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1. ### I Tangent Bundle and Level set

I believe problem is solved but I would like to receive some comments from the experts if I miss something. Set ##T## is given embedding in ##\mathbb{R}^4## and projection ##\pi## is very simple ##\pi(x_1,x_2,X_1,X_2)=(x_1,x_2)## For obvious reasons ##\pi(T)=S^1## Now I want to build the...
2. ### I Tangent Bundle and Level set

Yes. Using "physical intuition" everything is pretty clear but I would like to elaborate this simple example in a rigorous way. After even more thinking I come to the conclusion that this statement is not true. Providing such maps (##\pi## and ##\chi_\alpha##)implies that set ##T## is vector...
3. ### I Tangent Bundle and Level set

Book is Rudolph and Schmidt "Differential Geometry and Mathematical Physics"
4. ### I Tangent Bundle and Level set

First of all I would like to thank for all the answers. I think about this problem for a while again. On manifold there is homeomorphic map ##\kappa## from open subset ##U## to ##\mathbb{R}^n## and this is something touchable (don't know how to put it correctly. But it is quite clear how to...
5. ### I Tangent Bundle and Level set

Hello there! Reading the textbook on differential geometry I didn't get the commentary. In Chapter about vector bundles authors provide the following example Let ##M=S^1## be realized as the unit circle in ##\mathbb{R}^2##. For every ##x\in S^1##, the tangent space ##T_x S^1## can be identified...
6. ### B Inductive proof for multiplicative property of sdet

To add some detail of my struggle. I represent my super matrices according to suggested in the proof way ##\mathcal{M}=\left(\begin{matrix}M_{00}+\beta_L A_{00} & M_{01}+\beta_L A_{01} \\ M_{10}+\beta_L A_{10} & M_{11}+\beta_L A_{11}\end{matrix}\right)##...
7. ### B Inductive proof for multiplicative property of sdet

Hello! Reading Roger's book on supermanifolds one can find sketch of the proof for multiplicative property of super determinant. Which looks as follows All the words sounds reasonable however when it comes to the direct computation it turns out to be technical mess and I am about to give up. I...
8. ### B Definition of Super Lie Module

One more time thanks for your help!
9. ### B Definition of Super Lie Module

I do get the same results as you actually. But moving from line (8) to (9) you have used supercommutativity ##AB=(-1)^{|A|\, |B|}BA## but according to the textbook's definition algebra ##\mathbb{A}## is not necessarily super commutative it is just any super algebra. Perhaps author forgot to add...
10. ### B Definition of Super Lie Module

Hello! I have some troubles with the definition of the so called super Lie module. In Alice Rogers' textbook "Supermanifolds theory and applications" definition goes as follows Suppose that ##\mathbb{A}## is a super algebra and that #\mathfrak{u}# is a super Lie algebra which is also a super...
11. ### I Derivations and Derivatives

I also believe that all this machinery is because of this very specific definition that derivation works only on function that are defined globally on the whole manifold. Book is "Differential Geometry and Mathematical Physics" by Rudolph and Schmidt (very hard to read (for me) but rigorous).
12. ### I Derivations and Derivatives

I do understand what you are talking about. But I believe the reason is simply formal. If ##f## is of class ##C^k(M)##. Then l.h.s. of (1.4.19) is ##C^k## and I can apply derivation that maps ##C^k## functions to real numbers. However the remainder in Teylor's formula is of class ##C^{k-2}##...
13. ### I Derivations and Derivatives

Do you mean that one can prove this isomorphism ##T_m M=D_m M## without assuming that manifold is of ##C^\infty##? Indeed. Sorry for that here is the full proposition with the proof Proof of this proposition also uses this ##h##-thing. I used to think that it was made just to make the proof...
14. ### I Derivations and Derivatives

Hello! According to the attached proposition on ##C^\infty## manifold space of derivations ##D_m M## is isomorphic to Tangent space ##T_m M##. Cited here another proposition (1.4.5) states the following 1. For constant function ##D_m(f)=0## 2. If ##f\vert_U=g\vert_U## for some neighborhood...
15. ### B Exercise on quotient topology

I think I found rigorous way to solve second excercise. Homeomorphism should map closed sets to closed sets. Every single point in ##\mathbb{R}## is closed while ##\pi((a,b))## is open in ##\mathbb{R}/\sim## so there is no homeomorphism.
16. ### B Exercise on quotient topology

I believe I do not understand your exercise. I need to provide open set ##U_a\subset\mathbb{R}/\sim## which contains equivalence class of ##a##. I can take ##U_a=\pi((a-\varepsilon,a+\varepsilon))##, ##\pi^{-1}(U_a)=(a-\varepsilon,b)##. Then ##U_a## is open due to properties of canonical...
17. ### B Exercise on quotient topology

