Tau function (integrable systems)

Tau functions are an important ingredient in the modern mathematical theory of integrable systems, and have numerous applications in a variety of other domains. They were originally introduced by Ryogo Hirota[1] in his direct method approach to soliton equations, based on expressing them in an equivalent bilinear form.

The term tau function, or -function, was first used systematically by Mikio Sato[2] and his students[3][4] in the specific context of the Kadomtsev–Petviashvili (or KP) equation and related integrable hierarchies. It is a central ingredient in the theory of solitons. In this setting, given any -function satisfying a Hirota-type system of bilinear equations (see § Hirota bilinear residue relation for KP tau functions below), the corresponding solutions of the equations of the integrable hierarchy are explicitly expressible in terms of it and its logarithmic derivatives up to a finite order. Tau functions also appear as matrix model partition functions in the spectral theory of random matrices,[5][6][7] and may also serve as generating functions, in the sense of combinatorics and enumerative geometry, especially in relation to moduli spaces of Riemann surfaces, and enumeration of branched coverings, or so-called Hurwitz numbers.[8][9][10]

There are two notions of -functions, both introduced by the Sato school. The first is isospectral -functions of the SatoSegal–Wilson type[2][11] for integrable hierarchies, such as the KP hierarchy, which are parametrized by linear operators satisfying isospectral deformation equations of Lax type. The second is isomonodromic -functions.[12]

Depending on the specific application, a -function may either be: 1) an analytic function of a finite or infinite number of independent, commuting flow variables, or deformation parameters; 2) a discrete function of a finite or infinite number of denumerable variables; 3) a formal power series expansion in a finite or infinite number of expansion variables, which need have no convergence domain, but serves as generating function for certain enumerative invariants appearing as the coefficients of the series; 4) a finite or infinite (Fredholm) determinant whose entries are either specific polynomial or quasi-polynomial functions, or parametric integrals, and their derivatives; 5) the Pfaffian of a skew symmetric matrix (either finite or infinite dimensional) with entries similarly of polynomial or quasi-polynomial type. Examples of all these types are given below.

In the Hamilton–Jacobi approach to Liouville integrable Hamiltonian systems, Hamilton's principal function, evaluated on the level surfaces of a complete set of Poisson commuting invariants, plays a role similar to the -function, serving both as a generating function for the canonical transformation to linearizing canonical coordinates and, when evaluated on simultaneous level sets of a complete set of Poisson commuting invariants, as a complete solution of the Hamilton–Jacobi equation.

Tau functions: isospectral and isomonodromic

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A  -function of isospectral type is defined as a solution of the Hirota bilinear equations (see § Hirota bilinear residue relation for KP tau functions below), from which the linear operator undergoing isospectral evolution can be uniquely reconstructed. Geometrically, in the Sato[2] and Segal-Wilson[11] sense, it is the value of the determinant of a Fredholm integral operator, interpreted as the orthogonal projection of an element of a suitably defined (infinite dimensional) Grassmann manifold onto the origin, as that element evolves under the linear exponential action of a maximal abelian subgroup of the general linear group. It typically arises as a partition function, in the sense of statistical mechanics, many-body quantum mechanics or quantum field theory, as the underlying measure undergoes a linear exponential deformation.

Isomonodromic  -functions for linear systems of Fuchsian type are defined below in § Fuchsian isomonodromic systems. Schlesinger equations. For the more general case of linear ordinary differential equations with rational coefficients, including irregular singularities, they are developed in reference.[12]

Hirota bilinear residue relation for KP tau functions

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A KP (Kadomtsev–Petviashvili)  -function   is a function of an infinite collection   of variables (called KP flow variables) that satisfies the bilinear formal residue equation

  (1)

identically in the   variables, where   is the   coefficient in the formal Laurent expansion resulting from expanding all factors as Laurent series in  , and

 

As explained below in the section § Formal Baker-Akhiezer function and the KP hierarchy, every such  -function determines a set of solutions to the equations of the KP hierarchy.

Kadomtsev–Petviashvili equation

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If   is a KP  -function satisfying the Hirota residue equation (1) and we identify the first three flow variables as

 

it follows that the function

 

satisfies the   (spatial)  (time) dimensional nonlinear partial differential equation

  (2)

known as the Kadomtsev-Petviashvili (KP) equation. This equation plays a prominent role in plasma physics and in shallow water ocean waves.

