Alla cara memoria del mio Maestro, Boris Dubrovin
Abstract
In the first part of this paper, we give a new analytical proof of a theorem of C. Sabbah on integrable deformations of meromorphic connections on \({\mathbb P}^1\). This theorem generalizes a previous result of B. Malgrange to the case of connections admitting irregular singularities of Poincaré rank 1 with coalescing eigenvalues. In the second part of this paper, as an application, we prove that any semisimple formal Frobenius manifold (over \({\mathbb C}\)), with unit and Euler field, is the completion of an analytic pointed germ of a Dubrovin–Frobenius manifold. In other words, any formal power series, which provides a quasi-homogenous solution of WDVV equations and defines a semisimple Frobenius algebra at the origin, is actually convergent under no further tameness assumptions.
Similar content being viewed by others
Notes
A richer notion of complete CohFT on a given \((H,\eta )\) is also available, in which the datum is enriched to a family \((\Omega _{g,\mathfrak n})_{g,\mathfrak n}\) of k-linear tensors \(\Omega _{g,\mathfrak n}\in (H^*)^{\otimes \mathfrak n}\otimes _k H^\bullet (\overline{\mathcal M}_{g,\mathfrak n}; k)\), satisfying further compatibility properties, for any pair \((g,\mathfrak n)\) of non-negative integers in the stable range \(2g-2+\mathfrak n>0\). The prototypical example of a complete CohFT is provided by the Gromov–Witten theory of a smooth projective variety X. The corresponding formal Frobenius manifold attached to its genus zero sector is called quantum cohomology of X. See [51, 57] and Sect. 6 of this paper. Cohomological Field Theory
Here \(\mathfrak S_n\) denotes the symmetric group on a finite set with n elements.
I do not know any reference in the literature where a complete proof is given. I thank Yu.I. Manin for a friendly e-mail correspondence on this point. The current paper both recovers a proof of this known fact, and it also removes the tameness assumption.
Given a formal Frobenius manifold, the system (1.1) has coefficients in \(M_n({\mathbb C}[\![\varvec{t}]\!])\). Hence, for \(t=0\), we have a well-defined differential system with coefficients in \(M_n({\mathbb C})\).
Here, \({\mathbb C}\{\varvec{t}\}\) denotes the algebra of convergent power series in \(\varvec{t}\).
Recall that the index of a Fredholm operator T is the integer \(\mathrm{ind}\,T:=\dim \ker T-\dim \mathrm{coker}\,T\).
For this standard algebraic approach to Lie derivatives of tensors, see, e.g. [4].
In what follows, the musical isomorphisms with respect to the metric \(\eta \) will be denoted by \((-)^\flat \) and \((-)^\sharp \), respectively. If \(\xi \in \Gamma (TM)\), the 1-form \(\xi ^\flat \in \Gamma (T^*M)\) is defined by \(\xi ^\flat (X)=\eta (X,\xi )\), where \(X\in \Gamma (TM)\). Conversely, if \(\xi \in \Gamma (T^*M)\), the vector field \(\xi ^\sharp \in \Gamma (TM)\) is uniquely defined by the identity \(\xi (X)=\eta (X,\xi ^\sharp )\), where \(X\in \Gamma (TM)\). Thus, \((-)^\flat :\Gamma (TM)\rightarrow \Gamma (T^*M)\) and \((-)^\sharp :\Gamma (T^*M)\rightarrow \Gamma (TM)\) are mutually inverse. In components, these operations are also known as “lowering” and “raising” indices, respectively. These operations naturally extend to mixed tensors.
In fact, the resulting basis \((\pi _1',\ldots , \pi _n')\) is not just an \((h+1)\)-order idempotent, but even a \((2h+1)\)-order idempotent basis.
This means that each \(\Delta _2,\ldots ,\Delta _{r+1}\) intersects every effective curve class \(\beta \in \mathrm{Eff(X)}\) non-negatively.
Surely enough, such a list does not cover all the known cases of semisimple small quantum cohomologies available in the literature.
These include the equations in Part I of our proof of Theorem 5.1.
