User muon - MathOverflow

Stability of $T_{\mathbb{P}^2}$ and $\Omega_{\mathbb{P}^2}$?

2013-03-21T19:48:06Z

How can one prove that the tangent bundle $T_{\mathbb{P}^2}$ and its dual $\Omega_{\mathbb{P}^2}$ are stable vector bundles with respect to $\mathcal{O}_{\mathbb{P}^2}(1)$ ? Similarly, is it true that $T_{\mathbb{P}^n}$ and its dual $\Omega_{\mathbb{P}^n}$ are stable vector bundle for any $n \in \mathbb{N}$? By stable I mean polynomial stability.

Is $\pi_1(\widetilde{X/G})$ always finite if $\pi_1(X)$ is finite?

2013-04-21T07:46:38Z

Let $X$ be a smooth complex manifold with finite fundamental group. Suppose that a finite group $G$ acts on $X$ and let $\widetilde{X/G}$ be a resolution of singularities. Is $\pi_1(\widetilde{X/G})$ always finite?

I think this is true but don't know a way to prove or a reference.

Are rational sections of a vector bundle useful?

2013-01-17T06:28:39Z

Let $X$ be a complex manifold or variety and $L$ a line bundle on it. Given a rational section $s$ of $L$, we get a divisor $D=Div(s)$ and may recover $L$ as $\mathcal{O}(D)$. What about vector bundles? I think it makes sense to speak about rational section $t$ of a vector bundle $E$, as a section is locally a collection of functions. My question is, is this $t$ useful? For example, can we say anything about Chern classes of $E$ from $t$?

What do correlation functions compute in CFT?

2012-12-27T10:46:16Z

I would like to understand what correlation functions compute in Conformal Field Theory in mathematics. Let me begin with basic definitions. We define a free boson field $\phi(z)$ as a formal power series $$ \phi(z)=q+a_0\log(z)-\sum_{n\ne0}\frac{a_n}{n}z^{-n}, $$ where $q,a_n$ for $n\in \mathbb{Z}$ are operators satisfying relations $$ [a_n,q]=\delta_{n,0}, \ \ \ [a_m,a_n]=m\delta_{m+n,0}. $$ Here these operators acts on the $space$ generated by $vacuum$ vector $|0\rangle$ with properties $$ a_n |0\rangle=0\ (n\ge0), \ \ \langle0|q=\langle0|a_n=0\ (n<0), $$ where $\langle0|$ is the dual of the vacuum, i.e. $\langle0|0\rangle=1$. We then define the current $J(z)$ of $\phi(z)$ as a formal derivative $$ J(z)=\sum_{n\in \mathbb{Z}}a_nz^{-n-1}. $$

Here is my question. In CFT one is interested in $correlation$ $functions$ such as $$ \langle0|J(z_1)J(z_2)|0\rangle, \ \ \langle0|J(z_1)J(z_2)J(z_3)J(z_4)|0\rangle $$ What do they compute in this context?

In physics, correlation function like $\langle0|\phi(x)\phi(y)|0\rangle$ computes the possibility of the field $\phi(x)$ to become $\phi(y)$ etc where $x,y$ are coordinate in for example Minkowski space. How should one interprete the correlation function in our case?

Motivation of Virasoro algebra

2012-12-26T23:37:33Z

I have a question on definition/motivation of Virasoro algebra. Recall that Virasoro algebra is an infinite Lie algebra generated by elements $L_n$ $(n\in \mathbb{Z})$ and $c$ over $\mathbb{C}$ with relations $$ [L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}. $$ A typical explanation of this definition is the following.

Define vector fields $l_n=-z^n\frac{\partial}{\partial z}$ on $\mathbb{C}\setminus {0}$. They form a Lie algebra of infinitesimal conformal transformation $$ [l_m,l_n]=(m-n)l_{m+n}. $$ So the Virasoro algebra is a central extension of this algebra by $c$. $c$ is called the central charge.

My questions are

How can one see that the Lie algebra above is associate to infinitesimal conformal transformation?
What is the central charge $c$ intuitively? Why are we interested in such a central extension?

As to second question, I don't have enough physics background to check what the central charge $c$ means in physics literature.

