Fay Identities Research Articles

Fay Identities of Pfaffian Type for Hyperelliptic Curves

We establish identities of Pfaffian type for the theta function associated with twice or half the period matrix of a hyperelliptic curve. They are implied by the large size asymptotic analysis of exact Pfaffian identities for expectation values of ratios of characteristic polynomials in ensembles of orthogonal or quaternionic self-dual random matrices. We show that they amount to identities for the theta function with the period matrix of a hyperelliptic curve, and in this form we reprove them by direct geometric methods.

Basis decompositions of genus-one string integrals

One-loop scattering amplitudes in string theories involve configuration-space integrals over genus-one surfaces with coefficients of Kronecker-Eisenstein series in the integrand. A conjectural genus-one basis of integrands under Fay identities and integration by parts was recently constructed out of chains of Kronecker-Eisenstein series. In this work, we decompose a variety of more general genus-one integrands into the conjectural chain basis. The explicit form of the expansion coefficients is worked out for infinite families of cases where the Kronecker-Eisenstein series form cycles. Our results can be used to simplify multiparticle amplitudes in supersymmetric, heterotic and bosonic string theories and to investigate loop-level echoes of the field-theory double-copy structures of string tree-level amplitudes. The multitude of basis reductions in this work strongly validate the recently proposed chain basis and stimulate mathematical follow-up studies of more general configuration-space integrals with additional marked points or at higher genus.

Boson–Fermion correspondence of the multi-component constrained mKP hierarchy

We generalize the constrained modified KP (mKP) hierarchy to the multi-component version whose tau function can be expressed as the vacuum expectation value of Clifford operators using free fermions. Meanwhile, we extend the relevant theory of the Boson–Fermion correspondence in single component constrained mKP hierarchy to the multi-component version. This leads us to obtain the rational and soliton solutions of the multi-component constrained mKP hierarchy utilizing the Boson–Fermion correspondence. Furthermore, we give five sets of differential Fay identities and two sets of difference Fay identities of the multi-component mKP hierarchy.

Basis decompositions and a Mathematica package for modular graph forms

Modular graph forms (MGFs) are a class of non-holomorphic modular forms which naturally appear in the low-energy expansion of closed-string genus-one amplitudes and have generated considerable interest from pure mathematicians. MGFs satisfy numerous non-trivial algebraic- and differential relations which have been studied extensively in the literature and lead to significant simplifications. In this paper, we systematically combine these relations to obtain basis decompositions of all two- and three-point MGFs of total modular weight , starting from just two well-known identities for banana graphs. Furthermore, we study previously known relations in the integral representation of MGFs, leading to a new understanding of holomorphic subgraph reduction as Fay identities of Kronecker–Eisenstein series and opening the door toward decomposing divergent graphs. We provide a computer implementation for the manipulation of MGFs in the form of the Mathematica package ModularGraphForms which includes the basis decompositions obtained.

Darboux transformations and Fay identities for the extended bigraded Toda hierarchy**The first author is supported in part by Simons Foundation Grants 279074 and 584741.

The extended bigraded Toda hierarchy (EBTH) is an integrable system satisfied by the Gromov–Witten total descendant potential of with two orbifold points. We write a bilinear equation for the tau-function of the EBTH and derive Fay identities from it. We show that the action of Darboux transformations on the tau-function is given by vertex operators. As a consequence, we obtain generalized Fay identities.

On the modified KP hierarchy: Tau functions, squared eigenfunction symmetries and additional symmetries

In this paper, the modified KP hierarchy in the Kupershmidt–Kiso version is investigated starting from the tau functions. There are two tau functions for the modified KP hierarchy. We firstly give some properties about the tau functions, including Hirota bilinear identities and Fay identities. And it is showed that these two functions can be viewed as two different tau functions of the KP hierarchy. Then after constructing the spectral representations of the eigenfunction and the adjoint eigenfunctions, the expressions of all the squared eigenfunction potentials in the modified KP hierarchy are obtained. Next, the study of the squared eigenfunction symmetries and the additional symmetries is considered, whose actions on the two tau functions are derived. At last, the Virasoro additional symmetries in the constrained cases are given, also with the actions on the tau functions.

