### Abstract

We uncover and highlight relations between the M-branes in M-theory and various topological invariants: the Hopf invariant over Q, Z and Z2, the Kervaire invariant, the f-invariant, and the ν-invariant. This requires either a framing or a corner structure. The canonical framing provides a minimum for the classical action and the change of framing encodes the structure of the action and possible anomalies. We characterize the flux quantization condition on the C-field and the topological action of the M5-brane via the Hopf invariant, and the dual of the C-field as (a refinement of) an element of Hopf invariant two. In the signature formulation, the contribution to the M-brane effective action is given by the Maslov index of the corner. The Kervaire invariant implies that the effective action of the M5-brane is quadratic. Our study leads to viewing the self-dual string, which is the boundary of the M2-brane on the M5-brane worldvolume, as a string theory in the sense of cobordism of manifolds with corners. We show that the dynamics of the C-field and its dual are encoded in a unified way in the 4-sphere, which suggests the corresponding spectrum as the generalized cohomology theory describing the fields. The effective action of the corner is captured by the f-invariant, which is an invariant at chromatic level two. Finally, considering M-theory on manifolds with G_{2} holonomy, we show that the canonical G_{2} structure minimizes the topological part of the M5-brane action. This is done via the ν-invariant and a variant that we introduce related to the one-loop polynomial.

Original language | English (US) |
---|---|

Article number | 062304 |

Journal | Journal of Mathematical Physics |

Volume | 59 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2018 |

### Fingerprint

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*59*(6), [062304]. https://doi.org/10.1063/1.5007185

**Framed M-branes, corners, and topological invariants.** / Sati, Hisham.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 59, no. 6, 062304. https://doi.org/10.1063/1.5007185

}

TY - JOUR

T1 - Framed M-branes, corners, and topological invariants

AU - Sati, Hisham

PY - 2018/6/1

Y1 - 2018/6/1

N2 - We uncover and highlight relations between the M-branes in M-theory and various topological invariants: the Hopf invariant over Q, Z and Z2, the Kervaire invariant, the f-invariant, and the ν-invariant. This requires either a framing or a corner structure. The canonical framing provides a minimum for the classical action and the change of framing encodes the structure of the action and possible anomalies. We characterize the flux quantization condition on the C-field and the topological action of the M5-brane via the Hopf invariant, and the dual of the C-field as (a refinement of) an element of Hopf invariant two. In the signature formulation, the contribution to the M-brane effective action is given by the Maslov index of the corner. The Kervaire invariant implies that the effective action of the M5-brane is quadratic. Our study leads to viewing the self-dual string, which is the boundary of the M2-brane on the M5-brane worldvolume, as a string theory in the sense of cobordism of manifolds with corners. We show that the dynamics of the C-field and its dual are encoded in a unified way in the 4-sphere, which suggests the corresponding spectrum as the generalized cohomology theory describing the fields. The effective action of the corner is captured by the f-invariant, which is an invariant at chromatic level two. Finally, considering M-theory on manifolds with G2 holonomy, we show that the canonical G2 structure minimizes the topological part of the M5-brane action. This is done via the ν-invariant and a variant that we introduce related to the one-loop polynomial.

AB - We uncover and highlight relations between the M-branes in M-theory and various topological invariants: the Hopf invariant over Q, Z and Z2, the Kervaire invariant, the f-invariant, and the ν-invariant. This requires either a framing or a corner structure. The canonical framing provides a minimum for the classical action and the change of framing encodes the structure of the action and possible anomalies. We characterize the flux quantization condition on the C-field and the topological action of the M5-brane via the Hopf invariant, and the dual of the C-field as (a refinement of) an element of Hopf invariant two. In the signature formulation, the contribution to the M-brane effective action is given by the Maslov index of the corner. The Kervaire invariant implies that the effective action of the M5-brane is quadratic. Our study leads to viewing the self-dual string, which is the boundary of the M2-brane on the M5-brane worldvolume, as a string theory in the sense of cobordism of manifolds with corners. We show that the dynamics of the C-field and its dual are encoded in a unified way in the 4-sphere, which suggests the corresponding spectrum as the generalized cohomology theory describing the fields. The effective action of the corner is captured by the f-invariant, which is an invariant at chromatic level two. Finally, considering M-theory on manifolds with G2 holonomy, we show that the canonical G2 structure minimizes the topological part of the M5-brane action. This is done via the ν-invariant and a variant that we introduce related to the one-loop polynomial.

UR - http://www.scopus.com/inward/record.url?scp=85048797709&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85048797709&partnerID=8YFLogxK

U2 - 10.1063/1.5007185

DO - 10.1063/1.5007185

M3 - Article

VL - 59

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 6

M1 - 062304

ER -