So I come from a more mathematical background, and have a somewhat decent understanding of Gromov-Witten theory in algebraic geometry. I'm hoping to better understand one of the details in the equivalence between GW theory and A-model topological string theory. Let $X$ be a smooth Kahler manifold. In GW theory, we have a moduli space $\overline{\mathscr{M}}_{g,n}(X, \beta)$ of stable maps from possibly nodal marked curves of arithmetic genus $g$ into $X$ representing homology class $\beta$. This is a projective Deligne-Mumford stack, and we define invariants and partition functions using something called the virtual class. The projectivity is important in the mathematical side of the theory, and requires that we allow the domain curves to possibly have nodes. But I'm assuming a string theorist wouldn't want their worldsheets to degenerate to a point!
Quite simply, I'm wondering how physicists define correlation functions by integrating over non-compact moduli spaces of smooth domain curves in the A-twisted NLSM. I'm guessing there's some cleverness about replacing a curve which has nearly acquired a node, with some conformally-equivalent smooth curve. Maybe I can elaborate very briefly.
In the A-model topological field theory, we start by studying all continuous maps $\phi: C \to X$ where $C$ is a smooth genus $g$ curve. We can define correlation functions
$$\langle \mathcal{O}_{A_{1}} \cdots \mathcal{O}_{A_{n}}\rangle_{\beta} = \int_{\phi_{*}[C] = \beta} \mathcal{D}\phi \mathcal{D} \psi e^{-S}\mathcal{O}_{A_{1}} \cdots \mathcal{O}_{A_{n}}$$
where $A_{i}$ are $p$-forms on $X$, and we define the ghost number of $\mathcal{O}_{A_{i}}$ to be $p$. Because we have a fermionic symmetry $Q$, we can localize onto worldsheet instantons, i.e. localize onto the sector where $\phi$ is a holomorphic map. The ghost number anomaly tells us that these correlation functions vanish unless we have
$$\frac{1}{2} \sum_{i=1}^{n} \text{deg}(A_{i}) = \int_{\beta} c_{1}(X) + \text{dim}X(1-g).$$
Now, if there are no $\psi$ zero modes, it turns out we have a smooth moduli space $\mathscr{M}_{C}(X, \beta)$ of holomorphic maps from a fixed curve $C$ into $X$ landing in class $\beta$. The correlation functions are then computed as
$$\langle \mathcal{O}_{A_{1}} \cdots \mathcal{O}_{A_{n}}\rangle_{\beta} = e^{i \omega \cdot \beta} \int_{\mathscr{M}_{C}(X, \beta)} \text{ev}_{1}^{*}(A_{1}) \cup \cdots \cup \text{ev}_{n}^{*}(A_{n})$$
Okay...is this moduli space compact? If not how is this integral well-defined?
In addition, we then couple to worldsheet gravity to get the A-model topological string theory. The correlation functions then require integrating over $\mathscr{M}_{g,n}$. But again, I think this is the non-compactified moduli space, because I imagine string theorists don't want strings collapsing to a point. So how is this integration well-defined if we don't want to consider nodal curves?
If I had to take a wild guess, I would say that a curve which is about to acquire a node may be conformally-equivalent to a *smooth* curve with some long, skinny "neck" or something? And perhaps these integrands of interest are suppressed on such a configuration?
Q&A (4872)
