Monodromy Problem Research Articles

Monodromy of Picard–Fuchs system for a family of K3 toric hypersurfaces and fixed points of the Hilbert modular group for Q(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}(\\sqrt{2})$$\\end{document}

We solve the monodromy problem about the Picard–Fuchs system for the two-parameter family of the anticanonical hypersurfaces in the toric three-fold arising from a specific reflexive polytope. The members of the family are resolved to be K3 surfaces polarized by a lattice of signature (1, 17) with discriminant -8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-8$$\\end{document}. The Picard–Fuchs system consists of two hypergeometric differential equations of order 2 with the four-dimensional solution space. The independent variables are affine coordinates of the punctured weighted projective space P(1,2,3)\\{(1,0,0)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}(1,2,3)\\hspace{1.111pt}{\\setminus }\\hspace{1.111pt}\\{(1,0,0)\\}$$\\end{document} which forms a moduli space of the family. We construct the isomorphism from P(1,2,3)\\{(1,0,0)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}(1,2,3)\\hspace{1.111pt}{\\setminus }\\hspace{1.111pt}\\{(1,0,0)\\}$$\\end{document} to the symmetric Hilbert modular orbifold H2/⟨PSL2(OK),τ1⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^2/\\langle \ ext {PSL}_2(\\mathscr {O}_K),\ au _1\\rangle $$\\end{document} for K=Q(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K=\\mathbb {Q}(\\sqrt{2})$$\\end{document} and its inverse as a triplet of symmetric Hilbert modular forms. Here (1, 0, 0) corresponds to the cusp. As the fundamental set of solutions, we take a set of series convergent near the cusp. We obtain the generators of the monodromy group in GL4(Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {GL}_4(\\mathbb {Z})$$\\end{document} by deforming the domains for the period integrals expressing the solutions. We construct the isomorphism from the monodromy group to ⟨PSL2(OK),τ1⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \ ext {PSL}_2(\\mathscr {O}_K),\ au _1\\rangle $$\\end{document}. Using the monodromy, we obtain fixed points of the Hilbert modular group PSL2(OK)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {PSL}_2(\\mathscr {O}_K)$$\\end{document} and their respective isotropy groups, which are consistent with the results of the prior research taking an arithmetic approach.

Monodromy Problem and Tangential Center-Focus Problem for Products of Lines in General Position in P 2

Monodromy Problem and Tangential Center-Focus Problem for Products of Lines in General Position in P 2

Monodromy and Dulac's problem for piecewise analytical planar vector fields

Consider an analytical function f:V⊂R2→R having 0 as its regular value, a switching manifold Σ=f−1(0) and a piecewise analytical vector field X=(X+,X−), i.e. X± are analytical vector fields defined on Σ±={p∈V:±f(p)>0}. We characterize when the vector field X has a monodromic singular point in Σ, called Σ-monodromic singular point. Moreover, under certain conditions, we show that a Σ-monodromic singular point of X has a neighborhood free of limit cycles.

On symmetric solutions of the fourth q-Painlevé equation

The Painlevé equations possess transcendental solutions y(t) with special initial values that are symmetric under rotation or reflection in the complex t-plane. They correspond to monodromy problems that are explicitly solvable in terms of classical special functions. In this paper, we show the existence of such solutions for a q-difference Painlevé equation. We focus on symmetric solutions of a q-difference equation known as or and provide their symmetry properties and solve the corresponding monodromy problem.

Example of 4-pt non-vacuum [formula omitted] classical block

In this note we study a special case of the 4-pt non-vacuum classical block associated with the W3 algebra. We formulate the monodromy problem for the block and derive monodromy equations within heavy-light approximation. Fixing the remaining functional arbitrariness using parameters of the 4-pt vacuum W3 block, we compute the 4-pt non-vacuum W3 block function.

