previous version of this paper. I thank the organizers for the invitations to the March conference on moduli space and one-month stay at Simons center, where I met and briefly discussed with most of the groups working on virtual theories discussed in this article and got to listen to their talks. I would like to thank Lino Amorim, Yalong Cao, Cheol-Hyun Cho, Octav Cornea, Huijun Fan, Urs Fuchs, Kenji Fukaya, Andreas Gerstenberger, Gregory Ginot, Roman Golovko, Kei Irie, Janko Latschev, Maksim Maydanskiy, Dusa McDuff, Hiroshi Ohta, Frederic Paugam, Tim Perutz, Bai-Ling Wang, Shaofeng Wang, Katrin Wehrheim, Jinxin Xue, and Jingyu Zhao for questions, suggestions and/or discussions on virtual theories or math in general, and Yalong Cao, Roman Golovko, Kei Irie, Dominic Joyce, Kaoru Ono and Jake Solomon for helpful comments on the previous version of this paper. I thank the great writing environment between talks in Oberwolfach where a draft of this article was completed. Finally, I thank the excellent research atmosphere in IMJ-PRG, in particular the interactions in the Algebraic Analysis group. The author is currently supported by the ERC via the Starting Grant ERC StG-259118 managed by the CNRS.

Polyfold–Kuranishi correspondence
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The main result of my PhD 