Dubrovin conjecture and the second structure connection

Let $X$ be a smooth projective variety of complex dimension $D$. The Gromov–Witten (GW) invariants of $X$ are defined via the intersection theory on the moduli space of stable maps $\overline{\mathcal{M}}_{g,n}(X,d)$ (see Section 3.1 for more details). The structure of the GW invariants is best understood in genus $g=0$. Namely, the entire information is contained in a certain deformation of the classical cup product, known as the quantum cup product. Let us fix a homogeneous basis $\phi_{i}$ ($1\leq i\leq N$) of $H^{*}(X,\mathbb{C})$. Then the quantum cup product $\bullet$ is defined by

where the second sum is over all effective curve classes $d\in H_{2}(X,\mathbb{Z})$, $Q=(q_{1},\dots,q_{r})$ are the so-called Novikov variables, and $t=\sum_{i=1}^{N}t_{i}\phi_{i}\in H^{*}(X,\mathbb{C})$. The Novikov variables correspond to a choice of an ample basis $p_{1},\dots,p_{r}$ of $H^{2}(X,Z)\cap H^{1,1}(X,\mathbb{C})$ and $Q^{d}:=q_{1}^{\langle p_{1},d\rangle}\cdots q_{r}^{\langle p_{r},d\rangle}$. Let us assume that $\phi_{1}=1$ and $\phi_{i+1}=p_{i}$. Then the structure constants of the quantum cup product belong to the following ring of formal power series

where the fact that there is a dependence on $q_{i}e^{t_{i+1}}$ is a consequence of the so-called divisor equation (see [25]). The quantum cup product turns $QH(X,\mathbb{C}):=H^{*}(X,\mathbb{C}[\![Q,t]\!])$ into a commutative associative algebra known as the big quantum cohomology of $X$. We say that the quantum cohomology is semi-simple if $QH(X,\mathbb{C})$ is a semi-simple algebra, or equivalently the operators $\phi\bullet$ of quantum multiplication by $\phi\in H^{*}(X,\mathbb{C})$ are not nilpotent. In the limit $Q\to 0$, the quantum product becomes the classical cup product and since $\phi\cup$ is always nilpotent (for $\phi\neq 1$), we see that semi-simplicity is an indication that the manifold $X$ has sufficiently many rational curves. From that point of view it is very interesting to classify manifolds with semi-simple quantum cohomology. It is an observation of Alexey Bondal that semi-simplicity can be characterized using the language of derived categories. In his ICM talk in 1998, following Bondal’s ideas, Dubrovin was able to formulate a precise conjecture which is now known as the Dubrovin conjecture (see [9] and Conjecture 3.3.1). The goal of this paper is to answer a question which was raised by the second author in his joint work [26]. Namely, there is a conjectural description (see [26], Conjecture 1.6) of the so-called reflection vectors in quantum cohomology in terms of exceptional collections in the derived category. More precisely, Milanov–Xia were able to construct reflection vectors in the quantum cohomology of the blowup of a manifold relying only on certain vanishing results for GW invariants. The resulting formulas were very similar to the formulas for the central connection matrix conjectured by Galkin–Golyshev–Iritani in [13] and later on by Cotti–Dubrovin–Guzzetti [6]. The question is whether one can reformulate Dubrovin’s conjecture in terms of reflection vectors. As expected, the answer is yes and in this paper we would like to work out the precise relation between the reflection vectors and the monodromy data which enters the Dubrovin conjecture. Our main message is that by constructing a basis of reflection vectors in quantum cohomology one can obtain a proof of the $\Gamma$-Conjecture II of Galkin–Golyshev–Iritani (see [13]) or equivalently the refined Dubrovin conjecture of Cotti–Dubrovin–Guzzetti (see [6], Conjecture 5.2). In the rest of this introduction we would like to formulate our results.

