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Shifts of prepotentials (with an appendix by Michele Vergne)
by Nikita Nekrasov, Nicolo Piazzalunga, Maxim Zabzine
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We study the dynamics of supersymmetric theories in five dimensions obtained by compactifications of M-theory on a Calabi-Yau threefold X. For a compact X, this is determined by the geometry of X, in particular the Kahler class dependence of the volume of X determines the effective couplings of vector multiplets. Rigid supersymmetry emerges in the limit of divergent volume, prompting the study of the structure of Duistermaat-Heckman formula and its generalizations for non-compact toric Kahler manifolds. Our main tool is the set of finite-difference equations obeyed by equivariant volumes and their quantum versions. We also discuss a physical application of these equations in the context of seven-dimensional gauge theories, extending and clarifying our previous results. The appendix by M. Vergne provides an alternative local proof of the shift equation.
Published as SciPost Phys. 12, 177 (2022)
Author comments upon resubmission
thanks for your suggestions, which we implemented in v2.
Moreover, an appendix by M. Vergne has been added,
providing an alternative local proof of the shift equation.
List of changes
- add conclusion section
- add appendix by M. Vergne
- fix a few typos and clarify notation
Submission & Refereeing History
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Reports on this Submission
- Cite as: Anonymous, Report on arXiv:2111.07663v2, delivered 2022-04-13, doi: 10.21468/SciPost.Report.4921
This paper addresses an interesting and basic problem in topological string theory which was not clear before
The classical limit of the prepotential in topological string theory involves the triple intersection numbers of the Calabi-Yau threefold. However, when this threefold is non-compact, as it happens in local mirror symmetry, these intersection numbers are not really well defined. This paper studies this problem from the point of view of \epsilon-regularized volumes and the DH formula and proposes a mathematical framework to address this issue.
The new version of the draft fulfils the requested changes by including a succinct conclusion that clearly summarises the results of this paper. Better still, a new appendix is added to give an alternative proof to the shift equation, further strengthening the results of this paper. The publication of this paper is therefore highly recommended.