Hello! I have two related exercises I need help with 1. Partition the space ##\mathbb{R}## into the interval ##[a,b]##, and singletons disjoint from this interval. The associated equivalence ##\sim## is defined by ##x\sim y## if and only if either##x=y## or ##x,y\in[a,b]##. Then...
18. ### B Open sets in quotient topology

Many thanks! Can you provide some easy readable reference on this? Currently it looks like a pure magic of definitions
19. ### B Open sets in quotient topology

Can I put it this way? If the inverse image of the set ##U## of the canonical projection to the quotient is open then ##U## is open.
20. ### B Open sets in quotient topology

I am slightly confused. What kind of ##U##s are considered here? If open, then I completely fine with this since canonical projection is continuous by definition. Nonetheless, I believe I've asked about another thing. I want to build a manifold from this quotient therefore I need open...
21. ### B Open sets in quotient topology

Hello! Reading a textbook I found that authors use the same trick to show that subsets of quotient topology are open. And I don't understand why this trick is valid. Below I provide there example for manifold (Mobius strip) where this trick was used Quote from "Differential Geometry and...
22. ### A Hausdorff property of projective space

Authors did not provide any specific definition for ##\mathbb{K}_1## so I think ##\mathbb{K}_1=\mathbb{K}_1^1##. I believe it should be ##\min## in the definition of ##K_{\mathbf{x}}##.
23. ### A Hausdorff property of projective space

I do take ##\mathbb{R}^2##. Suppose $$\mathbf{x}=\left(\begin{matrix} 1 \\ 0 \end{matrix}\right),\;\; \mathbf{y}=\left(\begin{matrix} 0 \\ 1 \end{matrix} \right)$$ then this ##l## function is equal to ##\sqrt{2}##. On the other hand ##\max## in the definition of ##K_{\mathbf{x}}## takes values...

37. ### Grassmann integration

I know that ##\theta## anticommute, but how one deduce that ##d\theta## obey the same rule
38. ### Grassmann integration

Hi, everyone! I am trying to understand notation of this textbook http://arxiv.org/abs/hep-th/0108200 page 8, formulas 2.1.4 and 2.1.5 $$\int d \theta_\alpha \theta^\beta=\delta_\alpha^\beta$$ this could be found in any textbook the weird that from the above formula follows \int d^2...
39. ### Why is Lorentz Group in 3D SL(2, R)?

Equal sign means that they are isomorphic. Equal algebras doesn't mean that group are isomorphic. O(3) and SO(3) have the same Lie algebra, but they are not isomorphic. Exponential map from algebra to group gives only simply connencted part. One can not build smooth curve from matrices with...
40. ### Birth of strings

In string theory strings are fundamental, just electron or photon in SM
41. ### Why is Lorentz Group in 3D SL(2, R)?

Lorentz group in three dimensions is SO(2,1) and it is NOT isomorphic to SL(2,R). SL(2,R) - is spin group in three dimensions with the signature mentioned above. SO(2,1)=SL(2,R)/Z_2 Majorana spinors are real because they are Majorana))) Definition of Majorana spinor
42. ### String theory-connecting strings

In string theory Lorentz invariance preserved only in 26 dimension.
43. ### Transformation of auxiliary field

By construction :-) You impose several conditions on your action and then get how auxilary fields should transform. Just like with YM gauge field, you want gauge invarianceand from this you get transformation law
44. ### Infinitesimal SUSY transformation of SYM lagrangian

Lagrangian should be invariant off-shell, you are right. I am not an expert in this. But probably you need to introduce new field with purely algebraic equations of motion, like F in WZ model, just to cancel this term.
45. ### Infinitesimal SUSY transformation of SYM lagrangian

Does the extra term vanish on the equations of motion?
46. ### S-matrix in String Theory

I think I got the idea. Due to locality of Vertex Operators I can generate not just a state of a certain momenta but can actually put this state at a certain place. My problem was with understanding of constracting asymptotically free states which are needed in QFT as initial and final. In...
47. ### S-matrix in String Theory

This point is not clear. Vertex operators generate single particle/string states without interaction, so this states are obviously free. And these states are our string |in> and |out> states. How to proceed to S-matrix from this point? If I understand you right. If we look from state...
48. ### S-matrix in String Theory

Hi there! S-matrix is Path Integral with Vertex Operators inserted. I know how to compute Shapiro-Virasoro amplitude. So I don't have problems with calculations but with understanding. In this calculations formalism of 2-dimensional CFT is used. But there is no S-matrix in CFT, only...