Physical Mathematics Seminars

Abstract: Hermitian matrix models, together with their double-scaling limits, offer powerful frameworks for studying non-perturbative phenomena across string theory. Despite significant advances, exact global solutions remain elusive due to intricate analytic structures and complex phase behaviour. To address this, we construct local transseries solutions that capture both perturbative and non-perturbative contributions. Using a novel technique called diagonal framing, we reorganize transseries sectors to manifest background independence, improve convergence, and simplify resummation. These solutions deepen the understanding of non-perturbative effects by showing how transseries can be explicitly resummed into finite, fully non-perturbative results. Moreover, by determining all Stokes data, we promote the transseries to global exact solutions across all phases, laying the groundwork for studying phase transitions, dualities, and broader applications within string theory. 

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Abstract: The large-charge expansion is a powerful approach for analytically accessing strongly-coupled models at or near criticality. We focus on a sector of fixed and large global charge where it is possible to express observables as an expansion in inverse powers of the charge.

I will first introduce the approach using the simplest relativistic case, namely the O(2) model. Then I will discuss the large-charge expansion of non-relativistic models with applications to ultracold Fermi gases and nuclear physics. 

Abstract: Motivated by the recent construction of grey galaxy and Dual Dressed Black Hole solutions in AdS5 × S5, in this talk, I will present a conjecture relating to the large N entropy of supersymmetric states in N=4 Yang-Mills theory. We conjecture the existence of a large number of supersymmetric states which can be thought of as a non-interacting mix of supersymmetric black holes and supersymmetric `gravitons'. It predicts a microcanonical phase diagram of supersymmetric states and makes a sharp prediction for the supersymmetric entropy (as a function of 5 charges) in each phase. We also predict a large N formula for the superconformal index as a function of indicial charges and predict a microcanonical indicial phase diagram with nine distinct phases. It predicts agreement between the superconformal index and black hole entropy in one phase (so over one range of charges), but disagreement in other phases (and so at other values of charges).  

Abstract: The low-energy theory of 4d N=2 supersymmetric gauge theories is encoded in the Seiberg-Witten (SW) curve which is naturally related to an integrable system. For SU(2) gauge theories, in presence of Omega-background this integrable system is given by Painlevé equations. In this talk we will show that the Nekrasov partition functions of 4d SU(2) susy gauge theories on the blowup of spacetime are tau functions which solve the Painlevé equations in Hirota bilinear form. These solutions can be expressed as expansions in terms of the moduli of the quantum SW curve and the construction can be applied also to non-Lagrangian theories. Furthermore, these solutions have manifest modular properties which provide a natural non-perturbative completion of the corresponding topological string partition function and they directly lead to the BCOV holomorphic anomaly equations of the topological string.

Abstract: Anti-de Sitter space acts as an infra-red cutoff for asymptotically free theories, allowing interpolation between a weakly-coupled small-sized regime and a strongly-coupled flat-space regime. We discuss this interpolation in the context of Yang-Mills theories in AdS from the perspective of boundary conformal theories and its implications for the confinement/deconfinement transition. We find indications that at the transition a singlet scalar operator becomes marginal, destabilizing the deconfined phase existing at a small size and triggering a boundary renormalization group flow to a gapped, confined phase that smoothly connects to flat space.

Abstract: Joyce structures were introduced by T. Bridgeland in the context of the space of stability conditions Stab(X) of a 3d Calabi-Yau category X and its associated Donaldson-Thomas (DT) invariants. They are described by a certain C^*-family of non-linear flat connections on the tangent bundle T(Stab(X)) of Stab(X) and, under certain non-degeneracy conditions, encode a complex-hyperkähler structure on T(Stab(X)). On the other hand, the work of Gaiotto-Moore-Neitzke (GMN) shows how to construct from the Coulomb branch M of a 4d N=2 theory and its associated BPS indices, a real-hyperkähler (HK) geometry on the cotangent bundle T*M of M. This HK metric is the instanton corrected metric associated with the 3d theory obtained by compactifying the 4d N=2 theory on S^1. In this talk, I will try to reinterpret the GMN construction in a way that is similar to Joyce structures. The resulting structure is called a "special Joyce structure", and encodes an HK geometry on T*M. At least in the case where the BPS indices are uncoupled, the HK metric encoded by the corresponding special Joyce structure is shown to match the GMN HK metric. As a by-product of this description, we will see that the GMN HK geometry in the uncoupled case can be encoded in the geometry of the base M, together with a single function J on T^*M determined by the BPS indices. The function J seems to be related to a function previously studied by S. Alexandrov and B. Pioline in arXiv:1808.08479, which they call the "instanton generating function". This talk is based on arXiv:2403.00548.

Abstract : Local Calabi-Yau singularities are a playground for counting BPS states, and indeed, for toric varieties, the techniques such as the topological vertex, for computing Gopakumar-Vafa invariants are very powerful.

There is, however, an interesting class of *non-toric* CY local threefolds, of which  the conifold is a special case, that are constructed as ALE fibrations over the complex plane. They are known as higher-length flops. 

For those cases, my collaborators and I have stumbled upon a new duality frame that allowed us to compute GV invariants independently of previously known methods.

In this talk, I will explain the concept of “simultaneous partial resolutions”, which is a group-theoretic way of building such threefolds.

I will then show, that M-theory on such geometries can be dualized to IIA with intersecting D6-branes, where the GV invariants translate to elementary open string Ext^1 computations.

Finally, I will present some results for CY threefolds that do not admit crepant resolutions. In this case, we are essentially giving a new definition of GV invariants for threefolds that do not have compact curves.