Generalized monodromy method in gauge/gravity duality

The method of monodromy is an important tool for computing Virasoro conformal blocks in a two-dimensional Conformal Field Theory (2d CFT) at large central charge and external dimensions. In deriving the form of the monodromy problem, which defines the method, one needs to insert a degenerate operator, usually a level-two operator, into the corresponding correlation function. It can be observed that the choice of which degenerate operator to insert is arbitrary, and they shall reveal the same physical principles underlying the method. In this paper, we exploit this freedom and generalize the method of monodromy by inserting higher-level degenerate operators. We illustrate the case with a level-three operator, and derive the corresponding form of the monodromy problem. We solve the monodromy problem perturbatively and numerically; and check that it agrees with the standard monodromy method, despite the fact that the two versions of the monodromy problem do not seem to be related in any obvious way. The forms corresponding to other higher-level degenerate operators are also discussed. We explain the physical origin of the coincidence and discuss its implication from a mathematical perspective.

On the tau function of the hypergeometric equation

The monodromy map for a rank-two system of differential equations with three Fuchsian singularities is classically solved by the Kummer formulæ for Gauss’ hypergeometric functions. We define the tau-function of such a system as the generating function of the extended monodromy symplectomorphism, using an idea recently developed. This formulation allows us to determine the dependence of the tau-function on the monodromy data. Using the explicit solution of the monodromy problem, the tau-function is then explicitly written in terms of Barnes G-function. In particular, if the Fuchsian singularities are placed to 0, 1 and ∞, this gives the structure constants of the asymptotical formula of Iorgov–Gamayun–Lisovyy for solutions of Painlevé VI equation.

Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models

Virasoro conformal blocks are universal ingredients of correlation functions of two-dimensional conformal field theories (2d CFTs) with Virasoro symmetry. It is acknowledged that in the (classical) limit of large central charge of the Virasoro algebra and large external, and intermediate conformal weights with fixed ratios of these parameters Virasoro blocks exponentiate to functions known as Zamolodchikovs’ classical blocks. The latter are special functions which have awesome mathematical and physical applications. Uniformization, monodromy problems, black holes physics, quantum gravity, entanglement, quantum chaos, holography, mathcal{N} = 2 gauge theory and quantum integrable systems (QIS) are just some of contexts, where classical Virasoro blocks are in use. In this paper, exploiting known connections between power series and integral representations of (quantum) Virasoro blocks, we propose new finite closed formulae for certain multi-point classical Virasoro blocks on the sphere. Indeed, combining classical limit of Virasoro blocks expansions with a saddle point asymptotics of Dotsenko-Fateev (DF) integrals one can relate classical Virasoro blocks with a critical value of the “Dotsenko-Fateev matrix model action”. The latter is the “DF action” evaluated on a solution of saddle point equations which take the form of Bethe equations for certain QIS (Gaudin spin models). A link with integrable models is our main motivation for this research line. For instance, analogous quantities as the “DF on-shell action” appear in 2d CFT realization of the so-called Richardson’s solution of the reduced BCS model describing physics of ultra-small superconducting grains. Precisely, the Richardson solution and its particular limit characterizing the rational Gaudin model have known implementation in 2d CFT in terms of a free field representation of certain (perturbed and unperturbed) WZW blocks. The WZW analogue of the “DF on-shell action” appears here and plays a crucial role, it is a generating function for eigenvalues of quantum integrals of motion. An exploration of relationships between power expansions and Coulomb gas representations of the aforementioned WZW blocks could pave a way for new analytical tools useful in the study of models of the Richardson-Gaudin type.

Hyperbolic three-string vertex

We begin developing tools to compute off-shell string amplitudes with the recently proposed hyperbolic string vertices of Costello and Zwiebach. Exploiting the relation between a boundary value problem for Liouville’s equation and a monodromy problem for a Fuchsian equation, we construct the local coordinates around the punctures for the generalized hyperbolic three-string vertex and investigate their various limits. This vertex corresponds to the general pants diagram with three boundary geodesics of unequal lengths. We derive the conservation laws associated with such vertex and perform sample computations. We note the relevance of our construction to the calculations of the higher-order string vertices using the pants decomposition of hyperbolic Riemann surfaces.

