Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory

Abstract

Abstract:We study D-branes and Ramond-Ramond fields on global orbifolds of Type II string theory with vanishing H -flux using methods of equivariant K-theory and K-homology. We illustrate how Bredon equivariant cohomology naturally realizes stringy orbifold cohomology. We emphasize its role as the correct cohomological tool which captures known features of the low-energy effective field theory, and which provides new consistency conditions for fractional D-branes and Ramond-Ramond fields on orbifolds. We use an equivariant Chern character from equivariant K-theory to Bredon cohomology to define new Ramond-Ramond couplings of D-branes which generalize previous examples. We propose a definition for groups of differential characters associated to equivariant K-theory. We derive a Dirac quantization rule for Ramond-Ramond fluxes, and study flat Ramond-Ramond potentials on orbifolds.

Instantons, quivers and noncommutative Donaldson–Thomas theory

“…For this, rather than using the descent formula (3.12) from localization, we will define the instanton action of the D6 brane gauge theory on C 3 /Γ via the Wess-Zumino coupling of constant Ramond-Ramond fields C dual to fractional D0 branes (instantons); this enables the proper incorporation and weighting of the twisted sectors r = 0 in (3.46) to match with string theory expectations. Such fields decompose into twisted sectors of the closed string orbifold as [23,48,49,50]…”

“…For the example Γ = Z 3 considered in Section 3.2, the three-dimensional regular representation ρ(g) αβ = ζ α δ αβ naturally corresponds to a superposition of fractional instantons [49]. The corresponding Γ-equivariant Chern character is given by [50] ch…”

Adams operations in smoothK–theory

“…In this way by (20) we reduce the problem to the multiplicativity of y ‰ k in degree zero and (19). This finishes the proof of Theorem 3.1.…”

“…Motivated by applications in mathematical physics, in particular string theory, smooth extensions of other generalised cohomology theories, in particular of K -theory, have been considered for example by Moore and Witten [16], Freed [11; 10] and Szabo and Valentino [20]. The existence of smooth extensions of generalised cohomology theories has been shown by Hopkins and Singer [14].…”

Differential orbifold K-theory

“…[FMS07], [Wit98], [MM97]. For the theory on orbifolds one needs the corresponding generalization of differential Ktheory [SV10]. To serve this goal is one of the motivations of this paper.…”