# Uses for (Framed) E2 algebras twisted by braided monoidal structure

$$\newcommand{\C}{\mathcal{C}}$$ $$\newcommand{\g}{\mathfrak{g}}$$ If $$\C$$ is a monoidal category (not necessarily a symmetric monoidal category), it's possible to define the notion of an algebra object $$A$$ in $$\C$$, with multiplication operations $$A^{\otimes n} (:= A\otimes_\C A\otimes_\C \cdots\otimes_\C A)\to A.$$

Similarly, if $$\C$$ is a braided monoidal category (resp., a ribbon category), one can define a notion of $$E_2$$ DG algebra $$A$$ (resp., framed $$E_2$$ DG algebra $$A$$) "twisted" by $$\C$$, consisting of operations $$A^{\otimes n}\to A$$ compatible with braiding. (Note: I actually don't know a reference for this, but it follows from standard "homotopy field theory" arguments involving the Ran space.)

In particular, if $$\C$$ is a braided monoidal (or ribbon) category coming from an associator on a Lie algebra $$\g$$ (with choice of Casimir), there is a whole category of "associator-twisted" $$\g$$-equivariant $$E_2$$ (resp., framed $$E_2$$) algebras.

My question is whether algebras of this type have been encountered before. They feel very CFT-ish, and so I'm particularly curious about physics and knot theory applications. In particular, the framed variant should gove some kind of derived 2D TQFT-style invariants.

Any references would be useful. Thanks!

• Related reference: "Planar algebras in braided tensor categories" by myself, David Penneys, James Tener arxiv.org/abs/1607.06041 Apr 24, 2021 at 23:03
• @AndréHenriques Thank you! The notion of "ribbon braided operad" seems very close to what I am looking for. Is there a relationship between this "untwisted" notion of planar algebra and E2 algebras or (Koszul dually) Hopf algebras? Apr 24, 2021 at 23:52
• Curious - does one really need to invoke Ran spaces? I would have thought naively that for any operad $O$ in say topological spaces and an $O$-monoidal ($\infty$-)category $C$ we can discuss $O$-algebras in $C$? Apr 25, 2021 at 1:52
• These algebras go under the name "braided commutative algebras," or E_2 algebras in an E_2 monoidal category Apr 25, 2021 at 1:52
• @DavidBen-Zvi Good point! Yes, I think you're right: I think you can for example say that an O-algebra in C is an algebra of O in the (category, object) pair category over a given O-monoidal C Apr 25, 2021 at 20:09

• if you think of your braided tensor category $$C$$ as defining a 4d TFT (i.e. as an object in the Morita 4-category thereof), then $$E_2$$ algebras $$A$$ in $$C$$ are a special case of boundary conditions in the TFT (left modules over $$A$$ form a monoidal category over $$C$$, or Morita morphism to/from the unit). As such they are actually quite rare in say reps of quantum groups generically, and have to do with coisotropic subalgebras of $$\mathfrak g$$.
• if you think of $$C$$ as the value on the circle of a 3d TFT (eg as defined by a rational vertex algebra), you get $$E_2$$ algebras $$A$$ in $$C$$ as the value on the circle of boundary conditions for the TFT. Or you can look at interfaces of 3d TFTs, giving $$E_2$$-functors of $$E_2$$-categories of which I think your setup is a special case by passing to $$E_2$$-A-modules (maybe I'm getting turned around though). Or less generally - a concrete example of your setup is $$C=B-E_2-mod$$ and $$B\to A$$ an $$E_2$$-morphism - as such this appears all over CFT, when you have a homomorphism of algebras of observables (eg a map of rational vertex algebras).
From the TFT POV the difference between $$E_2$$ and framed $$E_2$$ is the difference between framed TFT and oriented TFT - the latter naturally produces framed $$E_2$$-algebras ( as usual the terminology is awful, for TFT definitely seems better to talk about "oriented 2-disc algebras" (framed $$E_2$$) and "framed 2-disc algebras" ($$E_2$$).
• What David is referring to in its first point is also known as an algebra over the Swiss-Cheese operad, in that case a pair of a monoidal category $M$ and a braided functor from $C$ to the Drinfeld center of $M$. If $A$ is an $E_2$-algebra in $C$ then $M=A-mod_C$ should have such a structure. As such it has to do do with quantization of shifted Lagrangian/coisotropic structures. In the particular case you mention this should indeed arise as quantizations of Lagrangians in the 2-shted symplectic stack $BG$. See eg Safronov-Melani arxiv.org/abs/1704.03201 Apr 25, 2021 at 9:03
• Examples of those typically arise in Etingof-Kazhdan quantization, e.g. if $G$ is the drinfeld double of a (quasi-)Poisson algebraic group $A$, then $O(A)$ is a braided commutative algebra in the Drinfeld category of $G$. As David says it boils down to the fact that the Lie algebra of $A$ is coisotropic in the Lie algebra of $G$ which, as shown by Pavel Safronov, is equivalent to the fact that $BA \rightarrow BG$ is shifted coisotropic. Apr 25, 2021 at 9:28