Quantum Baxter-Belavin R-matrices and multidimensional lax pairs for Painlevé VI

Quantum elliptic R-matrices satisfy the associative Yang-Baxter equation in Mat(N)⊗2, which can be regarded as a noncommutative analogue of the Fay identity for the scalar Kronecker function. We present a broader list of R-matrix-valued identities for elliptic functions. In particular, we propose an analogue of the Fay identities in Mat(N)⊗2. As an application, we use the ℤ N ×ℤ N elliptic R-matrix to construct R-matrix-valued 2N 2×2N 2 Lax pairs for the Painlevé VI equation (in the elliptic form) with four free constants. More precisely, the case with four free constants corresponds to odd N, and even N corresponds to the case with a single constant in the equation.

Planck constant as spectral parameter in integrable systems and KZB equations

We construct special rational ${\rm gl}_N$ Knizhnik-Zamolodchikov-Bernard (KZB) equations with $\tilde N$ punctures by deformation of the corresponding quantum ${\rm gl}_N$ rational $R$-matrix. They have two parameters. The limit of the first one brings the model to the ordinary rational KZ equation. Another one is $\tau$. At the level of classical mechanics the deformation parameter $\tau$ allows to extend the previously obtained modified Gaudin models to the modified Schlesinger systems. Next, we notice that the identities underlying generic (elliptic) KZB equations follow from some additional relations for the properly normalized $R$-matrices. The relations are noncommutative analogues of identities for (scalar) elliptic functions. The simplest one is the unitarity condition. The quadratic (in $R$ matrices) relations are generated by noncommutative Fay identities. In particular, one can derive the quantum Yang-Baxter equations from the Fay identities. The cubic relations provide identities for the KZB equations as well as quadratic relations for the classical $r$-matrices which can be halves of the classical Yang-Baxter equation. At last we discuss the $R$-matrix valued linear problems which provide ${\rm gl}_{\tilde N}$ Calogero-Moser (CM) models and Painleve equations via the above mentioned identities. The role of the spectral parameter plays the Planck constant of the quantum $R$-matrix. When the quantum ${\rm gl}_N$ $R$-matrix is scalar ($N=1$) the linear problem reproduces the Krichever's ansatz for the Lax matrices with spectral parameter for the ${\rm gl}_{\tilde N}$ CM models. The linear problems for the quantum CM models generalize the KZ equations in the same way as the Lax pairs with spectral parameter generalize those without it.

The multicomponent KP hierarchy: differential Fay identities and Lax equations

In this paper, we show that four sets of differential Fay identities of an N-component KP hierarchy derived from the bilinear relation satisfied by the tau function of the hierarchy are sufficient to derive the auxiliary linear equations for the wavefunctions. From this, we derive the Lax representation for the N-component KP hierarchy, which are equations satisfied by some pseudo-differential operators with matrix coefficients. Besides the Lax equations with respect to the time variables proposed in Date et al (1981 J. Phys. Soc. Japan 50 3806–12), we also obtain a set of equations relating different charge sectors, which can be considered as a generalization of the modified KP hierarchy proposed in Takebe (2002 Lett. Math. Phys. 59 157–72).

Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy

Recently the study of Fay-type identities revealed some new features of the DKP hierarchy (also known as "the coupled KP hierarchy" and "the Pfaff lattice"). Those results are now extended to a Toda version of the DKP hierarchy (tentatively called "the Pfaff-Toda hierarchy"). Firstly, an auxiliary linear problem of this hierarchy is constructed. Unlike the case of the DKP hierarchy, building blocks of the auxiliary linear problem are difference operators. A set of evolution equations for dressing operators of the wave functions are also obtained. Secondly, a system of Fay-like identities (difference Fay identities) are derived. They give a generating functional expression of auxiliary linear equations. Thirdly, these difference Fay identities have well defined dispersionless limit (dispersionless Hirota equations). As in the case of the DKP hierarchy, an elliptic curve is hidden in these dispersionless Hirota equations. This curve is a kind of spectral curve, whose defining equation is identified with the characteristic equation of a subset of all auxiliary linear equations. The other auxiliary linear equations are related to quasi-classical deformations of this elliptic spectral curve.

