User anirbit - MathOverflow

Green's function of coupled ODEs

2013-04-17T23:00:56Z

For functions $a(x)$ and $b(x)$ and "sources" $S_1(f,g)$, $S_2(f,g)$ and $S_3(f,g)$ lets say one has the differential equations for functions $f(x)$ and $g(x)$,

$f' + af + bg = S_1(f,g) + S_2(f,g)$

$g' + f = S_3(f,g)$

Now if $S_1= S_2 = S_3 = 0$ i.e all the sources are turned off then the system reduces to the differential equations, $g'' + ag' - bg = 0$ (by eliminating $f$ as $f = - g'$). For this 2nd order ODE for g let $G_g$ be the Green's function.

And let $f_{13}(x)$ and $g_{13}(x)$ be the solutions to the system when $S_1$ and $S_3$ are on but $S_2 = 0$.

(..if $f_0(x)$ and $g_0(x)$ are the solutions with all sources turned off then $f_{13}$ and $g_{13}$ can be thought to be given as a power-series in $f_0$ and $g_0$ respectively..)

Similarly when sources are all turned off, one can eliminate $g$ between the two equations to get a 2nd order ODE for f as, $f'' + (a b'/b)f'+(a'-(ab')/b-b)f=0$ and then let $G_f$ be the Green's function for this.

Given so much of data $G_g$, $G_f$, $f_{13}$ and $g_{13}$ can one build the full solutuions when all the 3 sources are on?
Will it be an approximation (how good?) if one says that the full $g$ is given by,

$g (x) = g_{13}(x) + \int _{0}^x G_g(x,x')S_2 (f_{13},g_{13}) dx'$

and similarly can one write (approximate?) that,

$f(x) = f_{13}(x) + \int _0 ^x G_f(x,x')S_2(f_{13},g_{13})dx'$

(..I guess that in the above the "=" should be replaced by "~" and one can pertrubatively develop a solution by taking the $g$ or $f$ obtained in the first pass and then using that again as the new $f_{13}$ and $g_{13}$ respectively..though such an iteration is a bit hard to make sense of given that $f_{13}$ and $g_{13}$ themselves are given as an infinite power-series - so it seems like for any truncation of that power-series one would have to still do infinite updates here!..)

Poles of products of Gamma functions

2013-04-10T20:50:46Z

I want to know if there can be a general statement about the poles (Laurent expansion) of such products of Gamma functions as a function of $p \in \mathbb{R}$ in the limit $\epsilon \rightarrow 0$,

$\Gamma[ \frac{(-1-2p)}{2} - \epsilon ]\Gamma[\frac{-1+p}{2} + \epsilon]\Gamma[\frac{(5+p)}{2} + \epsilon]$

For any given $p$ such that the $p$ dependent part of the argument is a negative integer one can do the usual Laurent expansion of the Gamma function in $\epsilon$ for each of the 3 factors and then multiply. But what can be said about the Laurent expansion in general as a function of $p$?

I wish one could write down the Laurent expansion in $\epsilon$ as a function of $p$!

One sees that there are these special cases like if $p$ is such that there are two integers $N$ and $M$ satisfying, $p= 1-2N^2 = -5 -2M^2$ then the later two Gamma function can have poles simultaneously. (like $N=2, M =1, p= -7$) Existence of such special $p$ naively seems to make things more tricky.

CFTs corresponding to affine Lie algebras

2013-03-26T22:22:51Z

I want to know how one can write down a CFT such that its conserved currents will satisfy some chosen (affine) Lie algebra $G$.

On the few pages leading up to page 192 in here one can see see the analysis of the CFT obtained in the compactified directions of closed bosonic strings.

From what one sees in these notes it seems that a CFT with the above properties will exist if it is put on a torus $\mathbb{R}^n/\Lambda$ where $n=rank (G)$ and $\Lambda =$root lattice of $G$.
Is the above correct? If it is correct then how does one write down the corresponding Lagrangian and the currents?
But in the string context where these attached notes are based one is forced to have simply laced $G$ and hence only the $A,D,E$ series. How would one do this for say the group $G_2$? (..being concerned with just a CFT and not connected to string theory..)
The restriction of being on the $A$, $D$, $E$ series is related to the fact that in string context one has to tune all the compactification radius to the same self-dual point. If $i$ indexes the compact directions , $1\leq i \leq n$ then the current operators are possibly like $:\partial _{z} X_i (z):$ and $:e^{i\vec{\alpha}.\vec{X}}:$ where the first set is one for each Cartan and the send set is one for each root $\alpha$. But I wonder where would the radius of the circles go in these currents.
- Finally where in this process can one tune the level of the affine Lie algebra? What choice fixes that?

cumulants and correlations

2013-03-14T17:01:46Z

I think the problem with this question is that it is assuming cumulant to be additive for any two variables. Which unlike for expectation values is not.

