Dubrovin Connection

This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.

In the previous post we saw that quantum cohomology \(QH^\ast(X)\) and the Jacobi ring \(\Jac(W_q)\) carry Frobenius algebra structures, and we embedded them into a Frobenius manifold structure in order to capture the naturality of their isomorphisms in the \(q\)-parameter direction. The data carried by a Frobenius manifold \(M\) consists of the following:

• The metric \(\eta\) (in \(QH^\ast(X)\) this is the Poincaré pairing) and the Levi-Civita connection derived from it,
• The product \(\circ\) of the Frobenius algebra (in \(QH^\ast(X)\) this is the quantum cup product),
• The Euler vector field \(E\) (encoding degree information)

and we verified that the associativity of \(\circ\) is expressed by the WDVV equation. (§Frobenius Manifolds, ⁋Proposition 7) The Dubrovin connection, which we treat in this post, goes a step further and shows that \(\eta\) and \(\circ\), or more precisely \(\nabla\) and \(\circ\), are deeply related to each other.

Dubrovin Connection

According to Dubrovin, \(\nabla\) and \(\circ\) are linked by a flat connection \(\nabla^z\) on \(M\times \mathbb{C}^\ast\) called the Dubrovin connection; this connection recovers \(\circ\) as \(z\rightarrow 0\) and \(\nabla\) as \(z\rightarrow\infty\). In order for this to make sense, we must somewhat justify what it means to treat \(\circ\) as a connection.

Generally, a connection is written in a local frame in the form \(\nabla_{\partial_\alpha} = \partial_\alpha + A_\alpha\), where \(A_\alpha\) is a connection \(1\)-form, i.e. an endomorphism on the fiber. ([Riemannian Geometry] §Connections, ⁋Definition 3) The key observation is that, for the product \(\circ\), considering the endomorphism \(\mathcal{C}_\alpha = \partial_\alpha \circ -\) of “multiplication by \(\partial_\alpha\)” in each direction \(\partial_\alpha\), its matrix entries are precisely the structure constants \(c_{\alpha\beta}^\gamma\) of the product. In other words, strictly speaking \(\circ\) itself is not a connection; rather, its structure constants play the role of Christoffel symbols. ([Riemannian Geometry] §Levi-Civita Connection, ⁋Proposition 6)

Indeed, choosing (flat) coordinates \(\{ t^\alpha \}\), we can immediately verify that

\[\mathcal{C}_\alpha(\partial_\beta) = \partial_\alpha \circ \partial_\beta = \sum_\gamma c_{\alpha\beta}^\gamma\, \partial_\gamma\]

Thus, considering the connection

\[\nabla^z_{\partial_\alpha} = \partial_\alpha + \frac{1}{z}\, \mathcal{C}_\alpha\]

that joins the two, we can organize them into a one-parameter family converging to the Levi-Civita connection \(\nabla\) as \(z \to \infty\) and diverging to the classical limit of the product \(\circ\) as \(z \to 0\), and when computing at \(z\rightarrow 0\) one simply rescales to extract the leading term of \(z\nabla^z_{\partial_\alpha} = z\partial_\alpha + \mathcal{C}_\alpha\) in the \(z \to 0\) limit. In any case, in this sense \(\nabla^z\) is a flat pencil of connections joining the two structures, and physically it is interpreted as the string coupling constant.

Definition 1 Let \(M\) be a Frobenius manifold and \(z \in \mathbb{C}^\ast\) an auxiliary complex parameter. Then the Dubrovin connection \(\nabla^z\) is the connection on the pullback bundle defined via the projection

\[\pr_1: M\times \mathbb{C}^\ast \rightarrow M\]

given for flat coordinates \(\{ t^\alpha \}\) on \(M\) by the formula

\[\nabla^z_{\partial_\alpha} = \partial_\alpha + \frac{1}{z}\, \mathcal{C}_\alpha, \qquad \mathcal{C}_\alpha(X) := \partial_\alpha \circ X\]