Taking further logarithmic derivatives of   gives an infinite sequence of functions that satisfy further systems of nonlinear autonomous PDE's, each involving partial derivatives of finite order with respect to a finite number of the KP flow parameters  . These are collectively known as the KP hierarchy.

Formal Baker–Akhiezer function and the KP hierarchy

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If we define the (formal) Baker-Akhiezer function   by Sato's formula[2][3]

 

and expand it as a formal series in the powers of the variable  

 

this satisfies an infinite sequence of compatible evolution equations

  (3)

where   is a linear ordinary differential operator of degree   in the variable  , with coefficients that are functions of the flow variables  , defined as follows

 

where   is the formal pseudo-differential operator

 

with  ,

 

is the wave operator and   denotes the projection to the part of   containing purely non-negative powers of  ; i.e. the differential operator part of   .

The pseudodifferential operator   satisfies the infinite system of isospectral deformation equations

  (4)

and the compatibility conditions for both the system (3) and (4) are

  (5)

This is a compatible infinite system of nonlinear partial differential equations, known as the KP (Kadomtsev-Petviashvili) hierarchy, for the functions  , with respect to the set   of independent variables, each of which contains only a finite number of  's, and derivatives only with respect to the three independent variables  . The first nontrivial case of these is the Kadomtsev-Petviashvili equation (2).

Thus, every KP  -function provides a solution, at least in the formal sense, of this infinite system of nonlinear partial differential equations.

Isomonodromic systems. Isomonodromic tau functions

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Fuchsian isomonodromic systems. Schlesinger equations

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Consider the overdetermined system of first order matrix partial differential equations

  (6)
  (7)

where   are a set of     traceless matrices,   a set of   complex parameters,   a complex variable, and   is an invertible   matrix valued function of   and  . These are the necessary and sufficient conditions for the based monodromy representation of the fundamental group   of the Riemann sphere punctured at the points   corresponding to the rational covariant derivative operator

 

to be independent of the parameters  ; i.e. that changes in these parameters induce an isomonodromic deformation. The compatibility conditions for this system are the Schlesinger equations[12]

  (8)

Isomonodromic  -function

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Defining   functions

  (9)

the Schlesinger equations (8) imply that the differential form

 

on the space of parameters is closed:

 

and hence, locally exact. Therefore, at least locally, there exists a function   of the parameters, defined within a multiplicative constant, such that

 

The function   is called the isomonodromic  -function associated to the fundamental solution   of the system (6), (7).

Hamiltonian structure of the Schlesinger equations

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Defining the Lie Poisson brackets on the space of  -tuples   of   matrices:

 
 

and viewing the   functions   defined in (9) as Hamiltonian functions on this Poisson space, the Schlesinger equations (8) may be expressed in Hamiltonian form as [13] [14]

 

for any differentiable function  .

Reduction of  ,   case to  

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The simplest nontrivial case of the Schlesinger equations is when   and  . By applying a Möbius transformation to the variable  , two of the finite poles may be chosen to be at   and  , and the third viewed as the independent variable. Setting the sum   of the matrices appearing in (6), which is an invariant of the Schlesinger equations, equal to a constant, and quotienting by its stabilizer under   conjugation, we obtain a system equivalent to the most generic case   of the six Painlevé transcendent equations, for which many detailed classes of explicit solutions are known.[15][16][17]

Non-Fuchsian isomonodromic systems

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For non-Fuchsian systems, with higher order poles, the generalized monodromy data include Stokes matrices and connection matrices, and there are further isomonodromic deformation parameters associated with the local asymptotics, but the isomonodromic  -functions may be defined in a similar way, using differentials on the extended parameter space.[12] There is similarly a Poisson bracket structure on the space of rational matrix valued functions of the spectral parameter   and corresponding spectral invariant Hamiltonians that generate the isomonodromic deformation dynamics.[13][14]

Taking all possible confluences of the poles appearing in (6) for the   and   case, including the one at  , and making the corresponding reductions, we obtain all other instances   of the Painlevé transcendents, for which numerous special solutions are also known.[15][16]

Fermionic VEV (vacuum expectation value) representations

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The fermionic Fock space  , is a semi-infinite exterior product space [18]

 

defined on a (separable) Hilbert space   with basis elements   and dual basis elements   for  .

The free fermionic creation and annihilation operators   act as endomorphisms on   via exterior and interior multiplication by the basis elements

 

and satisfy the canonical anti-commutation relations

 

These generate the standard fermionic representation of the Clifford algebra on the direct sum  , corresponding to the scalar product

 

with the Fock space   as irreducible module. Denote the vacuum state, in the zero fermionic charge sector  , as

 ,

which corresponds to the Dirac sea of states along the real integer lattice in which all negative integer locations are occupied and all non-negative ones are empty.

