Moduli spaces and Grassmannian. (English) Zbl 1276.14016

Liou, Jia-Ming (Frank); Schwarz, Albert S.

Lett. Math. Phys. 103, No. 6, 585-603 (2013).

Let \(\mathrm{Gr}(H)\) be the Sato Grassmannian associated with a polarized Hilbert space \(H =H_+\oplus H_-\) (for definition see [G. Segal and G. Wilson, Publ. Math., Inst. Hautes Étud. Sci. 61, 5–65 (1985; Zbl 0592.35112)] or [A. Pressley and G. Segal, Loop groups. Oxford Mathematical Monographs. Oxford: Clarendon Press. (1986; Zbl 0618.22011)]). It is an infinite-dimensional Banach manifold modeled on the space of compact operators from \(H_-\) to \(H_+\).
Let \(\hat{F}_{g,h}\) be the moduli space of quintuples \((C,p,z,L,\varphi )\) where \(C\) is a compact complex curve, \(z\) is a local coordinate at the point \(p\in C\), \(L\) is a line bundle over \(C\) having a trivialization \(\varphi\) at \(p\). This space can be mapped into the Sato Grassmanian by means of the Krichever map \(k\) (for details see Segal and Wilson [Zbl 0592.35112)] or [M. Mulase, in: Perspectives in mathematical physics. Proceedings of the conference on interface between mathematics and physics, held in Taiwan in summer 1992 and the special session on topics in geometry and physics, held in Los Angeles, CA, USA in winter of 1992. Boston, MA: International Press. Conf. Proc. Lect. Notes Math. Phys. 3, 151–217 (1994; Zbl 0837.35132)]). The torus \(S^1\) acts in a natural way on \(\hat{F}_{g,h}\) and on \(Gr (H)\) (see loc. sit. for details); this action commutes with the Krichever map and hence induces a homomorphism \(k^*\) on the equivariant cohomology.
Recall that lambda classes (see [D. Mumford, Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271–328 (1983; Zbl 0554.14008)]) are the Chern classes of the Hodge bundle over the moduli space \(M_g\) of Riemann surfaces of genus \(g\).
The authors of the paper under review introduce several generalizations of lambda classes (for example, certain pullbacks of the lambda classes or chern classes of the \(q\)-Hodge bundle) and make several calculations expressing the image of the generators of the equivariant cohomology ring of \(\mathrm{Gr}(H)\) under the Krichever map in terms of these generalized lambda classes. These calculations were used by authors in the preprint [the authors, Weierstrass cycles in moduli spaces and the Krichever map, arXiv:1207.0530] to give an estimate of the dimensions of Weierstrass cycles and explicit expressions for their cohomology classes.
As it was pointed out in the article, these calculations could be important in the analysis of the Grassmanian string theory suggested by the second author in [Commun. Math. Phys. 199, No. 1, 1–24 (1998; Zbl 0921.58078)].

Reviewer: Alexander Zheglov (Moskva)

MSC:

14D20	Algebraic moduli problems, moduli of vector bundles
14H10	Families, moduli of curves (algebraic)
14M15	Grassmannians, Schubert varieties, flag manifolds

Keywords:

Sato Grassmannian; Krichever map; lambda classes; moduli spaces; Schubert classes

Citations:

Zbl 0592.35112; Zbl 0618.22011; Zbl 0837.35132; Zbl 0554.14008; Zbl 0921.58078

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