Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 092, 21 pages      arXiv:2209.01062      https://doi.org/10.3842/SIGMA.2023.092

Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold

Felipe Reyes
SISSA, via Bonomea 265, Trieste, Italy

Received May 03, 2023, in final form November 06, 2023; Published online November 16, 2023

Abstract
We study the family of ordinary differential equations associated to a Dubrovin-Frobenius manifold along its caustic. Upon just loosing an idempotent at the caustic and under a non-degeneracy condition, we write down a normal form for this family and prove that the corresponding fundamental matrix solutions are strongly isomonodromic. It is shown that the exponent of formal monodromy is related to the multiplication structure of the Dubrovin-Frobenius manifold along its caustic.

Key words: Dubrovin-Frobenius manifolds; isomonodromic deformations; differential equations.

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References

  1. Arsie A., Buryak A., Lorenzoni P., Rossi P., Semisimple flat $F$-manifolds in higher genus, Comm. Math. Phys. 397 (2020), 141-197, arXiv:2001.05599.
  2. Arsie A., Buryak A., Lorenzoni P., Rossi P., Riemannian $F$-manifolds, bi-flat $F$-manifolds, and flat pencils of metrics, Int. Math. Res. Not. 2022 (2022), 16730-16778, arXiv:2104.09380.
  3. Basalaev A., Hertling C., 3-dimensional $F$-manifolds, Lett. Math. Phys. 111 (2021), 90, 50 pages, arXiv:2012.11443.
  4. Basalaev A., Takahashi A., On rational Frobenius manifolds of rank three with symmetries, J. Geom. Phys. 84 (2014), 73-86, arXiv:1401.3505.
  5. Cotti G., Dubrovin B., Guzzetti D., Local moduli of semisimple Frobenius coalescent structures, SIGMA 16 (2020), 040, 105 pages, arXiv:1712.08575.
  6. Dubrovin B., Geometry of $2$D topological field theories, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348, arXiv:hep-th/9407018.
  7. Dubrovin B., Differential geometry of the space of orbits of a Coxeter group, in Surveys in Differential Geometry: Integrable Systems, Surv. Differ. Geom., Vol. 4, International Press, Boston, MA, 1998, 181-211, arXiv:hep-th/9303152.
  8. Dubrovin B., Painlevé transcendents in two-dimensional topological field theory, in The Painlevé Property, CRM Ser. Math. Phys., Springer, New York, 1999, 287-412, arXiv:math.AG/9803107.
  9. Guzzetti D., Isomonodromic deformations along a stratum of the coalescence locus, J. Phys. A 55 (2022), 455202, 52 pages, arXiv:2111.02969.
  10. Hertling C., Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts Math., Vol. 151, Cambridge University Press, Cambridge, 2002.
  11. Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau $-function, Phys. D 2 (1981), 306-352.
  12. Saito K., Primitive forms for a universal unfolding of a function with an isolated critical point, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 775-792.
  13. Saito K., Takahashi A., From primitive forms to Frobenius manifolds, in From Hodge Theory to Integrability and TQFT tt*-Geometry, Proc. Sympos. Pure Math., Vol. 78, American Mathematical Society, Providence, RI, 2008, 31-48.
  14. Sibuya Y., Simplification of a system of linear ordinary differential equations about a singular point, Funkcial. Ekvac. 4 (1962), 29-56.
  15. Strachan I.A.B., Frobenius manifolds: natural submanifolds and induced bi-Hamiltonian structures, Differential Geom. Appl. 20 (2004), 67-99, arXiv:math.DG/0201039.

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