From now on we are going to assume that quantum cohomology is semi-simple and convergent. The latter means that there exists a domain $M\subseteq H^{*}(X,\mathbb{C})$, such that, the formal power series representing the structure constants are convergent. Let us introduce the following two linear operators:

and

where $\operatorname{deg}_{\mathbb{C}}(\phi):=k$ for $\phi\in H^{2k}(X,\mathbb{C})$ and $\cup$ is the classical cup product. The quantum differential equations are by definition the differential equations of the following system of ODEs:

where $J(t,z)\in H^{*}(X,\mathbb{C})$, $A_{i}(t)=\phi_{i}\bullet$ is the operator of quantum multiplication by $\phi_{i}$, and

is the Euler vector field. The differential equations (1)–(2) can be viewed also as the equations defining the horizontal sections of a connection on the vector bundle $TM\times\mathbb{C}^{*}\to M\times\mathbb{C}^{*}$ which is sometimes called quantum connection or Dubrovin connection.

The differential equation with respect to $z$ has two singularities: regular at $z=\infty$ and irregular at $z=0$. Near $z=\infty$ there is a geometric way to construct a fundamental solution. Namely, let us define $S(t,z)=1+S_{1}(t)z^{-1}+S_{2}(t)z^{-2}+\cdots$ where $S_{k}(t)\in\operatorname{End}(H^{*}(X,\mathbb{C}))$ are linear operators defined by

Then $S(t,z)z^{\theta}z^{-\rho}$ is a solution to the system (1)–(2).

The singularity at $z=0$ has an interesting Stokes phenomenon. Namely, suppose that $(u_{1},\dots,u_{N})$ are the canonical coordinates defined in a neighborhood of some semi-simple point $t^{\circ}\in M$. By definition, the quantum product and the Poincaré pairing become diagonal:

where $\Delta_{i}\in\mathcal{O}_{M,t^{\circ}}$ are some holomorphic functions. Let us define the linear map

We may think of $\Psi$ as a $N\times N$ matrix with entries $\Psi_{ai}=\sqrt{\Delta_{i}}\frac{\partial t_{a}}{\partial u_{i}}$. Let $U=\operatorname{diag}(u_{1},\dots,u_{N})$ be the diagonal matrix. There exists a unique formal asymptotic solution to (1)–(2) of the form $\Psi R(t,z)e^{U/z}$ where $R(t,z)=1+R_{1}(t)z+R_{2}(t)z^{2}+\cdots$, $R_{k}(t)$ are $N\times N$ matrices. The matrices $R_{k}(t)$ are determined uniquely by plugging in the ansatz $\Psi R(t,z)e^{U/z}$ into (1)–(2) and comparing the coefficients in front of the powers of $z$. This gives us a recursion relation for the coefficients $R_{k}(t)$ which turns out to have a unique solution. Moreover, the solution automatically satisfies $R(t,-z)^{t}R(t,z)=1$ where ^t denotes the standard transposition of matrices. Suppose that the semi-simple point $t^{\circ}$ is generic and that the coordinate neighbourhood is so small that $u_{i}\neq u_{j}$ for $i\neq j$. The rays in the $z$-plane of the form $\mathbf{i}(u_{i}-u_{j})\mathbb{R}_{>0}$ where $\mathbf{i}:=\sqrt{-1}$ and $i\neq j$, are called the Stokes rays. Let $0\in\ell\subset\mathbb{C}$ be a line not parallel to any of the Stokes rays. Let us fix an orientation of $\ell$ by choosing a unit vector $e^{\mathbf{i}\phi}\in\ell$, $\operatorname{Arg}(\phi)\in[0,2\pi)$. Following Dubrovin (see [10]) we will say that $\ell$ is an admissible oriented line. The line $\ell$ splits the $z$-plane into right $\Pi_{\rm right}$ and left $\Pi_{\rm left}$ half-planes. There exists unique solutions $X_{\rm left}(t,z)$ and $X_{\rm right}(t,z)$ to (1)–(2) holomorphic respectively for $z\in\Pi_{\rm right}$ and $z\in\Pi_{\rm left}$ which are asymptotic to $\Psi R(t,z)e^{U/z}$ as $z\to 0$. These solutions extend analytically in $z$ along the positive part $\ell_{+}$ of $\ell$. In particular, we obtain 3 holomorphic solutions to (1)–(2) along the positive half $\ell_{+}$ of $\ell$ which must be related as follows:

