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I am interested in the explicit computation of generating functions of rank 1 and higher rank Donaldson-Thomas (DT) invariants. In particular, I am interested in DT invariants of K3 fibered Calabi-Yau threefolds, and their modularity. Given a specific CY 3-fold and a fixed K3 lattice polarization, how can one systematically compute the DT invariants. What data of the CY 3-fold do I need? I am interested in not just smooth CYs, but smooth ones is a natural starting point, and I don't know how to handle those.

Are there any references, where such computations have been worked out, maybe in very simple cases.

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This answer might just be a list of references, but I hope it helps. The most explicit computations of which I am aware exploit a torus action on the Calabi-Yau 3-fold in question, where the calculus can be reduced to a combinatorial problem: for example, the famous paper Gromov–Witten theory and Donaldson–Thomas theory. I (MSN) by Maulik, Nekrasov, Okounkov, and Pandharipande. The same approach yields explicit computations for stable pairs invariants, as done by Pandharipande and Thomas in another paper, The 3-fold vertex via stable pairs. Finally, one more case where it might be possible to compute Donaldson-Thomas invariants is when you can describe the corresponding moduli space as a critical locus, and compute the perverse sheaf of vanishing cycles, as done by Dimca and Szendrői in The Milnor fibre of the Pfaffian and the Hilbert scheme of four points on $\mathbb C^3$. Joyce and Song have greatly developed the theory of invariants for local vanishing loci, but it might not be a good place to start to look for explicit computations.

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    $\begingroup$ When you link to MSN, you need to click the "Make Link" button on the specific review you want. This allows even non-subscribed users to see the article to which you're pointing. If instead you use the auto-generated link (which records the search you got there), then non-subscribed users cannot see it at all. I have edited accordingly. $\endgroup$ – LSpice Jun 3 at 22:33
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    $\begingroup$ Many thanks for the advice. I will make sure my links to MSN are available to non-subscribers in the future. $\endgroup$ – Aurelio Jun 4 at 8:44

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