at.algebraic topology – 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 $$A^{otimes n} (:= Aotimes_C Aotimes_C cdotsotimes_C A)to A.$$

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!

networking – Why are rollover/console port cables not twisted?

ever since I learned about console cables I felt there was something off about them, and only once I got one for a Cisco ASA did I notice what it was, it’s a completely flat cable, end to end!

I’m really curious as to why that is, and i haven’t been able to find anything online about it, mostly the difference between them and straight through cables.

homotopy theory – Using HoTT, why is twisted cohomology of BG group cohomology?

I’ve been reading Michael Shulman’s blog posts defining cohomology in homotopy type theory, and I’d like to understand (using HoTT) why cohomology of BG is group cohomology.

if I understand correctly, given a parametrized spectrum (i.e. a fibration by spectra) $E: X to mathsf{Spectra}$, we define the twisted cohomology of $X$ with coefficients in $E$ to be $H^n(X; E) equiv Vert prod_{x:X} Omega^{-n} E_0 Vert_0$.

In particular, if we have a parametrized family $V: X to mathsf{AbGroup}$ then we can compose with the Eilenberg-MacLane construction $H: mathsf{AbGroup} to mathsf{Spectra}$ to get a parametrized family of Spectra $HV: X to mathsf{Spectra}$. The cohomology $H^n(X; HV)$ is cohomology with local coefficients, which is the twisted version of ordinary cohomology.

Now if we consider the case $X = BG$ (i.e. $BG=K(G,1)$ ) for $G$ a set-group, then a parametrized family $V: BG to mathsf{AbGroup}$ is the same as a group representation of $G$, since given $g: bullet = bullet$, we get a path $g_*: V(bullet) = V(bullet)$. Now, if we consider the corresponding twisted cohomology $H^n(BG; HV) equiv Vert prod_{x:BG} K(V;n) Vert_0$, why do we get group cohomology?

For now let’s just consider $H^0(BG; HV) equiv Vert prod_{x:BG} V Vert_0 = prod_{x:BG} V$, where the second equality follows because $V$ is a set. In order to get group cohomology, it should be the case that any $v: prod_{x:BG} V$ encodes a $G$-invariant element of the $G$-representation. But it isn’t immediately obvious to me why this should be the case.

Any help would be greatly appreciated!

models – Is there a way to find .obj files of the twisted ones?

I am making a game about the twisted animatronics in FNaF, but I can’t put them in because I couldn’t find the .obj files anywhere on the internet, I did find one of them, but it wasn’t colored.

the files I need are .obj files of the following animatronics: Twisted Freddy, Twisted Bonnie, Twisted Foxy and Twisted wolf. you can search for these files for the internet, I will be thankful, or you can choose the hard way and make the models themselves. You can search up how they look like on the internet

reference request – Twisted affine Lie algebras, Lie bracket and normalized standard invariant form

I am reading the book: Infinite-Dimensional Lie Algebras (Kac) and the article: Affine Lie algebras and the Virasoro algebras I (Wakimoto). The formulas they wrote for the Lie bracket $(,)$, normalized standard invariant form $(|)$ of twisted affine Lie algebras of type $X_N^{(r)}$ are contradicted to each other:

Contradiction1: In the book, page 139, the bracket given by
enter image description here

but in the article, page 381, it is given by
enter image description here

Here $X(j)$ means $t^j otimes X$ and $c_s=rK/m$ (see the article to verify it). They are totally different.

Contradiction2: In the book, page 139, if the normalized standard invariant form is defined by
enter image description here

then it contradict to the Lie bracket in the same page,

enter image description here

since
$(d’| (t^i otimes x, t^j otimes y)) ne ((d’,t^i otimes x)| t^j otimes y)$

So, If are there anyone knows the right formulas for the Lie bracket and normalized standard invariant form for twisted affine Lie algebras mentioned in the Theorem 8.7 in the book of Kac?

ag.algebraic geometry – When is a twisted form coming from a torsor trivial?

Consider a sheaf of groups $G$, equipped with a left torsor $P$ and another left action $G$ on some $X$. Form the contracted product $P times^G X := (P times X)/sim$ where $sim$ is the antidiagonal quotient: $(g.p, x)sim (p, g.x)$.

Q1: When is $Ptimes^G X$ trivial? I.e., when do we have an isomorphism $P times^G X simeq X$?

Partial answer: $P times^G X simeq X$ over $(X/G)$ iff $P times (X/G)$ is a trivial torsor over the stack quotient $(X/G)$.

Proof: We can rewrite $P times^G X$ as a contracted product of two torsors $(P times (X/G))times^G_{(X/G)} X$. Then we contract with “$X^{-1}$” — the inverse to contracting with $X$ as a torsor over $(X/G)$ and we win. (as in B. Poonen’s Rational Points on Varieties, section 5.12.5.3)

Am I allowed to do this? This argument probably shouldn’t have to appeal to algebraic stacks and may be somewhat dubious.