This is annihilated by the following operators

 

The dual fermionic Fock space vacuum state, denoted  , is annihilated by the adjoint operators, acting to the left

 

Normal ordering   of a product of linear operators (i.e., finite or infinite linear combinations of creation and annihilation operators) is defined so that its vacuum expectation value (VEV) vanishes

 

In particular, for a product   of a pair   of linear operators, one has

 

The fermionic charge operator   is defined as

 

The subspace   is the eigenspace of   consisting of all eigenvectors with eigenvalue  

 .

The standard orthonormal basis   for the zero fermionic charge sector   is labelled by integer partitions  , where   is a weakly decreasing sequence of   positive integers, which can equivalently be represented by a Young diagram, as depicted here for the partition  .

 
Young diagram of the partition (5, 4, 1)

An alternative notation for a partition   consists of the Frobenius indices  , where   denotes the arm length; i.e. the number   of boxes in the Young diagram to the right of the  'th diagonal box,   denotes the leg length, i.e. the number of boxes in the Young diagram below the  'th diagonal box, for  , where   is the Frobenius rank, which is the number of elements along the principal diagonal.

The basis element   is then given by acting on the vacuum with a product of   pairs of creation and annihilation operators, labelled by the Frobenius indices

 

The integers   indicate, relative to the Dirac sea, the occupied non-negative sites on the integer lattice while   indicate the unoccupied negative integer sites. The corresponding diagram, consisting of infinitely many occupied and unoccupied sites on the integer lattice that are a finite perturbation of the Dirac sea are referred to as a Maya diagram.[2]

The case of the null (emptyset) partition   gives the vacuum state, and the dual basis   is defined by

 

Any KP  -function can be expressed as a sum

  (10)

where   are the KP flow variables,   is the Schur function corresponding to the partition  , viewed as a function of the normalized power sum variables

 

in terms of an auxiliary (finite or infinite) sequence of variables   and the constant coefficients   may be viewed as the Plücker coordinates of an element   of the infinite dimensional Grassmannian consisting of the orbit, under the action of the general linear group  , of the subspace   of the Hilbert space  .

This corresponds, under the Bose-Fermi correspondence, to a decomposable element

 

of the Fock space   which, up to projectivization, is the image of the Grassmannian element   under the Plücker map

 

where   is a basis for the subspace   and   denotes projectivization of an element of  .

The Plücker coordinates   satisfy an infinite set of bilinear relations, the Plücker relations, defining the image of the Plücker embedding into the projectivization   of the fermionic Fock space, which are equivalent to the Hirota bilinear residue relation (1).

If   for a group element   with fermionic representation  , then the  -function   can be expressed as the fermionic vacuum state expectation value (VEV):

 

where

 

is the abelian subgroup of   that generates the KP flows, and

 

are the ""current"" components.

Examples of solutions to the equations of the KP hierarchy

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Schur functions

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As seen in equation (9), every KP  -function can be represented (at least formally) as a linear combination of Schur functions, in which the coefficients   satisfy the bilinear set of Plucker relations corresponding to an element   of an infinite (or finite) Grassmann manifold. In fact, the simplest class of (polynomial) tau functions consists of the Schur functions   themselves, which correspond to the special element of the Grassmann manifold whose image under the Plücker map is  .

Multisoliton solutions

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If we choose   complex constants   with  's all distinct,  , and define the functions

 

we arrive at the Wronskian determinant formula

 

which gives the general  -soliton  -function.[3][4][19]

Theta function solutions associated to algebraic curves

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Let   be a compact Riemann surface of genus   and fix a canonical homology basis   of   with intersection numbers

 

Let   be a basis for the space   of holomorphic differentials satisfying the standard normalization conditions

 

where   is the Riemann matrix of periods. The matrix   belongs to the Siegel upper half space

 

The Riemann   function on   corresponding to the period matrix   is defined to be

 

Choose a point  , a local parameter   in a neighbourhood of   with   and a positive divisor of degree  

 

For any positive integer   let   be the unique meromorphic differential of the second kind characterized by the following conditions:

  • The only singularity of   is a pole of order   at   with vanishing residue.
  • The expansion of   around   is
     .
  •   is normalized to have vanishing  -cycles:
     

Denote by   the vector of  -cycles of  :

 

Denote the image of   under the Abel map  

 

with arbitrary base point  .

