Towards a Theory of Logarithmic GLSM Moduli Spaces

Abstract

In this article, we establish foundations for a logarithmic compactification of general GLSM moduli spaces via the theory of stable log maps [15,2,25]. We then illustrate our method via the key example of Witten's r-spin class. In the subsequent articles [17,16], we will push the technique to the general situation. One novelty of our theory is that such a compactification admits two virtual cycles, a usual virtual cycle and a "reduced virtual cycle".A key result of this article is that the reduced virtual cycle in the r-spin case equals to the r-spin virtual cycle as defined using cosection localization by Chang-Li-Li [13]. The reduced virtual cycle has the advantage of being C * -equivariant for a non-trivial C *action. The localization formula has a variety of applications such as computing higher genus Gromov-Witten invariants of quintic threefolds [27] and the class of the locus of holomorphic differentials [18].

The logarithmic gauged linear sigma model

“…Remark 1.4. In case of rank one E, the stability (1.5) is equivalent to a similar formation as in [22,Definition 4.9] using 0 P , see Remark 2.6. However, the latter does not generalize to the higher rank case, especially when E is non-splitting.…”

“…Giving the relative properness of stable log maps over underlying stable maps [1,20,33], establishing a proper moduli stack remains to be a rather difficult and technical step in developing our theory. An evidence is that the moduli of underlying R-maps fails to be universally closed [22,Section 4.4.6] in even most basic cases. The log structure of P plays an important role in the properness as evidenced by the subtle stability (1.5) which was found after many failed attempts.…”

“…However, the latter does not generalize to the higher rank case, especially when E is non-splitting. Consequently, we have to look for a stability of very different form, and a very different strategy for the proof of properness comparing to the intuitive proof in [22]. 1.3.5.…”

“…In our first paper [22], we developed a principalization of the boundary of the moduli of log maps, which provides a natural framework for extending cosections to the boundary of the logarithmic compactification. The simple but important r-spin case has been studied in [22] via the log compactification for maximal explicitness.…”

Structure of Higher Genus Gromov-Witten Invariants of Quintic 3-folds

“…It took yet another ten years for the authors' recent proof of the genus two BCOV conjecture [19]. The geometric input to our work on genus two is a construction of a certain reduced virtual cycle on an appropriate log compactification of the GLSM moduli space [7] (see also [9]). Its localization formula expresses the Gromov-Witten invariants of quintic 3-folds in terms of a graph sum of (rather mysterious) effective invariants and (rather well-understood) twisted GWinvariants of P 4 .…”