where $V_{+}$ and $C^{-1}$ are some constant matrices called respectively, the Stokes matrix and the central connection matrix. The refined Dubrovin conjecture (see [6], Conjecture 5.2) consists of 3 parts. First, the big quantum cohomology of $X$ is semi-simple if and only if the bounded derived category $D^{b}(X)$ has a full exceptional collection. The second part says that every admissible line $\ell$ determines a full exceptional collection $(E_{1},\dots,E_{N})$ which determines uniquely the Stokes matrix $V_{+}$ and the central connection matrix $C^{-1}$. Finally, the 3rd part of the conjecture gives very precise formulas for both $V_{+}$ and $C$, that is, the $(i,j)$-entry of $V_{+}$ is

and the $i$-th column of $C^{-1}$ is

where $\overline{D}\in\{0,1\}$ is the remainder of the division of $D$ by $2$, $\widehat{\Gamma}^{-}_{X}=\prod_{\delta}\Gamma(1-\delta)$ is the so-called gamma class of $X$, and $\operatorname{Ch}(E)=\sum_{\epsilon}e^{2\pi\mathbf{i}\epsilon}$ is the Chern character of $E$. Here the products are over the Chern roots $\delta$ and $\epsilon$ of respectively the holomorphic tangent bundle $TX$ and the complex vector bundle $E$. Just like in the case of $\Psi$, we think of $C^{-1}$ as a linear map $C^{-1}:\mathbb{C}^{N}\to H^{*}(X,\mathbb{C})$. We refer to Section 3.4 for more details.

The refined version of the conjecture still requires that the manifold $X$ is Fano. However, as it was pointed out by Arend Bayer in [3] (see also the recent work by Hiroshi Iritani [22]), by using the blowup operation we can construct many examples of non-Fano manifolds for which the first part of the Dubrovin conjecture holds. Moreover, the recent work by Milanov–Xia (see [26]) gives an indication that the blowup operation preserves the remaining two parts of the Dubrovin conjecture. Therefore, it is quite plausible that the Fano condition is redundant.

Suppose that the big quantum cohomology is semi-simple and convergent. The solutions to the quantum differential equations can be represented by complex oscillatory integrals of the following form:

where $m\in\mathbb{C}$ is a complex number and the semi-infinite integration cycle is chosen in such a way that the integral is convergent. It is easy to check that the above integral solves the quantum differential equations (1)–(2) iff the integrand $I^{(m)}(t,\lambda)$ satisfies the following system of ODEs:

This is a system of differential equations for the horizontal sections of a connection $\nabla^{(m)}$ on the trivial bundle

where

The hypersurface $\det(\lambda-E\bullet_{t})=0$ in $M\times\mathbb{C}$ is called the discriminant. The connection $\nabla^{(m)}$ is known as the second structure connection. In the case when $m=0$ or $-1$, the connection was used by Dubrovin to define the monodromy group of a Frobenius manifold (see [8]). However, it became clear shortly afterwards that it is important to study the entire family, that is, allow $m$ to be any complex number (see [24] and [11]).

The space of solutions to (4)–(5) is quite interesting. In the examples of mirror symmetry the second structure connection of quantum cohomology can be identified with a Gauss–Manin connection. Therefore, the solutions to (4)–(5) should be thought as period integrals. In particular, by using $\nabla^{(m)}$ we can introduce many of the ingreidents of Picard–Lefschetz theory. This was done by Dubrovin (see [11], Section 4). He called the solutions to (4)–(5) twisted periods because their properties are very similar to the period integrals in Givental’s twisted Picard–Lefschetz theory [15]. Motivated by the work of Givental in [17], the second author introduced in [27] the following fundamental solution to (4)–(5):

where

Note that both $I^{(m)}(t,\lambda)$ and $\widetilde{I}^{(m)}(\lambda)$ take values in $\operatorname{End}(H^{*}(X,\mathbb{C}))$. The second structure connection has a Fuchsian singularity at infinity, therefore the series $I^{(m)}(t,\lambda)$ is convergent and it defines a multi-valued analytic function in the complement to the discriminant. There are many ways to choose a fundamental solution but what makes the above choice special is the specific choice of building blocks, that is, the calibrated periods (7) while the standard approach would be monomials in $\lambda$. The existence of such decomposition follows from Givental’s formalism of quantized symplectic transformations and their actions on vertex operators (see [17], Section 5, especially Theorem 2). Although Dubrovin already knew that one can import concepts from singularity theory to quantum cohomology, somehow the above choice of a fundamental solution makes the parallel with singularity theory much more visible (at least to the authors).

