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The Hinich-Pridham-Lurie theorem on formal moduli problems says that unbounded $L_\infty$-algebras over some field are equivalently the (“infinitesimally cohesive”) infinitesimal $\infty$-group objects over the derived site over that field.
May we say anything about an analogous statement for infinitesimal group objects in the tangent $\infty$-topos of the $\infty$-topos over the site of smooth manifolds?
There exists for instance the $L_\infty$-algebra whose CE-algebra is
$\{ d h_3 = 0, d \omega_{2p+2} = h_3 \wedge \omega_{2p} | p \in \mathbb{Z} \} \,.$This looks like it wants to correspond to the smooth parameterized spectrum whose base smooth stack is the smooth group 2-stack $\mathbf{B}^2 U(1)$, and whose smooth parameterized spectrum is the pullback along $\mathbf{B}^2 U(1) \to \mathbf{B}GL_1(\mathbf{KU})$ of the canonical smooth parameterized spectrum over $\mathbf{B}GL_1(\mathbf{KU})$, for $\mathbf{KU}$ a smooth sheaf of spectra representing multiplicative differential KU-theory.
So it looks like it wants to be this way. How much more can we say?
So I have the following rough picture.
A group object in smooth parameterized spectra should be a sequence of group objects $E_\bullet$ in smooth $\infty$-stacks, each with a fixed retraction $G \to E_n \to G$ onto the base smooth group $\infty$-stack $G$, such that the canonical morphisms give equivalences $G \times_{E_{n+1}} G \overset{\simeq}{\to } E_{n}$.
Now under Lie differentiation each component group object $E_n$ and $X$ goes to a connective $L_\infty$-algebra $\mathfrak{h}_n$ and $\mathfrak{g}$, respectively, equipped with retractions of $L_\infty$-algebras $\mathfrak{g}\to \mathfrak{h}_n \to \mathfrak{g}$, subject to the condition that the canonical morphisms give equivalences $\mathfrak{g} \times_{\mathfrak{h}_{n+1}} \mathfrak{g} \stackrel{\simeq}{\to} \mathfrak{h}_n$.
So the main thing to show would be that such sequences of $\mathfrak{g}$-pointed connective $L_\infty$-algebras are equivalent in some way to certain unbounded $L_\infty$-algebras.
Do we need equivalence, or just a functor going in one direction?
I was hoping for an equivalence, but just a functor to unbounded $L_\infty$-algebras would be good, too.
So when analyzing T-duality in the “brane bouquet” then that unbounded $L_\infty$-algebra in #1 naturally appears. I’d like to say that the whole brane bouquet is the image under Lie derivation of something. In the case of T-duality the standard lore suggests that this something is a smooth/differential spectrum $\mathbf{KU}$ parameterized over $\mathbf{B}^2 U(1)$. So I’d be happy to see that the Lie derivation of that is the $L_\infty$-algebra in #1.
I’d like to say that the whole brane bouquet is the image under Lie derivation of something.
Are there compact Lie supergroups with superalgebras $\mathbb{R}^{m|n}$ ? E.g., is there a ’super $U(1)$’ for the odd line?
Ah, I see there’s $SU(1|1)$ in Compact forms of Complex Lie Supergroups, and then $SU(m|n)$ in general.
David C, why do you ask for compactness? The $\mathbb{R}^{m\vert n}$ that appear in the brane bouquet have the structure of super Minkowski super Lie algebras.
It’s interesting to ask for simple super Lie algebras of which these are contractions. This leads to the super anti de Sitter super Lie algebras.
I was thinking more about the story you told us in dcct:
At this point it is worth pausing for a second to note how the hallmark of quantum mechanics has appeared as if out of nowhere simply by applying Lie integration to the Lie algebraic structures in classical mechanics…
if we think of Lie integrating $\mathbb{R}$ to the interesting circle group $U(1)$ instead of to the uninteresting translation group $\mathbb{R}$, then the name of its canonical basis element $1 \in \mathbb{R}$ is canonically $``i''$, the imaginary unit…
the Poisson bracket, written in the form that makes its Lie integration manifest, indeed reads $[q, p] = i$. Since the choice of basis element of $i \mathbb{R}$ is arbitrary, we may rescale here the $i$ by any non-vanishing real number without changing this statement. If we write “$\hbar$” for this element, then the Poisson bracket instead reads $[q, p] = i \hbar$. This is of course the hallmark equation for quantum physics, if we interpret $\hbar$ here indeed as Planck’s constant, def. 6.4.156. We see it arises here merely by considering the non-trivial (the interesting, the non-simply connected) Lie integration of the Poisson bracket.
So that got me wondering about interesting ways to integrate $\mathbb{R}^{m|n}$, and whether there’s a compact choice in the odd direction.
There’s someone quantizing super symplectic spaces, but of course that’s already on nLab. Section 8 of Super symplectic geometry and prequantization looks for some kind of Lie supergroup.
Presumably already classically we should be using supermanifolds.
Coming back to #2, I suppose I should not ask for group objects with respect to the Cartesian product on parameterized spectra – which is the “external direct sum operation” – but instead the “external tensor product operation”. For $E$ a (sheaf of) ring spectra, then the canonical parameterized spectrum over $Pic(E)$ should be a lax monoidal functor
$(Pic(E),\otimes_{E}) \longrightarrow (Spectra, \wedge) \,.$Better probably to Lie differentiate that by precomposing with the canonical inclusions to get
$\mathbf{B} Lie( GL_1(E) ) \longrightarrow \mathbf{B} GL_1(E) \longrightarrow Pic(E) \longrightarrow (Spectra, \wedge) \,.$This is the same as an action of the $L_\infty$-algebra $Lie( GL_1(E) )$ on the smooth spectrum $E$. After real-ifying the latter to make it an $H \mathbb{R}$-module spectrum, hence equivalently an (unbounded) $\mathbb{R}$-chain complex, this should hence be an action of that $L_\infty$-algebra on that chain complex.
And that would be just the right kind of structure. For instance the example in #1 is the action $L_\infty$-algebra of $Lie ( \mathbf{B} U(1) )$ acting on the chain complex corresponding to $KU \otimes \mathbb{R}$.
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