Global Worldsheet Anomalies from M-Theory

Abstract

We present an M-theory proof of the anomaly of Freed and Witten which in general shifts the quantisation law for the U (1) gauge field on a D6-brane. The derivation requires an understanding of how fields on the D6-brane lift to Mtheory, together with a localisation formula which we prove using a U (1)-index theorem. We also show how the anomaly is related to the K-theory classification of Ramond-Ramond fields. In addition we discuss the M-theory origin of the D6-brane effective action, and illustrate the general arguments with a concrete example.

On flux quantization in F-theory

“…Then, the first order term in (26) gives the right D7-brane charge, namely 8G, while the second order term vanishes, compatibly with the fact that D5-branes are projected out by the orientifold involution. The third order term in (26) and the contribution (28) constitute the total induced D3-brane charge, which must be compensated by an equal number of explicitly added D3-sources in order to cancel the D3-brane tadpole. It is possible to single out two different contributions to this charge density:…”

“…First of all, the zeroth order term in (26) vanishes, indicating that no net charge of D9-brane is left after the tachyon condensation (stated in mathematical terms, this means that on the D9 − D9 system there is a class inK(X 3 )). Then, the first order term in (26) gives the right D7-brane charge, namely 8G, while the second order term vanishes, compatibly with the fact that D5-branes are projected out by the orientifold involution. The third order term in (26) and the contribution (28) constitute the total induced D3-brane charge, which must be compensated by an equal number of explicitly added D3-sources in order to cancel the D3-brane tadpole.…”

Holonomies of intersecting branes

“…Note that that very anomaly on a D6-brane can be directly related to the shifted flux of the M-theory four-form as was shown in [13]. The structure of the M-theory three-form is one of the intriguing current geometrical issues.…”

ABJ(M) and fractional M2's with fractional M2 charge

“…This is because the D6-brane carries a nontrivial Wilson loop. In [25] the author argued that the world-volume gauge field strength of a D6-brane is equal to the integral of the Mtheory 4-form field strength over a 4-cycle which is a circle-valued family of 3-cycles which are M-theory lifts from discs ending on the D6-brane. We will see below that the same correspondence holds for the connections.…”