Schedule for: 18w5025 - Tau Functions of Integrable Systems and Their Applications

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In this talk we show how to compute the asymptotic behavior of the recurrence coefficients for polynomials orthogonal with respect to a logarithmic weight w(x)dx = [log 2k/(1-x)] dx on (-1; 1), k > 1, and verify a conjecture of A. Magnus for these coefficients. We use Riemann{Hilbert/steepest-descent methods, but not in the standard way as there is no known parametrix for the Riemann{Hilbert problem in a neighborhood of the logarithmic singularity at x = 1. This is joint work with Oliver Conway.

We will give a brief overview of the results contained in joint work with Bertola and Korotkin. The moduli space of Schodinger Equations over $M_g$ can be equipped with a symplectic structure by choosing a base section and identifying with $T^*M_g$. We prove homological coordinates are Darboux coordinates and characterize base sections giving equivalent symplectic structures. In addition we show the monodromy map is a symplectomorphism for base section Schottky, Wringer, and Bergman. These results can be compared with Kawai '96 (more recently Loustou '15, Takhtajan '17) where the map is shown to be a symplectomorphism with base Bers.

The key to understanding the development of singularities in the solution of isomondromy problems on the Riemann sphere is whether the underlying bundle is trivial or not. For isomonodromy problems over curves of higher genus, the analogous notion is that of stability of the bundle. It turns out that isomonodromy deformations are a good source of transverse deformations away from the unstable loci, whether the connection in question is regular, or has regular or irregular singularities. The proof is fairly classical deformation theory; joint work with Biswas and Heu.

We present a procedure which allows one to integrate explicitly the class of (checkerboard) incircular nets. This class of privileged congruences of lines in the plane is known to admit a great variety of geometric properties. The parametrisation obtained in this manner is reminiscent of that associated with elliptic billiards. Connections with discrete confocal coordinate systems and the fundamental QRT maps of integrable systems theory are made. The formalism is based on the existence of underlying pencils of conics and quadrics which is exploited in a Laguerre geometric setting.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

(joint with M.Mazzocco and V.Roubtsov) Our goal is to provide an effective description for Poisson and quantum algebras of monodromies for $SL_k$ systems on Riemann surfaces $\Sigma_{g,s,n}$ of genus $g$, with $s>0$ holes and with $n>0$ bordered cusps on the boundaries of holes. We use the Fock-Goncharov coordinates for higher Teichmuller spaces associated with $SL_k$ data on these surfaces and show that we can derive Poisson and quantum commutation relations between all monodromy matrices from the basic relation in an ideal triangle $\Sigma_{0,1,3}$ using the groupoid property. The obtained quantum algebras have an R-matrix form predicted by Korotkin and Samtleben and satisfy all relations of Fock-Rosly algebras. In the semiclassical limit, these algebras generate the Goldman bracket.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Colour/kinematics duality in Yang-Mills theory has been discovered by Bern, Carrasco and Johansson in 2008. It is a property of tree-level amplitudes, and states (roughly) that YM amplitudes can be represented as "squares" of the Lie algebraic colour structure together with some mysterious "kinematics" that satisfies the same Jacobi identities as the colour. This property is intriguing because it suggests that YM has some hidden structure that is completely invisible in its usual Lagrangian formulation. This talk will explain the statement of the colour/kinematic duality, and summarise what is currently understood as to why this property holds. There is complete understanding of this duality in the self-dual (integrable) sector of the theory. There is also partial understand for the full theory, and surprisingly, the structure of the Drinfeld double of the Lie algebra of vector fields turns out to play a crucial role.

In this talk I shall discuss $\tau$-functions of Painlevé equations and their applications. In particular I shall discuss orthogonal polynomials with respect to semi-classical weights, which are generalisations of the classical weights and arise in applications such as random matrices. It is well-known that orthogonal polynomials satisfy a three-term recurrence relation. For some semi-classical weights the coefficients in the recurrence relation can be expressed in terms of Hankel determinants, which are $\tau$-functions that also arise in the description of special function solutions of Painlev\'e equations. The determinants arise as partition functions in random matrix models and the recurrence coefficients satisfy a discrete Painlev\'e equation.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The cluster integrable systems will be introduced using their combinatorial representation, as dimer partition functions on bipartite graphs, as well as integrable systems on the Poisson submanifolds in co-extended loop groups. The discrete integrable flows can be constructed as sequences of cluster mutations. At nonvanishing total co-extension they turn into deautonomized systems of the Painleve type. I am also going to discuss few advanced issues like their Lax representation and quantization. based on joint works with M.Bershtein, P.Gavrylenko and M.Semenyakin

