The Hamiltonian approach to isomonodromic deformation systems is extended to include generic rational covariant derivative operators on the Riemann sphere with irregular singularities of arbitrary Poincaré rank. The space of rational connections with given pole degrees carries a natural Poisson structure corresponding to the standard classical rational R-matrix structure on the dual space of the loop algebra . Nonautonomous isomonodromic counterparts of isospectral systems generated by spectral invariants are obtained by identifying deformation parameters as Casimir elements on the phase space. These are shown to coincide with higher Birkhoff invariants determining local asymptotics near to irregular singular points, together with the pole loci. Pairs consisting of Birkhoff invariants, together with the corresponding dual spectral invariant Hamiltonians, appear as “mirror images” matching, at each pole, the negative power coefficients in the principal part of the Laurent expansion of the fundamental meromorphic differential on the associated spectral curve with the corresponding positive power terms in the analytic part. Infinitesimal isomonodromic deformations are shown to be generated by the sum of the Hamiltonian vector field and an explicit derivative vector field that is transversal to the symplectic foliation. The Casimir elements serve as coordinates complementing those along the symplectic leaves, defining a local symplectomorphism between them. The explicit derivative vector fields preserve the Poisson structure and define a flat transversal connection, spanning an integrable distribution whose leaves may be identified as the orbits of a free Abelian local group action. The projection of infinitesimal isomonodromic deformation vector fields to the quotient manifold under this action gives commuting Hamiltonian vector fields corresponding to the spectral invariants dual to the Birkhoff invariants and the pole loci.
Skip Nav Destination
Hamiltonian structure of rational isomonodromic deformation systems
Article navigation
August 2023
Research Article|
August 07 2023
Hamiltonian structure of rational isomonodromic deformation systems
M. Bertola
;
M. Bertola
a)
(Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Writing – original draft, Writing – review & editing)
1
Centre de Recherches Mathématiques, Université de Montréal
, C. P. 6128, Succ. Centre Ville, Montréal, Quebec H3C 3J7, Canada
2
Department of Mathematics and Statistics, Concordia University
, 1455 de Maisonneuve Blvd. W., Montreal, Quebec H3G 1M8, Canada
3
SISSA, International School for Advanced Studies
, Via Bonomea 265, Trieste, Italy
Search for other works by this author on:
J. Harnad
;
J. Harnad
b)
(Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Writing – original draft, Writing – review & editing)
1
Centre de Recherches Mathématiques, Université de Montréal
, C. P. 6128, Succ. Centre Ville, Montréal, Quebec H3C 3J7, Canada
2
Department of Mathematics and Statistics, Concordia University
, 1455 de Maisonneuve Blvd. W., Montreal, Quebec H3G 1M8, Canada
b)Author to whom correspondence should be addressed: harnad@crm.umontreal.ca
Search for other works by this author on:
J. Hurtubise
J. Hurtubise
c)
(Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Writing – original draft, Writing – review & editing)
1
Centre de Recherches Mathématiques, Université de Montréal
, C. P. 6128, Succ. Centre Ville, Montréal, Quebec H3C 3J7, Canada
4
Department of Mathematics and Statistics, McGill University
, 805 Sherbrooke St. W., Montreal, Quebec H3A 0B9, Canada
Search for other works by this author on:
b)Author to whom correspondence should be addressed: harnad@crm.umontreal.ca
a)
E-mail: marco.bertola@concordia.ca
c)
E-mail: jacques.hurtubise@mcgill.ca
J. Math. Phys. 64, 083502 (2023)
Article history
Received:
January 14 2023
Accepted:
June 29 2023
Citation
M. Bertola, J. Harnad, J. Hurtubise; Hamiltonian structure of rational isomonodromic deformation systems. J. Math. Phys. 1 August 2023; 64 (8): 083502. https://doi.org/10.1063/5.0142532
Download citation file:
Sign in
Don't already have an account? Register
Sign In
You could not be signed in. Please check your credentials and make sure you have an active account and try again.
Sign in via your Institution
Sign in via your InstitutionPay-Per-View Access
$40.00
186
Views
Citing articles via
Related Content
The partition function of the two-matrix model as an isomonodromic τ function
J. Math. Phys. (January 2009)
Spectral curve and Hamiltonian structure of isomonodromic SU(2) Calogero–Gaudin system
J. Math. Phys. (August 2003)
Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: The s l 2 case
J. Math. Phys. (June 2020)
Elliptic Calogero–Moser systems and isomonodromic deformations
J. Math. Phys. (November 1999)
Schlesinger transformations for elliptic isomonodromic deformations
J. Math. Phys. (May 2000)