References
Anosov, D.V., Bolibruch, A.A.:The Riemann–Hilbert Problem, Asp. Math. E22, Vieweg, Braunschweig, vol. IX, p. 193 (1994)
Arsie, A., Buryak, A., Lorenzoni, P., Rossi, P.: Semisimple flat \(F\)-manifolds in higher genus, pp. 1–48. arXiv:2001.05599 [math.AG]
Arsie, A., Lorenzoni, P.: \(F\)-manifolds, multi-flat structures and Painlevé transcendents, pp. 1–69. arXiv:1501.06435v5 [math-ph]
Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications. Springer (1988)
Basalaev, A., Buryak, A.: Open WDVV Equations and Virasoro Constraints. Arnold Math. J. 5, 145–186 (2019). doi: 10.1007/s40598-019-00115-w
Bayer, A., Manin, Y.I.: (Semi)simple exercises in quantum cohomology. In: Proceedings of the Fano conference, Univ. Torino, Turin, pp. 143–173 (2004)
Benedetti, V., Manivel, l.: The small quantum cohomology of the Cayley Grassmannian, 19p. arXiv:1907.0751
Bothner, T.: On the origins of Riemann–Hilbert problems in mathematics, 56p. arXiv:2003.14374
Cotti, G., Dubrovin, B., Guzzetti, D.: Isomonodromy deformations at an irregular singularity with coalescing eigenvalues. Duke Math. J. 168(6), 967–1108 (2019). doi: 10.1215/00127094-2018-0059
Cotti, G., Dubrovin, B., Guzzetti, D.: Local moduli of semisimple Frobenius coalescent structures. Symmetry Integrability Geom. Methods Appl. 16, 040 (2020). doi: 10.3842/SIGMA.2020.040
Cotti, G., Dubrovin, B., Guzzetti, D.: Helix structures in quantum cohomology of Fano varieties, pp. 1–149. arXiv:1811.09235
Clancey, K.F., Gohberg, I.: Factorization of matrix functions and singular integral operators, Operator Theory: Advances and Applications, vol. 3. Birkhäuser Verlag, Basel - Boston, Mass (1981)
Cotti G., Guzzetti, D.: Analytic Geometry of Semisimple Coalescent Frobenius Structures, Random Matrices Theory Appl. 6(4), 1740004, 36 pp (2017)
G., Guzzetti, D.: Results on the Extension of Isomonodromy Deformations with a Resonant Irregular Singularity. Random Matrices Theory Appl. 07(4), 1840003, 27 pp (2018)
Ciolli, G.: Computing the quantum cohomology of some Fano threefolds and its semisimplicity. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 7(2), 511–517 (2004)
Ciolli, G.: On the quantum cohomology of some Fano threefolds and a conjecture of Dubrovin. Internat. J. Math. 16(8), 823–839 (2005)
Coates, T., Iritani, H.: On the convergence of Gromov-Witten potentials and Givental’s formula. Michigan Math. J. 64(3), 587–631 (2015). doi: 10.1307/mmj/1441116660
Cox, D.A., Katz, S.: Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs, vol. 68. Amer. Math. Soc, Providence, RI (1999)
Combe, N., Manin, Y.I.: \(F\)-manifolds and geometry of information. Bull. Lond. Math. Soc. 52, 777–792 (2020). doi: 10.1112/blms.12411
Cruz Morales, J.A., Mellit, A., Perrin, N., Smirnov, M.: On quantum cohomology of Grassmannians of isotropic lines, unfoldings of \(A_n\)-singularities, and Lefschetz exceptional collections. Ann. Inst. Fourier Grenoble 69(3), 955–991 (2019)
Chaput, P.-E., Manivel, L., Perrin, N.: Quantum cohomology of minuscule homogeneous spaces III?: semi-simplicity and consequences. Can. J. Math. 62(6), 1246–1263 (2010)
Cotti, G.: Quantum differential equations and helices, to appear in Geometric Methods in Physics XXXVIII. In: Kielanowski, P. et al. (eds.) Trends in Mathematics, 20p (2020). https://doi.org/10.1007/978-3-030-53305-2_3, arXiv:1911.11047
Chaput, P.E., Perrin, N.: On the quantum cohomology of adjoint varieties. Proc. Lond. Math. Soc. 103(2), 294–330 (2011)
Dubrovin, B.: Integrable systems in topological field theory. Nuclear Phys. B 379, 627–689 (1992)
Dubrovin, B.: Geometry of 2D topological field theories. In: Integrable Systems and Quantum Groups (Montecatini Terme: Lecture Notes in Math., vol. 1620. 1996, 120–348, Springer, Berlin (1993)
Dubrovin, B.: Geometry and analytic theory of Frobenius manifolds, Doc. Math. (1998), extra Vol. II, 315– 326. arXiv:math.AG/9807034
