### Abstract

We study the effects of having multiple Spin structures on the partition function of the spacetime fields in M-theory. This leads to a potential anomaly which appears in the eta invariants upon variation of the Spin structure. The main sources of such spaces are manifolds with nontrivial fundamental group, which are also important in realistic models. We extend the discussion to the Spin ^{c} case and find the phase of the partition function, and revisit the quantization condition for the C-field in this case. In type IIA string theory in 10 dimensions, the (mod 2) index of the Dirac operator is the obstruction to having a well-defined partition function. We geometrically characterize manifolds with and without such an anomaly and extend to the case of nontrivial fundamental group. The lift to KO-theory gives the α-invariant, which in general depends on the Spin structure. This reveals many interesting connections to positive scalar curvature manifolds and constructions related to the GromovLawsonRosenberg conjecture. In the 12-dimensional theory bounding M-theory, we study similar geometric questions, including choices of metrics and obtaining elements of K-theory in 10 dimensions by pushforward in K-theory on the disk fiber. We interpret the latter in terms of the families index theorem for Dirac operators on the M-theory circle and disk. This involves superconnections, eta forms, and infinite-dimensional bundles, and gives elements in Deligne cohomology in lower dimensions. We illustrate our discussion with many examples throughout.

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

Article number | 12500055 |

Journal | Reviews in Mathematical Physics |

Volume | 24 |

Issue number | 3 |

DOIs | |

State | Published - Apr 1 2012 |

### Fingerprint

### Keywords

- adiabatic limit
- anomalies
- AtiyahPatodiSinger index theorem
- Dirac operators
- eta form
- eta invariant
- fundamental group
- K-theory
- M-theory
- partition function
- positive scalar curvature
- spin structures
- Spin structures

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

^{c}structures in the m-theory partition function.

*Reviews in Mathematical Physics*,

*24*(3), [12500055]. https://doi.org/10.1142/S0129055X12500055

**Geometry of Spin and Spin ^{c} structures in the m-theory partition function.** / Sati, Hisham.

Research output: Contribution to journal › Review article

^{c}structures in the m-theory partition function',

*Reviews in Mathematical Physics*, vol. 24, no. 3, 12500055. https://doi.org/10.1142/S0129055X12500055

^{c}structures in the m-theory partition function. Reviews in Mathematical Physics. 2012 Apr 1;24(3). 12500055. https://doi.org/10.1142/S0129055X12500055

}

TY - JOUR

T1 - Geometry of Spin and Spin c structures in the m-theory partition function

AU - Sati, Hisham

PY - 2012/4/1

Y1 - 2012/4/1

N2 - We study the effects of having multiple Spin structures on the partition function of the spacetime fields in M-theory. This leads to a potential anomaly which appears in the eta invariants upon variation of the Spin structure. The main sources of such spaces are manifolds with nontrivial fundamental group, which are also important in realistic models. We extend the discussion to the Spin c case and find the phase of the partition function, and revisit the quantization condition for the C-field in this case. In type IIA string theory in 10 dimensions, the (mod 2) index of the Dirac operator is the obstruction to having a well-defined partition function. We geometrically characterize manifolds with and without such an anomaly and extend to the case of nontrivial fundamental group. The lift to KO-theory gives the α-invariant, which in general depends on the Spin structure. This reveals many interesting connections to positive scalar curvature manifolds and constructions related to the GromovLawsonRosenberg conjecture. In the 12-dimensional theory bounding M-theory, we study similar geometric questions, including choices of metrics and obtaining elements of K-theory in 10 dimensions by pushforward in K-theory on the disk fiber. We interpret the latter in terms of the families index theorem for Dirac operators on the M-theory circle and disk. This involves superconnections, eta forms, and infinite-dimensional bundles, and gives elements in Deligne cohomology in lower dimensions. We illustrate our discussion with many examples throughout.

AB - We study the effects of having multiple Spin structures on the partition function of the spacetime fields in M-theory. This leads to a potential anomaly which appears in the eta invariants upon variation of the Spin structure. The main sources of such spaces are manifolds with nontrivial fundamental group, which are also important in realistic models. We extend the discussion to the Spin c case and find the phase of the partition function, and revisit the quantization condition for the C-field in this case. In type IIA string theory in 10 dimensions, the (mod 2) index of the Dirac operator is the obstruction to having a well-defined partition function. We geometrically characterize manifolds with and without such an anomaly and extend to the case of nontrivial fundamental group. The lift to KO-theory gives the α-invariant, which in general depends on the Spin structure. This reveals many interesting connections to positive scalar curvature manifolds and constructions related to the GromovLawsonRosenberg conjecture. In the 12-dimensional theory bounding M-theory, we study similar geometric questions, including choices of metrics and obtaining elements of K-theory in 10 dimensions by pushforward in K-theory on the disk fiber. We interpret the latter in terms of the families index theorem for Dirac operators on the M-theory circle and disk. This involves superconnections, eta forms, and infinite-dimensional bundles, and gives elements in Deligne cohomology in lower dimensions. We illustrate our discussion with many examples throughout.

KW - adiabatic limit

KW - anomalies

KW - AtiyahPatodiSinger index theorem

KW - Dirac operators

KW - eta form

KW - eta invariant

KW - fundamental group

KW - K-theory

KW - M-theory

KW - partition function

KW - positive scalar curvature

KW - spin structures

KW - Spin structures

UR - http://www.scopus.com/inward/record.url?scp=84859135995&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84859135995&partnerID=8YFLogxK

U2 - 10.1142/S0129055X12500055

DO - 10.1142/S0129055X12500055

M3 - Review article

VL - 24

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

SN - 0129-055X

IS - 3

M1 - 12500055

ER -