Abstract:  I will review basic properties of W-algebras, in particular the WN family and the associated Winfinity algebra. These algebraic structures show up at many places in mathematical physics. Winfinity admits two different descriptions: the traditional description starts from the Virasoro algebra of 2d conformal field theory and extends it by local conserved currents of higher spin. The description discovered more recently is the Yangian description manifesting the integrable structure of the algebra. The map between these two pictures is non-trivial, but can be understood by using the Maulik-Okounkov 'instanton' R-matrix as a bridge between these two pictures. R-matrix allows for construction of infinite sets of commuting quantities and there are two different sets of Bethe ansatz equations known that can be used to diagonalize these higher conserved quantities. In particular, one of these are very similar to Bethe equations of the simplest Heisenberg XXX SU(2) spin chain, while the other generalizes the equations for Calogero-like particles.

Abstract: Generating series of HOMFLYPT polynomials colored by symmetric representations have been found to coincide with partition functions of motivic Donldson-Thomas invariants of symmetric quivers, after a suitable identification of variables. I will discuss an interpretation of this relation based on string theory, where quivers encode interactions of M2 branes mediated by an M5 brane. 

Invariance of this picture under deformations leads to a generalization of the knots-quivers correspondence corroborated by wall-crossing type phenomena associated with skein relations among M2 brane boundaries. A generalization to skein valued curve counts, corresponding to counts of M2 on a stack with multiple M5 branes, will be discussed. Based on joint works with Ekholm and Kucharski and ongoing work with Ekholm and Nakamura.

Abstract: It is well known that perturbative series encode information about many – but not necessarily all -- nonperturbative effects and that both can be unified in a so-called ‘transseries’. Such transseries are universally present in physics, but relatively few examples are known in closed form. In this talk I will discuss one-dimensional quantum mechanical oscillators and demonstrate how one obtains a transseries solution for the energy spectrum which contains all instanton corrections in closed form. I’ll subsequently elucidate the resurgent structure of this transseries using alien calculus and demonstrate that in more nontrivial models, these transseries are not ‘minimally resurgent’, but factorize into two distinct transseries structures. This talk is based on joint work with Marcel Vonk.

Abstract:  

The 't Hooft model, 2d QCD in the large N limit, offers a unique playground for exploring the dynamics of confinement in gauge theories.  In this talk, based on joint works with S. Komatsu, I will illustrate that the Fredholm integral equation ('t Hooft equation) determining the masses of mesons in the model is equivalent to finding solutions to a TQ-Baxter equation.  This reformulation of the problem illustrates a rich analytical structure of the spectrum in the complex plane of the quark masses, and makes possible to extract systematic analytical expansions for spectral determinants, energy levels, and wavefunctions. I will comment on possible connections between our techniques and TS/ST correspondence. 

Remarkably, this integrable structure is unique to the 't Hooft model, but persists also in the broad class of theories called generalized QCD, obtained by replacing the gluon kinetic term with a BF coupling plus a potential for the B-field. I will show that, also in this case, the associated 't Hooft equation is recasted into a TQ-equation with a transfer matrix given in closed form for any given potential. Applying the techniques developed for the 't Hooft model to these theories allows for a novel systematic study of the spectrum in the complex space of the couplings. 

ABSTRACT: Given a Riemann surface C and a central charge c, one can define the notion of Virasoro conformal block. Virasoro conformal blocks capture universal features of conformal field theory on C. I will describe a new scheme for constructing Virasoro conformal blocks at central charge c=1, by relating them to simpler "abelian" objects, namely conformal blocks for the Heisenberg algebra, on a branched double cover of C. The key new ingredient is a spectral network on the surface C. Some particularly important Virasoro conformal blocks at c=1 are also known as "tau functions", and I will explain what abelianization tells us about them. This is joint work in progress with Qianyu Hao, inspired by work of Coman-Longhi-Pomoni-Teschner, Iwaki, Marino and others.

Abstract:  In the presence of confinement, the Einstein field equations are expected to exhibit turbulent dynamics. The simplest example of such behaviour is described by the AdS instability conjecture, put forward by Dafermos and Holzegel in 2006; this conjecture suggests that generic small perturbations of the AdS initial data lead to the formation of trapped surfaces when reflecting boundary conditions are imposed at conformal infinity. However, whether a similar scenario also holds in the more complicated case of the exterior region of an asymptotically AdS black hole spacetime has been the subject of debate.

In this talk, we will discuss a rigorous proof of the AdS instability conjecture for Einstein--scalar field system reduced under spherical symmetry. We will also show that weak turbulence does emerge in the dynamics of a quasilinear toy model for the vacuum Einstein equations on the Schwarzschild-AdS exterior spacetimes for an open and dense set of black hole mass parameters. The latter is joint work with Christoph Kehle.

Abstract: A central problem in quantum gravity is to get a quantitative microscopic interpretation of the Bekenstein-Hawking entropy of black holes. In type II strings compactified on a Calabi-Yau manifold, BPS black hole microstates are realized by bound states of D-branes wrapped along complex submanifolds, or in mathematical terms by "stable coherent sheaves". String dualities predict that suitable generatingseries of indices counting such BPS states have modular properties, although the mathematical origin of modularity is still mysterious. I will explain some recent progress in computing these BPS indices using relations to topological string theory and wall-crossing, and present strong evidence that modularity is indeed at work. Based on [arXiv:2301.08066] in collaboration with S. Alexandrov, S. Feyzbakshsh, A. Klemm and T. Schimannek, and sequel to appear.

Abstract: The resurgent structures of the first fermionic spectral trace of local P2 (computed separately by C. Rella and by M. Mariño and J. Gu) are simple yet very rich. In this talk, I will focus on their conjectural relationship with quantum modular forms and possible applications. This is based on a joint project with C. Rella.