Here \(\mathcal{C}_\alpha\) is the endomorphism \(\partial_\alpha\circ-\) defined above. The remaining connection component in the \(z\) direction is given by the formula

\[\nabla^z_{\partial_z} = \partial_z - \frac{1}{z^2}E\circ(-) + \frac{1}{z}\mu\]

Here \(E\) is the Euler vector field (§Frobenius Manifolds, ⁋Definition 5), and \(\mu\) is the grading operator, defined by \(\mu(\partial_\alpha) = (d_\alpha - d/2)\, \partial_\alpha\) from the half-degree \(d_\alpha = \tfrac{1}{2}\deg\sigma^\alpha\) of the cohomology class \(\sigma^\alpha\) corresponding to the flat coordinate \(t^\alpha\) and the conformal dimension \(d\).

Recall that in §Frobenius Manifolds, ⁋Definition 5, when defining a Frobenius manifold, we introduced \(E\) in order to encode the grading structure of the Frobenius algebra at each point. Specifically,

\[\Lie_E(\circ)=\circ,\qquad \Lie_E(\eta)=(2-d)\eta\]

reflect the fact that the quantum product respects degree and that the Poincaré pairing is concentrated in top degree, respectively. In particular, in the case of §Frobenius Manifolds, ⁋Proposition 9, which is our main object of interest, if the Euler vector field \(E\) in the above formula generates the grading by rescaling coordinates on the base \(M\), then \(\mu\) is the same grading viewed as an endomorphism on the fiber \(T_tM \cong H^\ast(X)\), and they are related by

\[\mu = \frac{2-d}{2}I - \nabla E\]

Here \(I\) is the identity matrix on \(H^\ast(X)\). To see this more intuitively, writing it out directly in flat coordinates for \(\nabla\), \(E\) is

\[E = \sum_\alpha (1-d_\alpha)t^\alpha \partial_\alpha + \text{(constant terms)}\]

so we know that \(\nabla E\) has eigenvalue \(1-d_\alpha\) corresponding to the eigenvector \(\partial_\alpha\). Substituting this into \(\mu = \frac{2-d}{2}I - \nabla E\),

\[\mu(\partial_\alpha) = \frac{2-d}{2} - (1-d_\alpha) = d_\alpha - d/2\]

so \(\mu\) also has eigenvalue \(d_\alpha-d/2\). The shift by \(\frac{2-d}{2}\) is to make \(\mu\) skew-symmetric with respect to \(\eta\), arising from the fact that \(\eta\) pairs a class of degree \(d_\alpha\) with one of degree \(d - d_\alpha\).

The most important property of the Dubrovin connection is that it is flat for every \(z\). To verify this, let us compute the curvature \([\nabla^z_{\partial_\alpha}, \nabla^z_{\partial_\beta}]\). In flat coordinates \(\nabla^z_{\partial_\alpha} = \partial_\alpha + z^{-1}\mathcal{C}_\alpha\), and \([\partial_\alpha, \partial_\beta] = 0\), while by the Leibniz rule \([\partial_\alpha, \mathcal{C}_\beta] = \partial_\alpha \mathcal{C}_\beta\) (the endomorphism obtained by differentiating the components of \(\mathcal{C}_\beta\)), so

\[[\nabla^z_{\partial_\alpha}, \nabla^z_{\partial_\beta}] = [\partial_\alpha + z^{-1}\mathcal{C}_\alpha,\ \partial_\beta + z^{-1}\mathcal{C}_\beta] = \frac{1}{z}\,(\partial_\alpha \mathcal{C}_\beta - \partial_\beta \mathcal{C}_\alpha) + \frac{1}{z^2}\,[\mathcal{C}_\alpha, \mathcal{C}_\beta]\]

For this curvature to vanish for all \(z\), the coefficients of \(z^{-1}\) and \(z^{-2}\) must each vanish; the \(z^{-1}\) term vanishes by the potentiality of \(\mathcal{C}\), namely \(\partial_\alpha\mathcal{C}_\beta = \partial_\beta\mathcal{C}_\alpha\), and the \(z^{-2}\) term vanishes by the associativity of the product, \([\mathcal{C}_\alpha, \mathcal{C}_\beta] = 0\).

