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

Hisham Sati

    Research output: Contribution to journalReview article

    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 languageEnglish (US)
    Article number12500055
    JournalReviews in Mathematical Physics
    Volume24
    Issue number3
    DOIs
    StatePublished - Apr 1 2012

    Fingerprint

    Spin Structure
    M-Theory
    Partition Function
    partitions
    Dirac Operator
    K-theory
    Fundamental Group
    Anomaly
    geometry
    Eta Invariant
    Positive Scalar Curvature
    Index Theorem
    Type Theory
    String Theory
    anomalies
    Obstruction
    operators
    Well-defined
    Cohomology
    Quantization

    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

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

    In: Reviews in Mathematical Physics, Vol. 24, No. 3, 12500055, 01.04.2012.

    Research output: Contribution to journalReview article

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