The monodromy problem for hyperelliptic curves

We study the Dynkin diagrams associated to the monodromy of direct sums of polynomials. The monodromy problem asks under which conditions on a polynomial, the monodromy of a vanishing cycle generates the whole homology of a regular fiber. We consider the case y4+g(x), which is a generalization of the results of Christopher and Mardešić about the monodromy problem for hyperelliptic curves. Moreover, we solve the monodromy problem for direct sums of fourth degree polynomials.

2-Parameter $$\tau $$-Function for the First Painlevé Equation: Topological Recursion and Direct Monodromy Problem via Exact WKB Analysis

We show that a 2-parameter family of $\tau$-functions for the first Painlev\'e equation can be constructed by the discrete Fourier transform of the topological recursion partition function for a family of elliptic curves. We also perform an exact WKB theoretic computation of the Stokes multipliers of associated isomonodromy system assuming certain conjectures.

Multi-centered higher spin solutions from {mathcal{W}}_N conformal blocks

Motivated by the question of bulk localization in holography, we study the problem of constructing multi-centered solutions in higher spin gravity which describe point particles in the interior of AdS3. In the Chern-Simons formulation these take into account the backreaction after adding Wilson line sources. We focus on chiral solutions where only the left-moving sector is excited. In that case it is possible to choose a gauge where the dynamical variables are a set of Toda fields living in the bulk. The problem then reduces to solving the {{mathcal{A}}_N}_{-1} Toda equations with delta function sources, which in turn requires solving an associated monodromy problem. We show that this monodromy problem is equivalent to the monodromy problem for a particular {mathcal{W}}_N vacuum conformal block at large central charge. Therefore, knowledge of the {mathcal{W}}_N vacuum block determines the multi-centered solution. Our calculations go beyond the heavy-light approximation by including the backreaction of all higher spin particles.

Probing beyond ETH at large c

We study probe corrections to the Eigenstate Thermalization Hypothesis (ETH) in the context of 2D CFTs with large central charge and a sparse spectrum of low dimension operators. In particular, we focus on observables in the form of non-local composite operators {mathcal{O}}_{mathrm{obs}}(x)={mathcal{O}}_L(x){mathcal{O}}_L(0) with hL ≪ c. As a light probe, {mathcal{O}}_{mathrm{obs}}(x) is constrained by ETH and satisfies {leftlangle {mathcal{O}}_{mathrm{obs}}(x)rightrangle}_{h_H}approx {leftlangle {mathcal{O}}_{mathrm{obs}}(x)rightrangle}_{mathrm{micro}} for a high energy energy eigenstate |hH〉. In the CFTs of interests, {{leftlangle {mathcal{O}}_{mathrm{obs}}(x)rightrangle}_h}_{{}_H} is related to a Heavy-Heavy-Light-Light (HL) correlator, and can be approximated by the vacuum Virasoro block, which we focus on computing. A sharp consequence of ETH for {mathcal{O}}_{mathrm{obs}}(x) is the so called “forbidden singularities”, arising from the emergent thermal periodicity in imaginary time. Using the monodromy method, we show that finite probe corrections of the form mathcal{O}left({h}_L/cright) drastically alter both sides of the ETH equality, replacing each thermal singularity with a pair of branch-cuts. Via the branch-cuts, the vacuum blocks are connected to infinitely many additional “saddles”. We discuss and verify how such violent modification in analytic structure leads to a natural guess for the blocks at finite c: a series of zeros that condense into branch cuts as c → ∞. We also discuss some interesting evidences connecting these to the Stoke’s phenomena, which are non-perturbative e−c effects. As a related aspect of these probe modifications, we also compute the Renyi-entropy Sn in high energy eigenstates on a circle. For subsystems much larger than the thermal length, we obtain a WKB solution to the monodromy problem, and deduce from this the entanglement spectrum.

Poles of Painlevé IV Rationals and their Distribution

We study the distribution of singularities (poles and zeros) of rational solutions of the Painlev\'e IV equation by means of the isomonodromic deformation method. Singularities are expressed in terms of the roots of generalised Hermite $H_{m,n}$ and generalised Okamoto $Q_{m,n}$ polynomials. We show that roots of generalised Hermite and Okamoto polynomials are described by an inverse monodromy problem for an anharmonic oscillator of degree two. As a consequence they turn out to be classified by the monodromy representation of a class of meromorphic functions with a finite number of singularities introduced by Nevanlinna. We compute the asymptotic distribution of roots of the generalised Hermite polynomials in the asymptotic regime when $m$ is large and $n$ fixed.