I guess that its true that for any 2 random variables $X$ and $Y$ whether or not independent it follows that $\langle X + Y \rangle = \langle X\rangle + \langle Y \rangle$. (..in what follows I am often binomially expanding $(X + a(X^2 - \langle X^2 \rangle))^n$ inside the $\langle \rangle$ and using that $\langle X^p + X^q \rangle = \langle X^p \rangle + \langle X^q \rangle$

Then I guess that the following conjecture is true that,

If $X$ is a Gaussian random variable then for $n>3$ there are no linear terms in "a" in $\langle (X + a(X^2 - \langle X^2 \rangle))^n\rangle_c$

One way I can think of proving this is to use the following theorem (hopefully correct!) that for any random variable $Y$ the following is true,

$\langle Y^n \rangle_c = \sum _{\pi} (\vert \pi \vert - 1)!(-1)^{\vert \pi \vert -1}\prod _{B \in \pi} \langle Y^{\vert B\vert}\rangle$

where $\pi$ runs over all partiions of the set ${1,2,3..,n }$ and $B$ are the subsets in a partiton $\pi$ and $\vert \pi \vert $ is the numnber of parts in $\pi$ and $\vert B\vert$ is the size of the part $B$.

If the above formula is correct then I would like to know of the proof of it!

I guess this is the "inverse" of the statement that,

$\langle Y^n \rangle = \sum _ \pi \prod _{B \in \pi} \langle Y^{\vert B\vert}\rangle_c$

I can only check the conjecture by plugging in specific values of $n$ into it for $Y = X + a(X^2 - \langle X^2 \rangle)$ and then using the above expansions with the properties of a Gaussian variable that $\langle X^{odd}\rangle = 0$ and $\langle X^{2n}\rangle = \frac{(2n)!}{n!2^n}\langle X^2 \rangle ^n$. (..or by going into some integral representation of the expectation value (and making a mess!)...) I would like to know of the proof which is purely algebraic.

Also in any of these formulas the terms in the RHS depend only on $\vert B\vert$ (and not on what exactly is in the parts of a given partition) and hence one needs a combinatorial count of how many ways can a set of $n$ distinct objects be divided into $k$ parts of size $r_i$ (..some given set of $k$ integers such that $\sum_{i=1}^k r_i = n$..). Given such a set of $r_i$s if $a_i$ of them are of size $i$ then I think the number of such partitions is,

$\frac{n!}{(\prod_{j=1}^k r_j !)(\prod_{i=1}^n a_i!)}$

For various special cases of $n$ I have checked that the above count is correct but I would like to know if there is some elegant way to derive or understand this by connecting it to say Bell numbers or Faa di Bruno's formula etc. I would want to know the answer to this.

Normal Ordering with Vertex Operators in Conformal Field Theory

2010-06-03T06:59:30Z

The "definition" of the normal ordering in CFT looks a bit vague to me.

I found the definition in terms of exponentiated functional derivative pretty opaque.

Also in this context it might help if someone can give a reference or if there is a short explanation to understand how the Operator Product Expansion is derived using products of normal ordered operators.

I don't see the conceptual framework in which these ideas fit together. Some of the books I looked at gave a very disparate view as a collection of some complicated formulas.

Let me give a precise example of the kind of calculation that I am stuck with,

Refer to these lecture notes

I can understand equation 4.26 of this but not the next 4 equations that seem to follow from it leading to 4.28.

It would be helpful if someone can decrypt the calculation.

In light of the kinds of references that came in as responses, I think it would help if I make the problematic calculation a little more explicit.

