The f-invariant and index theory

Bunke, Ulrich; Naumann, Niko

doi:10.1007/s00229-010-0351-7

Cited by 11 publications

(16 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The set of 2-framings 2 F is an affine space with translation group π 3 (SO(6)) ∼ = Z. This is related to String structures [14] and applied to describe the C-field and the M2-brane in [57] [58]. The main connection is via the quantization condition of the C-field [81], which implies that C 3 is essentially (but not literally) the difference of two Chern-Simons forms; see [57].…”

Section: Stable Framingmentioning

confidence: 99%

See 1 more Smart Citation

Framed M-branes, corners, and topological invariants

Sati

2018

Journal of Mathematical Physics

View full text Add to dashboard Cite

We uncover and highlight relations between the M-branes in M-theory and various topological invariants: the Hopf invariant over Q, Z and Z 2 , 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 M5brane 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 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. *

show abstract

Section: Stable Framingmentioning

confidence: 99%

“…We will see that there is a framing of the M2-brane worldvolume M 3 which is, in a sense, preferred. In [57] [58] the canonical String structure of [53] [14] was highlighted as the one preferred by the M2-brane. Here we elaborate, highlighting some dynamical aspects.…”

Section: )mentioning

confidence: 99%

Framed M-branes, corners, and topological invariants

Sati

2018

Journal of Mathematical Physics

View full text Add to dashboard Cite

show abstract

“…The invariant we will associate to this is a framed cobordism invariant, and hence depends only on the framed cobordism class. Using the results of [19], the data that refines the corner M 10 into a representative of a framed cobordism class [M 10 ] ∈ Ω (U,fr) 2 12 , and which interestingly matches the physical setting (see diagram (3.1)), is the following 1. A decomposition T M 10 ∼ = T 0 M 10 ⊕ T 1 M 10 of framed bundles.…”

Section: The Heterotic Theory As a Cornermentioning

confidence: 90%

“…We have used the f -invariant at the level of cohomology classes. We now show that the refined version, the geometric f -invariant [19,83], also captures part of the dynamics and anomalies of the heterotic corner.…”

Section: Chromatic Level 2: Heterotic Corners and The F -Invariantmentioning

confidence: 92%

M-Theory with Framed Corners and Tertiary Index Invariants

Sati¹

2014

SIGMA

View full text Add to dashboard Cite

Abstract. The study of the partition function in M-theory involves the use of index theory on a twelve-dimensional bounding manifold. In eleven dimensions, viewed as a boundary, this is given by secondary index invariants such as the Atiyah-Patodi-Singer eta-invariant, the Chern-Simons invariant, or the Adams e-invariant. If the eleven-dimensional manifold itself has a boundary, the resulting ten-dimensional manifold can be viewed as a codimension two corner. The partition function in this context has been studied by the author in relation to index theory for manifolds with corners, essentially on the product of two intervals. In this paper, we focus on the case of framed manifolds (which are automatically Spin) and provide a formulation of the refined partition function using a tertiary index invariant, namely the f -invariant introduced by Laures within elliptic cohomology. We describe the context globally, connecting the various spaces and theories around M-theory, and providing a physical realization and interpretation of some ingredients appearing in the constructions due to Bunke-Naumann and Bodecker. The formulation leads to a natural interpretation of anomalies using corners and uncovers some resulting constraints in the heterotic corner. The analysis for type IIA leads to a physical identification of various components of eta-forms appearing in the formula for the phase of the partition function.

show abstract

“…Remark 12.2. In [BN10] we gave an intrinsic formula for the f -invariant in terms of a sequence of η-invariants of Dirac operators twisted by bundles derived from the tangent bundle. In order to interpret this as an example of the intrinsic formula 10.3 it would be necessary to translate this construction to a construction with a complementary bundle.…”

mentioning

confidence: 99%

The Universal -Invariant for Manifolds With Boundary

Bunke

2015

The Quarterly Journal of Mathematics

View full text Add to dashboard Cite

We extend the theory of the universal η-invariant to the case of bordism groups of manifolds with boundaries. This allows the construction of secondary descendants of the universal η-invariant. We obtain an interpretation of Laures' f -invariant as an example of this general construction. As an aside we improve a recent result by Han-Zhang about the modularity of a certain formal power series of η-invariants.

show abstract

The f-invariant and index theory

Cited by 11 publications

References 16 publications

Framed M-branes, corners, and topological invariants

Framed M-branes, corners, and topological invariants

M-Theory with Framed Corners and Tertiary Index Invariants

The Universal -Invariant for Manifolds With Boundary

Contact Info

Product

Resources

About