00 - 10:45
Given a section of a bundle with quadratic form, we construct a specialisation map between K-groups of matrix factorisations of the quadratic function from the space to the normal cone of the zero locus. When the section is isotropic, the quadratic function becomes zero and the construction recovers usual specialisation map in Fulton-MacPherson's intersection theory. When a log Calabi-Yau 4-fold (X,D) is given, we apply the construction to define a pullback from sheaves on D to ones on X after assuming the space of sheaves on D is a critical locus globally. For a Calabi-Yau 4-fold X, we artificially define "the space of sheaves on D" to be a point so that its structure sheaf pulls back to the virtual structure sheaf of the space of sheaves on X. This is a joint work in progress with Dongwook Choa and Richard Thomas.
Talk 12
Stable envelope for critical loci
Yehao Zhou (SIMIS, Shanghai)
Saturday, November 1, 2025
108-320, 11:00 - 11:45
In this talk we will introduce a generalization of Maulik-Okounkov’s stable envelopes to equivariant critical cohomology. In the case of a tripled quiver variety with standard cubic potential, this reduces to MO’s stable envelope for the Nakajima variety of the corresponding doubled quiver along the dimensional reduction. We define non-abelian stable envelopes for quivers with potentials following a similar construction of Aganagic-Okounkov, and use them to relate critical COHAs to the abelian stable envelopes. RTT formalism leads to natural (shifted) (super) Yangian action on the critical cohomology of quiver varieties with potentials. This talk is based on joint work with Yalong Cao, Andrei Okounkov, and Zijun Zhou.