Moreover, the following proposition shows that the flatness of these connections is exactly equivalent to these two conditions. These were the axioms of a Frobenius manifold (§Frobenius Manifolds, ⁋Definition 5), and therefore the \(M\)-direction flatness of \(\nabla^z\) is not merely a by-product of adjusting moduli, but can be said to be the Frobenius structure itself.

Proposition 2 Consider the connection \(\nabla^z\) on a Frobenius manifold \(M\) (Definition 1). Under the assumption that the product \(\circ\) is commutative, \(\nabla^z\) being flat in the \(M\)-directions (i.e. among the \(\partial_\alpha\) directions) for every \(z\) is equivalent to the following two conditions both holding.

1. Potentiality: \(\partial_\alpha\, c_{\beta\gamma}^\delta = \partial_\beta\, c_{\alpha\gamma}^\delta\). That is, \(c_{\alpha\beta}^\delta\) is the third derivative of some potential \(F\).
2. Associativity (WDVV): \([\mathcal{C}_\alpha, \mathcal{C}_\beta] = 0\), or in components \(\sum_\delta c_{\alpha\beta}^\delta\, c_{\delta\gamma}^\epsilon = \sum_\delta c_{\beta\gamma}^\delta\, c_{\alpha\delta}^\epsilon\). That is, \(\circ\) is associative.

Proof

We have already shown one direction above, so we need only check the converse. Suppose \(\nabla^z\) is flat for all \(z \in \mathbb{C}^\ast\). The curvature

\[[\nabla^z_{\partial_\alpha}, \nabla^z_{\partial_\beta}] = \frac{1}{z}\,(\partial_\alpha\mathcal{C}_\beta - \partial_\beta\mathcal{C}_\alpha) + \frac{1}{z^2}\,[\mathcal{C}_\alpha, \mathcal{C}_\beta]\]

is a Laurent polynomial in \(z^{-1}\) and \(z^{-2}\), so its vanishing for all \(z\) is equivalent to the two coefficients vanishing separately. The vanishing of the \(z^{-1}\) coefficient is exactly the first condition \(\partial_\alpha\mathcal{C}_\beta = \partial_\beta\mathcal{C}_\alpha\), and writing the vanishing of the \(z^{-2}\) coefficient, \([\mathcal{C}_\alpha, \mathcal{C}_\beta] = 0\), in components gives \(\sum_\delta (c_{\alpha\delta}^\epsilon c_{\beta\gamma}^\delta - c_{\beta\delta}^\epsilon c_{\alpha\gamma}^\delta) = 0\), which under the assumption that \(\circ\) is commutative is exactly associativity, i.e. the WDVV equation. (§Frobenius Manifolds, ⁋Proposition 7)

On the other hand, flatness in the \(z\)-direction, \([\nabla^z_{\partial_z}, \nabla^z_{\partial_\alpha}] = 0\), requires the condition that the Euler vector field \(E\) and the grading operator \(\mu\) are compatible with the product, i.e. the homogeneity (or conformal) condition of the Frobenius manifold. Since this condition is already built into our definition as the fourth condition in §Frobenius Manifolds, ⁋Definition 5, in our definition we obtain the full flatness of \(\nabla^z\) including the \(z\)-direction.

D-module

A connection \(\nabla\) is essentially a tool for differentiating sections. When we view a vector bundle as an \(\mathcal{O}_X\)-module, the only operation we have is multiplication by functions, but once differentiation is available, that bundle becomes an object receiving the action of differential operators in addition to functions, namely a \(\mathcal{D}_X\)-module, and flatness is required for the definition to be meaningful.

