The Wess-Zumino-Witten term was first introduced in the low energy σ-model which describes pions, the Goldstone bosons for the broken flavor symmetry in quantum chromodynamics. We introduce a new definition of this term in arbitrary gravitational backgrounds. It matches several features of the fundamental gauge theory, including the presence of fermionic states and the anomaly of the flavor symmetry. To achieve this matching, we use a certain generalized differential cohomology theory. We also prove a formula for the determinant line bundle of special families of Dirac operators on 4-manifolds in terms of this cohomology theory. One consequence is that there are no global anomalies in the Standard Model (in arbitrary gravitational backgrounds).

0. Introduction. Let Mn be a n-dimensional minimally immersed submanifold of M, Q> 1. Throughout this paper M is taken to be one of the simply connected space forms with curvature 1, 0, or-1, ie Mfn+ f sn+ f, Rn+ f, or Hn+ f. Given a point p EM, let rp (x) be the dis-tance function on M, we denote the restriction of rp to M as the extrinsic distance function on M. For any a> 0, we define the extrinsic ball cen-tered at p with radius a by

In this paper we prove that given a point p ∈ Mn, where Mn is a closed Riemannian manifold of dimension n, the length of a shortest geodesic loop lp(Mn) at this point is bounded above by 2nd, where d is the diameter of Mn. Moreover, we show that on a closed simply connected Riemannian manifold Mn with a nontrivial second homotopy group there either exist at least three geodesic loops of length less than or equal to 2d at each point of Mn, or the length of a shortest closed geodesic on Mn is bounded from above by 4d.

We obtain a residue formula for an obstruction to the existence of coupled Kähler–Einstein metrics described by Futaki–Zhang. We apply it to an example studied separately by Futaki and Hultgren which is a toric Fano manifold with reductive automorphism, does not admit a Kähler–Einstein metric but still admits coupled Kähler–Einstein metrics.

In this paper, we show that any compact Kähler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a Kähler–Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold X homotopic to a compact Riemannian manifold with negative sectional curvature, for any almost complex structure J compatible with the symplectic form, there is no non-constant J-holomorphic entire curve f:C→X.