The n-Category Café

Ezra Getzler speaks about joint work with Kai Behrend.

Here a brief and incomplete note on what he said.

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Let $A$ be a dg-algebra. Form the simplicial set

$$N_\bullet A := MC(C^\bullet(\Delta[\bullet]), A)$$

of Maurer-Cartan elements $\alpha$

$$\delta \alpha + D \alpha + \alpha^2 = 0$$

on the $A$-valued cochains on a simplicial set, where $\delta$ is the simplicial differential and $D$ is the “internal” differential on $A$.

Claim: This simplicial set is a quasi-category.

If $A$ is an ordinary algebra, then this is the nerve of the corresponding multiplicative monoid.

(ed.: Hence it makes sense to write $B A := N(A)$.)

Consider now the maximal Kan complex inside this, we may think of it as the delooping of the general linear group with coefficient in $A$

$$B GL(A) := core(N(A)) \subset N(A) \,.$$

Theorem: This also works for differential graded Banach algebras.

Application: for $E \to X$ a homomorphic vector bundle, its algebra of forms

$$A := \oplus_i \Omega^{0,i}(X, End(E))$$

is a dg-Banach algebra. Hence we have an application to Kuranishi deformation theory.

Bruno Valette speaks on joint work with Gabriel Drummond-Cole on Minimal models for the BV-operad.

Here are brief and incomplete notes from the talk.

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Theorem (Baranikov, Kontsevich 98, Manin 99)

Consider a dg-BV-algebra

$$(A, d, \cdot, \Delta) \,,$$

where $(A,d)$ is a chain complex, “$\cdot$” a binary commutative product , $\Delta$ an order 2 operator of degree +1 (or -1 depending on convetion), which satisfies the “$d \Delta$-condition”

$$ker d \cap ker \Delta \cap (Im d \oplus Im \Delta) = Im(d \Delta)$$

($\Rightarrow \Delta = 0$ on $H_\bullet(A, d)$).

Then there exists on the underlying homology groups $H_\bullet(A,d)$ a natural Frobenius manifold structure.

Let $\bar \mathcal{M}_{0,n+1}$ be the moduli space of surfaces with $n$-inputs

$H_\bullet ( \bar \mathcal{M}_{0,n+1})$-algebra

$$dim \bar \mathcal{M} = 2(n+2)$$

Theorem (Kadishvili). Homotopy transfer theorem

Let

$$(A,d) \stackrel{h}{\to} (A,d) \stackrel{\overset{p}{\to}}{\underset{i}{\leftarrow}} (H, d_H)$$

be a homotopy retract $i,p$ with homotopy exhibited by $h$.

$$i p - id_A = h d_A \pm d_A h \,.$$

Now let $\mu : A \times A \to A$ be an associative product on $A$. This Induces a product $p \circ \mu \circ i \otimes i$ on $H$, which is in general not associative.

But the given $h$ can be used to equip it with the structure of an A-infinity algebra.

By means of this one can give a construction that rectifies any $A_\infty$-algebra to a dg-algebra.

Task: do something similar for C-infinity algebra in order to have rectification for the BV-operad.

idea: use Koszul duality to pass from the BV-operad to a dg-cooperad.

Theorem (Valette): There is a homotopy transfer for infinity-BV-algebras.

Moreover, there are minimal models for the dg-BV-operad. There is an explicit construction using the homotopy transfer theorem for $C_\infty$-algebras. They are unique up to isomorphism.

One finds: where in the Baranikov-Kontsevich theorem above one finds Frobenius manifolds, here one finds, of course, homotopy Frobenius manifolds.

Camilo Arias Abad speaks about joint work with Florian Schätz.

Here a quick and incomplete note.

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In the first half of the talk he reviews the A-infinity refinement of the de Rham theorem, which over a smooth manifold $X$ provides an equivalence of $A_\infty$-algebras

$$\Omega^\bullet(X) \stackrel{\simeq}{\to} C^\bullet(X, \mathbb{R})$$

from the de Rham algebra of differential forms under the wedge product and the singular cochains, which becomes an $A_\infty$-algebra under the cup product.

In his context, Camilo thinks of ∞-representations of ∞-groupoids $X$ on (∞,1)-vector spaces, hence ∞-functors

$$\rho : X \to (\infty,1)Vect$$

alternatively as $A_\infty$-homomorphisms from the linearization of $X$ (the formal dual to the cochain $A_\infty$-algebra on the underlying simplicial set) to the the $A_\infty$-category of chain complexes

$$\tilde \rho: \mathbb{R}X \to dgVect \,.$$

He says “representation up to homotopy” for this.