This has to do with what are called "Vertex Operators" in CFT given as $:e^{ikX(z)}:$ where $::$ is the notation for normal ordering and $k$ is some scalar and $X$ is a conformally invariant free Bosonic field. Then I would like to understand the derivation of this equality,

(all expressions are understood to be valid under the Feynman Path Integral)

$:\partial X(z)\partial X(z)::e^{ikX(w)}: = -\frac{k^2\alpha ^2}{4}\frac{:e^{ikX(w)}:}{(z-w)^2}-ik\alpha\frac{:\partial X(z) e^{ikX(w)}:}{(z-w)}$

where we have $X(z)X(w) = -\frac{\alpha}{2}ln \vert z - w \vert$

and what would be the similar simplification of

$:e^{ikX(z)}::e^{ikX(w)}: = ?$

Some more elaboration on what about normal ordering I am concerned about.

The problem is that I can't these books give an honest definition of what it means to "normal order" operators in CFT. Like there is a very clean definition in rest of QFT whose relation to time-ordering is given by the Wick's Theorem. Here in CFT one is supposed to understand that while normal ordering a string of operators inserted at different points on the space-time one is subtracting away from the product every possible way in which one or more pairs of insertion points can coincide and produce a singularity

Like if A,B,C,D are 4 different Bosonic operators say inserted at 4 different space-time points. Then one would define normal ordering as,

$:ABCD: = ABCD - (AB):CD: - (AC):BD: - (AD):BC:-(BC):AD:-(BD):AC:$ $$-(CD):AB:-(AB)(CD)-(AC)(BD)-(AD)(BC)$$

where () denotes the correlation function of the operators.

Now the point is whether one is supposed to take the above kind of equations as being just well-motivated definition or is there is anything more fundamental from which it is derivable?

There is definitely an issue about defining difference of two divergent expressions here.

Can't find Segal's papers in CFT

2010-06-17T19:35:50Z

At various places I have seen people referring to Segal's papers in CFT as the "standard definition" of the subject. These seem to have become classics in this field. But I can't locate them on the net. I searched on arxiv,spires,Google/Scholar,AMS etc.

The papers I am referring to are (especially the fist one),

G. Segal, The definition of conformal field theory, in: Differential geometrical methods in theoretical physics (Como, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 250, Kluwer Acad. Publ., Dordrecht, 1988, 165-171
G. Segal, Two-dimensional conformal field theories and modular functors, in: Proceedings of the IXth International Congress on Mathematical Physics, Swansea, 1988, Hilger, Bristol, 1989, 22-37.
G. Segal, The definition of conformal field theory, preprint, 1988; also in: Topology, geometry and quantum field theory, ed. U. Tillmann, London Math. Soc. Lect. Note Ser., Vol. 308. Cambridge University Press, Cambridge, 2004, 421-577.

and on similar lines this paper by Moore and Seiberg

G. Moore and N. Seiberg, Lectures on RCFT, in: Physics, geometry, and topology (Banff, AB, 1989), ed. H.C. Lee, NATO Adv. Sci. Inst. Ser. B Phys., 238, Plenum, New York, 1990, 263-361.

One possibility is that these are only available in the proceedings mentioned and hence only as books in some library. Are there no online copies of these?

A question about a formal power series manipulation

2012-10-05T15:53:39Z

I want to find a function $f(x,y)$ which can satisfy the following equation,

$\prod _{n=1} ^{\infty} \frac{1+x^n}{(1-x^{n/2}y^{n/2})(1-x^{n/2}y^{-n/2})} = exp [ \sum _{n=1} ^\infty \frac{f(x^n,y^n)}{n(1-x^{2n})}]$

I would like to know how this is solved.

(..though I landed into this through a different route (calculating Witten Index!), such expressions also occur in finite dimensional representation theory where a generating function for the character of the (anti)symmetric powers of a representation (the LHS) is written as a Plethystic exponential (the RHS) of the original (generally fundamental) representation...)

One can perturbatively check that the following function satisfies the above equation,

$f(x,y) = \sqrt{x} (\sqrt{y} + 1/\sqrt{y}) + x (1 + y + 1/y) + x^{3/2} (y^{3/2} + 1/y^{3/2}) + x^2 (y^2 + 1/y^2)$

$\quad\quad\quad\quad\quad + \frac{(x y)^{5/2} }{(1 - \sqrt{x y})}(1 - 1/y^2) + \frac{(x/y)^{5/2}}{(1 - \sqrt{x/y})} (1 - y^2) $

The paper doesn't state any proof or explanation for how this was obtained but the above is order-by-order in $x$ checkable to be right after truncating the original equation at any finite value of $n$. (...I don't know how to check this keeping the full sum/product over $n$..)

Now I tried to do something obvious but it didn't work!