It has been shown by Bertola and Cafasso [Comm. Math. Phys. 352, 2017] that the Kontsevich matrix integral, which is the formal KdV tau function generating Witten intersection numbers on the moduli space of Riemann surfaces, has an interpretation as isomonodromic tau function. I will present this result and some close generalizations of it (open intersection numbers, r-spin intersection numbers, Gromov-Witten invariants of the Riemann sphere) along with applications of this isomodromic approach (effective generating functions, Virasoro constraints). Based on joint work with M. Bertola [arXiv:1711.03360].

My talk is the sequel of the talk of A. Marshakov. We discuss the solutions of the difference equations which appears as deatonomization of discrete flows. First we give a solution of the autonomous equations in terms of theta functions. Then solve deatonomized equations in terms of Nekrasov partition functions. This lead to q-deformation of the CFT formulas which appears in the talk of P. Gavrylenko. If time permits we also discuss quantization of these solutions. Based on joint works with A. Marshakov and P. Gavrylenko

Proof of the power series formula for the $q$-Painleve III tau function Gamayun-Iorgov-Lisovyy proposed the formula for Painleve tau functions as a Fourier series of conformal blocks. Two years ago q-deformation of this formula for Painleve III(D_8) was conjectured. This conjecture is equivalent to the bilinear relations on q-Virasoro conformal blocks. In my talk I will present the proof of these bilinear relations. The proof is based on introducing auxiliary tau functions which also have representation as the sum of conformal blocks. I will also discuss relations between these auxiliary tau functions and ABJ spectral determinants, GOE ensemble and Riemann-Hilbert problem. Based on joint work with Mikhail Bershtein.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We will assign a tau function to the Riemann-Hilbert problem set on a union of non-intersecting smooth closed curves with generic jump matrix. The main focus will be on the one-circle case, relevant to the analysis of Painlevé VI equation and its degenerations to Painlevé V and III. The tau functions in question will be defined as block Fredholm determinants of integral operators with explicit integrable kernels. It will be shown that the conventional Jimbo-Miwa-Ueno definition is recovered in the isomonodromic setting. As an application, I will explain how Fredholm determinants can be used to compute Dyson-Widom type constant in the asymptotics of the generic Painlevé VI tau function.

In the first part of the talk I will explain how principal minor expansion of the Fredholm determinant gives rise to explicit combinatorial formula for the general isomonodromic tau function. This formula with be also identified with the dual Nekrasov partition function. In the second part of the talk we will realize N*N isomonodromic tau function as a vacuum expectation value of some explicitly constructed vertex operators in the N-fermionic CFT. It will be also shown that this representation gives the same combinatorial formula, which will be identified with a series over W_N conformal blocks at c=N-1. Based on joint works with M. Cafasso, N. Iorgov, O. Lisovyy, A. Marshakov

Multiparametric families of hypergeometric $\tau$-functions of KP or Toda type serve as generating functions for weighted Hurwitz numbers, providing weighted enumerations of branched covers of the Riemann sphere. A graphical interpretation of the weighting is given in terms of constellations mapped onto the covering surface. The theory is placed within the framework of topological recursion, with the Baker function at ${\bf t} ={\bf 0}$ shown to satisfy the quantum spectral curve equation, whose classical limit is rational. A basis for the space of formal power series in the spectral variable is generated that is adapted to the Grassmannian element associated to the $\tau$-function. Multicurrent correlators are defined in terms of the $\tau$-function and shown to provide an alternative generating function for weighted Hurwitz numbers. Fermionic VEV representations are provided for the adapted bases, pair correlators and multicurrent correlators. Choosing the weight generating function as a polynomial, and restricting the number of nonzero ``second'' KP flow parameters in the Toda $\tau$-function to be finite implies a finite rank covariant derivative equation with rational coefficientw satisfied by a finite ``window'' of adapted basis elements. The pair correlator is shown to provide a Christoffel-Darboux type finite rank integrable kernel, and the WKB series coefficients of the associated adjoint system are computed recursively, leading to topological recursion relations for the generators of the weighted Hurwitz numbers. Based on joint work with: Alexander Alexandrov, Guillaume Chapuy and Bertrand Eynard.

