# Elliptic $\infty$-line bundles over $B G$

Theorem 5.2 in Jacob Lurie's "Survey of Elliptic Cohomology" (pdf) states the equivalence of two maps

$$B G \longrightarrow B \mathrm{GL}_1(A)$$

for $A$ an $E_\infty$-ring carrying an oriented derived elliptic curve $E \to \mathrm{Spec}(A)$.

The theorem is stated for $G = \mathrm{Spin}$. That's the case of relevance in the paragraph directly following its statement. But, in view of the whole 2-equivariance story, the statement of the theorem as such would make sense for more general $G$ equipped with a homomorphism $G \to O$, as in the discussion on the preceding page.

First question: Might this statement still be true for more general $G\to O$? (What's the idea of the proof, anyway?)

More in detail, one of the two maps is the composite

$$B G \to B O \stackrel{J}{\longrightarrow} B \mathrm{GL}_1(\mathbb{S}) \longrightarrow B \mathrm{GL}_1(A)$$

where $J$ denotes the $J$-homomorphism and $\mathbb{S}$ the sphere spectrum. (Here I am giving a nominally different but hopefully straightforwardly equivalent description of what is indicated on p. 38 of Lurie's note.) The other one is the restriction of the derived theta-line bundle

$$\theta \;\colon\; \mathrm{Loc}_{\mathrm{Spin}}(E) \longrightarrow \mathbf{B} \mathbb{G}_m$$

along the global point inclusion $\mathrm{Spec}(A)\to \mathrm{Loc}_{\mathrm{Spin}}(E)$ that picks the trivial local system/flat connection (that's again my slight refomulation which I believe is straightforwardly equivalent, but check -- my $\mathrm{Loc}_{\mathrm{Spin}}(E)$ is Lurie's "$M_{\mathrm{Spin}}$").

Regarding the first map, for $G = \mathrm{Spin}$ and for the special case $A = \mathrm{tmf}$ this is discussed in section 8 of Ando-Blumberg-Gepner arXiv:1002.3004 and crucially equivalent to the $\mathrm{tmf}$-line bundle which is associated via the String-orientation $\sigma$ to the "Chern-Simons 3-bundle" classified by $\tfrac{1}{2}p_1$, i.e. to the map

$$B \mathrm{Spin} \stackrel{\tfrac{1}{2}p_1}{\longrightarrow} B^4 \mathbb{Z} \stackrel{\tilde \sigma}{\longrightarrow} B \mathrm{GL}_1(\mathrm{tmf}) \,.$$

I suppose the argument there straightforwardly generalizes from $A = \mathrm{tmf}$ to any $A$ as above. But as a sanity check:

Second question: is that right?

In conlusion I am wondering:

Third question: In which generality in $E_\infty$-rings $A$ carrying oriented derived elliptic curves and in compact simply connected simple Lie groups $G$, is it true that the restriction of the derived theta line bundle along the $\mathrm{Spec}(A)$-point inclusion is equivalent to the $A$-line bundle classified by

$$B G \stackrel{c_2}{\longrightarrow} B^4 \mathbb{Z} \stackrel{\tilde \sigma}{\longrightarrow} B \mathrm{GL}_1(A) \,.$$

?

(Where now $c_2$ denotes the generator of $H^4(B G, \mathbb{Z}) \simeq \mathbb{Z}$.)