$\prod_{n=1}^{\infty} \frac{ (1+x^n) }{1+x^n -x^{\frac{n}{2}} \left(y^{\frac{n}{2}} + y^{-\frac{n}{2}}\right) } = \exp \left[ \sum_{n=1}^{\infty} \frac{ I_{ST}(x^n,y^n) } {n (1-x^{2n}) } \right] $

$\Rightarrow \sum_{n=1}^{\infty} \left[ \ln (1+x^n) - \ln(1-(\sqrt{xy})^n) - \ln\left(1- \left(\sqrt{\frac{x}{y}}\right)^n\right) \right] = \sum_{n=1}^\infty \frac{I_{ST}(x^n,y^n)} {n(1-x^{2n})} $

Now we expand the logarithms and we have,

$\sum_{n=1}^{\infty} \left[ \sum_{a=1}^{\infty} (-1)^{a+1} \frac{x^{na}}{a} + \sum_{b=1}^{\infty} \frac{ (\sqrt{xy})^{nb} } {b} + \sum_{c=1}^{\infty} \frac{ (\sqrt{x/y})^{nc} }{c} \right] = \sum _{n=1}^\infty
\frac{f(x^n,y^n)} {n(1-x^{2n})}$

$\Rightarrow \sum _{a=1} ^{\infty} \frac{1}{a} \left[ \sum _{n=1} ^{\infty} \left( (-1)^{a+1}x^{na} + (xy)^{\frac{na}{2}} + \left(\frac{x}{y}\right)^{\frac{na}{2}} \right) \right] = \sum _{n=1} ^\infty \frac{f(x^n,y^n)} {n(1-x^{2n})}$

By exchanging $a$ and $n$ (relabeling on the LHS) and matching the patterns on both sides and picking out the $n=1$ term one sees that one way this equality can hold is if,

$f(x,y) = (1-x^2) \sum _{a=1} ^{\infty} \left[ x^a + (xy)^{\frac{a}{2}} + (\frac{x}{y})^{\frac{a}{2}} \right]$

$\Rightarrow f (x,y) = (1-x^2) \left(-1 + \frac{1}{1-x} -1 + \frac{1}{1-\sqrt{xy}} - 1 + \frac{1}{1-\sqrt{\frac{x}{y}} } \right)$

But this solution is neither the one above which could be perturbatively checked to be true nor does it satisfy the original equation! Why?

After doing a series expansion of the above (using Series on Mathematica) one sees that this above derived equation (2) differs from (1) in having just one extra term of $x^2$. (...I would like to know what is wrong in the derivation that gives (2) this one extra term compared to the non-derivable but perturbatively checked correct answer (1)...)

Representation theory of (anti)self-dual tensors

2012-07-13T20:48:13Z

I am using usual physics notations and I guess the physics motivations of this question are obvious.

Let a basis of the $SO(n,m)$ Lie algebra be denoted by $S^{\mu \nu}$ and the Lie algebra be, $[S^{\mu \nu},S^{\lambda \rho}] = i(g^{\mu \rho}S^{\nu \lambda} + g^{\nu \lambda}S^{\mu \rho} - g^{\mu \lambda}S^{\nu \rho} - g^{\nu \rho}S^{\mu \lambda})$ where $g$ is the matrix $diag(-1,-1,..m-times..,-1,1,1,..n-times..,1)$.

At least in the $m=1$ (Lorentz) case it is true that if one can find a set of matrices $\Gamma^\mu$ such that they satisfy the "corresponding" Clifford algebra, $[\Gamma^\mu, \Gamma ^\nu] = 2g^{\mu \nu}$ then a representation of the $SO(n,1)$ Lie algebra is given by $S^{\mu \nu} = \frac{i}{4}[\Gamma ^\mu , \Gamma ^\nu]$

Does the above construction work for arbitrary $m$ , especially $m = 0,2$ and what is the global understanding for why this should work?
The representation $S^{\mu \nu} = \frac{i}{4}[\Gamma ^\mu , \Gamma ^\nu]$ is how the $SO(n,1)$ Lie algebra acts on its spinorial representations. (..those representations whose weights are given by a $[\frac{n+1}{2}]$ tuple of $\pm \frac{1}{2}$..)

What is the $m \neq 1$ generalization of the above? (...one case that I have often seen used are these two earlier questions of mine..)
Now my main question is to understand how the above construction - at least for the most familiar case of $n=3, m=1$ - gives an alternative to using the language of tensors.

