The Sheffield Geometry and Physics Seminar

The variety of commuting matrices is an important space in algebraic geometry, and has been studied from various perspectives. The stack of commuting matrices M is the same as the stack of zero dimensional sheaves in the plane, and can be used to define various Hall algebras that act on cohomologies of the Hilbert scheme of points in the plane. In this talk, I will talk about cohomologies of the stack M. First, I will recall results of Davison and Davison-Meinhardt about the Borel-Moore homology of M. These results are proved via an analysis of the BPS sheaves of points in the three dimensional affine space. Next, I will discuss (partial) analogues of these results for the K-theory and the category of coherent sheaves on M. The central objects are a categorical replacement of the BPS sheaves. The talk is based on joint results with Yukinobu Toda. 

Recently, a relationship between BPS invariants of four-dimensional supersymmetric QFTs (equivalently, Donaldson-Thomas invariants of certain 3CY triangulated categories) and the Eynard-Orantin topological recursion (which computes invariants of "spectral curves'' originally appearing in the theory of matrix models), was observed for a class of fundamental examples.  We review both formalisms and explain how to modify the topological recursion to obtain the "$\beta$-deformed'' or "refined" topological recursion when the initial data is sufficiently nice. For the simplest such examples, we show how the corresponding free energies can be expressed in terms of a new collection of refined BPS invariants which, unlike the unrefined case, do not seem to have appeared in the Donaldson-Thomas theory to date.  Based on joint works with K. Osuga.

The Alday-Gaiotto-Tachikawa (AGT) conjecture in physics predicts a relationship between 2d conformal field theories and certain 4d gauge theories. A precise mathematical version (proved by Maulik-Okounkov, Schiffmann-Vasserot and others) states that the equivariant cohomology of the moduli space of instantons (4d side) is a module of a certain W-algebra (2d side), and that the fundamental class of the moduli space is a Whittaker vector in the W-algebra module, known as the Gaiotto vector. I will show how one can realize this Gaiotto vector as the partition function of an Airy structure, and thereby relate it to the topological recursion formalism of Eynard and Orantin. This means that one can compute the Nekrasov instanton partition function (which is the norm squared of the Gaiotto vector) using topological recursion techniques. Time permitting, I will discuss some possible applications (all work-in-progress) of this relationship including extensions to Argyres-Douglas theories, relations to Hurwitz theory and matrix models, and connections to integrability.

The talk is based on joint work with Vincent Bouchard, Gaetan Borot and Thomas Creutzig.

I will report on joint work in progress with Luca Giovenzana. I will describe some developments in the abstract theory of quasi del Pezzo pseudolattices, before showing how this theory arises naturally in the contexts of type II degenerations of K3 surfaces and elliptically fibred K3 surfaces. This can be thought of as a manifestation of mirror symmetry; I will discuss what it could tell us about mirror symmetry for K3 surfaces and the 2-dimensional Fano/LG correspondence.

In recent years various unifying frameworks for understanding 2d integrable field theories have emerged. In this talk I will review the approach based on 4d Chern-Simons theory, due to Costello and Yamazaki, and describe recent progress towards extracting general 2d integrable fields theories from 4d Chern-Simons theory.

Placing D3-branes at conical Calabi-Yau threefold singularities produces many AdS5/CFT4 duals. Recent progress in differential geometry has produced a technique, called K-stability, to recognize which singularities admit conical Calabi-Yau metrics. On the other hand, the algebraic technique of noncommutative crepant resolutions (NCCRs) has been developed to associate a quiver to a singularity. In favorable situations (such as the hypersurface case), producing an NCCR is equivalent to finding suitable matrix factorizations of the hypersurface. I will put together K-stability and NCCRs to produce new AdS5/CFT4 pairs, beyond the well-known toric setup.

