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Degenerate Riemann–Hilbert–Birkhoff problems, semisimplicity, and convergence of WDVV-potentials

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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.

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Notes

  1. 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

  2. Here \(\mathfrak S_n\) denotes the symmetric group on a finite set with n elements.

  3. 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.

  4. 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 \(M_n({\mathbb C})\).

  5. Here, \({\mathbb C}\{\varvec{t}\}\) denotes the algebra of convergent power series in \(\varvec{t}\).

  6. We warn the reader that in the exposition of [57], the isomonodromic system (1.1) is replaced by a Fuchsian one obtained by applying a (formal) Laplace transform, see [57, Ch. II.§1-3].

  7. Recall that the index of a Fredholm operator T is the integer \(\mathrm{ind}\,T:=\dim \ker T-\dim \mathrm{coker}\,T\).

  8. For this standard algebraic approach to Lie derivatives of tensors, see, e.g.   [4].

  9. 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.

  10. 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.

  11. This means that each \(\Delta _2,\ldots ,\Delta _{r+1}\) intersects every effective curve class \(\beta \in \mathrm{Eff(X)}\) non-negatively.

  12. Surely enough, such a list does not cover all the known cases of semisimple small quantum cohomologies available in the literature.

  13. These include the equations in Part I of our proof of Theorem 5.1.

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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).

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    Giordano Cotti

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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

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