Like to give probably the most used example - if $F_{\mu \nu}$ is a $4$-dimensional antisymmetric rank $2$ tensor then one defines the quantities $F_{\alpha \beta}$ and $\bar{F} _ {\dot{\alpha} \dot{\beta}}$ s.t $F_{\alpha \beta} = (S^{\mu \nu}) _ {\alpha \beta} F_{\mu \nu}$ and $\bar{F} _ {\dot{\alpha} \dot{\beta}} = (S^{\mu \nu})_{\dot{\alpha} \dot{\beta}} F_{\mu \nu}$

What exactly is the group/representation theoretic meaning behind defining these $F_{\alpha \beta}$ and $\bar{F}_{\dot{\alpha} \dot{\beta}}$ ?
In what sense is knowing the $F_{\alpha \beta}$ and $\bar{F}_{\dot{\alpha} \dot{\beta}}$ equivalent to knowing the $F_{\mu \nu}$?
In what sense is $F_{\alpha \beta}$ and $\bar{F}_{\dot{\alpha} \dot{\beta}}$ the self-dual ($\frac{1}{2}(F + *F)$) and the anti-self-dual ($\frac{1}{2}(F - *F)$) parts of the tensor $F$?
- I guess from here one can also explain why the self-dual part is thought to be in the $(1,0)$ representation and the anti-self-dual part is thought to be in the $(0,1)$ representation of either the $SL(2,\mathbb{C}) \times SL(2,\mathbb{C})$ (..isometry group of the complexified Minkowski space..) or of $SU(2) \times SU(2)$ (..in in Euclidean four dimensional space..)
- Is there an analogue of the $F_{\alpha \beta}$ and $\bar{F}_{\dot{\alpha} \dot{\beta}}$ for general $SO(n,m)$ and arbitary dimension higher rank tensors ?

The Chern-Simons/Wess-Zumino-Witten correspondence

2012-07-01T22:00:04Z

I have often seen a relationship being alluded to between these two theories but I am unable to find any literature which proves/derives/explains this relationship.

I guess in the condensed matter physics literature this is the "same" thing which is referred to when they say that one has propagating chiral bosons on the boundary of the manifold if there is a Chern-Simons theory defined in the interior("bulk")

Let me quote (with some explanatory modifications) from two papers two most important aspects of the relationship that is alluded to,

"...It is well known that any Chern-Simons theory admits a boundary which carries a chiral WZW model; however these degrees of freedom are not topological (the partition function of Chern-Simons theory coupled to such boundary degrees of freedom depends on the conformal class of the metric on the boundary)..."
"...In general if the pure Chern-Simons theory (of group $G$) at level k is formulated on a Riemann furface then the number of zero-energy states equals the number of conformal blocks of the WZW model of $G$ at level $k' = k - \frac{h}{2}$ ($h=$the quadratic Casimir of G in the adjoint representation)..(and when the Riemann surface is a torus) the number of conformal blocks is equal to the number of representations of $\hat{G}$ at level $k'$.."

I would like to know of a reference(s) (hopefully pedagogical/introductory!) which explains/proves/derives the above two claims. (..I looked through various sections of the book by Toshitake Kohno on CFT which deals with similar stuff but I couldn't identity these there..may be someone could just point me to the section in that book which may be explains the above claims but may be in some different garb which I can't recognize!..)

Completeness of the future null infinity in defining a black hole

2011-01-09T19:21:36Z

I am using these lectures by Rodnianksi and Dafermos as the reference for this question.

In third point in the list on the top of page 19 they emphasize the importance of completeness of the future null infinity in being able to define a black hole through a Penrose diagram. I guess by completeness they mean geodesic completeness.

What is the issue that is being hinted at?

In the statement just below the second diagram on page 21 they seem to be able to read off that the part of the future null infinity that is intersected by the maximal Cauchy development of their chosen Cauchy surface is incomplete.

I could not understand how this is obvious.

This observation is what they are using to motivate Christodoulou's thinking of incompleteness of future null infinity as a defining criteria of naked singularity.

How does this relate to the other viewpoint of Christodoulou that naked singularity is characterized by the non-compactness of the intersection of the past of the future null infinity with the Cauchy surface? (This paper for reference)

Conformal Killing spinors

2012-05-25T00:48:43Z

In general I would like to know about the significance of conformal Killing spinors (especially keeping in mind supersymmetric theories on curved space-time).