Let us choose a base point $t^{\circ}\in M$, such that, $\operatorname{Re}u_{i}(t^{\circ})\neq\operatorname{Re}u_{j}(t^{\circ})$ for $i\neq j$. Then $\operatorname{Re}u_{i}(t)\neq\operatorname{Re}u_{j}(t)$ for $i\neq j$ for all $t$ sufficiently close to $t^{\circ}$. Let $\lambda^{\circ}$ be a positive real number, such that, $\lambda^{\circ}>|u_{i}(t^{\circ})|$ for all $i$. We define the $m$-twisted period vectors $I_{a}^{(m)}(t,\lambda):=I^{(m)}(t,\lambda)a$ where $a\in H^{*}(X,\mathbb{C})$ and the value depends on the choice of a reference path avoiding the discriminant from $(t^{\circ},\lambda^{\circ})$ to $(t,\lambda)$. Note that at $(t^{\circ},\lambda^{\circ})$ the only ambiguity is in the choice of the value for the calibrated periods, that is, we need to specify a branch of $\log\lambda$ when $\lambda$ is close to $\lambda^{\circ}$. Since $\lambda^{\circ}$ is a positive real number we simply take the principal branch of the logarithm.

Let us introduce the following pairings $h_{m}:H^{*}(X,\mathbb{C})\times H^{*}(X,\mathbb{C})\to\mathbb{C}$

Using the differential equations of $\nabla^{(\pm m)}$ it is easy to check that $h_{m}(a,b)$ is independent of $t$ and $\lambda$. It turns out that there is an explicit formula for $h_{m}$ in terms of the Hodge grading operator $\theta$ and the nilpotent operator $\rho$. Let us recall the so-called Euler pairing

As a byproduct of the proof of Theorem 2.6.1 we will get the following simple formula:

where $q:=e^{\pi\mathbf{i}m}$. The above formula shows that $h_{m}$ is the analogue of the $\mathbb{Z}[q,q^{-1}]$-bilinear intersection form in twisted Picard–Lefschetz theory (see [15], Section 3). Furthermore, let us fix a reference path from $(t^{\circ},\lambda^{\circ})$ to a point $(t,\lambda)$ sufficiently close to a generic point $b$ on the discriminant. The local equation of the discriminant near $b$ has the form $\lambda=u_{i}(t)$ where $u_{i}(t)$ is an eigenvalue of $E\bullet$. It turns out that the set of all $a\in H^{*}(X,\mathbb{C})$, such that, $I^{(m)}_{a}(t,\lambda)$ is analytic at $\lambda=u_{i}(t)$ is a codimension 1 subspace of $H^{*}(X,\mathbb{C})$. Suppose that $m\nin\tfrac{1}{2}+\mathbb{Z}$, then there is a 1-dimensional subspace of vectors $\beta\in H^{*}(X,\mathbb{C})$, such that, $(\lambda-u_{i}(t))^{m+1/2}I^{(m)}_{\beta}(t,\lambda)$ is analytic at $\lambda=u_{i}(t)$ and the value at $\lambda=u_{i}(t)$ belongs to $\mathbb{C}\Psi(e_{i})$ where $\Psi$ is the map (3). Therefore, for the given reference path and an arbitrary choice of $\log(\lambda-u_{i}(t))$ there is a uniquely defined vector $\beta=\beta(m)$, such that,

where the coefficient in front of $\Psi(e_{i})$ is such that $h_{m}(\beta(m),\beta(-m))=q+q^{-1}$. Note that the choice of a reference path and a branch of $\log(\lambda-u_{i}(t))$ determines $\beta(m)$ for all $m\nin\tfrac{1}{2}+\mathbb{Z}$, that is, we have a map

