String structures associated to indefinite Lie groups

Abstract

String structures have played an important role in algebraic topology, via elliptic genera and elliptic cohomology, in differential geometry, via the study of higher geometric structures, and in physics, via partition functions. We extend the description of String structures from connected covers of the definite-signature orthogonal group O(n) to the indefinite-signature orthogonal group O(p, q), i.e. from the Riemannian to the pseudo-Riemannian setting. This requires that we work at the unstable level, which makes the discussion more subtle than the stable case. Similar, but much simpler, constructions hold for other noncompact Lie groups such as the unitary group U(p, q) and the symplectic group Sp(p, q). This extension provides a starting point for an abundance of constructions in (higher) geometry and applications in physics.

Twisted Cohomotopy Implies M-Theory Anomaly Cancellation on 8-Manifolds

“…(see, e.g., [HN12, Sec. 17.2]) and similarly for higher connected covers (see [SS19]). Therefore, we might call Cohomotopy with coefficients in S p × S p , and twisted by generalized geometry, generalized Cohomotopy (not to be confused with older terminology [Ja62]).…”

Variations of rational higher tangential structures

“…Given the discussions in previous sections, the descriptions of the higher connected covers Fivebranepp, qq Q and Ninebranepp, qq rationally will follow analogously. Note that this is in stark contrast with the calculations in [SS15] where the various torsion groups arising notoriously in the indefinite case made the extension to Fivebrane and Ninebrane not immediately possible. (ii) This rationalization is trivial, i.e.…”

“…Note, however, that an extension to path connected spaces with general (not necessarily nilpotent) π 1 is possible (see [FHT15]). This allows us, for instance, to start our Whitehead tower (Example 3 below) with SOpnq or BOpnq, as in [SS15].…”

“…For degree 3, i.e. for Stringpp, qq, these are characterized in [SS15]. The homotopy groups encountered there are complicated, but upon rationalizing the problem becomes much more tractable: If p, q ă 3 then the problem is trivial.…”