If $\epsilon$ and the $\bar{\epsilon}$ are the holomorphic and the anti-holomorphic conformal Killing spinors then apparently the following are true,

In a conformally flat background it satisfies $D_\mu \epsilon = \gamma _ \mu \kappa$ for arbitrary spinor $\kappa$ and on a flat background it is identically sastified by $\epsilon = \xi_1 - x^\mu \gamma _\mu \xi_2$ for $\xi_1, \xi_2$ being arbitrary spinors.
On a $S^3$ with radius $r$ there are apparently $4$ independent anti-holomorphic ones (how?)and they split into $2$ satisfying $D_\mu \bar{\epsilon} = \pm \frac{i}{2r} \gamma_\mu \bar{\epsilon}$
If the $S^3$ above is replaced by $S^2 \times \mathbb{R}$ then apparently the equation changes into $D_\mu \bar{\epsilon} = \pm \frac{1}{2r} \gamma_\mu \gamma_3 \bar{\epsilon}$where $r$ is now the radius of $S^2$.

I would be glad if someone can explain or give a reference which has a quick explanation for this.

Also I want to know how the representations of the isometry group of $S^2$ i.e $S0(3)$ somehow help "classify" the solutions. (like I think the claim is that each of the set of two actually lies in a two dimensional irreducible representation of $SO(3)$). I would like to know of the explanation for this classification and if there is a general representation theory idea which works always (like also may be in the first two cases?)

About the quantum spectrum of a certain potential.

2012-05-07T00:35:19Z

Intuitively one understands that if one is solving the Schroedinger's equation for energies $E$ such that $\{ x \vert U(x)\leq E \}$ is compact (..is there a weaker criteria?..) then the spectrum will turn out to be discrete and the wave-functions will decay exponentially for large values of $x$. What is the most rigorous statement and proof of this?

I want to focus on one potential,

$U = \vert G(s_i)\vert ^2 + \vert p \vert ^2 \sum _ {i} \vert \frac{\partial G}{\partial s_i} \vert ^2 + \frac{e^2}{2}(\sum _i \vert s_i \vert ^2 - n^2 \vert p \vert^2 - r ) + 2\vert \sigma \vert ^2 (\sum _i \vert s_i \vert ^2 + n^2 \vert p \vert^2) $

where $e$ is a real constant, $s_i$, $p$ and $\sigma$ are complex and $r$ is a real field and $G$ is a degree $n$ transverse homogeneous function in $s_i$.

Now apparently the following claims are true,

If $r = 0$ then for any value of $\sigma$, the range of $s_i$, $p$ where $U(s_i,p) \leq E$ is true is compact and hence the spectrum is discrete.
If $\sigma \neq 0$ then for any fixed non-zero value of $r$, the region of $s_i$ and p where $U(s_i,p;r) \leq E$ is true is compact for small enough values of $E$ and hence the spectrum is discrete for low-lying values of $E$ below some $E_{critical}$.

(..the above is apparently motivated by the fact that at $s_i=p=0$, $U$ becomes constant and equal to $\frac{e^2 r^2}{2}$ and hence independent of $\sigma$..so apparently if one goes above some critical value of $E$ the spectrum is continuous thanks to field configurations with large $\vert \sigma \vert$ but at $s_i = p =0$... )

My reading of the literature is that the above two claims are true independent of the topology of the space on which the fields are valued though in a case of interest one wants the theory to be on a circle and hence I guess one wants to think of $s_i$, $p$, $\sigma$ to be maps from $S^1$ to $\mathbb{C}$ or $\mathbb{R}$. If the theory is on a circle then semiclassically apparently the estimate for $E_{critical}$ is $\frac{e^2 r^2}{2} 2\pi R$ where $R$ is the radius of the circle.

I would be glad if someone can help justify the above three claims.

This paper is the reference for this question

Wightman fields vs local functionals vs operators

2012-01-29T23:35:19Z

In QFT literature one wants to look at $n-$point correlation functions of "operators" inserted at $x$ say, $\cal{O}(x)$ and if $\phi_i$ are the fields then the quantity one has in mind is written as, $<\cal{O}(x)\phi_1(x_1)\phi_2(x_2)..\phi_n(x_n)>$ and this is defined as a path-integral. Typically there are going to be short-distance singularities if any of the two $x_i$ start coinciding. (and that was the topic of my last question)

Though I have done numerous calculations of calculating such correlation functions in various QFTs, it remains unclear to me as to at the fundamental level where does one draw the line between an "operator" and a "field". After quantization aren't all fields actually operators or more precisely Hilbert space operator valued fields on the space-time? This is the confusion that I would like to clarify here.