Q2: If I have one isomorphism $P times^G X simeq X$, can I choose another one that lies over $(X/G)$? Or at least is $G$-equivariant?

Q3: Is there a natural way to write the triviality of such a twisted form?

I first thought $P times^G X simeq X$ iff $P$ was trivial, which is clearly false for trivial actions on $X$. Then I was excited to have the pullback $* to BG$ represent triviality of the twisted form $P times^G X$ as well as the torsor $P$. Is there a natural representative of the sheaf of isomorphisms between $P times^G X$ and $X$?

These can all be sheaves, although I’m primarily interested in $G = GL_n, PGL_n, SL_n$, etc. acting on $X = mathbb{A}^n, mathbb{P}^n$ as appropriate. More ambitious is $G = text{Aut}(X)$ for even simple $X$. I’d be happy with answers in any level of generality.

Due Diligence Statement: I’m a novice in the area of “twisted forms” of varieties, so I apologize if the above is evident or obtuse. I checked all the “similar questions” listed here and couldn’t find an answer.

unity – How do I rotate a twisted upper body towards mouse pointer position?

I have created a Third Person Controller.
The camera is behind the player:

enter image description here

I would like to make it so that the player aims at the mouse pointer position.

To do that, I use the following code to rotate the chest towards the position:

        var mousePos = Input.mousePosition;
        mousePos.z = 10; // Make sure to add some "depth" to the screen point 
        var aim = Camera.main.ScreenToWorldPoint(mousePos);
        Chest.LookAt(aim);

At first I wondered why it doesn’t work as expected. The chest wasn’t rotated towards the target.
Then I noticed that the chest is “twisted”.

It can be seen well when observed from above:

enter image description here

I would like to learn how to handle this in the smartest way.

Should I add a vector to the “aim” vector to compensate for the twist or is there a better way that I don’t know yet?

Thank you.

rt.representation theory – Twisted screening operators and twisted free-field realizations of $mathcal{W}_n$ algebras

Let $mathcal{g}=mathcal{sl}_{n+1}$ and I am interested in the principal $mathcal{W}$-algebra of $mathcal{g}$ at self-dual level i.e. $k=- h ^{vee} +1$, usually denoted by $mathcal{W}_n$. Now these VOAs can be realized as subalgebras inside the rank $n$ Heisenberg (free boson) VOA. It can also be realized as the intersection of the kernels of screening operators,
$$mathcal{W}_n cong cap mathrm{Ker} Q_{alpha}$$
where the screening operators are obtained as integrals of vertex operators for every simple root $alpha$ of $mathcal{g}$ (scaled by some number $k_{alpha}$,
$$Q_{alpha}= int exp left( k_{alpha}alpharight).$$

In particular these screening operators map highest weight Fock space states to singular vectors of $mathcal{W}_n$. I am interested in “twisted” generalization of this picture. For simplicity let $mathcal{g}=mathfrak{sl}_3$. Let $H$ be a rank 2 Heisenberg algebra. Then we can construct a $mathbb{Z}_2$-twisted Fock module, $M$ (See for example Doyon for the definition of twisted modules.) generated by integer and half-integer modes ${J_{n}}_{n in mathbb{Z}/2}$ instead of just integer modes.

Question:
Is there a generalization of the above picture for twisted modules?

  1. Can I construct screening operators from twisted vertex operators and in what sense will these yield singular vectors?
  2. Does the intersection of these twisted screenings produce a free-field realization of $mathcal{W}_3$ inside the Heisenberg algebra?

mesh gets twisted wildly when trying to use Unreal’s mannequin’s skeleton

I have modelled a simple mannequine and made a skeleton for it in Blender. As far as I can judge, this skeleton copies the Unreal’s standard mannequine’s skeleton perfectly…

All the hierarchy and bone names are the same, and Unreal also does not complain when I import this mesh and use Unreal’s skeleton asset for it.

However, when I try to play a preview animation on my mesh, it gets twisted wildly.

This is a normal state:

enter image description here

… and this happens when I play an animation on this asset:

enter image description here

I thought it was due to different joint initial transforms, so I tried exporting from Blender with varying bone axes (x-axis along the bone, z-axis along the bone etc.) but it did not help. There is neither imporevement nor mere difference when I change that. Can you tell me possible reasons?

Please find the .fbx here:

https://drive.google.com/file/d/1v1X8t7l-6lwQragO8E0mtC1chdT2p1m4/view?usp=sharing

gr.group theory – Some sort of twisted group homomorphism

Start with a group $G$ that acts on a set $X$, and a second group $H$. We want to consider functions $varphi: G times X to H$ such that $varphi(g g’, x) = varphi(g, g’x) varphi(g’, x)$ for all $g$, $g’$, and $x$. Note that if $X$ is singleton (or if $G$ acts trivially), then $varphi$ is essentially just an ordinary group homomorphism. Is there a name for a function like this?