We relate the isomonodromic tau functions with corresponding classical actions for all Painlevé equations. Such relation provides differential identities, required for asymptotic analysis of corresponding tau functions. We notice similar differential identities for general isomonodromic tau functions. We also present the Hamiltonian structure for the isomonodromic deformations corresponding to Painlevé equations and we conjecture such structure for general isomonodromic deformations. Joint work with Alexander Its.

For smooth symbols, the constant in the Szego-Widom Limit Theorem for determinants of finite block Toeplitz matrices can be described as a determinant of certain operator. The description in the scalar case can be made very explicit, but in the non-scalar case this is no longer true. This talk will focus on some examples where the constant, in the two by two case, can be made explicit. These will include the case of rational symbols, symbols from the special unitary group, the special linear group, and some combinations of these. Factorizations for such symbols will also be described. These are certain triangular factorizations and what are called root subgroup factorizations.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I shall present some recent results of our research project with P.G. Grinevich (LITP, RAS). We associate real algebraic-geometric data à la Krichever to any real regular multiline soliton solution of the Kadomtsev-Petviashvili II (KP) equation. These solutions correspond to a certain finite dimensional reduction of the Sato Grassmannian and their asymptotic behavior is known to be classified in terms of the combinatorial structure of the totally non-negative part of real Grassmannians $Gr^{TNN} (k,n)$. In our construction, using Postnikov classification of totally nonnegative Grassmannians, to any network representing a given soliton data in $Gr^{TNN} (k,n)$, we uniquely associate a rational degeneration of an M--curve of genus g and a real and regular degree g KP divisor on it. In particular, if we use the Le-network, then g is minimal and is equal to the dimension of the positroid cell to which the soliton datum belongs.

A purely numerical approach to compact Riemann surfaces starting from plane algebraic curves is presented. The critical points of the algebraic curve are computed via a two-dimensional Newton iteration. The starting values for this iteration are obtained from the resultants with respect to both coordinates of the algebraic curve and a suitable pairing of their zeros. A set of generators of the fundamental group for the complement of these critical points in the complex plane is constructed from circles around these points and connecting lines obtained from a minimal spanning tree. The monodromies are computed by solving the de ning equation of the algebraic curve on collocation points along these contours and by analytically continuing the roots. The collocation points are chosen to correspond to Chebychev collocation points for an ensuing Clenshaw?Curtis integration of the holomorphic differentials which gives the periods of the Riemann surface with spectral accuracy. At the singularities of the algebraic curve, Puiseux expansions computed by contour integration on the circles around the singularities are used to identify the holomorphic differentials. The Abel map is also computed with the Clenshaw-Curtis algorithm and contour integrals. A special approach is presented for hyperelliptic curves in Weierstrass normal form.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A famous result of Hitchin is that the moduli space of Higgs bundles on a compact Riemann surface has the structure on an algebraically integrable system, known as the Hitchin system. Central to the study of Hitchin systems is the construction of an associated family of spectral curves. It turns out that Hitchin spectral curves are equipped with exactly the right structure needed to perform topological recursion, yielding Eynard-Orantin invariants. The natural question that arises is to find a geometric meaning of the Eynard-Orantin invariants for Hitchin spectral curves. In this talk I will give a (very) partial answer in terms of Special Kahler geometry (for the genus zero invariants) and Bergman tau functions (for genus one invariants). This talk is based on joint work with Zhenxi Huang.

