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Let $W$ be the canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on a symplectic 4-manifold $(M, \omega)$, with a compatible $J$ and $g$, so \begin{equation} {W_ + } = {T^{0,0}}{M^*} \oplus {T^{0,2}}{M^*}, \;\;\;... \end{equation} and the determinant line bundle $L = T^{0,2}M^*$.

Now I want to define a connection on $W$. I learn about basic general theory on spinor bundle, and it says a connection on $W$ consists of the Levi-civita connection $\nabla_g$ and a connection $A$ on $L$. In particular, after I have a $\nabla_g$, I still need to saparately define the $A$, which does not induce from $\nabla_g$.

But I see in Sergeev's lecture on Vortex and Seiberg-Witten equations, that in the case of $(M, \omega, J, g)$ one can modify $\nabla_g$ by adding a term with Nijenhuis tensor to give a connection to $L = T^{0,2}M^*$, and therefore a full connection on $W$.

  1. Am I right about the general theory that connection $A$ on $L$ is not prescribed by $g$-connection?

  2. In the special case of canonical $\operatorname{Spin}^\mathbb{C}$-bundle on $(M, \omega, J, g)$, is $\nabla_g$ prescribing a canonical connection on $L$, contrasting with the general theory?

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  • $\begingroup$ If $T^{0,2}M^*$ means what I think, it is not a line bundle. $\endgroup$ – abx Apr 1 '14 at 15:36
  • $\begingroup$ @abx: Sorry, I forgot to say $M$ is four-dimensional. $\endgroup$ – Lelouch Apr 1 '14 at 16:14
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  1. You're correct that in general there is no natural choice of a reference $\mathrm{Spin}^c$ connection on an arbitrary manifold.

  2. The reason there is a natural choice of a $\mathrm{Spin}^c$ connection on a symplectic $4$-manifold $(M, \omega)$ is that there is an isomorphism $$W^+ = \wedge^{0,0} T^\ast M \oplus \wedge^{0,2} T^\ast M \cong \varepsilon_{\mathbb{C}} \oplus K^{-1},$$ where $\varepsilon_{\mathbb{C}}$ is the trivial complex line bundle over $M$ and $K = \wedge^2_{\mathbb{C}} T^\ast M$ is the canonical bundle of $(M,J)$. The Levi-Civita connection on $(M,g)$ induces a canonical connection on $K^{-1}$, and we can have the connection on the $\varepsilon_{\mathbb{C}}$ factor be the trivial connection. In this was the Levi-Civita gives rise to a canonical choice of connection on the canonical spinor bundle of a symplectic manifold.

One place you can see the derivation of the natural connection on $K^{-1}$ is Lemma 4.4 (Lemma 4.5 computes the corresponding Dirac operator) of An Introduction to the Seiberg-Witten Equations on Symplectic Manifolds by Hutchings and Taubes.

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  • $\begingroup$ Thank you. I came across Lemma 4.4 before but was not able to fully appreciate the importance. I'll try again. $\endgroup$ – Lelouch Apr 2 '14 at 2:01
  • $\begingroup$ The Lemma 4.4 says it wants to prove there is a $A_0$ such that ${\nabla _{\mathbf{A}_0}} u_0 \in \Omega^1(X, K^{-1})$; should it be trying to prove ${\nabla _{\mathbf{A}_0}} u_0 = 0 \in \Omega^1(X, K^{-1}) $? $\endgroup$ – Lelouch Apr 2 '14 at 2:14

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