At this point, I don't have any intuition and have trouble in digesting the concept. I would really appreciate your help.

Global dimensions of non-commutative rings

2012-08-01T02:33:00Z

This is related to my previous question: When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau? I now would like to know the global dimension of the ring $R=\mathbb{C}\langle x_1,\dots,x_n\rangle/I$, where I is the two-sided ideal generated by $x_ix_j=a_{ij}x_jx_i$ for some $a_{ij}=a_{ji}^{-1}\in \mathbb{C}$ for $1≤i,j≤n$. Recall that the global dimension $gl\dim(S)$ of a ring $S$ is defined to be the supremum of the set of projective dimensions of all left $S$-modules.

In case your ring $R$ is defined by a quiver with relations, there seem some techniques. Are there another approach to compute $gl\dim(R)$? Moreover, are there any standard way to compute $gl\dim(S)$ when $S$ is non-commutative?

There are many characterization of $gl\dim(R)$ when $R$ is commutative and I am aware of similar questions, such as Commutative Ring of Finite Global Dimension.

Boundedness of modules on AS regular algebras

2012-11-16T07:10:47Z

Let $k$ be an algebraically closed field and $A$ be an Artin-Shelter regular $k$-algebra. Fix a numerical polynomial $H(t)$. I would like to know whether or not semi-stable f.g. graded $A$-modules with Hilbert polynomial $H(t)$ is bounded. More precisely we regard g.f. graded modules $M$ and $N$ as isomorphic is they are isomorphic up to torsion modules. Semi-stability and boundedness are defined in the same manner as commutative case.

The boundedness may not be hold for general AS regular algebras, but then I would like to know for which class of algebra this kind of boundedness result hold. I would appreciate any reference, comments and suggestion.

Strongly Noetherian propoerty. When is the tensor $A\otimes_{k}B$ Noetherian for Noetherian rings $A$ and $B$?

2012-09-26T17:10:04Z

Let $k$ be a field. It is well-known that $A\otimes_{k}B$ is not necessarily Noetherian even if $k$-algebras $A$ and $B$ are Noetherian. For example $\mathbb{R}\otimes_{\mathbb{Q}}\mathbb{R}$.

When is the tensor $A\otimes_{k}B$ Noetherian for Noetherian "commutative" $k$-algebras $A$ and $B$?
What if $A$ is noncommutative? Are there any good criteria for $A\otimes_{k}B$ to be Noetherian? If $B$ is a finitely generated $k$-algebra, Hilbert's basis theorem implies that $A\otimes_{k}B$ is again Noetherian. So we need to check this with quite nasty $B$.

My primary motivation to ask these questions is the second question. Such ring $A$ is called a "strongly Noethrian ring" and has a lot of good properties, but I don't know many examples. Moreover I realized that things are not very clear even in commutative case and I need to understand commutative case first. I would appreciate it if experts on MO could let me know good criteria for this property and provide me with examples.

Rings I have in my mind are weakly noncommutative in the sense that they are commutative up to scalar multiplication such as quantum planes and their $good$ hypersurfaces.

Does Castelnuovo-Mumford regularity hold for this $\mathbb{C}$-algebra$?

2012-11-05T10:48:43Z

Let $R$ be a noncommutative finitely generated $\mathbb{C}$-algebra such that its center $S$ is smooth (in commutative sense) and $R$ is finite over $S$. Is there Castelnuovo-Mumford regularity theorem for this kind of $\mathbb{C}$-algebra $R$? I have $$ R=\mathbb{C}\langle x,y,z \rangle/(xy=\zeta_1yx,yz=\zeta_2zy,zx=\zeta_3xz,x^3+y^3+z^3), $$ where $\zeta_i$ is a 3rd root of unity. I am aware of CM regularity theorem for AS regular algebra but not sure if it works for this example. Thank you in advance.

Balanced dualizing complex vs rigid dualizing complex?

2012-11-05T10:27:57Z

In noncommutative projective geometry, there is a counterpart of dualizing complex in commutative world. It seems to me that they are called either a balanced dualizing complex or rigid dualizing complex. I am aware that they are different objects but cannot really understand how they are different. I would appreciate it if someone kindly explain to me the difference.