(ed.: Maybe “strong homotopy representations” would be more in line with historical convention?)

(Talk almost but not quite over, but my laptop battery is dying right now.)

Pavol Ševera talks about joint work with Stefan Sakáloš.

Here are the notes I took in the first 15 or so minutes, stating the problem. The rest of the talk is detailed algebra giving the solution, which I don’t feel like reproducing here.

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The following problem of quantization of Lie bialgebras was already discussed/partly solved by Enriques-Halbent ‘08 and Etingof-Kazhdan 96.

Given a quasi-bialgabra $H$ consider the monoidal category $H Mod$ of its modules.

The tensor product on $H Mod$ comes from the coproduct

$$\Delta : H \to H \otimes H$$

on $H$ and the associator in $H Mod$ corresponds to an element

$$\Phi_H \in H^{\otimes^3}$$

Let now

$$H = \mathcal{U}\mathfrak{g}[ [\hbar ] ]$$

be a universal enveloping algebra as a $k[ [\hbar] ]$-module

Let the skew-symmetric part of $\Delta$ be

$$\delta : \mathfrak{g} \to \wedge^2 \mathfrak{g}$$

The associator element corresponds now to some

$$\phi \in \wedge^3 \mathfrak{g} \,.$$

So we have a Lie algebra $\mathfrak{g}$, and some additional structure $\delta$ and $\phi$ on

Call this a Lie quasi-bialgebra.

Then

$$d = \mathfrak{g}\oplus \mathfrak{g}^* ,$$

is a Lie algebra and the canonical pairing

$$\langle -,-\rangle$$

is an invariant polynomial on it.

Task: given this Lie algebraic data $\mathfrak{g} \subset d$, like this, (re)construct $H$.

Thomas Willwacher recalls how Maurer-Cartan elements serve to twist all kinds of higher algebras and asks: can we generally say which algebras over which operads have twistings by MC elements?

I didn’t take notes. Notice that MC elements are secretly homomorphisms, see here.

On Tuesday, I speak about

$\infty$-Chern-Simons Functionals (pdf slides).

I am still fiddling with the slides a bit. All comments are welcome.

Chris Rogers spoke about higher geometric quantization of 2-plectic geometry.

He provides pdf slides accompanying the talk.

I am fond of the idea that Chris presents, this looks like it is going in the right direction. Eventually one should look into how to lift not only the categorical degree of the prequantum line bundle but also that of the base space, to have a notion of geometric quantization of symplectic infinity-groupoids.

Konrad Waldorf speaks about joint work with Thomas Nikolaus.

The first part of the talk is about nonabelian obstruction theory in terms of lifting bundle gerbes, and lifting bundle 2-gerbes, making use of aspects of their article

Four equivalent versions of non-abelian gerbes (arXiv:1103.4815)

on the various equivalent incarnations of 2-groupoids of principal 2-bundles and bundle gerbes.

The second part is about transgression of principal 2-bundles to ordinary principal bundles over the loop space of their base space.

Are there any surprising new directions being taken by anyone, or does it feel like the talks fit nicely inside your view of things?

André Henriques speaks about a spin-off idea of his joint work with Chris Douglas and Arthur Bartels.

The idea of the talk is to start with 3d Chern-Simons theory regarded as an extended TQFT

$$Z_{CS} : Bord_3 \to 3 Vect$$

and then “compactify” it on the interval $I$ by making sense of the expression

$$(\int_I Z_CS) : \Sigma \mapsto Z_{CS}(\Sigma \times I) \,.$$

If one imagines that one can choose “conformal boundary conditions” on the boundaries $\partial (\Sigma \times I)$, then something like this should give the 2d CFT called the WZW model, regarded as an “extended CFT”.

One formalization of this instance of the holographic principle is what underlies the FRS formalism. André is pushing here towards a more systematic understanding of this process from the point of view of extended QFT, reminiscent (to me) of what I once talked about at 2-functorial CFT.

Unfortunately the talk ended too early.

Ingo Runkel talks about his joint work with Alexei Davydov and Liang Kong.

Since this, too, appeared in our book, I won’t produce notes here but just point you there.

R. Meyer indicated in his talk a construction of a resolution of any topological groupoid that may fail to be degreewise Hausdorff by a topological 2-hypergroupoid that is.

I would think that there is a construction that does this very generally – for all $\infty$-groupoids modeled on any site that has small coproducts and fiber products (such as Hausdorff spaces).

This follows from a slight variant of Dugger’s resolution in the projective model structure on simplicial presheaves. The argument is written out as prop. 2.1.49 here.