Using that $\partial_{\lambda}I^{(m)}(t,\lambda)=I^{(m+1)}(t,\lambda)$ we get that this map is periodic: $\beta(m+1)=\beta(m).$ It will follow from our results that $\beta$ is a trigonometric polynomial, that is, $\beta\in H^{*}(X,\mathbb{C}[q^{2},q^{-2}])$. If we change the value of the logarithmic branch $\log(\lambda-u_{i}(t))\mapsto\log(\lambda-u_{i}(t))+2\pi\mathbf{i}$, then $\beta(m)\mapsto-q^{-2}\beta(m)$. Therefore, for a fixed reference path the value of $\beta(m)$ is fixed up to a factor in the spiral $(-q^{-2})^{\mathbb{Z}}=\{(-1)^{k}q^{-2k}\ |\ k\in\mathbb{Z}\}$. We will say that $\beta$ is a twisted reflection vector corresponding to the given reference path. We usually supress the dependence on the logarithmic branch if the choice is irrelevant or it is clear from the context. The following formula for the local monodromy of $\nabla^{(m)}$ justifies our terminology:

that is, the local monodromy is a complex reflection whose fixed points locus is the hyperplane orthogonal to $\beta(-m)$. We refer to Section 2.3 for more details and for more general settings, i.e., we can introduce twisted reflection vectors for any semi-simple Frobenius manifold.

We continue to work in the settings from the previous two sections. Let $\ell$ be an admissible oriented line (see Section 1.2) with orientation $e^{\mathbf{i}\phi}$. By definition $\eta=\mathbf{i}e^{\mathbf{i}\phi}$ is not parallel to any of the differences $u_{i}(t^{\circ})-u_{j}(t^{\circ})$ for $i\neq j$. We will refer to $\eta$ as an admissible direction. Our choice of a reference point $(t^{\circ},\lambda^{\circ})$ is such that the real line with its standard orientation is an admissible oriented line. The corresponding admissible direction is $\eta^{\circ}:=\mathbf{i}$. Any other admissible direction $\eta$ will be equipped with a reference path to $\eta^{\circ}$ or equivalently, we fix an analytic branch of $\log$ in a neighbourhood of $\eta$. Finally, we consider only $t\in M$ sufficiently close to $t^{\circ}$, such that, $\eta$ is an admissible direction for $t$, that is, $\eta$ is not parallel to $u_{i}(t)-u_{j}(t)$ for $i\neq j$.

Let us construct a system of reference paths $C_{1}(\eta),\dots,C_{N}(\eta)$ corresponding to $\eta$. Each $C_{i}(\eta)$ starts at $\lambda=u_{i}$, approaches the circle $|\lambda|=\lambda^{\circ}$ in the direction of $\eta$, after hitting the circle at some point $\lambda^{i}(\eta)$ the path continues clockwise along the circle arc from $\lambda^{i}(\eta)$ to $\lambda^{\circ}(\eta):=-\mathbf{i}\eta\lambda^{\circ}$, and finally by continuously deforming the direction $\eta$ to $\eta^{\circ}$ the path connects $\lambda^{\circ}(\eta)$ and $\lambda^{\circ}(\eta^{\circ})=\lambda^{\circ}$ – see Figure 1. Note that if $\lambda\in C_{i}$ is sufficiently close to $u_{i}$, then $\lambda-u_{i}=s\eta$ for some positive real number $s$ and we have a natural choice of a logarithmic branch: $\log(\lambda-u_{i}):=\ln(s)+\log(\eta)$. Therefore, as it was explained in Section 1.3, we may choose a twisted reflection vector $\beta_{i}(m)$. In other words, each admissible direction determines a set of twisted reflection vectors $(\beta_{1}(m),\dots,\beta_{N}(m))$. Furthermore, the admissible direction determines the following order of the eigenvalues $u_{1},\dots,u_{N}$ of $E\bullet$: we say that $u_{i}<u_{j}$ if $u_{j}$ is on the RHS of the line through $u_{i}$ parallel to $\eta$ where RHS means that we have to stand at $u_{i}$ and look in the direction $\eta$. For example, for the standard admissible direction $\eta^{\circ}=\mathbf{i}$, $u_{i}<u_{j}$ would mean that $\operatorname{Re}(u_{i})<\operatorname{Re}(u_{j})$. We will refer to the order as the lexicographical order determined by $\eta$. Let us assume that the enumeration of the eigenvalues $u_{1},\dots,u_{N}$ is according to the lexicographical order, that is, $u_{i}<u_{j}$ iff $i<j$. Our main result can be stated as follows.