Dubrovin, B.: Painlevé transcendents in two-dimensional topological field theory. In: The Painlevé Property, pp. 287–412. Springer, New York, CRM Ser. Math. Phys (1999)
Duren, P.L.: The Theory of \(H^p\) Spaces. Academic Press, New York and London (1970)
Dijkgraaf, R., Verlinde, H., Verlinde, E.: Topological strings in \(d < 1\). Nuclear Phys. B 352(1), 59–86 (1991)
Deift, P., Zhou, X.: Perturbation theory for infinite dimensional integrable systems on the line: a case study. Acta Math. 188, 163–262 (2002)
Deift, P., Zhou, X.: A priori \(L^p\) estimates for solutions of Riemann-Hilbert problems. Int. Math. Res. Notices 40, 2121–2154 (2002)
Fokas, A.S., Its, A.R., Kapaev, A.A., Novokshenov, VYu.: Painlevé transcendents: The Riemann–Hilbert approach, Mathematical Surveys and Monographs, vol. 128. American Mathematical Society, Providence, RI (2006)
Fokas, A.S., Zhou, X.: On the solvability of Painlevé \(II\) and \(IV\). Commun. Math. Phys. 144(3), 601–622 (1992)
Givental, A.: Gromov-Witten invariants and quantization of quadratic hamiltonians. Mosc. Math. J 1(4), 551–56 (2001)
Galkin, S., Golyshev, V., Iritani, H.: Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures. Duke Math. J. 165(11), 2005–2077 (2016)
Gohberg, I., Goldberg, S., Kaashoek, M.A.: Classes of Linear Operators I. Birkhäuser, Basel, Boston (1990)
Galkin, S., Mellit, A., Smirnov, M.: Dubrovin’s conjecture for \(IG(2, 6)\). Int. Math. Res. Not. 18,8847–8859 (2015)
Gramsch, B.: Meromorphie in der Theorie der Fredholmoperatoren mit Anwendungen auf elliptische Differentialoperatoren. Math. Ann. 188, 97–112 (1970)
Gohberg, I., Sigal, E.I.: An operator generalization of the logarithmic residue theorem and the theorem of Rouché. Math. USSR Sb. 13, 603–625 (1971)
Guzzetti, D.: Inverse problem and monodromy data for three-dimensional Frobenius manifolds. Math. Phys. Anal. Geom. 4, 245–291 (2001)
Hertling, C.: Frobenius Manifolds and Moduli Spaces for Singularities, Cambridge Tracts in Mathematics, vol. 151. Cambridge University Press, Cambridge (2002)
Hertling, C., Manin, Y.I.: Weak Frobenius manifolds. Int. Math. Res. Notices 6, 277–286 (1999). Preprint math.QA/9810132
Hertling, C., Manin, Y.I., Teleman, C.: An update on semisimple quantum cohomology and F-manifolds. Tr. Mat. Inst. Steklova 264 (2009), Mnogomernaya Algebraicheskaya Geometriya, 69–76; translation in Proc. Steklov Inst. Math. 264 (2009), no. 1, 62–69
Iritani, H.: Convergence of quantum cohomology by quantum Lefschetz. J. Reine Angew. Math. 610, 29–69 (2007)
Its, A.R.: The Riemann-Hilbert problem and integrable systems. Not. Am. Math. Soc. 50(11), 1389–1400 (2003)
Its, A.R.: Large \(N\) asymptotics in random matrices: the Riemann-Hilbert approach. In: Harnad, J. (ed.) Random Matrices. Random Processes and Integrable Systems. CRM Ser. Math. Phys, Springer, New York (2011)
Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients I. Physica 2D, 306–352 (1981)
Kaballo, W.: Meromorphic generalized inverses of operator functions. Indagationes Mathematicae 23(4), 970–994 (2012)
Kedlaya, K.: Good formal structures for flat meromorphic connections, I: surfaces. Duke Math. J. 154(2), 343–418 (2010). arXiv:0811.0190
Kedlaya, K.: Good formal structures for flat meromorphic connections, II: excellent schemes. J. Amer. Math. Soc. 24(1), 183–229 (2011). arXiv:1001.0544
Kontsevich, M., Manin, Y.I.: Gromov-Witten classes, quantum cohomology, and enumerative geometry. Comm. Math. Phys. 164, 525–562 (1994)
Lee, Y.P.: Quantum \(K\)-theory: foundations. Duke Math. J. 121(3), 389–424 (2004)
Losev, A., Manin, Y.I.: Extended modular operad. In: Hertling, C., Marcolli, M. (eds.) Frobenius Manifolds, pp. 181–211. Vieweg & Sohn Verlag, Wiesbaden (2004)
Malgrange, B.: Déformations de systèmes différentiels et microdifférentiels. Séminaire E.N.S. Mathématique et Physique (L. Boutet de Monvel, A. Douady & J.-L. Verdier, eds.), Progress in Math., vol. 37, pp. 351–379. Birkhäuser, Basel, Boston (1983)
