Special and general types
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
Special notions
Variants
differential cohomology
Extra structure
Operations
Theorems
spin geometry, string geometry, fivebrane geometry …
rotation groups in low dimensions:
see also
A string structure on a manifold is a higher version of a spin structure. A string structure on a manifold with spin structure given by a Spin group-principal bundle to which the tangent bundle is associated is a lift of the classifying map through the third nontrivial step in the Whitehead tower of to a String group-principal bundle:
A lift one further step through the Whitehead tower is a Fivebrane structure.
This has generalizations to the smooth context, where instead of the topological String-group one uses the String Lie 2-group.
Let be an -dimensional topological manifold.
Its tangent bundle is canonically associated to a -principal bundle, which is in turn classified by a continuous function
from to the classifying space of the orthogonal group .
A String structure on is the choice of a lift of this map a few steps through the Whitehead tower of .
The manifold “is string” if such a lift exists.
This means the following:
there is a canonical map from the classifying space of to that of that represents the generator of the cohomology . The classifying space of the group is the homotopy pullback
Namely using the homotopy hypothesis (which is a theorem, recall), we may identify with the one object groupoid whose space of morphisms is and similarly for . Then the map in question is the one induced from the group homomorphism that sends orientation preserving elements in to the identity and orientation reversing elements to the nontrivial element in .
there is a canonical map representing the generator of . The classifying space of the group is the homotopy pullback
there is a canonical map The classifying space of the group is the homotopy pullback
there is a canonical map The classifying space of the group is the homotopy pullback
Let the ambient (∞,1)-topos by ETop∞Grpd or Smooth∞Grpd. Write for a topological manifold or smooth manifold of dimension , respectively.
Write for the string 2-group, a 1-truncated ∞-group object in .
The 2-groupoid of (topological or smooth) string structures on is the hom-space of cocycles , or equivalently that of (topological or smooth) -principal 2-bundles:
Write in for the topological or smooth refinement of the first fractional Pontryagin class (see differential string structure for details on this).
The 2-groupoid of string structure on is the homotopy fiber of : the (∞,1)-pullback
By definition of the string 2-group we have the fiber sequence . The hom-functor preserves every (∞,1)-limit, hence preserves this fiber sequence.
Given a spin structure we say that the string structures extending this spin-structure is the homotopy fiber of the projection from observation :
(…)
The 2-groupoid of string structures is the homotopy fiber of
over the trivial cocycle. Followowing the general logic of twisted cohomology the 2-groupoids over a nontrivial cocycle may be thought of as that of twisted string structures.
The Pontryagin class refines to the smooth first fractional Pontryagin class . That leads to differential string structures.
(…)
The space of choices of string structures extending a given spin structure are as follows
if it is empty: ;
if it is .
In particular the set of equivalence classes of string structures lifting is the cohomology set
Apply the pasting law for (∞,1)-pullbacks on the diagram
The outer diagram defines the loop space object of . Since commutes with forming loop space objects (see fiber sequence for details) we have
One can reformulate an
structure in terms of the existence of a certain class in abelian cohomolgy on the total space of the given principal bundle. This decomposition is a special case of th general Whitehead principle of nonabelian cohomology.
Let be a manifolds with spin structure . Write for the corresponding spin group-principal bundle.
Then a string structure lifting is a cohomology class such that the restriction of the class to any fiber is a generator of .
This kind of definition appears in (Redden, def. 6.4.2).
Every string structure in the sense of def. induces a string structure in the sense of def. .
Consider the pasting diagram of (∞,1)-pullbacks
This uses repeatedly the pasting law for -pullbacks. The map appears by decomposing the homotopy pullback of the point along into a homotopy pullback first along and then along using the given String structure. This is the cocycle for a -principal 2-bundle on the total space of the -principal bundle: a bundle gerbe.
The rest of the diagram is constructed in order to prove the following:
string structure
(all hooks are homotopy fiber sequences)
The relevance of String structures was originally recognized in the physics of spinning strings (super string theory) in higher dimensional generalization of the theory (from half a century earlier) of spin structures for spinning particles, therefore the name string structure.
The coinage “string structure” is due to p. 585 (shown on the right) of:
where this structure is understood as a de-transgression (both the naming and this insight are often incorrectly attributed to various mathematicians) of the required “spin structure on loop space”, due to
Edward Witten: The Index of the Dirac Operator in Loop Space, in Elliptic Curves and Modular Forms in Algebraic Topology (Princeton, N.J., Sep 1986), Lecture Notes in Mathematics 1326, Springer (1988) [doi:10.1007/BFb0078045]
Edward Witten, Elliptic Genera and Quantum Field Theory, Commun. Math. Phys. 109 (1987) 525–536 [doi:10.1007/BF01208956]
which arises as an anomaly cancellation condition on the worldsheet of the superstring‘s (specifically the heterotic string’s) sigma-model, analogous to how the condition of spin structure may be understood (Witten 1985) as an anomaly cancellation on the worldline of spinning particles such as superparticles.
See also:
The understanding of Killingback’s string structure equivalently as the condition to lifting the structure group of the tangent bundle from a spin group to a string group (or string 2-group) was introduced or at least popularized by:
The relation between the two pictures is further analyzed for instance in
A mathematically rigorous formulation of Killingback 1987‘s original argument, now using differential K-theory, is given by
in turn reviewed in:
The definition of string structures by degree-3 classes on the total space of the spin bundle (also already mentioned by Killingback 1987) is used in
Corbett Redden, Canonical metric connections associated to string structures PhD Thesis (2006) [pdf]
Redden & Waldorf at Oberwolfach Workshop, June 2009 – Friday, June 12
Discussion of the moduli stack of twisted differential string structures:
and its appearance on M5-branes under Hypothesis H:
An explicit cocycle construction of the essentially unique string 2-group-principal 2-bundle lift of the tangent bundle of the 5-sphere is given in
Discussion for indefinite (Lorentzian) signature is in
More discussion of relation to spin structures on loop spaces is in
Alessandra Capotosti, From String structures to Spin structures on loop spaces, Ph.D. thesis, Università degli Studi Roma Tre, Rome, April 2016
Konrad Waldorf, String structures and loop spaces, in: Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2312.12998]
A study of (flat) string structures encoded in the bicategory of flat 2-group bundles on an oriented surface via weak representations of the fundamental group is in
Last revised on May 29, 2025 at 08:24:59. See the history of this page for a list of all contributions to it.