Global Stringy Orbifold Cohomology, K-Theory and de Rham Theory

Abstract

There are two approaches to constructing stringy multiplications for global quotients. The first one is given by first pulling back and then pushing forward. The second one is given by first pushing forward and then pulling back. The first approach has been used to define a global stringy extension of the functors K ₀ and K ^top by Jarvis–Kaufmann–Kimura, A* by Abramovich–Graber–Vistoli, and H* by Chen–Ruan and Fantechi–Göttsche. The second approach has been applied by the author in the case of cyclic twisted sector and in particular for singularities with symmetries and for symmetric products. The second type of construction has also been discussed in the de Rham setting for Abelian quotients by Chen–Hu. We give a rigorous formulation of de Rham theory for any global quotient from both points of view. We also show that the pull–push formalism has a solution by the push–pull equations in the setting case of cyclic twisted sectors. In the general, not necessarily cyclic case, we introduce ring extensions and treat all the stringy extension of the functors mentioned above also from the second point of view. A first extension provides formal sections and a second extension fractional Euler classes. The formal sections allow us to give a pull–push solution while fractional Euler classes give a trivialization of the co-cycles of the pull–push formalism. The main tool is the formula for the obstruction bundle of Jarvis–Kaufmann–Kimura. This trivialization can be interpreted as defining the physics notion of twist fields. We end with an outlook on applications to singularities with symmetries aka. orbifold Landau–Ginzburg models.