Malgrange, B.: Sur les déformations isomonodromiques, II. Séminaire E.N.S. Mathématique et Physique. In: Boutet de Monvel, L., Douady, A., Verdier, J.-L. (eds.) Progress in Math., vol. 37, pp. 427–438. Birkhäuser, Basel, Boston (1983)
Malgrange, B.: Deformations of differential systems. II. J. Ramanujan Math. Soc. 1, 3–15 (1986)
Manin, Y.I.: Frobenius manifolds, Quantum Cohomology, and Moduli Spaces. Amer. Math. Soc, Providence, RI (1999)
Manin, Y.I.: \(F\)-manifolds with flat structure and Dubrovin’s duality. Adv. Math. 198(1), 5–26 (2005)
Mochizuki, T.: Good formal structure for meromorphic flat connections on smooth projective surfaces, in Algebraic Analysis and Around (Kyoto, : Advanced Studies in Pure Math., vol. 54, Math. Soc. Japan, Tokyo 2009, 223–253 (June 2007). arXiv:0803.1346
Mochizuki, T.: Wild Harmonic Bundles and Wild Pure Twistor \(D\)-modules, Astérisque, vol. 340. Société Mathématique de France, Paris (2011)
Mochizuki, T.: Stokes structure of a good meromorphic flat bundle. Journal de l’Institut mathématique de Jussieu 10(3), 675–712 (2011)
Mochizuki, T.: Holonomic D-modules with Betti structure, Mém. Soc. Math. France (N.S.), vol. 138–139, Société Mathématique de France (2014). arXiv:1001.2336
Mikhlin, S.G., Prössdorf, S.: Singular Integral Operators. Springer, New York (1980)
Muskhelishvili, N.I.; Singular Integral Equations. Woolters–Noordhoff Publishing (1972)
Nowicki, A.: Commutative bases of derivations in polynomial and power series rings. J. Pure Appl. Algebra 40, 275–279 (1986)
Pandharipande, R.: Cohomological field theory calculations. In: Proceedings of the International Congress of Mathematicians (ICM 2018). https://doi.org/10.1142/9789813272880_0031
Perrin, N.: Semisimple quantum cohomology of some Fano varieties. arXiv:1405.5914
Sabbah, C.: Équations différentielles à points singuliers irréguliers en dimension 2. Ann. Inst. Fourier (Grenoble) 43, 1619–1688 (1993)
Sabbah, C.: Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2, Astérisque, vol. 263. Société Mathématique de France, Paris (2000)
Sabbah, C.: Isomonodromic deformations and Frobenius manifolds, Universitext, Springer & EDP Sciences (2007) (in French: 2002)
Sabbah, C.: Integrable deformations and degenerations of some irregular singularities, accepted for publication in Publ. RIMS Kyoto Univ. 35 p. arXiv:1711.08514v3
Sardanashvily, G.: Lectures on Differential Geometry of Modules and Rings, Lambert Academic Publishing, Saarbrücken (2012). arXiv:0910.1515
Teleman, C.: The structure of \(2d\) semi-simple field theories. Inventiones Mathematicae 188(3), 525–588 (2012)
Trogdon, T., Olver, S.: Riemann-Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions. Society for Industrial and Applied Mathematics, SIAM), Philadelphia PA (2016)
Vekua, N.P.: Systems of singular integral equations, P. Noordhoff, Ltd., Groningen (1967). Translated from the Russian by A.G. Gibbs and G.M. Simmons. Edited by J.H. Ferziger
Witten, E.: On the structure of the topological phase of two-dimensional gravity. Nuclear Phys. B 340(2–3), 281–332 (1990)
Zhou, X.: The Riemann-Hilbert problem and inverse scattering. SIAM J. Math. Anal. 20(4), 966–986 (1989). doi: 10.1137/0520065
Funding
Funding was provided by the Engineering and Physical Sciences Research Council (Grant No. EP/P021913/2), Hausdorff Research Institute for Mathematics (JTP Fellowship “New Trends in Representation Theory”), and by the Fundação para a Ciência e a Tecnologia, FCiências da Universidade de Lisboa (Grant No. PTDC/MAT-PUR/ 30234/2017).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Cotti, G. Degenerate Riemann–Hilbert–Birkhoff problems, semisimplicity, and convergence of WDVV-potentials. Lett Math Phys 111, 99 (2021). https://doi.org/10.1007/s11005-021-01427-9
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11005-021-01427-9