In this talk I shall discuss recent work with Tom Coates and Sara Veneziale in which we successfully recover the dimension of a Fano variety X directly from the regularized quantum period of X via machine learning. We apply machine learning to the question: does the quantum period of a Fano variety X know the dimension of X? Note that there is as yet no theoretical understanding of this. We show that machine learning techniques can recover the dimension with > 80% accuracy, demonstrating that machine learning can pick out structure from complex mathematical data in situations where we lack a theoretical understanding. It also gives positive evidence for the assertion (which is proven for smooth Fanos in low dimensions, but unknown in general) that the quantum period of a Fano variety determines that variety. 

 Fix a Calabi-Yau 3-fold X of Picard rank one satisfying the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda, such as the quintic 3-fold. I will first describe explicit formulae relating rank zero Donaldson-Thomas (DT) invariants to Pandharipande-Thomas (PT) invariants using wall-crossing with respect to weak Bridgeland stability conditions on X. As applications, I will find sharp Castelnuovo-type bounds for PT invariants, and explain how combining these explicit formulae with S-duality in physics enlarges the known table of Gopakumar-Vafa (GV) invariants. The second part is joint work with string theorists Sergei Alexandrov, Albrecht Klemm, Boris Pioline and Thorsten Schimannek. 

I will discuss how to use Laurent inversion, a technique coming from mirror symmetry which constructs toric embeddings, to study the local structure of the K-moduli space of a K-polystable toric Fano variety. More specifically, starting from a given toric Fano 3-fold X of anticanonical volume 28 and Picard rank 4, and combining a local study of its singularities with the global deformation provided by Laurent inversion, we are able to conclude that the K-moduli space is rational around X. This is joint work with Andrea Petracci.

 FJRW theory is an enumerative theory built from characteristic classes corresponding to the moduli space of W-spin curves, a natural generalisation of higher spin curves. They can be interpreted as the enumerative geometry of gauged Landau-Ginzburg models. We construct an enumerative theory for an open version of FJRW invariants, over the moduli space of W-spin discs, rather than compact Riemann surfaces. We then will build the mirror to the original Landau-Ginzburg model as a generating function of open FJRW invariants, and prove a mirror symmetry statement. This is joint work with Mark Gross and Ran Tessler. 

Inspired by constructions in Calabi-Yau compactifications of type IIA/B string theory, we explain how to construct quaternionic-Kähler (QK) manifolds from certain special variations of BPS structures. We furthermore specify a subclass of such QK metrics admitting a rather non-trivial SL(2,Z) action by isometries, related to S-duality in type IIB string theory.  Along the way, we comment on relations to the TBA equations from Gaiotto-Moore-Neitzke, and joint work with M. Alim, A. Saha and J. Teschner. This is joint work with V. Cortés (arXiv:2105.09011, arXiv:2306.01463) based on several works in the physics literature by S. Alexandrov, D. Persson,  B. Pioline, F. Saueressig, S. Vandoren and many more. 

Celebrated work of Bridgeland and Smith shows a correspondence between quadratic differentials on Riemann surfaces and stability conditions on certain 3-Calabi--Yau triangulated categories. Part of this correspondence is that finite-length trajectories of the quadratic differential correspond to categories of semistable objects of a fixed phase. Categories of semistable objects have an associated Donaldson--Thomas invariant which, in some sense, counts the objects in the category. Work of Iwaki and Kidwai predicts particular values for these Donaldson--Thomas invariants for different types of finite-length trajectories, based on the output of topological recursion. The Donaldson--Thomas invariants produced by the category of Bridgeland and Smith do not always match these predictions. However, we show that if one replaces this category by the category recently studied by Christ, Haiden, and Qiu, then one does obtain the Donaldson--Thomas invariants matching the predictions. This is joint work with Omar Kidwai.  

We revisit the connection between cluster integrable systems and q-difference Painlevé equations. In this relation, the Painlevé dynamics is interpreted as the deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations. On the other hand, the cluster quivers coincide with BPS quivers of 5d gauge theories. The partition functions of the corresponding theories solve q-difference Painlevé equations. It appears that in order to include Painlevé equations with the largest E7 and E8 symmetry into this relation it is necessary to extend Goncharov-Kenyon integrable systems by  their Hamiltonian reductions. 

