A method of deforming G-structures

We consider deformations of G-structures via the right action on the frame bundle in a base-point-dependent manner. We investigate which of these deformations again lead to G-structures and in which cases the original and the deformed G-structures define the same instantons. Further, we construct a bijection from connections compatible with the original G-structure to those compatible with the deformed G-structure and investigate the change of intrinsic torsion under the aforementioned deformations. Finally, we consider several examples.Comment: 14 pages; v3: references added, published in Journal of Geometry and Physic

Transgression of D-branes

Closed strings can be seen either as one-dimensional objects in a target space or as points in the free loop space. Correspondingly, a B-field can be seen either as a connection on a gerbe over the target space, or as a connection on a line bundle over the loop space. Transgression establishes an equivalence between these two perspectives. Open strings require D-branes: submanifolds equipped with vector bundles twisted by the gerbe. In this paper we develop a loop space perspective on D-branes. It involves bundles of simple Frobenius algebras over the branes, together with bundles of bimodules over spaces of paths connecting two branes. We prove that the classical and our new perspectives on D-branes are equivalent. Further, we compare our loop space perspective to Moore-Segal/Lauda-Pfeiffer data for open-closed 2-dimensional topological quantum field theories, and exhibit it as a smooth family of reflection-positive, colored knowledgable Frobenius algebras

Principal ∞\infty-Bundles and Smooth String Group Models

We provide a general, homotopy-theoretic definition of string group models within an $\infty$-category of smooth spaces, and we present new smooth models for the string group. Here, a smooth space is a presheaf of $\infty$-groupoids on the category of cartesian spaces. The key to our definition and construction of smooth string group models is a version of the singular complex functor, which assigns to a smooth space an underlying ordinary space. We provide new characterisations of principal $\infty$-bundles and group extensions in $\infty$-topoi, building on work of Nikolaus, Schreiber, and Stevenson. These insights allow us to transfer the definition of string group extensions from the $\infty$-category of spaces to the $\infty$-category of smooth spaces. Finally, we consider smooth higher-categorical group extensions that arise as obstructions to the existence of equivariant structures on gerbes. We show that these extensions give rise to new smooth models for the string group, as recently conjectured in joint work with M\"uller and Szabo.Comment: 44 pages, v2: Lemmas 4.15 and 4.16 improve

∞\infty-Bundles

Higher bundles are homotopy coherent generalisations of classical fibre bundles. They appear in numerous contexts in geometry, topology and physics. In particular, higher principal bundles provide the geometric framework for higher-group gauge theories with higher-form gauge potentials and their higher-dimensional holonomies. An $\infty$-categorical formulation of higher bundles further allows one to identify these objects in contexts outside the worlds of smooth manifolds or topological spaces. This article reviews the theory of $\infty$-bundles, focussing on principal $\infty$-bundles, and surveys several of their applications. It is an invited contribution to the Topology section in the second edition of the Encyclopedia of Mathematical Physics.Comment: 24 pages, several diagram

The R\mathbb{R}-Local Homotopy Theory of Smooth Spaces

Simplicial presheaves on cartesian spaces provide a general notion of smooth spaces. We define a corresponding smooth version of the singular complex functor, which maps smooth spaces to simplicial sets. We exhibit this functor as one of several Quillen equivalences between the Kan-Quillen model category of simplicial sets and a motivic-style $\mathbb{R}$-localisation of the (projective or injective) model category of smooth spaces. These Quillen equivalences and their interrelations are powerful tools: for instance, they allow us to give a purely homotopy-theoretic proof of a Whitehead Approximation Theorem for manifolds. Further, we provide a functorial fibrant replacement in the $\mathbb{R}$-local model category of smooth spaces. This allows us to compute the homotopy types of mapping spaces in this model category in terms of smooth singular complexes. We explain the relation of our fibrant replacement functor to the concordance sheaves introduced recently by Berwick-Evans, Boavida de Brito, and Pavlov. Finally, we show how the $\mathbb{R}$-local model category of smooth spaces formalises the homotopy theory on sheaves used by Galatius, Madsen, Tillmann, and Weiss in their seminal paper on the homotopy type of the cobordism category.Comment: 59 pages; improved exposition, in particular clarifications in Sections 1, 2 and 6; added references; added Appendix