Universal Whitham hierarchy, dispersionless Hirota equations and multicomponent KP hierarchy

The goal of this paper is to identify the universal Whitham hierarchy of genus zero with a dispersionless limit of the multicomponent KP hierarchy. To this end, the multicomponent KP hierarchy is (re)formulated to depend on several discrete variables called “charges”. These discrete variables play the role of lattice coordinates in underlying Toda field equations. A multicomponent version of the so called differential Fay identity are derived from the Hirota equations of the τ -function of this “charged” multicomponent KP hierarchy. These multicomponent differential Fay identities have a well-defined dispersionless limit (the dispersionless Hirota equations). The dispersionless Hirota equations turn out to be equivalent to the Hamilton–Jacobi equations for the S -functions of the universal Whitham hierarchy. The differential Fay identities themselves are shown to be a generating functional expression of auxiliary linear equations for scalar-valued wave functions of the multicomponent KP hierarchy.

Functional representations of integrable hierarchies

We consider a general framework for integrable hierarchies in Lax form and derive certain universal equations from which `functional representations' of particular hierarchies (like KP, discrete KP, mKP, AKNS), i.e. formulations in terms of functional equations, are systematically and quite easily obtained. The formalism genuinely applies to hierarchies where the dependent variables live in a noncommutative (typically matrix) algebra. The obtained functional representations can be understood as `noncommutative' analogs of `Fay identities' for the KP hierarchy.

Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains

We study the integrable structure of the Dirichlet boundary problem in two dimensions and extend the approach to the case of planar multiply-connected domains. The solution to the Dirichlet boundary problem in the multiply-connected case is given through a quasiclassical tau-function, which generalizes the tau-function of the dispersionless Toda hierarchy. It is shown to obey an infinite hierarchy of Hirota-like equations which directly follow from properties of the Dirichlet Green function and from the Fay identities. The relation to multi-support solutions of matrix models is briefly discussed.

Pfaff $\tau$ -functions

Consider the two-dimensional Toda lattice, with certain skew-symmetric initial condition, which is preserved along the locus \(s=-t\) of the space of time variables. Restricting the solution to \(s=-t\), we obtain another hierarchy called Pfaff lattice, which has its own tau function, being equal to the square root of the restriction of 2D-Toda tau function. We study its bilinear and Fay identities, W and Virasoro symmetries, relation to symmetric and symplectic matrix integrals and quasiperiodic solutions.

Ernst equation, Fay identities and variational formulas on hyperelliptic curves

We present a unified approach to theta-functional solutions of the stationary axisymmetric Einstein equations in vacuum. Using Fay's trisecant identity and variational formulas on hyperelliptic Riemann surfaces, we establish formulas for the metric functions, the Ernst potential and their derivatives.

Generating functions for intersection numbers on moduli spaces of curves

Using the connection between intersection theory on the Deligne-Mumford spaces ℳ¯g,n and the edge scaling of the GUE matrix model, we express the n-point functions for the intersection numbers as n-dimensional error-function-type integrals and also give a derivation of Witten's KdV equations using the higher Fay identities of Adler, Shiota, and van Moerbeke.

The KP hierarchy in Miwa coordinates

A systematic reformulation of the KP hierarchy by using continuous Miwa variables is presented. Basic quantities and relations are defined and determinantal expressions for Fay's identities are obtained. It is shown that in terms of these variables the KP hierarchy gives rise to a Darboux system describing an infinite-dimensional conjugate net.

Chiral fermions on a compact Riemann surface: A sheaf cohomology analysis

We prove that the correlation functions of a system of chiral fermions on a compact Riemann surface are determined by postulating their behaviour at coincident points and a principle of maximal analyticity. The proof proceeds by a reformulation as a problem of sheaf cohomology. Wick's theorem and the Fay identities are rigorous consequences of our analysis.