Classes of bundles
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The notion of an -module bundle is a categorification/homotopification of the notion of a module bundle/vector bundle, where fields and rings are replaced by ∞-rings and modules by ∞-modules; a central notion in parameterized stable homotopy theory.
Recall that for a field, a vector space is a -module, and a vector bundle over a space is classified by a morphism Mod with Mod regarded as an object in the relevant topos. For instance for discrete or flat vector bundles is the category Vect of vector spaces. There is the subcategory of 1-dimensional -vector bundles, and morphisms that factor as are -line bundles. In the discrete case the vector space of sections of the vector bundle classified by is the colimit .
These statements categorify in a straightforward manner to the case where is generalized to a commutative ∞-ring: an E-∞ ring or ring spectrum . Modules are replaced by module spectra and colimits by homotopy colimits.
The resulting notion of -vector bundles plays a central role in many constructions in orientation in generalized cohomology, twisted cohomology and Thom isomorphisms.
Further generalization of the concept leads to (∞,n)-vector bundles: an -module over an E-∞-ring is an object of the (∞,n)-category , where we are iteratively forming module -categories over the monoidal -category of -modules, times.
Discrete -vector bundles
We discuss -vector bundles internal to the (∞,1)-topos ∞Grpd Top. Since we are discussing objects with geometric interpretation, we are to think of this as the -topos of discrete ∞-groupoids.
Discussion of -vector bundles internal to structured (non-discrete) -groupoids is below.
-Modules and -Module bundles
Assume in the following choices
In this form this appears as (ABG def. 3.7). Compare this to the analogous definition at principal ∞-bundle.
-Lines and -line bundles
for the full sub-(∞,1)-category on the -lines : on those -modules that are equivalent to as an -module. The full subcatgeory of on morphisms factoring through this inclusion we call the -catgeory of -line -bundles.
This appears as (ABG def. 3.12), (ABGHR 08, 7.5).
Let be an A-∞ ring spectrum.
For the underlying A-∞ space and the ordinary ring of connected components, writ for its group of units.
Then the ∞-group of units of is the (∞,1)-pullback in
This appears in (ABG, 3.6) (p. 10). See also (ABGHR 08, section 6).
For the moment see twisted cohomology for more on this.
Sections and twisted cohomology
The -module of (dual) sections of an -module bundle is the (∞,1)-colimit over this functor
The corresponding spectrum of sections is the -dual
This is (ABG, def. 4.1) and (ABG, p. 15), (ABG11, remark 10.16).
Because for the sphere spectrum, is indeed the classical Thom spectrum of the spherical fibration given by :
For the sphere spectrum, an -line bundle – hence a spherical fibration, and any other -ring with canonical inclusion , the Thom -module of the composite is the classical Thom spectrum of tensored with :
This is (ABGHR 08, theorem 4.5).
Trivializations and orientations
For an -line -bundle, its ∞-groupoid of trivializations is the -groupoid of lifts
For the canonical inclusion and a -line bundle, we say that an -orientation of is a trivialization of the associated -line bundle .
That this encodes the notion of orientation in A-cohomology is around (ABGHR 08, 7.32).
Every trivialization/orientation of an -line -bundle induces an equivalence
of the -module of sections of / the Thom A-module of with the generalized A-homology-spectrum of :
This appears as (ABGHR 08, cor. 7.34).
Therefore if is not trivializable, we may regard its -module of sections as encoding -twisted A-cohomology:
For an -line -bundle, the -twisted A-homology of is
The -twisted A-cohomology is
Structured -vector bundles
We discuss now -vector bundles in more general (∞,1)-toposes.
The string topology operations on a compact smooth manifold may be understood as arising from a sigma-model quantum field theory with target space whose background gauge field is a flat -line -bundle which is -oriented over , hence trivializabe over (for instance for the Eilenberg-MacLane spectrum this may be the sphereical fibration of Thom spaces induced from the tangent bundle if the manifold is oriented in the ordinary sense).
By prop. 1 this implies that the space of states of the -model is the -homology spectrum of , and that for every suitable surface with incoming and outgoing boundary components the mapping space span
acts by path integral as a pull-push transform on these spaces of states
A systematic discussion of discrete -module bundles has a precursor in
(discussing the string orientation of tmf) and is then discussed in more detail in the triple of articles
Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra and Thom spectra (arXiv:0810.4535)
Matthew Ando, Andrew Blumberg, David Gepner, Twists of K-theory and TMF, in Robert S. Doran, Greg Friedman, Jonathan Rosenberg, Superstrings, Geometry, Topology, and -algebras, Proceedings of Symposia in Pure Mathematics vol 81, American Mathematical Society (arXiv:1002.3004)
Matthew Ando, Andrew Blumberg, David Gepner, Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map (arXiv:1112.2203)
The last of these explains the relation to
A streamlined version of (ABGHR 08) appears as
Lecture notes on these articles are in
- Ben Knudsen, Scott Slinker, Paul VanKoughnett, Brian Williams, and Dylan Wilson, Thom spectra reading course (web)