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

Given a plane nodal cubic D, how many straight lines are there meeting D in exactly one point? This question can be treated using classical techniques in Algebraic Geometry which return the correct answer (three). However, just a slight generalisation of this exercise (e.g. the enumeration of higher degree and higher genus curves maximally tangent to D) renders standard techniques essentially useless. In my talk I will present a correspondence which translates questions about maximally tangent curves to D phrased in the framework of logarithmic Gromov-Witten theory to the Gromov-Witten theory of local P1 which is solved completely by the topological vertex. This is a special instance of a logarithmic-local correspondence for so called bicyclic pairs proven in joint work with Michel van Garrel and Navid Nabijou. 

Twistor theory gives a powerful, and unifying, way to understand many (and conjecturally all) integrable systems. However certain well-known examples have failed to be incorporated into this scheme, but certain limiting cases of such systems do. Thus the KP equation does not fit, but the dispersionless KP equation does; the Joyce equation (which encodes DT invariants) fits, but its quantum counterpart (which encodes qDT invariants) does not.

An alternative way to think about this is to regard such equations as a deformation of its dispersionless counterpart. This borrows ideas from the deformation quantization programme: how to deform objects while preserving their essential features – in this case, their integrability.

The talk will cover initial ideas of how one can deform hyperKahler geometry, and potentially twistor theory, to incorporate such systems.

We give a friendly introduction to K-stability, and the motivation behind it. We will see how to study and completely describe all one-dimensional components of the K-moduli of smooth Fano 3-folds. And we will finish giving some specific examples for family 3.12 (blow-up of a disjoint line and twisted cubic on P^3). This result is in collaboration with Abban, Cheltsov, Denisova, Kaloghiros, Jiao, Martinez-Garcia and Papazachariou. 

The Gromov-Witten invariants of projective spaces are not enumerative in positive genera. The reason is geometric: the moduli space of genus-g stable maps has several irreducible components, which contribute in the form of lower-genera GW invariants. In genus one, Vakil and Zinger constructed a blow-up of the moduli space of stable maps and used it to define reduced Gromov-Witten invariants, which correspond to curve-counts in the main component. I will present a new definition of all-genera reduced Gromov-Witten invariants of complete intersections in projective spaces using desingularizations of sheaves. This is joint work with E. Mann, C. Manolache and R. Picciotto and can be found in arXiv:2310.06727. 

I’ll show that the topological recursion free energy of a family of elliptic spectral curves (which is related to Painlevé I through discrete Fourier transform) has a series expansion

when a parameter tends to be large, and its leading term is written by the Bernoulli number.

This shows the so-called conifold gap property in the above example.

I’ll also explain a potential application of the result to the resurgence property.

(Based on on-going joint work with N. Iorgov, O. Lisovyy and Y. Zhuravlov.)

Quantizing the mirror curve of a toric Calabi-Yau threefold gives rise to quantum-mechanical operators. Their fermionic spectral traces produce factorially divergent power series in the Planck constant and its inverse, which are conjecturally captured by the Nekrasov-Shatashvili and standard topological strings via the TS/ST correspondence. In this talk, I will discuss a general conjecture on the resurgence of these dual asymptotic series, and I will present a proven exact solution in the case of the first spectral trace of local P^2. A remarkable number-theoretic structure underpins the resurgent properties of the weak and strong coupling expansions and paves the way for new insights relating them to quantum modular forms. Finally, I will mention how these results fit into a broader paradigm linking resurgence and quantum modularity. This talk is based on arXiv:2212.10606 and further work in progress with V. Fantini. 

Joint work with Inder Kaur. When studying dynamics of birational automorphisms on a variety, it is natural to focus on those which are “primitive”, roughly meaning not coming from lower dimensional varieties. For Calabi—Yau varieties, Oguiso gave a useful criterion for primitivity of a map in terms of the associated linear map on cohomology. This talk will explain how to upgrade Oguiso’s criterion slightly by also keeping track of convex geometry, and give a new example of a primitive birational automorphism in dimension 3.