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  Integrating over non-compact moduli spaces in A-model TFT

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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?   

asked May 19 in Theoretical Physics by Benighted (270 points) [ no revision ]

Do you have more details on the holomorphic structure of Mc ? and what is the form of the holonomy functions ?

The holonomies functions choice is somehow a free parameter of the GW theory. To get consistent results, some tried to replace the holonomies which are given in terms of Schur functions by analog functions, ie from Macdonald ( dual ) and/or refined Chern-Simons T.. It is an active field of research. In example , if you have 40' to 1h to spend, see the video curse of Andrea Brini Chern-Simons theory, mirror symmetry and the topological recursion. You may start at 25'50.

I hadn't heard about this at all, thanks!  I'll look into it.  

1 Answer

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The moduli space $\mathcal{M}_C(X,\beta)$ is not compact in general. Even in the simplest case where $C$ is a sphere (complex projective line, unique complex structure), you can have "bubbling" phenomenon, requiring the addition of nodal curves to get a compact moduli space (Example: $X=\mathbb{P}^2$, smooth conic degenerating to union of two lines).

The subtlety is in "we can localize onto worldsheet instantons". If the fixed locus of the fermionic symmetry is not compact, the localization is not completely obvious. The standard way to prove the localization is to say that, up to the topological term $\omega.\beta$, the action S is $Q$-exact and so can be rescaled by some parameter t. In the limit t goes to infinity, the path integral should localize on the $Q$-fixed locus. But if the $Q$-fixed locus is not compact, part of the measure of the path integral can escape to infinity. (Finite-dimensional toy model: if $f_n$ is a sequence of functions on a non-compact space, it is possible to have  $\lim_{n \rightarrow +\infty} \int f_n \neq \int \lim_{n \rightarrow +\infty} f_n$ because part of the mass "escapes to infinity"). The compactification of the moduli space of holomorphic maps by allowing nodal curves is necessary exactly for this reason.

In the physics definition of the A-model, C is smooth and correlation functions are defined by an integral over some infinite dimensional space of fields. If the Q-fixed locus were compact, this integral would localize to the Q-fixed locus. But as the Q-fixed locus is not compact, the measure can espace to infinity and to not loose it, one needs to compactify the Q-fixed locus by allowing nodal curves, which is the usual mathematical definition of stable maps.

Exactly the same happens after coupling to topological gravity.

In short, if one does not want to consider nodal curves, the correct definition, used by physicists, is the path integral one, which is not yet mathematically well-defined.

answered May 20 by 40227 (5,000 points) [ no revision ]

@40227 Thanks a lot.  Unless I'm missing the point, I feel like your answer doesn't indicate there is some physical argument making that integral over $\mathscr{M}_{C}(X, \beta)$ in my OP well-defined.  You seem to say the full path integral is fine physically, but the localization isn't well-defined if the space isn't compact.  So why do people even write an equation like that integral in my OP?  That shows up, for example, in Chapter 16 of the big mirror symmetry book. Is this subtelty discussed anywhere, like in Witten's original paper on mirror manifolds and topological field theories?  

Indeed, the correct formula would involve an integral over the moduli space of stable maps (unless $\mathcal{M}_C(X,\beta)$ is already compact which could happen in very specific situations). I think that people write equations like that because they are slightly sloppy.

The short description I have given is identical to the top of page 147 of https://arxiv.org/abs/hep-th/9309140

Some related phenomenon is discussed in https://arxiv.org/abs/hep-th/9207094

Thanks a lot for your help.  It's unfortunate there isn't a purely physics way of compactifying.  If the target is a Calabi-Yau threefold the theory is conformal right?  So does it definitely not work to take a limit of curves converging to a nodal configuration and somehow replace the sequence by a conformally-equivalent sequence of smooth curves which instead of acquiring a node, converge to an infinitely elongated "neck" or something?  

Perhaps this is complete nonsense, but if something like this were true, you would at least feel much more comfortable writing $\overline{\mathscr{M}}_{g,n}(X, \beta)$ in physics discussions.  

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