# Donaldson-Thomas Theory and "Quantum Foam" for Mathematicians

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Let $$X$$ be a smooth, projective Calabi-Yau threefold. From an algebro-geometric perspective, the Donaldson-Thomas invariants $$\text{DT}_{\beta, n}(X)$$ are virtual counts of ideal sheaves on $$X$$ with trivial determinant. They are defined as the degree of the virtual cycle of the moduli space of such sheaves. The Donaldson-Thomas partition function is the generating function

$$Z_{\text{DT}}(X) = \sum_{\beta \in H_{2}(X, \mathbb{Z})} \sum_{n \in \mathbb{Z}} \text{DT}_{\beta, n}(X) Q^{\beta} q^{n}.$$

In physics, the DT invariants have a number of interpretations. I more or less understand that they are virtual counts of BPS states of D2-D0 branes inside a single D6-brane, and the DT partition function can be thought of as a BPS black hole partition function with no D4-branes. I also have some partial understanding that the DT invariants are related to the statistical mechanics of crystal melting.

The perspective I'm interested in here is DT theory related to "quantum foam." This is obviously a physics term, and it refers to changes in the topology of spacetime at small scales. This has always puzzled me, since in DT theory we have one fixed $$X$$, but I believe the idea is the following:

As algebraic geometers, if we're given a sheaf of ideals $$\mathcal{I}$$ on $$X$$, we can form the blowup $$\text{Bl}_{\mathcal{I}}(X)$$. This certainly has a different topology than $$X$$, and I believe the idea is that instead of counting ideal sheaves as you do in DT theory, you can count line bundles but on blowups of $$X$$. This, I think, is the topology change involved in quantum foam. See the original source (https://arxiv.org/pdf/hep-th/0312022.pdf) as well as page 10 of (https://arxiv.org/pdf/0912.1509.pdf).

So my question is, to what extent (if any) do we have a rigorous mathematical understanding of this procedure? I mean, we have a nice moduli scheme of sheaves in DT theory, but somehow at each point, we want to blow up $$X$$ along that ideal sheaf? By equation (2.3) in (https://arxiv.org/pdf/hep-th/0312022.pdf) I believe the hope is to write the DT partition function as

$$Z_{\text{DT}}(X) = \sum q^{\text{ch}_{3}} \prod_{i} Q_{i}^{\int_{C_{i}} \text{ch}_{2}}$$

where is sum is over blowups of $$X$$ (?) and the Chern characters are of the pullback of the ideal sheaf under the blowup? If anyone could help me understand any of this from a mathematical perspective, I'd really appreciate it.

This post imported from StackExchange MathOverflow at 2024-09-03 14:41 (UTC), posted by SE-user Benighted
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