Karl-Henning Rehren reviews the construction and properties of the DHR category associated with a local net of observables.

Arturo Prat-Waldron reviews first differential cohomology and differential K-theory, and then reviews the relation between Supersymmetric field theories and generalized cohomology.

(ed.: It was long expected that if one takes the relation between (1,1)-dimensional Euclidean field theories and K-theory and instead of dividing out concordance of QFTs over some space $X$, just divides out equivalences, one gets differential K-theory. The claim of Arturo and Alexander Kahle is maybe that they can now show this.)

Fei Han reviews the setup of Supersymmetric field theories and generalized cohomology and then passes for the latter to equivariant cohomology. The statement is that for the former this means to pass to gauged field theory of sorts.

Peter Teichner reviews ideas of Supersymmetric field theories and generalized cohomology.

Ulrich Bunke talks about joint work with David Gepner.

The claim here is: a deep theorem by Beilinson about the “Borel regulator” for the cyclotomic field which may look a bit mysterious, apparently has a natural interpretation in the differential cohomology refinement of the corresponding algebraic K-theory.

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Let $R$ be any number ring. The running example to keep in mind is the the cyclotomic field

$$R = \mathbb{Z}/ (1 + \xi + \cdots + \xi^{p-1}) \,.$$

Write then

$$B GL(R)$$

for the classifying space for $R$-module bundles and

$$B GL(R) \stackrel{Quillen constr}{\hookrightarrow} (B GL(R))^+$$

for the Quillen construction, such that for

$K R$ the algebraic K-theory spectrum, the one whose homotopy groups are the algebraic K-theory groups $K_i(R)$

$$K_i(R) \simeq \pi_i (B GL(R))^*$$

we have

$$\Omega^\infty K R \simeq B GL(R)^+ \times K_0(R) \,.$$

Now for any choice $\sigma : R \hookrightarrow \mathbb{C}$, which for the running example: would be given by $\xi \mapsto \mu \in U(1)$ a $p$th root of unity, and a bundle $V \otimes_\sigma \mathbb{C} \to B GL(R)$, an observation by S. Götte gives a corresponding bundle $\mathcal{V}$ over the +-construction fitting into

$$\array{ V \otimes_\sigma \mathbb{C} &\to& \mathcal{V} \\ \downarrow && \downarrow \\ B GL(R) &\hookrightarrow& (B GL(R))^+ }$$

and such that $\mathcal{V}$ is locally filtered.

The Borel regulator construction starts with a K-class

$$x \in K_{2n-1}(R)$$

classified by a map

$$x : S^{2n-1} \to B GL(R)^+$$

and considers on the pullback bundle

$$\array{ x^* \mathcal{V} \\ \downarrow \\ S^{2n-1} }$$

a choice of connection $\nabla$ that preserves the locally filtered structure.

Choose also a hermitean structure and let $\nabla^*$ be the corresponding adjoint connection

Then the Borel regulator of $x$ is the expression

$$r_\sigma(x) = \int_{S^{2n-1}} ch_{2n-1}(\nabla^* , \nabla) \in \mathbb{R} \,,$$

where

$$ch_{2n-1}(\nabla^*, \nabla) \in \Omega^{2n-1}(S^{2n-1})$$

is the relative Chern-Simons form between the two connections.

This defines a map

$$r: K_{2n-1}(R) \to \mathbb{R}$$

and the big question is: what is the image of this map? This is complicated, and the answer in general is not known.

But a theorem of Beilinson says that for $R$ the cyclotomic field as above, there exist $q \in \mathbb{Q}$ and $x \in K_{2n-1}(R)$ such that

$$r_\sigma(x) = q Re\left[ \frac{1}{(2 \pi i)^n} Li_{2 n -1}(\mu) \right]$$

where

$$Li_k(z) := \sum_{n \geq 1}^\infty \frac{z^n}{n^k}$$

is a “higher logarithm”.

The proof that Beilinson gives is complicated. The claim in the following is: this is also a consequence of a statement that naturally ought to be true in index theory for differental algebraic K-theory.

Index theorem is about comparison of two pushforwards, one analytic, one homotopy theoretic.

Fix a bundle $\pi : W \to B$ with compact fibers and being a proper submersion (no need for any kind of orientation in the following).