To start off let me cite some of the definitions that I have seen in this regard.

For the quantities labelled as $\phi$, as used above, one seems to use two terms - "local functionals" and "Wightman fields", namely,
- A "local functional" at $x$ is a function of the fields and finitely many derivatives of the fields evaluated at $x$, like $\phi(x)$, $\phi^2(x)$ but NOT $\phi(x) + \phi(2x)$
- "Wightman fields" $\phi(x)$ are distributions on the Minkowski space ($V$) with values in the space of operators on the subspace $\cal{D} \subset \cal{H}$ (the Hilbert space of multiparticle states). This means that for any Schwartz function $f$ on $V$, $\phi(f)$ is an honest operator $\phi(f)$ on $\cal{D}$.

It is not clear to me whether "local functionals" and "Wightman fields" are the same things of if there is a natural way to pass between the two things above but I feel that in literature these terms are used interchangeably.

For the quantities $\cal{O}$ I seem to see two statements,
- that $\cal{O}(x)$ does not act as an operator (or an operator valued distribution) on any reasonable subspace of the Hilbert space.
- that since one wants to deal with products of operators at different space-time points, its better to think in terms of "smeared" quantities like $\cal{O}(f) = \int f(x)\cal{O}(x) d^nx $ where $f$ is a compactly supported smooth function on $V$. Then the statement is that, "..the product $\cal{O}(f)\cal{O}(f')$ exists in the sense of correlation functions if and only if the "operator" $\cal{O}(x)$ (an operator valued distribution) is actually an honest operator i.e matrix elements of $\cal{O}(f)$ are matrix elements of some operator on $\cal{D}$.."

Like in the first pair of points, here too there seems to be some interchangeability between the notion of $\cal{O}(x)$ and $\cal{O}(f)$ and they seem quite analogous to the corresponding $\phi(x)$ and $\phi(f)$! Where is the difference?

What exactly is an "operator valued distribution" that $\cal{O}(x)$ usually perhaps is not (first statement of the above pair) but it seem to have to be if products like $\cal{O}(f)\cal{O}(f')$ have to be defined (last statement of the above pair)?

It would be very helpful if someone can disentangle the above (and may be also my linked previous question!) and explain the difference between the notions of a "local functional", "Wightman field" and "operator".

Doing geometry using Feynman Path Integral?

2010-03-27T07:36:05Z

I have often heard in the folk-lore that Feynman Path Integral can be used to compute geometric invariants of a space.

Coming from a background of studying Quantum Field Theory from the books like that of Weinberg, I have myself used Feynman Path Integrals to compute scattering of particles.

Earlier I had done courses in Riemannian Geometry and these days I am also doing courses in Algebraic Topology and hence I think it would be very educative if I can see how exactly the calculation of topological invariants that one does here are related to Feynman's ideas.

It would be helpful if someone can give me references which explain (hopefully starting with simple examples!) how one can use path integrals in geometry.

Answer by Anirbit for The current status of the Birch & Swinnerton-Dyer Conjecture

2012-01-29T23:53:57Z

I have technically no clue about this subject but what I glean from "coffee-table discussions" is that this paper is a recent progress about BSD. Or may be that can be said collectively about all the 3 papers that have been written by this friend/college-mate of mine and Manjul Bhargava.

Correlation functions of complex operators

2012-01-20T22:44:27Z

One defines the "scaling dimension" (as opposed to "engineering dimension") of an operator $\cal{O}$ as $[\cal{O}]$ such that if $\cal{O}(t^{-1}x) = t^{[\cal{O}]}\cal{O}(x)$ then the Lagrangian in which $\cal{O}$ appears would be scale invariant.

Unlike for "engineering dimensions" it seems that the value of scaling dimensions (even classically!) can't be derived from just looking at the operator but one seems to need to know the Lagrangian in which it appears so that the "right" $[\cal{O}]$ can be assigned to preserve scale-invariance.

For example - how else does one explain that the "engineering dimension" of $m^2\phi$ is $3$ whereas its "scaling dimension" is $1$? (same as that of $\phi$) (..the above obviously follows if I think of the term to be occurring in a $2+1$-dimensional Lagrangian and ask as to what should the scaling dimensions be so that the Lagrangian is scale-invariant..but something doesn't look very intuitive..)