Weak nonlinear asymptotic solutions for the fourth order analogue of the second Painlevé equation

The fourth-order analogue of the second Painleve equation is considered. The monodromy manifold for a Lax pair associated with the P 2 2 equation is constructed. The direct monodromy problem for the Lax pair is solved. Asymptotic solutions expressed via trigonometric functions in the Boutroux variables along the rays ϕ = $$\frac{2}{5}$$ π(2n + 1) on the complex plane have been found by the isomonodromy deformations technique.

Multi-centered AdS3 solutions from Virasoro conformal blocks

We revisit the construction of multi-centered solutions in three-dimensional anti-de Sitter gravity in the light of the recently discovered connection between particle worldlines and classical Virasoro conformal blocks. We focus on multi-centered solutions which represent the backreaction of point masses moving on helical geodesics in global AdS3, and argue that their construction reduces to a problem in Liouville theory on the disk with Zamolodchikov-Zamolodchikov boundary condition. In order to construct the solution one needs to solve a certain monodromy problem which we argue is solved by a vacuum classical conformal block on the sphere in a particular channel. In this way we construct multi-centered gravity solutions by using conformal blocks special functions. We show that our solutions represent left-right asymmetric configurations of operator insertions in the dual CFT. We also provide a check of our arguments in an example and comment on other types of solutions.

Many-point classical conformal blocks and geodesic networks on the hyperbolic plane

We study the semiclassical holographic correspondence between 2d CFT n-point conformal blocks and massive particle configurations in the asymptotically AdS3 space. On the boundary we use the heavy-light approximation in which case two of primary operators are the background for the other (n-2) operators considered as fluctuations. In the bulk the particle dynamics can be reduced to the hyperbolic time slice. Although lacking exact solutions we nevertheless show that for any n the classical n-point conformal block is equal to the length of the dual geodesic network connecting n-3 cubic vertices of worldline segments. To this end, both the bulk and boundary systems are reformulated as potential vector fields. Gradients of the conformal block and geodesic length are given respectively by accessory parameters of the monodromy problem and particle momenta of the on-shell worldline action represented as a function of insertion points. We show that the accessory parameters and particle momenta are constrained by two different algebraic equation systems which nevertheless have the same roots thereby guaranteeing the correspondence.

Deformed SW curve and the null vector decoupling equation in Toda field theory

It is shown that the deformed Seiberg-Witten curve equation after Fourier transform is mapped into a differential equation for the AGT dual 2d CFT cnformal block containing an extra completely degenerate field. We carefully match parameters in two sides of duality thus providing not only a simple independent prove of the AGT correspondence in Nekrasov-Shatashvili limit, but also an extension of AGT to the case when a secondary field is included in the CFT conformal block. Implications of our results in the study of monodromy problems for a large class of n’th order Fuchsian differential equations are discussed.

Monodromy of a class of analytic generalized nilpotent systems through their Newton diagram

Newton diagram of a planar vector field allows to determine whether a singular point of an analytic system is a monodromic singular point. We solve the monodromy problem for the nilpotent systems and we apply our method to a wide family of systems with a degenerate singular point, so-called generalized nilpotent cubic systems.

On the monodromy problem for the four-punctured sphere

We consider the monodromy problem for the four-punctured sphere in which the character of one composite monodromy is fixed, by looking at the expansion of the accessory parameter in the modulus x directly, without taking the limit of the quantum conformal blocks for an infinite central charge. The integrals that appear in the expansion of the Volterra equation involve products of two hypergeometric functions to first order and up to four hypergeometric functions to second order. It is shown that all such integrals can be computed analytically. We give the complete analytical evaluation of the accessory parameter to first and second order in the modulus. The results agree with the evaluation obtained by assuming the exponentiation hypothesis of the quantum conformal blocks in the limit of infinite central charge. Extension to higher orders is discussed.