Delocalized Chern character for stringy orbifold K-theory (Q2855930)

An orbifold \(\mathcal X\) is a Hausdorff topological space endowed with an atlas of quotients \(U_x = \tilde{U}_x/G_x\) of Euclidean space by a finite local group \(G_x\), the de Rham cohomology \(H^*_{\mathrm{dR}}(\mathcal X)\) of which is isomorphic to the singular cohomology of the underlying topological space \(X = |\mathcal X|\). To an effective orbifold \(\mathcal X\) one associate a an inertia orbifold \(\tilde{\mathcal X}\), which is a canonical non-effective orbifold and is a disjoint union of twisted sectors (connected components) \(\mathcal X_{(g)}\) of dimension \((g)\) running over the set of conjugacy classes of local groups. Chen-Ruan cohomology \(H^*_{\mathrm{CR}}(\mathcal X) = (H^*_{\mathrm{dR}}(\tilde{\mathcal X}, \mathbb C), \circ_{\mathrm{CR}})\) is endowed with a product \(\circ_{\mathrm{CR}}\) is well defined. For almost complex orbifolds, with immersion \(e: \tilde{\mathcal X} \to \mathcal X\) one associates the obstruction bundle\(E^{[2]}\) over \(\tilde{\mathcal X}^{[2]} = \tilde{\mathcal X} \times_e \tilde{\mathcal X} \) which is a disjoint union of twisted sectors \(\mathcal X_{(g_1,g_2)}\), included into \(\mathcal X\), following the commutative diagram of inclusions \(\begin{tikzcd}\mathcal X_{(g_1,g_2)} \rar["e_{12}"] & \mathcal X_{(g_1g_2)} \rar["e_{(g_1g_2)}"] & \mathcal X\end{tikzcd}\) and two anothers \(\begin{tikzcd}\mathcal X_{(g_1,g_2)} \rar["e_{1}"] & \mathcal X_{(g_1)} \rar["e_{(g_1)}"] & \mathcal X\end{tikzcd}\) and \(\begin{tikzcd}\mathcal X_{(g_1,g_2)} \rar["e_{2}"] & \mathcal X_{(g_2)} \rar["e_{(g_2)}"] & \mathcal X\end{tikzcd}\). One defines then the normal bundle \(\mathcal N_e\) as the disjoint union of normal of each immersed \(\mathcal X_{(g_1,g_2)}\) in \(\mathcal X\) with a unitary action \(\Phi\) of local groups on \(\mathcal N_e\), as the subbundle of the tangent bundle to \(\mathcal X\). For an almost complex orbifold \(\mathcal X\) the direct sum of the obstruction bundle \(E^{[2]}\) and the normal bundle \(\mathcal N\) admits a decomposition NEWLINE\[CARRIAGE_RETURNNEWLINEE^{[2]} \oplus \mathcal N = e_1^*\mathcal N_{e,\Phi} + e_2^*\mathcal N_{e,\Phi} + e_{12}^*\mathcal N_{e,\Phi^{-1}} CARRIAGE_RETURNNEWLINE\]NEWLINE in the group \(K^0(\tilde{\mathcal X}^{[2]}) \otimes \mathbb Q = \bigoplus_{(g_1,g_2)} K^0(\mathcal X_{(g_1,g_2)}) \otimes \mathbb Q\) (Theorems 1.1, 3.4). The delocalized Chern characters were defined as NEWLINE\[CARRIAGE_RETURNNEWLINEch_{\mathrm{deloc}} : K^*_{\mathrm{orb}}(\mathcal X) \to H^*(\tilde{\mathcal X}) \to H^*(\tilde{\mathcal X},\mathbb C) = \bigoplus _{(g)} H^*(\mathcal X_{(g)},\mathbb C).CARRIAGE_RETURNNEWLINE\]NEWLINE In this paper a modification of delocalized Chern characters NEWLINE\[CARRIAGE_RETURNNEWLINE\widetilde{ch}_{\mathrm{deloc}} = \mathcal T(\mathcal N_e, \Phi) \wedge ch_{\mathrm{deloc}} : K^*_{\mathrm{orb}}(\mathcal X) \to H^*_{\mathrm{CR}}(\mathcal X, *_{\mathrm{CR}})CARRIAGE_RETURNNEWLINE\]NEWLINE with a stringy product on K-theory are introduced and show that is is a ring isomorphism (Theorems 1.2, 3.8). As an application, the authors studied this stringy product on the equivariant K-theory of finite group with conjugation action. In this case, the stringy product is different from the Pontryagin product, known as the fusion product in the string theory.