nLab cubical structure on a line bundle

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.

(Breen 83, Hopkins 94, section 4, Ando-Hopkins-Strickland 01, def. 2.40)

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.

(Hopkins 94, theorem 6.1, 6.2, Ando-Hopkins-Strickland 01, corollary 2.50)

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.

Related concept

• theta characteristic

References

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)