Ring completion of rig categories

Baas, Nils A.; Dundas, Bjørn Ian; Richter, Birgit; Rognes, John

doi:10.1515/crelle.2012.024

Cited by 11 publications

(59 citation statements)

References 23 publications

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“…Here PM denotes the based path space of M , 3 ΩM the based loop space, and D 2 M denotes the space of based disks f : D 2 → M in M . 4 The map PM → M is generically a surjective submersion, but for the map D 2 M → ΩM to be a surjective submersion we must require 1connectedness of M . The horizontal arrows restrict a loop to its first and second halves, reversing the orientation of the second half in order to obtain a based map.…”

Section: Geometric Quantisationmentioning

confidence: 99%

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Fluxes, bundle gerbes and 2-Hilbert spaces

Bunk

Szabo

2017

Lett Math Phys

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We elaborate on the construction of a prequantum 2-Hilbert space from a bundle gerbe over a 2-plectic manifold, providing the first steps in a programme of higher geometric quantisation of closed strings in flux compactifications and of M5-branes in C-fields. We review in detail the construction of the 2-category of bundle gerbes and introduce the higher geometrical structures necessary to turn their categories of sections into 2-Hilbert spaces. We work out several explicit examples of 2-Hilbert spaces in the context of closed strings and M5-branes on flat space. We also work out the prequantum 2-Hilbert space associated with an M-theory lift of closed strings described by an asymmetric cyclic orbifold of the SU(2) WZW model, providing an example of sections of a torsion gerbe on a curved background. We describe the dimensional reduction of M-theory to string theory in these settings as a map from 2-isomorphism classes of sections of bundle gerbes to sections of corresponding line bundles, which is compatible with the respective monoidal structures and module actions.

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Section: Geometric Quantisationmentioning

confidence: 99%

“…In the following all mapping spaces are assumed to consist of maps having appropriate sitting instants to make all the necessary gluing of maps smooth. 4 The base-point of D 2 ⊂ R 2 is the point (1, 0), such that S 1 ֒→ D 2 is a map of pointed manifolds.…”

Section: Geometric Quantisationmentioning

confidence: 99%

Fluxes, bundle gerbes and 2-Hilbert spaces

Bunk

Szabo

2017

Lett Math Phys

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“…We know from [4] that there is a chain of simplicial rig categories R ∼ ←− ZR −→R such that (i) R ← ZR becomes a weak equivalence on realization;…”

Section: The Spine Of the Argument Giving A Proof Of Theorem 11mentioning

confidence: 99%

“…This means that the somewhat complicated construction ofR from [4] may be safely forgotten once we know it exists.…”

Section: The Spine Of the Argument Giving A Proof Of Theorem 11mentioning

confidence: 99%

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Stable bundles over rig categories

Baas¹,

Dundas

Richter

et al. 2011

Journal of Topology

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The point of this paper is to prove the conjecture that virtual 2-vector bundles are classified by K(ku), the algebraic K-theory of topological K-theory. Hence, by the work of Ausoni and the fourth author, virtual 2-vector bundles give us a geometric cohomology theory of the same telescopic complexity as elliptic cohomology. The main technical step is showing that for well-behaved small rig categories R (also known as bimonoidal categories), the algebraic K-theory space, K(HR), of the ring spectrum HR associated to R is equivalent to K(R) Z × |BGL(R)| + , where GL(R) is the monoidal category of weakly invertible matrices over R. The title refers to the sharper result that BGL(R) is equivalent to BGL(HR). If π0R is a ring, this is almost formal, and our approach is to replace R by a ring completed version,R, provided by Baas, Dundas, Richter, and Rognes [J. reine angew. Math., to appear] with HR HR and π0R the ring completion of π0R. The remaining step is then to show that 'stable R-bundles' and 'stableR-bundles' are the same, which is done by a hands-on contraction of a custom-built model for the difference between BGL(R) and BGL(R).

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On genus one mirror symmetry in higher dimensions and the BCOV conjectures

Eriksson

Montplet

Mourougane

2022

Forum of Mathematics, Pi

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The mathematical physicists Bershadsky–Cecotti–Ooguri–Vafa (BCOV) proposed, in a seminal article from 1994, a conjecture extending genus zero mirror symmetry to higher genera. With a view towards a refined formulation of the Grothendieck–Riemann–Roch theorem, we offer a mathematical description of the BCOV conjecture at genus one. As an application of the arithmetic Riemann–Roch theorem of Gillet–Soulé and our previous results on the BCOV invariant, we establish this conjecture for Calabi–Yau hypersurfaces in projective spaces. Our contribution takes place on the B-side, and together with the work of Zinger on the A-side, it provides the first complete examples of the mirror symmetry program in higher dimensions. The case of quintic threefolds was studied by Fang–Lu–Yoshikawa. Our approach also lends itself to arithmetic considerations of the BCOV invariant, and we study a Chowla–Selberg type theorem expressing it in terms of special $\Gamma $ -values for certain Calabi–Yau manifolds with complex multiplication.

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Ring completion of rig categories

Cited by 11 publications

References 23 publications

Fluxes, bundle gerbes and 2-Hilbert spaces

Fluxes, bundle gerbes and 2-Hilbert spaces

Stable bundles over rig categories

On genus one mirror symmetry in higher dimensions and the BCOV conjectures

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