Тау-функции являются важным элементом современной теории интегрируемых систем и имеют множество приложений во множестве других областей. Первоначально они были введены Риого Хиротой [1] в его подходе к солитонным уравнениям прямым методом , основанном на их выражении в эквивалентной билинейной форме. Термин функция Тау , или -функция , впервые был систематически использован Микио Сато [2] и его учениками [3] [4] в конкретном контексте уравнения Кадомцева – Петвиашвили (или КП) и связанных с ним интегрируемых иерархий. Это центральный компонент теориисолитоны . Тау-функции также появляются как статистические суммы матричной модели в спектральной теории случайных матриц и могут также служить производящими функциями в смысле комбинаторики и перечислительной геометрии , особенно в отношении пространств модулей римановых поверхностей и перечисления разветвленных покрытий , или так называемые числа Гурвица .
Есть два понятия -функции, оба введены школой Сато . Первый - это изомонодромные -функции
. [5] Во - вторых, -функциям Сато - Сегэл типа -Wilson [2] [6] для интегрируемых иерархий, таких как иерархии КП, которые параметризованных линейных операторов , удовлетворяющих изоспектральные деформационные уравнения Лакса типа.
-Функции изоспектрального типа является решением билинейных уравнений Хироты, из которых линейный оператор претерпевает эволюцию изоспектральную может быть однозначно реконструированных. Геометрический в Сато [2] и Сегал -Wilson [6] смысл, это значение определителя интегрального оператора Фредгольма , интерпретируются как ортогональная проекция элемента надлежащим образом определенной (бесконечномерного) грассманова многообразия на происхождение , поскольку этот элемент эволюционирует под действием линейной экспоненты максимальной абелевой подгруппы общей линейной группы. Обычно возникает как статистическая суммав смысле статистической механики , квантовой механики многих тел или квантовой теории поля , поскольку основная мера подвергается линейной экспоненциальной деформации.
Соотношение билинейных вычетов Хироты для KP- функций [ править ]
КП ( Кадомцева – Петвиашвили ) -функция
- это функция бесконечного числа переменных потока КП, которая удовлетворяет следующему билинейному уравнению формального вычета
( 1 )
одинаково в переменных, где - коэффициент в формальном разложении Лорана, полученный в результате разложения всех факторов в виде ряда Лорана в , и
Уравнение Кадомцева-Петвиашвили [ править ]
Если - КП- функция, удовлетворяющая уравнению вычетов Хироты ( 1 ), и мы идентифицируем первые три переменные потока как
следует, что функция
удовлетворяет размерному нелинейному уравнению в частных производных
( 2 )
известное как уравнение Кадомцева-Петвиашвили (КП) , которое играет важную роль в физике плазмы и в океанских волнах на мелководье.
Дальнейшие логарифмические производные от дают бесконечную последовательность функций, которые удовлетворяют дальнейшим системам нелинейных автономных УЧП, каждая из которых включает частные производные конечного порядка по конечному числу параметров потока КП . Все вместе они известны как иерархия КП .
Формальная функция Бейкера-Ахиезера и иерархия КП [ править ]
Если мы определим (формальную) функцию Бейкера-Ахиезера
формулой Сато
и разложим его формальным рядом по степеням переменной
это удовлетворяет бесконечной последовательности согласованных эволюционных уравнений
( 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 ,
where
is the wave operator and denotes the projection to the part of containing
purely non-negative powers of ; i.e. to 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
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.
Consider the overdetermined system of first order matrix partial differential equations
(5)
(6)
where are a set of traceless matrices, a set of complex parameters and 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 [5]
Defining the functions
the Schlesinger equations 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 -functionassociated to the fundamental solution of the system (5), (6).
For non-Fuchsian systems, with higher order poles, the generalized monodromy data include Stokes parameters 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.[5]
Fermionic VEV (vacuum expectation value) representations[edit]
The fermionic Fock space , is a semi-infinite exterior product space
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
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 diagonal elements.
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
Then any KP -function can be expressed as a sum
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 Plucker 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
Plucker map
where is a basis for the subspace and denotes projectivization of
an element of .
The Plucker coordinates satisfy an infinite set of bilinear
relations, the Plucker relations, defining the Plücker embedding
into the projecivization 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.
Multisoliton solutions[edit]
If we choose complex constantswith 's all distinct, , and define the functions
we arrive at the Wronskian determinant formula
which gives the general N {\displaystyle N}
-soliton solution.[3][4]
Theta function solutions associated to algebraic curves[edit]
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 θ {\displaystyle \theta } 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:
Matrix model partition functions as KP -functions[edit]
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.[7]Then satisfies the bilinear Hirota residue equation (1), and hence is a -function of the KP hierarchy.[8]
-functions of hypergeometric type. Generating function for Hurwitz numbers[edit]
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
is a double KP -function, both in the and the variables, known as a function of hypergeometric type.[9]
In particular, choosing
for some small parameter , denoting the corresponding content product coefficient as and setting , the resulting -function can be equivalently expanded as
(7)
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 (7) thus shows that
the (formal) KP hypergeometric -function corresponding to the content
product coefficients is a generating
function, in the combinatorial sense, for simple Hurwitz numbers.[10][11]
References[edit]
^R. Hirota, "Reduction of soliton equations in bilinear form", Physica D, Nonlinear Phenomena18 , 161-170 (1986)
^ a b c dM. Sato, "Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds", Kokyuroku, RIMS, Kyoto Univ., 30–46 (1981).
^ a bE. Date, M. Jimbo, M. Kashiwara and T. Miwa, "Operator approach to the Kadomtsev-Petviashvili equation III". J. Phys. Soc. Jap.50 (11): 3806–3812 (1981). doi:10.1143/JPSJ.50.3806.
^ a bM. Jimbo and T. Miwa, "Solitons and infinite-dimensional Lie algebras", Publ. Res. Inst. Math. Sci., 19(3):943–1001 (1983).
^ a b cM. Jimbo, T. Miwa, and K. Ueno, "Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coefficients I", Physica D,2, 306–352 (1981)
^ a bG. Segal, G, Wilson, G., "Loop groups and equations of KdV type", Inst. Hautes Etudes Sci. Publ. Math., 6 (61), 5–65 (1985)
^M.L. Mehta, "Random Matrices", 3rd ed.,
vol. 142 of Pure and Applied Mathematics, Elsevier, Academic Press, ISBN 9780120884094 (2004).
^S. Kharchev, A. Marshakov, A. Mironov, A. Orlov, A. Zabrodin, "Matrix models among integrable theories:
Forced hierarchies and operator formalism", Nucl. Phys. B366, 569-601 (1991).
Dickey, L.A. (2003), "Soliton Equations and Hamiltonian Systems", vol. 26 of Advanced Series in Mathematical Physics. World Scientific Publishing Co., Inc., River Edge, NJ, 2nd ed.
Harnad, J.; Balogh, F. (2021), "Tau functions and Their Applications", Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, U.K.CS1 maint: discouraged parameter (link)
Hirota, R. (2004), "The Direct Method in Soliton Theory", Cambridge University Press, Cambridge , U.K.
Jimbo, M.; Miwa, T. (1999), "Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras", Cambridge University Press, Cambridge , U.K., Cambridge Tracts in Mathematics, 135CS1 maint: discouraged parameter (link)
Kodama, Y. (2017), KP Solitons and the Grassmannians: Combinatorics and Geometry of Two-Dimensional Wave Patterns, Springer Briefs in Mathematical Physics, Springer Nature