Null Kähler Geometry and Isomonodromic Deformations

We construct the normal forms of null-Kähler metrics: pseudo-Riemannian metrics admitting a compatible parallel nilpotent endomorphism of the tangent bundle. Such metrics are examples of non-Riemannian holonomy reduction, and (in the complexified setting) appear on the space of Bridgeland stability conditions on a Calabi–Yau threefold. Using twistor methods we show that, in dimension four—where there is a connection with dispersionless integrability—the cohomogeneity-one anti-self-dual null-Kähler metrics are generically characterised by solutions to Painlevé I or Painlevé II ODEs.

Theory space of one unitary matrix model and its critical behavior associated with Argyres–Douglas theory

The lowest critical point of one unitary matrix model with cosine plus logarithmic potential is known to correspond with the [Formula: see text] Argyres–Douglas (AD) theory and its double scaling limit derives the Painlevé II equation with parameter. Here, we consider the critical points associated with all cosine potentials and determine the scaling operators, their vacuum expectation values (vevs) and their scaling dimensions from perturbed string equations at planar level. These dimensions agree with those of [Formula: see text] AD theory.

A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants

We consider Fredholm determinants of matrix Hankel operators associated to matrix versions of the n-th Airy functions. Using the theory of integrable operators, we relate them to a fully noncommutative Painlevé II hierarchy, defined through a matrix-valued version of the Lenard operators. In particular, the Riemann-Hilbert techniques used to study these integrable operators allows to find a Lax pair for each member of the hierarchy. Finally, the coefficients of the Lax matrices are explicitly written in terms of the matrix-valued Lenard operators and some solutions of the hierarchy are written in terms of Fredholm determinants of the square of the matrix Airy Hankel operators.

A degenerate Gaussian weight connected with Painlevé equations and Heun equations

In this paper, we study the recurrence coefficients of a deformed Hermite polynomials orthogonal with respect to the weight [Formula: see text] where [Formula: see text] and [Formula: see text]. It is an extension of Chen and Feigin [J. Phys. A., Math. Gen. 39 (2006) 12381–12393]. By using the ladder operator technique, we show that the recurrence coefficients satisfy a particular Painlevé IV equation and the logarithmic derivative of the associated Hankel determinant satisfies the Jimbo–Miwa–Okamoto [Formula: see text] form of the Painlevé IV. Furthermore, the asymptotics of the recurrence coefficients and the Hankel determinant are obtained at the hard-edge limit and can be expressed in terms of the solutions to the Painlevé XXXIV and the [Formula: see text]-form of the Painlevé II equation at the soft-edge limit, respectively. In addition, for the special case [Formula: see text], we obtain the asymptotics of the Hankel determinant at the hard-edge limit via semi-classical Laguerre polynomials with respect to the weight [Formula: see text], which reproduced the result in Charlier and Deano, [Integr. Geom. Methods Appl. 14(2018) 018 (p. 43)].

Multicritical points of unitary matrix model with logarithmic potential identified with Argyres–Douglas points

In our recent publications, the partition function of the Gross–Witten–Wadia unitary matrix model with the logarithmic term has been identified with the [Formula: see text] function of a certain Painlevé system, and the double scaling limit of the associated discrete Painlevé equation to the critical point provides us with the Painlevé II equation. This limit captures the critical behavior of the [Formula: see text], [Formula: see text], [Formula: see text] supersymmetric gauge theory around its Argyres–Douglas 4D superconformal point. Here, we consider further extension of the model that contains the [Formula: see text]th multicritical point and that is to be identified with [Formula: see text] theory. In the [Formula: see text] case, we derive a system of two ODEs for the scaling functions to the free energy, the time variable being the scaled total mass and make a consistency check on the spectral curve on this matrix model.

Half-Space Stationary Kardar–Parisi–Zhang Equation

We study the solution of the Kardar–Parisi–Zhang (KPZ) equation for the stochastic growth of an interface of height h(x, t) on the positive half line, equivalently the free energy of the continuum directed polymer in a half space with a wall at $$x=0$$ x = 0 . The boundary condition $$\partial _x h(x,t)|_{x=0}=A$$ ∂ x h ( x , t ) | x = 0 = A corresponds to an attractive wall for $$A<0$$ A < 0 , and leads to the binding of the polymer to the wall below the critical value $$A=-1/2$$ A = - 1 / 2 . Here we choose the initial condition h(x, 0) to be a Brownian motion in $$x>0$$ x > 0 with drift $$-(B+1/2)$$ - ( B + 1 / 2 ) . When $$A+B \rightarrow -1$$ A + B → - 1 , the solution is stationary, i.e. $$h(\cdot ,t)$$ h ( · , t ) remains at all times a Brownian motion with the same drift, up to a global height shift h(0, t). We show that the distribution of this height shift is invariant under the exchange of parameters A and B. For any $$A,B > - 1/2$$ A , B > - 1 / 2 , we provide an exact formula characterizing the distribution of h(0, t) at any time t, using two methods: the replica Bethe ansatz and a discretization called the log-gamma polymer, for which moment formulae were obtained. We analyze its large time asymptotics for various ranges of parameters A, B. In particular, when $$(A, B) \rightarrow (-1/2, -1/2)$$ ( A , B ) → ( - 1 / 2 , - 1 / 2 ) , the critical stationary case, the fluctuations of the interface are governed by a universal distribution akin to the Baik–Rains distribution arising in stationary growth on the full-line. It can be expressed in terms of a simple Fredholm determinant, or equivalently in terms of the Painlevé II transcendent. This provides an analog for the KPZ equation, of some of the results recently obtained by Betea–Ferrari–Occelli in the context of stationary half-space last-passage-percolation. From universality, we expect that limiting distributions found in both models can be shown to coincide.