The Witten top Chern class via $$K$$-theory.(English)Zbl 1117.14008

In the formulation of the E. Witten conjecture [in: Topological methods in modern mathematics. Proc. symp. hon. John Milnor’s sixtieth birthday, Stony Brook, USA 1991, 235–269 (1993; Zbl 0812.14017)] concerning the relation of higher KdV hierarchies to higher spin curves, the Witten top Chern class plays a crucial role.
In the paper the author provides a straightforward construction of the Witten top Chern class using $$K$$-theory. A. Polishchuk and A. Vaintrob [in: Advances in algebraic geometry motivated by physics. Proc. AMS spec. sess. Univ. Massa., Lowell, MA, USA 2000. Contemp. Math. 276, 229–249 (2001; Zbl 1051.14007)] gave an algebraic construction of this class. However the construction presented in the paper is simpler and allows one to avoid usage of bivariant intersection theory and Mac-Pherson graph construction. The paper is an extended version of the previous note by the author [in: Gromov-Witten theory of spin curves and orbifolds. AMS spec. sess., San Francisco, USA 2003. Contemp. Math. 403, 21–29 (2006; Zbl 1105.14032)].

MSC:

 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14H10 Families, moduli of curves (algebraic)
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References:

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