Definition 3 On a complex manifold \(B\), the sheaf of rings \(\mathcal{D}_B\) of differential operators on \(B\) is the sheaf of operators generated by the structure sheaf \(\mathcal{O}_B\) and vector fields, i.e. derivations on \(\mathcal{O}_B\) ([Commutative Algebra] §Differentials, ⁋Definition 1). In this case, a vector field \(\partial\) and a function \(f\) satisfy the relation

\[[\partial, f] = \partial(f)\]

An \(\mathcal{O}_B\)-module \(\mathcal{M}\) equipped with a \(\mathcal{D}_B\)-action is called a \(\mathcal{D}_B\)-module.

The \(\mathcal{O}_B\)-module structure on \(\mathcal{M}\) is generally thought of as multiplication by a function \(f\in \mathcal{O}_B\). Then for any section \(s\in \mathcal{M}\), the relation \([\partial, f]=\partial(f)\) yields the Leibniz rule

\[\partial(f s) = (\partial f)\, s + f\, \partial s \qquad (f \in \mathcal{O}_B,\ s \in \mathcal{M})\]

as can be verified. As a concrete example, given a vector bundle \(E\rightarrow B\) equipped with a flat connection \(\nabla\), defining the action of a vector field \(\partial\) to be \(\nabla_\partial\) makes \(E\) a \(\mathcal{D}_B\)-module, and in this example the flatness of \(\nabla\), \([\nabla_\partial, \nabla_{\partial'}] = \nabla_{[\partial, \partial']}\), becomes exactly the commutator relation required by the definition of a \(\mathcal{D}_B\)-module.

In particular, since we have seen that the Dubrovin connection \(\nabla^z\) is flat (Proposition 2), we can verify from this that \(\pr_1^\ast TM\) becomes a \(\mathcal{D}_{M\times \mathbb{C}^\ast}\)-module. This is called the quantum \(D\)-module. Here, writing the horizontal sections of the \(\mathcal{D}\)-module, i.e. functions satisfying \(\nabla^z s=0\), as \(s=\sum_\alpha s^\alpha\partial_\alpha\) using flat coordinates, we obtain the differential equation

\[\partial_\alpha s^\beta + \frac{1}{z} \sum_\gamma c_{\alpha\gamma}^\beta\, s^\gamma = 0\]

which is called the quantum differential equation. Since the above equation is a first order linear ODE, once a base point \(b_0\) and initial condition \(s(b_0)\) are given, the solution is uniquely determined along a path, and because \(\nabla^z\) is flat, this parallel transport is independent of the path, giving a well-defined horizontal section in a simply connected neighborhood of \(b_0\). Therefore the solution space of the above QDE is \(\dim_\mathbb{C} M\)-dimensional, matching the rank of the bundle, and the \(\pi_1\) of the base \(M \times \mathbb{C}^\ast\) acts on this solution space via parallel transport, yielding a monodromy representation. In particular, a loop in the \(z\)-direction (\(\pi_1 \cong \mathbb{Z}\)) gives monodromy around the irregular singularities at \(z = 0, \infty\), and a loop in the \(q\)-direction of the Novikov parameter \(q = e^t\) gives quantum monodromy.

A representative example of a Frobenius manifold is \(M = H^\ast(X, \mathbb{C})\), which encodes deformations of quantum cohomology. (§Frobenius Manifolds, ⁋Proposition 9) Let us examine the Dubrovin connection in this case in detail. First, the base manifold is \(M \times \mathbb{C}^\ast\), and we can write a point of this manifold as \((t,z)\in M\times \mathbb{C}^\ast\). By definition, the fiber over a point \((t,z)\) is \((\pr_1^\ast TM)_{(t,z)}\cong T_tM\), and since \(M\) was a vector space from the outset, this fiber is canonically isomorphic to \(H^\ast(X, \mathbb{C})\), and the Frobenius manifold structure is given by endowing each of these with the big quantum product \(\circ_t\) defined by \(t\). On this bundle the Dubrovin connection is given by \(\nabla^z_{\partial_\alpha} = \partial_\alpha + z^{-1}\mathcal{C}_\alpha\), where \(\mathcal{C}_\alpha = \partial_\alpha \circ_t -\) is the endomorphism given by multiplication by the big quantum product. The quantum \(D\)-module corresponding to big quantum cohomology is what we obtain by taking the whole \(M = H^\ast(X)\) as the base in this way. As we saw in the previous post, our primary object of interest is the deformation in the \(H^2\) direction, which corresponds to small quantum cohomology among these, so in particular, letting \(\{T_\alpha\}\) be a basis of \(H^2(X)\), the connection constant in this direction is \(\mathcal{C}_a=T_a\qtimes -\). Therefore the QDE in this direction is