In the running example we look at lens spaces

$$S^{2n-1}/{\mathbb{Z}_p} \simeq L_p^{2n+1} \to \mathbb{C}P^n \,.$$

Consider local systems of $R$-modules on $W$ and $B$. The claim is then that there is a diagram as follows, to be explained now:

$$\array{ Loc(W) &\stackrel{cycle}{\to}& \hat K R^0(W) \\ {}^{\mathllap{analytic transfer}}\downarrow &\neArrow_{\simeq}^{higher analytic torsion}& \downarrow^{\mathrlap{Becher-Gottlieb transfer}} \\ Loc(B) &\stackrel{cycle}{\to}& \hat K R^0(B) }$$

Here $\hat K R$ is the differential K-theory spectrum

$Loc(B)$ is isomorphism classes of geometrically constant sheaves of finitely generated projective $R$-modules on $B$,

where the “geometric” is choices of hermitean metrics on the complexification of the sheaves (which are sheaves of sections of complexes of flat vector bundles)

In the example we have

$$\pi_1(L_p^{2n+1}) = \mathbb{Z}_p$$

take $V$ te rank 1, free, with holonomy $\xi$

Now definition of $\hat K R$ following Hopkins-Singer.

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[to be polished from here on, no time now]

We have the spectrum $K R$ and can form smash product with the Moore spectrum, the result being equivalent to an Eilenberg-MacLane spectrum

$$K R \to K R \wedge M \mathbb{R} \stackrel{\simeq}{\to} H A \,.$$

where $A_\bullet = K\bullet(R) \otimes \mathbb{R}$

give a de Rham construction for this

$$H(\Omega(B,A)) \stackrel{de Rham theorem}{\to} H(A)^B$$

form the homotopy pullback

$$\array{ \hat K R^0(B) &\to& H(\sigma^{\geq 0} \Omega(B,A)) \\ {}^{\mathllap{I}}\downarrow && \downarrow \\ K R^B &\to& H(\Omega(B,A)) }$$

$$K R^{-1} \stackrel{r}{\to} \Omega^{-1}(B,A)/im D \stackrel{a}{\to} \hat K R^0(B)$$

the kernel of $a$ is the Borel regulator from before

task: give geometric models for elements in this hopullback

Theorem: there exists a cycle map

$$cycl : Loc \to \hat K R$$

$g^V$ is the geometry

$$I(cycle(V, g^V)) = [V]$$

$$in K R^0(B) [M \stackrel{V}{\to} B GL(R) \to \Omega^\infty K R]$$

$$R(cycle(V, g^V)) = contains ch(\nabla^* , \nabla)$$

now

$$\hat B G$$

transfer choose Riem structure on $\Pi$

pushforward induced from the following morphism of diagrams

$$\array{ K R^W &\to& H(\Omega(W,A)) &\leftarrow& H(\Sigma^{\geq 0} \Omega(B,A)) \\ \downarrow &\neArrow_{canonical}& \downarrow^{\int_{W/R} \wedge e^{\nabla}} && \downarrow^{\int (-) \wedge ...} \\ K R^B &\to& H(\Omega(W,A)) &\leftarrow& H(\Sigma^{\geq 0} \Omega(B,A)) }$$

analytic transfer

$$(V, g^V) \mapsto (R^i \pi_* V, g_{L^R}^V)$$

Conjecture:

$$cycle(transfer^{analytic}(V, g^V)) - \hat tr(cycle(V, G^V)) = a(\tau)$$

$$\tau$$

is higher analytic torsion form.

John Huerta spoke about his thesis work.

This has been mentioned here many times before, but let me suggest to look at it from the following perspective.

The Green-Schwarz action for sigma-models with supermanifold target spaces is strictly of (coset) higher WZW-model type. See here.

What John calles the superstring 2-group is the total space of the corresponding WZW-gerbe / WZW-2-bundle over the super-translation group. Maybe it would eventually be better to call it the GS-superstring 2-group to distinguish it from the role of the string 2-group for the heterotic string, which is analogously the WZW gerbe on $Spin$ and the joint generalization, which will be a WZW gerbe on all of the super-Poincaré-Lie algebra.

By the way, one may wonder why the M5-brane does not appear in the brane scan (the NS5-brane does). I guess it’s clear: the necessary 7-cocycle that is missing on the $(10,1)$d super Poincaré-Lie algebra does reappear on its “$\mathfrak{m}2\mathfrak{brane}_{gs}$” super Lie 3-algebra extension only. It is the 7-cocycle that defines the supergravity Lie 6-algebra.

Alberto Cattaneo spoke about joint work with Pavel Mnev and Nicolai Reshitikhin.

They are aiming to get a kind of extended quantization from the BV-formalism by taking the boundary term in the variation of the action into account.

Parts of the discussion reminded me a bit of what I was talking about with Igor Khavkine here.

Pdf slides for the talk are available, but I am not sure if I may link to them.