P.S. I am reading this paper by M. van den Bergh

Diophantine theory of homogeneous cubic polynomials

2012-07-31T04:33:51Z

Arithmetic of quadratic forms over $\mathbb{Z}$ (or lattices theory) has received much attention and there are many applications in broad area of mathematics (such as intersection forms on fourfolds). I now wonder whether or not a similar theory for cubic forms can be developed. I recently found a beautiful theorem about binary cubic forms:

Theorem (B. N. Delone and D. K. Faddeev,W.-T. Gan, B. H. Gross, and G. Savin) There is a canonical bijection between isomorphism classes of cubic rings and the set of $GL_{2}(\mathbb{Z})$-equivalence classes of integral binary cubic forms. Under this bijection, the discriminant of a cubic ring is equal to the discriminant of the corresponding binary cubic form.

I don't know how useful this theorem is because I don't know how difficult to classify cubic rings. My question is, are there any classification theory when the number of variable is small? I would appreciate it if anyone could give me a reference for recent development of cubic forms.

Point modules of quantum projective space $\mathbb{P}^n$

2012-10-11T01:42:57Z

Let $A$ be a quantum $\mathbb{P}^n$ defined by $$ A=\mathbb{C}\langle x_1,x_2,\dots,x_{n+1}\rangle/(x_ix_j-r_{ij}x_jx_i)_{1\le i < j\le n+1}. $$ I would like to know the set $X$ of isomorphism classes of point modules for $A$. Here a point module is a cyclic graded right $A$-module $M$ such that each graded piece of $M$ is one-dimensional.

The set $Y$ of isomorphism classes of of point modules for the quantum $\mathbb{P}^2$ $$ \mathbb{C}\langle x_1,x_2,x_3\rangle/(x_ix_j-r_{ij}x_jx_i)_{1\le i < j\le 3} $$ is known to be $\mathbb{P}^2$ or a union of three lines in $\mathbb{P}^2$.

It seems known that $X$ is projective for $n=4$, but does anyone know explicit description of $X$? I tried to compute $X$ in a similar manner as the quantum $\mathbb{P}^2$ case, but it seems quite complicated. I would also appreciate it if someone give me a good description for higher dimensional $n$ case (especially $n=3,4$).

Thank you very much.

Complex manifold $X$ with $\dim H^0(X,\Omega^{\dim_{\mathbb{C}} X})>2$

2012-10-11T08:30:50Z

Are there any complex surface or threefold $X$ with $$ \dim H^0(X,\Omega^{\dim_{\mathbb{C}} X})>2? $$ I am asking this because I don't know any example while there are complex curves of genus greater than one. I guess that there are no such example. If so, could someone kindly explain why? Any counter example is also welcome.

Edit My question turns out to be a silly question. Please ignore this.

How to understand $Ext(\mathcal{O}_{Y},\mathcal{O}_{Z})$ for subvarieties $Y,Z\subset X$?

2012-09-30T09:26:51Z

By standard homological algebra we know that $Ext(A,B)$ of $R$-modules classifies certain equivalence classes of short exact sequences $0\rightarrow B\rightarrow C \rightarrow A \rightarrow 0$ of $R$-modules, where $R$ is a commutative ring. I now would like to understand this fact in geometry.

Let $X$ be a variety (or a scheme if you want), how should I understand $Ext(O_{Y},O_{Z})$ for subvarieties $Y,Z\subset X$? Of course it classifies extensions of $O_{Z}$ by $O_{Y}$, but are there any geometric or intuitive way to understand $Ext(O_{Y},O_{Z})$?
More generally are there any geometric way to understand $Ext(\mathcal{E},\mathcal{F})$ for coherent $O_X$-modules $\mathcal{E},\mathcal{F}$?

I would appreciate any idea about "seeing" these extensions.