In fact our result is more general. The above theorem can be formulated in the settings of semi-simple Frobenius manifolds. Under an additional technical assumption, i.e., we assume that the Frobenius manifold has a calibration for which the grading operator is a Hodge grading operator (see Definition 2.1.2), we prove that the conclusions of the above theorem remain true (see Theorem 2.6.1).

Using Theorem 1.4.1 we can answer the question raised in [26]. Following the analogy with singularity theory (see [1, 12]), we introduce the concept of a distinguished system of reference paths (see Definition 3.5.1). Let us recall the Iritani’s integral structure map (see [21]) $\Psi_{Q}:K_{0}(X)\rightarrow H^{*}(X,\mathbb{C})$ defined by

where $Q=(q_{1},\dots,q_{r})$ are the Novikov variables corresponding to an ample basis $p_{1},\dots,p_{r}$ of $H^{2}(X,\mathbb{Z})\cap H^{1,1}(X,\mathbb{C})$. We have the following relation:

which justifies why we refer to (9) as the Euler pairing.

Theorem 1.4.2 is proved in Section 3.5 (see Theorem 3.5.1). The definition of a reflection vector is the same as the definition of a twisted reflection vector except that we require $m\in\mathbb{Z}$. Since the fundamental group of $\mathbb{C}\setminus{\{u_{1}^{\circ},\dots,u_{N}^{\circ}\}}$ is generated by simple loops corresponding to the reference paths $C_{i}(\eta)$ where $\eta$ is an admissible direction, we have the following interesting corollary.

For the proof, thanks to the braid group action on the set of distinguished systems of reference paths (see Section 3.5), we may assume that $\beta_{1},\dots,\beta_{N}$ are the reflection vectors corresponding to the reference paths $C_{1}(\eta),\dots,C_{N}(\eta)$ for some admissible direction $\eta$. Note that every twisted reflection vector is obtained from some $\beta_{i}$ by a sequence of local monodromy transformations $M_{j}^{\pm 1}$ ($1\leq j\leq N$) where $M_{j}$ is the monodromy transformation corresponding to the simple loop associated with $C_{j}(\eta)$. According to Theorem 1.4.1, part a), the matrix of $M_{j}$ (resp. $M_{j}^{-1}$) in the basis $\beta_{1},\dots,\beta_{N}$ is upper-triangular and the only entry depending on $q$ is in position $(j,j)$, that is, it is equal to $-q^{-2}$ (resp. $-q^{2}$). In addition, if the refined Dubrovin conjecture holds, then since $\langle\beta_{i},\beta_{j}\rangle=\chi(F_{i},F_{j})\in\mathbb{Z}$, we get that the entries of $M_{j}^{\pm 1}$ belong to $\mathbb{Z}[q^{2},q^{-2}]$.

Let us point out that the full exceptional collection $(F_{1},\dots,F_{N})$ in Theorem 1.4.2 is not the same as the full exceptional collection $(E_{1},\dots,E_{N})$ in the refined Dubrovin conjecture. The reason is that the objects $E_{i}$ correspond to oscillatory integrals in which the integration paths are rays with direction $-\eta$ while $F_{i}$ correspond to reference paths going in the opposite direction $\eta$. Changing the admissible direction from $\eta$ to $-\eta$ means that one has to perform a certain sequence of mutations in order to get from one exceptional collection to the other one. It turns out that the sequence of mutations that we need is well known in the theory of derived categories, i.e., this is the same sequence used to define the left Koszul dual of an exceptional collection. The precise statement is that up to a shift by $\tfrac{D-\overline{D}}{2}$ the exceptional collection $(F_{1},\dots,F_{N})$ is the left Koszul dual to $(E_{N}^{\vee},\dots,E_{1}^{\vee})$. The moral is that although there is some freedom in defining a Stokes matrix and a central connection matrix of the quantum connection, the statement of the refined Dubrovin conjecture is independent of the choices that we make. The different full exceptional collection that one might get due to the discrepancy of the definitions are related by mutations in the derived category. Let us point out that even Dubrovin himself did not use the definitions consistently – the central connection matrix in [10, 11] is the inverse of the central connection matrix in [6].