It is well-known that gap probabilities of some discrete probabilistic models can be computed using discrete Painlev\’e equations. However, choosing correct Painlev\’e coordinates and matching the model with the standard Painlev\’e dynamics is a highly non-trivial problem. In this talk we show how this can be done using the geometric tools of Sakai’s theory for the model of boxed plane partitions with generalized wights suggested by Borodin, Gorin, and Rains. An important feature of our result is its consistency with the degeneration schemes for the weights matching the degeneration scheme for discrete Painlev\’e equations. (joint work with Alisa Knizel)

We derive the expansion of the partition function of the Hermitian matrix model in the two cut regime. We calculate explicitly the first few terms of the expansion confirming the formula obtained by B. Eynard. This is a joint work with K. McLaughlin and T. Claeys

We construct the first known example of a cohomological field theory that takes values not only in the tautological cohomology ring of the moduli space, but also in the non-tautological part. This is a joint work with Rahul Pandharipande.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In the context of two dimensional conformal field theories (CFT), we review some analytical results describing the entanglement of disjoint intervals. In particular, we consider the Renyi entropies and on the moments of the partial transpose, which provide respectively the entanglement entropy and the logarithmic negativity through some replica limits. These analytic expressions are obtained as the partition function of the CFT model on some particular singular higher genus Riemann surfaces constructed through the replica method. For simple models like the compactified free boson and the Ising model, explicit expressions in terms of Riemann theta functions are presented. Numerical calculations on different lattice models which support the analytic results are also discussed.

In earlier work with Harold Widom, we obtained formulas for the probability in the asymmetric simple exclusion process (ASEP) that the mth particle from the left is at site x at time t. These formulas were expressed in general as sums of multiple integrals and, for the case of step initial condition, as an integral involving a Fredholm determinant. In recent work, these results were generalized to the case where the mth particle is the left-most one in a contiguous block of L particles. For the KPZ regime with step initial condition, we determine the conditional probability (asymptotically as t->infinity) that a particle is the beginning of an L-block, give that it is at site x at time t. Using a duality argument, we obtain the analogous result for a gap of G unoccupied sites between the particle at x and the next one. This is joint work with Harold Widom.

The multi-variate sigma function associated with an (n,s) curve has a series expansion whose coefficients are polynomials of coefficients of the defining equation of the curve. Therefore the sigma function has a well defined limit when the curve degenerates in any way. In this talk we study the degeneration of a hyperelliptic curve of genus g to a curve of genus g-1. Using the tau function and the Sato Grassmannian we show that the limit of the corresponding sigma function can be expressed as a sum of genus g-1 sigma functions. This gives an alternative proof of the results obtained by J. Bernatska and D. Leykin, who used the linear differential equations satisfied by the sigma function. This is a joint work with J. Bernatska and V. Enolski.

Some of the main results of \cite{CDG} (see also \cite{Eretico1} for a synthetic exposition with examples), concerning non-generic isomonodromy deformations of a certain linear differential system with irregular singularity and coalescing eigenvalues, are discussed from the point of view of Pfaffian systems adopted in \cite{guz}, making a distinction between weak and strong isomonodromic deformations. The results are motivated by the problem of extending to coalescent structures the analytic theory of Frobenius manifolds \cite{CDG1}, \cite{Eretico2}, \cite{C-et-all}. \vskip 1 cm $$\begin{thebibliography}{99} \bibitem{CDG} G. Cotti, B. Dubrovin. D. Guzzetti: Isomonodromy Deformations at an Irregular Singularity with Coalescing Eigenvalues. arXiv:1706.04808 (2017). To appear in Duke Math. J. \vskip 0.3 cm \bibitem{CDG1} G. Cotti, B. Dubrovin. D. Guzzetti: Local Moduli of Semisimple Frobenius Coalescent Structures. arXiv:1712.08575 (2017). \vskip 0.3 cm \bibitem{Eretico2} G. Cotti, D. Guzzetti: Analytic geometry of semisimple coalescent Frobenius structures. Random Matrices Theory Appl. 6 (2017), no. 4, 1740004, 36 pp. \vskip 0.3 cm \bibitem{Eretico1} G. Cotti, D. Guzzetti: Results on the Extension of Isomonodromy Deformations to the case of a Resonant Irregular Singularity. Random Matrices Theory Appl. (2018) https://doi.org/10.1142/S2010326318400038. \vskip 0.3 cm \bibitem{guz} D. Guzzetti: Notes on non-generic Isomonodromy Deformations. arXiv:1804.05688 (2018) \vskip 0.3 cm \bibitem{C-et-all} G. Cotti et al. Helix Structures, Quantum Cohomology of Fano Varieties and Monodromy Data. To appear \end{thebibliography}$$

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.