Then the following is a KP  -function:[20]

 .

Matrix model partition functions as KP  -functions

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Let   be the Lebesgue measure on the   dimensional space   of   complex Hermitian matrices. Let   be a conjugation invariant integrable density function

 

Define a deformation family of measures

 

for small   and let

 

be the partition function for this random matrix model.[21][5] Then   satisfies the bilinear Hirota residue equation (1), and hence is a  -function of the KP hierarchy.[22]

 -functions of hypergeometric type. Generating function for Hurwitz numbers

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Let   be a (doubly) infinite sequence of complex numbers. For any integer partition   define the content product coefficient

 ,

where the product is over all pairs   of positive integers that correspond to boxes of the Young diagram of the partition  , viewed as positions of matrix elements of the corresponding   matrix. Then, for every pair of infinite sequences   and   of complex vaiables, viewed as (normalized) power sums   of the infinite sequence of auxiliary variables

  and  ,

defined by:

 ,

the function

  (11)

is a double KP  -function, both in the   and the   variables, known as a  -function of hypergeometric type.[23]

In particular, choosing

 

for some small parameter  , denoting the corresponding content product coefficient as   and setting

 ,

the resulting  -function can be equivalently expanded as

  (12)

where   are the simple Hurwitz numbers, which are   times the number of ways in which an element   of the symmetric group   in   elements, with cycle lengths equal to the parts of the partition  , can be factorized as a product of    -cycles

 ,

and

 

is the power sum symmetric function. Equation (12) thus shows that the (formal) KP hypergeometric  -function (11) corresponding to the content product coefficients   is a generating function, in the combinatorial sense, for simple Hurwitz numbers.[8][9][10]