Moduli space of modules over non-commutative rings

2012-09-13T23:39:11Z

Let $X=Proj(A)$ be a projective scheme, one can the moduli space of coherent sheaves on $X$ with fixed Hilbert polynomial and stability. Since coherent sheaves on $X$ are all obtained as the sheafification $\widetilde{M}$ of a graded $A$-module $M$, it is reasonable to ask whether we can construct the moduli space of sheaves on non-commutative space, i.e. the moduli space of graded right $B$-modules for some $good$ non-commutative graded ring $B$ with some fixed data. I am pretty sure that some works in this direction have been done (what condition on $B$ and what data of modules should be fixed, etc).

Could anyone tell me a reference or paper which discuss the construction? Thank you very much.

Edit I have for example a quantum plane in my mind.

Deformation of modules over noncommutaitve rings

2012-09-10T09:33:35Z

Let $M$ be a finitely generated module over a commutative ring $R$. The first order deformation of module $M$ is parametrized by $Ext^{1}(M,M)$ and the obstruction is parametrized by $Ext^{2}(M,M)$. Is there a similar story for noncommutative $R$? I don't expect this to be true for any noncommutative rings but wonder if this is still true for some $good$ ones. I would appreciate any reference suggestion, comments, and ideas.

Edit I would like naively to compute the tangent space of the moduli space $X$ of module with some data if it exists. At $M\in X$, the tangent space can be understood as the set of extensions of the $R$-module $M$ to some $R\otimes_{k} k[\epsilon]/(\epsilon^2)$-module. The obstruction is defined in the same manner.

Online Number Theory Video?

2012-02-22T21:30:05Z

Are there any graduate level number theory course available on line ? The only video series I am aware of are some MSRI videos, and Ted Chinburg's courses http://www.math.upenn.edu/~ted/noframes.html, which are hard to see sometimes. I am also looking for good video lectures on algebraic k-theory. Thank you for your sharing information :)

Comment by Muon

2013-04-22T04:42:06Z

Thank you for the nice answer, Francesco. I was a bit surprised to know that my question was not as easy as I had initially expected.

Comment by Muon

2013-03-21T23:20:43Z

Thank you for the answer, but could you explain a bit more? I don't know why it is enough to check $H^0(T_{\mathbb{P}^2}(-2))=0$. I am not familiar with this field.

Comment by Muon

2013-02-27T01:25:05Z

Could you explain what you mean by Ward-Takahashi identity in your question, please?

Comment by Muon

2013-02-11T00:40:12Z

Doesn't your example have fixed locus $f=0$?

Comment by Muon

2013-01-19T10:13:06Z

Thank ou for the answer.

Comment by Muon

2012-12-28T17:50:11Z

@Kelly What is "moment" of the partition function?

Comment by Muon

2012-12-28T07:29:31Z

Thank you for the detailed answer providing physic background. I don't fully understand your answer, but now get a feeling about what CFT try to understand.

Comment by Muon

2012-12-28T02:54:03Z

Thank you for the answer, Igor. I am now happy to know that holomorphicity may be interpreted as a field equation. May I ask what the fields $\phi(z)$ and $J(z)$ stands for in this context? Is $\langle0| J(z)J(w)|0\rangle$ the amplitude to observe a particle at $z$ after a particle is created at $w$ or something?

Comment by Muon

2012-12-28T02:48:17Z

@Jeff I would rather want to know the physical interpretation. It would be nice if you could provide an answer. By the way, I don't know if there is any "purely mathematical interpretation" of correlation function.

Comment by Muon

2012-12-28T02:45:40Z

@user1540 My question is a bit vague. I would like to know both the complex coordinates and the physical meaning of $J(z)$, and they must be related. My problem is that I don't have any intuition behind what I am computing.

Comment by Muon

2012-12-27T22:53:46Z

Thanks you for letting us know the projective representation point of view. That makes sense.

Comment by Muon

2012-12-27T06:30:24Z

I don't quite understand what you try to mean in the second comment. I would appreciate it if you could post a bit more detail as an answer.

Comment by Muon

2012-12-27T06:29:05Z

Thanks for your writing "why $c=D$" above.

Comment by Muon

2012-12-27T00:48:59Z

Thank you for the answer, Chris. May I ask why multiplication by the unit vector is related to the space dimension?

Comment by Muon

2012-12-27T00:42:42Z

@Abdelmalek Thanks for pointing out the typo.