Some geometric properties of the Witten genus

Dessai, Anand

doi:10.1090/conm/504/09877

Cited by 9 publications

(13 citation statements)

References 27 publications

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“…] be the Witten genus, see [5,19], and section 6 below. By definition, if φ W x = 0, then x ∈ Ω string * has infinite order.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…and each element x 4k with k ≥ 4 is represented by a manifold W 4k which is a total space of a geometrical CaP 2 -bundle π k : W 4k → L 4k−16 , see [5] and section 6 below. We consider a transfer map T string : Ω string…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…Here we recall necessary facts concerning the Cayley projective plane CaP 2 . We follow the constructions due to A. Dessai [5]. Let F 4 denote the 52-dimensional compact simple sporadic Lie group.…”

Section: 3mentioning

confidence: 99%

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Highly connected manifolds of positive $p$-curvature

Botvinnik

Labbi

2014

Trans. Amer. Math. Soc.

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Abstract. We study and in some cases classify highly connected manifolds which admit a Riemannian metric with positive p-curvature. The p-curvature was defined and studied by the second author in [8,9,10]. It turns out that the positivity of p-curvature could be preserved under surgeries of codimension at least p + 3 . This gives a key to reduce a geometrical classification problem to a topological one, in terms of relevant bordism groups and index theory. In particular, we classify 3 -connected manifolds with positive 2 -curvature in terms of the bordism groups Ω spin *, Ω string *, and by means of α -invariant and Witten genus φ W . Here we use results from [5], which provide appropriate generators of the ring Ω string *

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“…] be the Witten genus, see [5,19], and section 6 below. By definition, if φ W x = 0, then x ∈ Ω string * has infinite order.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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Highly connected manifolds of positive $p$-curvature

Botvinnik

Labbi

2014

Trans. Amer. Math. Soc.

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“…(4) Behavior in fiber bundles. The A-genus vanishes on any smooth fiber bundle of closed oriented manifolds provided that the fiber is a Spin manifold and the structure group is a compact connected Lie group which acts smoothly and non-trivially on the fiber [52].…”

Section: H Satimentioning

confidence: 99%

GEOMETRY OF SPIN ANDSPIN^cSTRUCTURES IN THE M-THEORY PARTITION FUNCTION

Sati

2012

Rev. Math. Phys.

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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 Spinccase 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 Gromov–Lawson–Rosenberg 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.

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“…A version of the Witten genus can be described by requiring the rational first Pontrjagin class to vanish; see e.g. [De99][CHZ11] (and references therein), where a similar definition of a rational structure is used. There, a Spin manifold M is a rational BOx8y manifold if and only if p 1 pM q is a torsion class.…”

Section: Introductionmentioning

confidence: 99%

Variations of rational higher tangential structures

Sati

Wheeler

2018

Journal of Geometry and Physics

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The study of higher tangential structures, arising from higher connected covers of Lie groups (String, Fivebrane, Ninebrane structures), require considerable machinery for a full description, especially for connections to geometry and applications. With utility in mind, in this paper we study these structures at the rational level and by considering Lie groups as a starting point for defining each of the higher structures, making close connection to p i -structures. We indicatively call these (rational) Spin-Fivebrane and Spin-Ninebrane structures. We study the space of such structures and characterize their variations, which reveal interesting effects whereby variations of higher structures are arranged to systematically involve lower ones. We also study the homotopy type of the gauge group corresponding to bundles equipped with the higher rational structures that we define.

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Some geometric properties of the Witten genus

Abstract: Abstract. We give a survey on geometric properties of the Witten genus. The survey focuses on relations between the Witten genus, group actions and positive curvature.

Cited by 9 publications

References 27 publications

Highly connected manifolds of positive $p$-curvature

Highly connected manifolds of positive $p$-curvature

GEOMETRY OF SPIN ANDSPIN^cSTRUCTURES IN THE M-THEORY PARTITION FUNCTION

Variations of rational higher tangential structures

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Some geometric properties of the Witten genus

Abstract: Abstract. We give a survey on geometric properties of the Witten genus. The survey focuses on relations between the Witten genus, group actions and positive curvature.

Cited by 9 publications

References 27 publications

Highly connected manifolds of positive $p$-curvature

Highly connected manifolds of positive $p$-curvature

GEOMETRY OF SPIN ANDSPINcSTRUCTURES IN THE M-THEORY PARTITION FUNCTION

Variations of rational higher tangential structures

Contact Info

Product

Resources

About

GEOMETRY OF SPIN ANDSPIN^cSTRUCTURES IN THE M-THEORY PARTITION FUNCTION