Finally, let us comment on the proofs. The relation between the quantum connection and its Laplace transform was studied by many people. In particular, the relation between Stokes multipliers and the monodromy data of the Laplace transform of the quantum connection is well known thanks to the work of Balsar–Jurkat–Lutz [2]. There is also a recent paper by Guzzetti (see [18]) who was able to remove some technical conditions from the main result in [2]. Although, we do not directly use any results from [2], the ideas for all proofs come from there, except for the formula for the connection matrix (see Theorem 2.6.1, part a) whose proof follows the ideas of Dubrovin (see [11], Theorem 4.19). Our results should not be very surprising to the experts. Especially, the work of Galkin–Golyshev–Iritani [13] and Dubrovin [11] contain almost all ideas and results necessary to prove Theorems 1.4.1 and 1.4.2. In some sense we could have written much shorter text. Nevertheless, in order to avoid gaps in the arguments due to misquoting results, we decided to have a self contained text independent of the results in [2, 11, 13].

We are thankful to Alexey Bondal for many useful discussions on the Dubrovin conjecture and especially for pointing out to us the notion of left and right Koszul dual of an exceptional collection. The first author also thanks Jin Chen and Mauricio Romo for many interesting discussions on the $\Gamma$-conjectures and Dubrovin conjecture. This work is supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan and by JSPS Kakenhi Grant Number JP22K03265. The first author is also supported by the President’s International Fellowship Initiative of the Chinese Academy of Sciences.

Let $M$ be a complex manifold and $\mathcal{T}_{M}$ be the sheaf of holomorphic vector fields on $M$. Suppose that $M$ is equipped with the following structures:

1. (F1)

   A non-degenerate symmetric bilinear pairing

   $$(\,\cdot\,,\,\cdot\,):\mathcal{T}_{M}\otimes\mathcal{T}_{M}\to\mathcal{O}_{M}.$$

2. (F2)

   A Frobenius multiplication: commutative associative multiplication

   $$\cdot\bullet\cdot:\mathcal{T}_{M}\otimes\mathcal{T}_{M}\to\mathcal{T}_{M},$$

   such that $(v_{1}\bullet w,v_{2})=(v_{1},w\bullet v_{2})$ $\forall v_{1},v_{2},w\in\mathcal{T}_{M}$.

3. (F3)

   A unit vector field: global vector field $\mathbf{1}\in\mathcal{T}_{M}(M)$, such that,

   $$\mathbf{1}\bullet v=v,\quad\nabla^{\rm L.C.}_{v}\mathbf{1}=0,\quad\forall v\in\mathcal{T}_{M},$$

   where $\nabla^{\rm L.C.}$ is the Levi–Civita connection of the pairing $(\cdot,\cdot)$.

4. (F4)

   An Euler vector field: global vector field $E\in\mathcal{T}_{M}(M)$, such that, there exists a constant $D\in\mathbb{C}$, called conformal dimension, and

   $$E(v_{1},v_{2})-([E,v_{1}],v_{2})-(v_{1},[E,v_{2}])=(2-D)(v_{1},v_{2})$$

   for all $v_{1},v_{2}\in\mathcal{T}_{M}$.