References

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  1. ^ Hirota, Ryogo (1986). "Reduction of soliton equations in bilinear form". Physica D: Nonlinear Phenomena. 18 (1–3). Elsevier BV: 161–170. Bibcode:1986PhyD...18..161H. doi:10.1016/0167-2789(86)90173-9. ISSN 0167-2789.
  2. ^ a b c d e Sato, Mikio, "Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds", Kokyuroku, RIMS, Kyoto Univ., 30–46 (1981).
  3. ^ a b c Date, Etsuro; Jimbo, Michio; Kashiwara, Masaki; Miwa, Tetsuji (1981). "Operator Approach to the Kadomtsev-Petviashvili Equation–Transformation Groups for Soliton Equations III–". Journal of the Physical Society of Japan. 50 (11). Physical Society of Japan: 3806–3812. Bibcode:1981JPSJ...50.3806D. doi:10.1143/jpsj.50.3806. ISSN 0031-9015.
  4. ^ a b Jimbo, Michio; Miwa, Tetsuji (1983). "Solitons and infinite-dimensional Lie algebras". Publications of the Research Institute for Mathematical Sciences. 19 (3). European Mathematical Society Publishing House: 943–1001. doi:10.2977/prims/1195182017. ISSN 0034-5318.
  5. ^ a b Akemann, G.; Baik, J.; Di Francesco, P. (2011). The Oxford Handbook of Random Matrix Theory. Oxford: Oxford University Press. ISBN 978-0-19-957400-1.
  6. ^ Dieng, Momar; Tracy, Craig A. (2011). Harnad, John (ed.). Random Matrices, Random Processes and Integrable Systems. CRM Series in Mathematical Physics. New York: Springer Verlag. arXiv:math/0603543. Bibcode:2011rmrp.book.....H. doi:10.1007/978-1-4419-9514-8. ISBN 978-1461428770. S2CID 117785783.
  7. ^ Harnad, J.; Balogh, F. (2021). Tau functions and Their Applications, Chapts. 11-12. Cambridge Monographs on Mathematical Physics. Cambridge, U.K.: Cambridge University Press. doi:10.1017/9781108610902. ISBN 9781108610902. S2CID 222379146.
  8. ^ a b Pandharipande, R. (2000). "The Toda Equations and the Gromov–Witten Theory of the Riemann Sphere". Letters in Mathematical Physics. 53 (1). Springer Science and Business Media LLC: 59–74. doi:10.1023/a:1026571018707. ISSN 0377-9017. S2CID 17477158.
  9. ^ a b Okounkov, Andrei (2000). "Toda equations for Hurwitz numbers". Mathematical Research Letters. 7 (4). International Press of Boston: 447–453. arXiv:math/0004128. doi:10.4310/mrl.2000.v7.n4.a10. ISSN 1073-2780. S2CID 55141973.
  10. ^ a b Harnad, J.; Balogh, F. (2021). Tau functions and Their Applications, Chapts. 13-14. Cambridge Monographs on Mathematical Physics. Cambridge, U.K.: Cambridge University Press. doi:10.1017/9781108610902. ISBN 9781108610902. S2CID 222379146.
  11. ^ a b Segal, Graeme; Wilson, George (1985). "Loop groups and equations of KdV type". Publications mathématiques de l'IHÉS. 61 (1). Springer Science and Business Media LLC: 5–65. doi:10.1007/bf02698802. ISSN 0073-8301. S2CID 54967353.
  12. ^ a b c d Jimbo, Michio; Miwa, Tetsuji; Ueno, Kimio (1981). "Monodromy preserving deformation of linear ordinary differential equations with rational coefficients". Physica D: Nonlinear Phenomena. 2 (2). Elsevier BV: 306–352. doi:10.1016/0167-2789(81)90013-0. ISSN 0167-2789.
  13. ^ a b Harnad, J. (1994). "Dual Isomonodromic Deformations and Moment Maps into Loop Algebras". Communications in Mathematical Physics. 166 (11). Springer: 337–365. arXiv:hep-th/9301076. Bibcode:1994CMaPh.166..337H. doi:10.1007/BF02112319. S2CID 14665305.
  14. ^ a b Bertola, M.; Harnad, J.; Hurtubise, J. (2023). "Hamiltonian structure of rational isomonodromic deformation systems". Journal of Mathematical Physics. 64 (8). American Institute of Physics: 083502. arXiv:2212.06880. Bibcode:2023JMP....64h3502B. doi:10.1063/5.0142532.
  15. ^ a b Fokas, Athanassios S.; Its, Alexander R.; Kapaev, Andrei A.; Novokshenov, Victor Yu. (2006), Painlevé transcendents: The Riemann–Hilbert approach, Mathematical Surveys and Monographs, vol. 128, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3651-4, MR 2264522
  16. ^ a b Conte, R.; Musette, M. (2020), The Painlevé handbook, second edition, Mathematical physics studies, Switzerland: Springer Nature, ISBN 978-3-030-53339-7
  17. ^ Lisovyy, Oleg; Tykhyy, Yuriy (2014). "Algebraic solutions of the sixth Painlevé equation". Journal of Geometry and Physics. 85: 124–163. arXiv:0809.4873. Bibcode:2014JGP....85..124L. doi:10.1016/j.geomphys.2014.05.010. S2CID 50552982.
  18. ^ Kac, V.; Peterson, D.H. (1981). "Spin and wedge representations of infinite-dimensional Lie Algebras and groups". Proc. Natl. Acad. Sci. U.S.A. 58 (6): 3308–3312. Bibcode:1981PNAS...78.3308K. doi:10.1073/pnas.78.6.3308. PMC 319557. PMID 16593029.
  19. ^ Harnad, J.; Balogh, F. (2021). Tau functions and Their Applications, Chapt. 3. Cambridge Monographs on Mathematical Physics. Cambridge, U.K.: Cambridge University Press. doi:10.1017/9781108610902. ISBN 9781108610902. S2CID 222379146.
  20. ^ Dubrovin, B.A. (1981). "Theta Functions and Nonlinear Equations". Russ. Math. Surv. 36 (1): 11–92. Bibcode:1981RuMaS..36...11D. doi:10.1070/RM1981v036n02ABEH002596. S2CID 54967353.
  21. ^ M.L. Mehta, "Random Matrices", 3rd ed., vol. 142 of Pure and Applied Mathematics, Elsevier, Academic Press, ISBN 9780120884094 (2004)
  22. ^ Kharchev, S.; Marshakov, A.; Mironov, A.; Orlov, A.; Zabrodin, A. (1991). "Matrix models among integrable theories: Forced hierarchies and operator formalism". Nuclear Physics B. 366 (3). Elsevier BV: 569–601. Bibcode:1991NuPhB.366..569K. doi:10.1016/0550-3213(91)90030-2. ISSN 0550-3213.
  23. ^ Orlov, A. Yu. (2006). "Hypergeometric Functions as Infinite-Soliton Tau Functions". Theoretical and Mathematical Physics. 146 (2). Springer Science and Business Media LLC: 183–206. Bibcode:2006TMP...146..183O. doi:10.1007/s11232-006-0018-4. ISSN 0040-5779. S2CID 122017484.

Bibliography

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