[Paper Review] Moduli spaces of Einstein-Hermitian generalized connections over generalized Kahler manifolds of symplectic type

Ryushi Goto|arXiv (Cornell University)|Jul 11, 2017

Geometry and complex manifolds17 references3 citations

TL;DR

This paper introduces Einstein-Hermitian generalized connections on generalized Kähler manifolds of symplectic type using a moment map framework, showing that their moduli spaces arise as Kähler quotients. The deformation complex is elliptic, ensuring the smooth part of the moduli space is a finite-dimensional Kähler manifold, with curvature defined via a $d$-closed pure spinor $\psi = e^{b + \sqrt{-1}\omega}$, and Kähler–Ricci solitons and co-Higgs bundles are shown to be special cases.

ABSTRACT

From a view point of the moment map, we shall introduce the notion of Einstein-Hermitian generalized connections over a generalized Kähler manifold of symplectic type. We show that moduli spaces of Einstein-Hermitian generalized connections arise as the Kähler quotients. The deformation complex of Einstein-Hermitian generalized connections is an elliptic complex and it turns out that the smooth part of the moduli space is a finite dimensional Kähler manifold. The canonical line bundle over a generalized Kähler manifold of symplectic type has the canonical generalized connection and its curvature coincides with "the scalar curvature as the moment map" which is defined in the previous paper [Goto_2016]. Kähler-Ricci solitons provide examples of Einstein-Hermitian generalized connections and Einstein Hermitian co-Higgs bundles are also discussed.

Motivation & Objective

• To extend the moment map framework from Kähler geometry to generalized Kähler geometry by defining Einstein-Hermitian generalized connections.
• To construct moduli spaces of such connections as Kähler quotients using the unitary gauge group action.
• To define a curvature for generalized connections via a $d$-closed, non-degenerate, pure spinor $\psi = e^{b + \sqrt{-1}\omega}$, enabling the moment map interpretation.
• To show that the deformation complex of Einstein-Hermitian generalized connections is elliptic, implying the moduli space is a finite-dimensional Kähler manifold.
• To identify Kähler–Ricci solitons and Einstein–Hermitian co-Higgs bundles as special cases of the framework.

Proposed method

• Define a generalized connection $\mathcal{D}^\mathcal{A} = D^A + V$ as a differential operator on a vector bundle $E$, with $D^A$ an ordinary connection and $V$ an $\text{End}(E)$-valued vector field.
• Introduce curvature $\mathcal{F}_\mathcal{A}(\psi)$ via the spin representation of $\text{End}(E) \otimes \wedge^\bullet(TM \oplus T^*M)$ acting on a $d$-closed pure spinor $\psi = e^{b + \sqrt{-1}\omega}$, given by $\mathcal{F}_\mathcal{A}(\psi) = F_A \cdot \psi + d^{D_A}(V \cdot \psi) + \frac{1}{2}[V,V] \cdot \psi$.
• Equip the affine space $\widetilde{\mathcal{GM}}(E,h)$ of generalized Hermitian connections with a natural Kähler structure, with symplectic form $\omega_{\widetilde{\mathcal{GM}}}(\dot{\mathcal{A}}_1, \dot{\mathcal{A}}_2) = -\int_M \text{tr} \langle \mathcal{J}_\psi \dot{\mathcal{A}}_1, \dot{\mathcal{A}}_2 \rangle_{T \oplus T^*} \text{vol}_M$.
• Define the moment map $\mu: \widetilde{\mathcal{GM}}(E,h) \to \mathfrak{u}(E)^*$ for the unitary gauge group action, identifying the Einstein–Hermitian condition as the zero locus of the moment map.
• Use the $b$-field transformation $\text{Ad}_{e^{-b}}$ to relate generalized connections on $\psi = e^{b + \sqrt{-1}\omega}$ to those on $e^{\sqrt{-1}\omega}$, showing invariance of the Einstein–Hermitian condition under $b$-field shifts.
• Derive the Einstein–Hermitian condition as $\sqrt{-1}\Lambda_\omega F_A + \sum_i [V_i^{1,0}, (V_i^{1,0})^*] = \lambda \text{id}_E$ and generalize it to $\sqrt{-1}\Lambda_\omega(F_A - d^A(\text{ad}_b V) + \frac{1}{2}[\text{ad}_b V, \text{ad}_b V]) + \sum_i [V_i^{1,0}, (V_i^{1,0})^*] = \lambda \text{id}_E$.

Experimental results

Research questions

• RQ1 How can the moment map framework from Kähler geometry be generalized to generalized Kähler manifolds of symplectic type?
• RQ2 What is the correct curvature definition for generalized connections that supports a moment map interpretation?
• RQ3 How do moduli spaces of Einstein–Hermitian generalized connections arise as Kähler quotients?
• RQ4 What is the role of the $b$-field in transforming the Einstein–Hermitian condition across generalized Kähler structures?
• RQ5 Under what conditions do Kähler–Ricci solitons and Einstein–Hermitian co-Higgs bundles emerge from the generalized connection framework?

Key findings

• The moduli space of Einstein–Hermitian generalized connections is shown to be a Kähler quotient, inheriting a Kähler structure from the ambient space of generalized Hermitian connections.
• The deformation complex of Einstein–Hermitian generalized connections is elliptic, implying the smooth part of the moduli space is a finite-dimensional Kähler manifold.
• The curvature of a generalized connection is defined via the spin representation of $\text{End}(E) \otimes \wedge^\bullet(TM \oplus T^*M)$ acting on a $d$-closed pure spinor $\psi = e^{b + \sqrt{-1}\omega}$, yielding $\mathcal{F}_\mathcal{A}(\psi) = F_A \cdot \psi + d^{D_A}(V \cdot \psi) + \frac{1}{2}[V,V] \cdot \psi$.
• The canonical generalized connection on the canonical line bundle has curvature equal to the scalar curvature as a moment map, generalizing the result in [Go5].
• Kähler–Ricci solitons over ordinary Kähler manifolds are shown to be examples of Einstein–Hermitian generalized connections when $\psi = e^{\sqrt{-1}\omega}$.
• In the case $b = \omega$, the generalized Einstein–Hermitian condition reduces to a $\text{End}(E)$-valued Kähler–Ricci soliton equation, establishing a direct link to geometric flows.