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Hey I am a bit confused because when reading physics papers I encounter the free energy as a generating function of topological invariants as integrals over certain compactified moduli spaces. For example the Gromov-Witten invariants for the A-model or the Donaldson-Thomas invariants when considering the B-model. In other papers which are usually more math-oriented these generating functions are considered as partition function. From physics however I know that the partition function is the exponentiated free energy. Similar holds for the Nekrasov instanton partition function, which is the weighted sum over the inverse of Euler characteristics of the tangent bundle at the fixed-point set of the usual torus action on the moduli space. In this case however it is said that the log of the partition function gives the free energy of a twisted gauge theory. Can someone pls explain why these differences are made? Why is the generating function a free energy in one case and in the other it is termed as partition function?

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I don't know much about those particular models, so this might not be a suitable answer, but from what I understand, partition function $Z$ and free energy $F$ (or effective actions $W$) are related by $Z= e^{-F}$. Both are generating functions, but they don't generate the same things.

For example in QFT, the partition function $Z[J]$ generates $n$-points function, while the effective action $W = -\log Z$ generates connected $n$-point functions.

This distinction seems to apply similarly to the models you are refering to (see this answer for example, on the Gromov-Witten invariants).

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