\[z\, q_a \partial_{q_a} s = -\,(T_a \qtimes s), \qquad a = 1, \ldots, r\]

and as we examined above, the solution space of this equation is \(\dim_\mathbb{C} H^\ast(X)\)-dimensional. Givental’s \(J\)-function, which gives the explicit fundamental solution of this system, carries the A-side data. Thus, when the connection is the Dubrovin connection \(\nabla^z\), its quantum \(D\)-module carries the A-model data of \(X\), and it is also called the A-model \(D\)-module.

Since from the manifold \(M\times \mathbb{C}^\ast\) we have kept only the torus in the \(H^2\) direction of \(M\), the effective base on the A-side that we now treat is the \((r+1)\)-dimensional algebraic torus

\[M_A := (\mathbb{C}^\ast)^r \times \mathbb{C}^\ast_z = \operatorname{Spec}\mathbb{C}[q_1^{\pm}, \ldots, q_r^\pm, z^\pm]\]

The fiber over this \(M_A\) is still the same as the fiber of \(\pr_1^\ast TM\), namely \(H^\ast(X)\), and therefore the bundle over it is given by the formula

\[H_A=H^\ast(X)\otimes_\mathbb{C}\mathbb{C}[q^\pm, z^\pm]=H^\ast(X, \mathbb{C}[z^\pm, q^\pm])\]

This is called the A-model state space.

Similarly, in the next post we will show that the Jacobi rings \(\Jac(W_q)\) defined by the mirror dual \(\check{X}\) of \(X\) can also be made into fibers over a suitable manifold \(M_B\). Moreover, there exists a Gauss-Manin connection \(\nabla^{GM}\) making this into a \(\mathcal{D}\)-module, and we will prove that the sections of this \(D\)-module constitute the B-model state space \(H_B\). Then our mirror symmetry statement is elevated to the following claim.

Conjecture 4 (Mirror theorem, \(D\)-module form) For a mirror pair \((X, \check{X})\), there exists a mirror isomorphism between the A-model state space \(H_A\) and the B-model state space \(H_B\) introduced above, such that \(\Phi\) is compatible with the Dubrovin connection and the Gauss-Manin connection:

\[\Phi \circ \nabla^z = \nabla^{GM} \circ \Phi\]

It is worth noting carefully that this conjecture is not, strictly speaking, a proven fact, but rather a guiding philosophy. It has been proved separately for various mirror pairs, for example the case of Calabi-Yau hypersurfaces in toric varieties, proved by Givental, was historically the first, after which it was extended to toric Fano varieties, and then further extended to toric stacks by Coates-Corti-Iritani-Tseng. A generalization in a slightly different direction replaces toric varieties by homogeneous spaces, in particular by partial flag varieties \(G/P\). In this direction, the LG superpotentials for Grassmannians and flag varieties were first constructed physically by Eguchi-Hori-Xiong, and studied Lie-theoretically by Rietsch, and exploring this is one of the main aims of this series.

Meanwhile, mirror symmetry takes many forms besides this \(\mathcal{D}\)-module isomorphism, and one of the most representative is Kontsevich’s homological mirror symmetry. This formulates a mirror pair as an equivalence of derived categories \(D^b\mathrm{Fuk}(X) \simeq D^b\Coh(\check{X})\), and has been proved in the cases of elliptic curves, abelian varieties, K3 surfaces, Calabi-Yau hypersurfaces, etc., and SYZ mirror symmetry is the framework that geometrically realizes this by viewing a mirror pair as the dual of a special Lagrangian torus fibration.

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References

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