Note that the complex manifold $TM\times\mathbb{C}^{*}$ has a structure of a holomorphic vector bundle with base $M\times\mathbb{C}^{*}$: the fiber over $(t,z)\in M\times\mathbb{C}^{*}$ is $T_{t}M\times\{z\}\cong T_{t}M$ which has a natural structure of a vector space. Given the data (F1)-(F4), we define the so called Dubrovin connection on the vector bundle $TM\times\mathbb{C}^{*}$

where $z$ is the standard coordinate on $\mathbb{C}^{*}=\mathbb{C}\setminus{\{0\}}$, where $v\bullet$ is an endomorphism of $\mathcal{T}_{M}$ defined by the Frobenius multiplication by the vector field $v$, and where $\theta:\mathcal{T}_{M}\to\mathcal{T}_{M}$ is an $\mathcal{O}_{M}$-modules morphism defined by

Near $z=\infty$ the Dubrovin connection has a fundamental solution of the following form:

where $\delta$ is a diagonalizable operator, $\rho$ is a nilpotent operator, and the operator-valued series $S(t,z)=1+S_{1}(t)z^{-1}+\cdots$, $S_{k}\in\operatorname{End}(\mathcal{T}_{M})$ satisfies the symplectic condition $S(t,z)S(t,-z)^{T}=1$, where ^T is transposition with respect to the Frobenius pairing. It can be proved that $\delta$ coincides with the semi-simple part $\theta_{s}$ of the grading operator $\theta$ in the Jordan–Chevalley decomposition $\theta=\theta_{s}+\theta_{n}$, where the operators $\theta_{s}$ and $\theta_{n}$ are uniquely determined by the following 3 conditions:

1. (i)

   Commutativity: $[\theta_{s},\theta_{n}]=0$.

2. (ii)

   The operator $\theta_{s}$ is diagonalizable.

3. (iii)

   The operator $\theta_{n}$ is nilpotent.

Moreover, the operator $\rho=-\theta_{n}+\sum_{l=1}^{\infty}\rho_{l}$ where $\rho_{l}\neq 0$ for finitely many $l$ and $[\delta,\rho_{l}]=-l\rho_{l}$. For more details we refer to [28], Section 1.3.1. Following Givental [16] we will refer to the pair $(S(t,z),\rho)$ as calibration of $M$. Sometimes we will drop $\rho$ from the notation and say that $S(t,z)$ is the calibration.

Note that if $\theta$ is a Hodge grading operator, then $\delta=\theta$ and $\rho_{l}=0$ for all $l\neq 1$. The fundamental solution takes the form $X(t,z)=S(t,z)z^{\theta}z^{-\rho}$. From now on we will consider only Frobenius manifolds with a fixed calibration, such that, $\theta$ is a Hodge grading operator. The problem that we will be interested in is local, so we will further assume that $M$ has a global flat coordinate system $(t_{1},\dots,t_{N})$.

Let us fix a base point $t^{\circ}\in M$. Put $\phi_{i}:=\partial/\partial t_{i}|_{t^{\circ}}$ $(1\leq i\leq N)$, then $\{\phi_{i}\}_{i=1}^{N}$ is a basis of the reference tangent space $H:=T_{t^{\circ}}M$. The flat vector fields $\partial/\partial t_{i}$ ($1\leq i\leq N$) provide a trivialization of the tangent bundle $TM\cong M\times H$. This allows us to identify the Frobenius multiplication $\bullet$ with a family of associative commutative multiplications $\bullet_{t}:H\otimes H\to H$ depending analytically on $t\in M$. The operator $\theta:\mathcal{T}_{M}\to\mathcal{T}_{M}$ defined above preserves the subspace of flat vector fields. It induces a linear operator on $H$, known to be skew symmetric with respect to the Frobenius pairing $(\ ,\ )$.

There are two flat connections that one can associate with the Frobenius structure. The first one is the Dubrovin connection – defined above. The Dubrovin connection in flat coordinates takes the following form:

where $z$ is the standard coordinate on $\mathbb{C}^{*}=\mathbb{C}-\{0\}$ and for $v\in\Gamma(M,\mathcal{T}_{M})$ we denote by $v\bullet:H\to H$ the linear operator of Frobenius multiplication by $v$.

We will be interested also in the second structure connection

where $m\in\mathbb{C}$ is a complex parameter. This is a connection on the trivial bundle