Gerbes in Geometry, Field Theory, and Quantisation

This is a mostly self-contained survey article about bundle gerbes and some of their recent applications in geometry, field theory, and quantisation. We cover the definition of bundle gerbes with connection and their morphisms, and explain the classification of bundle gerbes with connection in terms of differential cohomology. We then survey how the surface holonomy of bundle gerbes combines with their transgression line bundles to yield a smooth bordism-type field theory. Finally, we exhibit the use of bundle gerbes in geometric quantisation of 2-plectic as well as 1- and 2-shifted symplectic forms. This generalises earlier applications of gerbes to the prequantisation of quasi-symplectic groupoids.Comment: 37 page

The 2-Hilbert Space of a Prequantum Bundle Gerbe

We construct a prequantum 2-Hilbert space for any line bundle gerbe whose Dixmier-Douady class is torsion. Analogously to usual prequantisation, this 2-Hilbert space has the category of sections of the line bundle gerbe as its underlying 2-vector space. These sections are obtained as certain morphism categories in Waldorf's version of the 2-category of line bundle gerbes. We show that these morphism categories carry a monoidal structure under which they are semisimple and abelian. We introduce a dual functor on the sections, which yields a closed structure on the morphisms between bundle gerbes and turns the category of sections into a 2-Hilbert space. We discuss how these 2-Hilbert spaces fit various expectations from higher prequantisation. We then extend the transgression functor to the full 2-category of bundle gerbes and demonstrate its compatibility with the additional structures introduced. We discuss various aspects of Kostant-Souriau prequantisation in this setting, including its dimensional reduction to ordinary prequantisation.Comment: 97 pages; v2: minor changes; Final version to be published in Reviews in Mathematical Physic

Instantons on conical half-flat 6-manifolds

We present a general procedure to construct 6-dimensional manifolds with SU(3)-structure from SU(2)-structure 5-manifolds. We thereby obtain half-flat cylinders and sine-cones over 5-manifolds with Sasaki-Einstein SU(2)-structure. They are nearly Kahler in the special case of sine-cones over Sasaki-Einstein 5-manifolds. Both half-flat and nearly Kahler 6-manifolds are prominent in flux compactifications of string theory. Subsequently, we investigate instanton equations for connections on vector bundles over these half-flat manifolds. A suitable ansatz for gauge fields on these 6-manifolds reduces the instanton equation to a set of matrix equations. We finally present some of its solutions and discuss the instanton configurations obtained this way.Comment: 1+32 pages, 1 figure, v2: 6 references added, v2 accepted for publication in JHE

Fluxes, bundle gerbes and 2-Hilbert spaces

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 program 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 to 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.Comment: 38 pages; v2: Exposition improved, references added; Final version published in Letters in Mathematical Physic

Higher Geometric Structures on Manifolds and the Gauge Theory of Deligne Cohomology

We study smooth higher symmetry groups and moduli $\infty$-stacks of generic higher geometric structures on manifolds. Symmetries are automorphisms which cover non-trivial diffeomorphisms of the base manifold. We construct the smooth higher symmetry group of any geometric structure on $M$ and show that this completely classifies, via a universal property, equivariant structures on the higher geometry. We construct moduli stacks of higher geometric data as $\infty$-categorical quotients by the action of the higher symmetries, extract information about the homotopy types of these moduli $\infty$-stacks, and prove a helpful sufficient criterion for when two such higher moduli stacks are equivalent. In the second part of the paper we study higher $\mathrm{U}(1)$-connections. First, we observe that higher connections come organised into higher groupoids, which further carry affine actions by Baez-Crans-type higher vector spaces. We compute a presentation of the higher gauge actions for $n$-gerbes with $k$-connection, comment on the relation to higher-form symmetries, and present a new String group model. We construct smooth moduli $\infty$-stacks of higher Maxwell and Einstein-Maxwell solutions, correcting previous such considerations in the literature, and compute the homotopy groups of several moduli $\infty$-stacks of higher $\mathrm{U}(1)$- connections. Finally, we show that a discrepancy between two approaches to the differential geometry of NSNS supergravity (via generalised and higher geometry, respectively) vanishes at the level of moduli $\infty$-stacks of NSNS supergravity solutions.Comment: 102 pages; comments welcom