# nLab cubical structure on a line bundle

Contents

cohomology

### Theorems

#### Bundles

bundles

fiber bundles in physics

# Contents

## Idea

A cubical structure on a complex line bundle over an abelian group is a certain trivialization of a certain induced line bundle on the 3-fold Cartesian product (“cube”) of the group which is constructed in a kind of cubical generalization of the polarization identity formula for quadratic forms.

Over formal groups associated with complex oriented cohomology theories cubical structures encode orientation in generalized cohomology.

## Definition

###### Definition

Given a circle group-principal bundle/complex line bundle $\mathcal{L}$ on an abelian group $A$, write $\Theta(\mathcal{L})$ for the line bundle on $G^3$ which is given by the formula

$\Theta(\mathcal{L})_{x,y,z} = \mathcal{L}_{x+y+z} \otimes \mathcal{L}_{x+y}^{-1} \otimes \mathcal{L}_{x+z}^{-1} \otimes \mathcal{L}_{y+z}^{-1} \otimes \mathcal{L}_x \otimes \mathcal{L}_y \otimes \mathcal{L}_z \otimes \mathcal{L}_0^{-1} \,.$
###### Definition

A cubical structure on $\mathcal{L}$ is a trivializing section $s$ of $\Theta(\mathcal{L})$ such that

1. $s(0,0,0) = 1$

2. $s(x_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)}) = s(x_1, x_2, x_3)$

3. $s(w+x,y,z) s(w,x,z) = s(w,x + y, z) s(x,y,z)$

for all elements of $A$ as indicated, and for all permutations $\sigma$ of three elements. Here the equalities are equalities of sections after applying the canonical isomorphisms of complex lines on both sides.

###### Remark

The canonical isomorphsms hidden in def. are:

1. $\mathcal{L}_0^{\otimes 3} \otimes (\mathcal{L}_0^{-1})^{\otimes 3} \to 1$ the canonical map exhibiting $\mathcal{L}_0^{-1}$ as the inverse (dual object) of $\mathcal{L}_0$:

2. etc.

There is the following further refinement.

###### Definition

In the situation of def. , if the line bundle $\mathcal{L}$ is equipped with a natural “symmetry”

$t \colon \mathcal{L}_x \stackrel{\simeq}{\longrightarrow} \mathcal{L}_{-x}$

then a $\Sigma$-structure on $\mathcal{L}$ is a cubical structure, def. , such that in addition

$s(x,y , -x-y) = 1 \,.$

## Examples

### Relation to orientations in complex-oriented cohomology theory

For $E$ a multiplicative weakly periodic complex orientable cohomology theory, we have that $Spec E^0(B U\langle 6\rangle)$ is naturally equivalent to the space of cubical structures on the trivial line bundle over the formal group of $E$.

In particular, homotopy classes of morphisms of E-infinity ring spectra $MU\langle 6\rangle \to E$ from the Thom spectrum to $E$, and hence universal $MU\langle 6\rangle$-orientations (see there) of $E$ are in natural bijection with these cubical structures.

This way for instance the string orientation of tmf has been constructed. See there for more on this.

### On the 11-dimensional Chern-Simons term

The 11-dimensional Chern-Simons action functional in 11-dimensional supergravity gives a line bundle $L$ on the space of supergravity C-fields whose $\Theta^3(L)$ is the transgression of the cup product in ordinary differential cohomology of three factors. It seems that each trivialization of the class of the supergravity C-field induces a “cubical” trivialization of $\Theta^3(L)$ as above, and hence a cubical structure on $L$. See at cubical structure in M-theory for more on this.

An early reference discussing the relation with theta functions is

• Lawrence Breen, Fonctions thêta et théorème du cube, Springer Lecture Notes in Mathematics 980 (1983). (MR0823233).

In relation to orientation in generalized cohomology cubical structures have been prominently discussed in

• Michael Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Basel), Birkhäuser, 1995, 554–565. MR 97i:11043 (pdf)

• Matthew Ando, Michael Hopkins, Neil Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001) 595–687 MR1869850 (pdf)