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).

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.

ct.category theory – Existence of twisted metaplectic categories

The paper Classification of metaplectic modular categories by Ardonne-Cheng-Rowell-Wang (2016) mentions (in Section 3) the Grothendieck ring for the metaplectic modular categories, i.e. $SO(N)_2$, $N>1$ odd (see below).

Now by exchanging $1$ and $Z$ in lines (2) and (3), we get a new family of fusion rings. 

Question 1: Are there (unitary) fusion categories corresponding to this new family?
(which could be called twisted metaplectic)

Question 2: Can this procedure be extended to other fusion categories?


enter image description here

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?

graphics and networks – Discretizing a line (twisted) and showing how the points were determined

Suppose I have a (twisted) line, for example

2-x==y && x< 1
3-2x == y && 1.5>=x>= 1

I want to take this crooked line and represent it discretize it (that is, I want to show points along this line)

I can do this with discretize region, for example

r=ImplicitRegion(2-x==y && x<1 , {x,y});
r2=ImplicitRegion(3-2x == y && 1.5>=x>= 1,{x,y});
DiscretizeRegion(RegionUnion(r,r2))

This will give me a graph of the line and some discrete points (I would rather not include the line here, but that is not too important)

  • How can I show how these points are determined?
  • How can I control how these points are determined?

I suppose there is a mesh function or something similar, and if so, I could control how the points are determined by specifying the mesh, but I could not solve this.

Alternatively, perhaps it is better to use something like MeshRegion?


An explanation of what I want to do:

I want to take a function (a twisted line in this case, but the function doesn't really matter).

So, I want to build a grid consisting of points generated by the intersection of vertical and horizontal grid lines, spaced at size intervals $ d $

Next, I want to find the intersection of these points and the function, and show only these points, as well as the underlying grid that generated them

  • (I'm fine if I don't visualize the grid if it gets in the way of the visualization, as long as I can generate a separate chart that shows the grid that was used. Basically, I want to be able to visualize how the points are selected)

I realize that I can write a custom code to do this. If this is the route you want to suggest, I'd rather try to write the code myself to improve in math, unless I have a particularly elegant (or short) solution to suggest.

I also realize that I could do the above with a mesh, where the mesh cells are points. If this is possible, I would like to see how to do it, since I don't understand well enough how the mesh works (and the mesh cells, etc.).

nt. number theory – Functional equation of the L function of the twisted triple product

Leave $ mathbb {E} = E_1 times E_2 times E_3 $ denote the product of three elliptical curves on $ mathbb {Q} $ first level $ p $ and consider the $ p $-adical representation of Galois $$ V_p ( mathbb {E}) = H ^ 1_ {et} (E_ {1 / bar { mathbb {Q}}}, mathbb {Q} _p) otimes H ^ 1_ {et} (E_ {2 / bar { mathbb {Q}}}, mathbb {Q} _p) otimes H ^ 1_ {et} (E_ {3 / bar { mathbb {Q}}}, mathbb {Q} _p). $$ We denote by $ L ( mathbb {E}, s) = L (V_p ( mathbb {E}), s) $ the associated triple product $ L $-function. It has a functional equation centered on $ s = $ 2 with global sign equal to $ a_p (E_1) a_p (E_2) a_p (E_2) in { pm 1 } $ (cf. Gross-Kudla & # 39; 92). Here, $ a_p (E_i) $ denotes the $ p $-th Fourier coefficient of weight 2 normalized new level form $ Gamma_0 (p) $ associated to $ E_i $ by modularity

Leave $ chi $ be a Dirichlet character module $ p $ and denote by $ L ( mathbb {E} otimes chi, s) $ the $ L $-function attached to the Galois representation $ V_p ( mathbb {E}) otimes chi $. My question is: what is the functional equation of $ L ( mathbb {E} otimes chi, s) $ And what is your global sign?

Thanks in advance for any assistance.

dg. differential geometry – Definition of deformed and twisted geometries

In Yong Wang's Multiply Twisted Products article, general definitions for so-called deformed Y twisted Products are delivered. The notion of deformed The products seem to be a fairly standard definition, first presented by O & # 39; Neill. I wonder if the definition of twisted products (and the generalizations of distorted and twisted products) in the previous document is also standard. Many theoretical physics articles sometimes talk about deformed Y twisted Space times in a fairly manual way, so I am never really sure if they are talking about a mathematical definition or if they simply want to sound elegant.

For example, Compère speaks in a general document about the Kerr / CFT correspondence about the following metric

begin {align}
mathrm {d} s ^ 2 = J left (1+ cos ^ 2 theta right) left (-r ^ 2dt ^ 2 + frac {dr ^ 2} {r ^ 2} + d theta ^ 2 right) + frac {4 sin ^ 2 theta} {1+ cos ^ 2 theta} left (d phi + rdt right) ^ 2 ,,
end {align}

be a "deformed and twisted product of $ AdS_2 times S ^ 2 $ "(J is only scaling the metric.) This seems to be a different definition of a twisted product, since in the role of twisted Wang it is a generalization of deformation. Also, I cannot understand that the previous metric is $ AdS_2 times S ^ 2 $, not even for $ theta = frac { pi} {2} $ a "twisted product of $ AdS_2 $ and a circle of constant radius ", since the suggested geometry should not have terms outside the diagonal such as $ mathrm {d} phi mathrm {d} t $ in the metric Are I missing coordinate transformations that make it obvious? What is a standard definition for a twisted product of Pseudo-Riemannian varieties?