Let $f\!=\!(u\colon (\Sigma,p_1,\ldots,p_k) \to X)$ be an element of the moduli space of genus $g$ $k$-marked degree $A$ $J$-holomorphic maps $\mathcal{M}_{g,k}(X,A,J)$. For simplicity assume $C=(\Sigma,p_1,\ldots,p_k)$ is stable.

In the algebraic (or holomorphic) setting, deformation long exact sequence of $\mathcal{M}_{g,k}(X,A,J)$ around $f$ has the form

\begin{align} 0\to \operatorname{Def}(u) &\to \operatorname{Def}(f) \to \operatorname{Def}(C) \\ \to \operatorname{Obs}(u) &\to \operatorname{Obs}(f) \to 0\;; \end{align}

see Section 24 of "Mirror Symmetry" book by Hori, Katz, Klemm, etc.

In the analytical (symplectic) setting, the first column corresponds to kernel and cokernel of linearization of Cauchy Riemann operator $$ D_u\bar\partial\;\colon \Gamma(u^*TX)\to \Gamma(\Omega^{0,1}_\Sigma\otimes u^*TX); $$ i.e. $\operatorname{Def}(u)=\ker(D_u\bar\partial)$ and $\operatorname{Obs}(u)=\operatorname{coker}(D_u\bar\partial)$.

$\operatorname{Def}(C)$ is as in the algebraic case: $$ \operatorname{Def}(C)=H^1(T\Sigma(-p_1\cdots -p_k)) $$

Question 0: Does such long exact sequence even always exist in the analytical setting?

Question 1: What is the analytical description of the map $\operatorname{Def}(C)\to \operatorname{Obs}(u)$? Whether the answer to Q0 is Yes or No, this map should be naturally definable.

Question 2: What are the analytical descriptions of $\operatorname{Obs}(f)$ and $\operatorname{Def}(f)$?

Question 3: Do you know of any reference where this sequence is explained for the analytical setup of GW theory?

Comments: In the case of no-marked points, $\operatorname{Def}(C)=H^1_{\bar\partial}(T\Sigma)$ and we can get the map via $du:T\Sigma \to TX$. In the case there are marked points, the map should be similar, I just have hard time visualizing it. In question 2, if the map is immersion and no-marked points the spaces are similar to that of $u$ with $N_{u(\Sigma)}X$ instead of $TX$.

4more comments