I would like to know what is the special difficulty that is faced in defining $2-$point correlation functions of $\cal{O}$ if it is real? (..as opposed to when they are complex like in the next question - thought not that obvious either!..)
For complex $\cal{O}$ it "follows" that $<\cal{O}(x)\cal{O}^*(y)> \sim \vert x - y \vert ^{-2[ \cal{O}]}$ It is clearly consistent with definitions of the scaling dimension but is there a "derivation" for this? I have often seen the statement that the above short-distance behaviour follows from "reflection positivity" (..ala Wightman axioms..) I would like to know of some explanations.

Reducibility (or not) of algebraic curves

2011-10-13T01:19:52Z

[ I am a bit clueless about why this question is getting downvotes!? I put it up with a genuine serious interest and I don't seem to be making any egregious error either - apart from those unsure sentences which I have made with a "?" in the bracket. Please explain if something is terribly wrong with this question! Is this question too elementary for this forum? ]

Here by a "curve" I shall tend to think of algebraic curves in $\mathbb{CP}^2$

Is reducibility or not of a curve a question of whether the defining equation factorizes (necessarily into linear factors?) or is something more demanded from the factors? If the curve is thought of as a monic (in $y$) element in $\mathbb{C}[x][y]$ (which it can always be) has even $1$ root isn't that sufficient to say that it is reducible?

The degree of an algebraic curve will be the highest degree homogeneous component in it and hence if it has a triple point that would imply that the third degree term is the only term. Hence further if this is an ordinary triple point that would mean that this only term (of third degree) has 3 distinct roots and hence the curve is reducible. Is the argument right?

I would like to understand the other related such statements that I face like - a fourth degree curve with 4 singular (whether or not ordinary? whether or not distinct?) is also reducible, that if a fourth degree irreducible curve has 3 distinct singular points then they are necessarily double.

As the framing itself suggests, I am not sure of the statements and would like to know what is the precise statement that is correct and why.

Thought of as the monic polynomial (as in the first bullet point) if the curve has $0$ discriminant then it will have repeated roots. Is that then equivalent to saying that irreducibility implies that the discriminant is not identically $0$? (..or is some further work required?..)(I guess the converse is not true - a non-zero discriminant curve can still be reducible?--I guess so..)
What is the meaning of an "ordinary singular" point on a curve? I am aware of the notion of an "ordinary k-tuple" point. (...is it true that for $k>1$ such a point has to be singular?..seems so..)
What is the general way to connect reducibility or not of a curve with the fact as to whether or not it has singular points or how many of them does it have?

Motivation behind defining the Ramification Divisor

2011-10-04T22:06:35Z

I would like to understand what exactly is the motivation for defining the notion of a ramification divisor of a function.

As I see the definition,

If $f$ is a meromrophic function between two Riemann surfaces - say $X$ and $X'$ then let $\nu_p(f)$ be the ramification (or order) of the function $f$ at $p$. Basically if one is working in local coordinates such that $z(p)=0$ then $f$ in a neighbourhood of $p$ looks like $f=z^{\nu_p}h(z)$ where $h(z)$ is a holomorphic function which is never $0$ in a neighbourhood of $p$.

In the above definition of ramification, can the function $h$ be always set to unity? By choosing coordinate in$X'$ such that $f(p)=0$? (...I am not sure..)
Does anything in the above definition depend on $X$ or $X'$ being compact?

Now for a similar map $f$ one defines its ramification divisor ($R_f$) as $R_f = \sum _{p \in X} (\nu_f(p) - 1)p$

Its not clear to me whether people define ramification divisors for meromorphic functions too since i almost seem to see the texts exclusively using it in the case of non-constant holomorphic functions. I would be glad if someone can clarify this...may be I am missing something very basic.
Also this definition almost exclusively seems to be used when $X$ and $X'$ are compact Riemann surfaces. Is that somehow necessary?

{I guess in all this discussion one has to keep in mind that a holomorphic function on a Riemann surface and a holomorphic function between two Riemann surfaces are defined "differently" - as i see it. I guess there is no analogue of Liouville's theorem in the later case.}

Why that "-1" in the definition? Is $\nu_p(f)$ always greater than $1$ ?
Let $q \in X'$ and let $p_1$ be a pre-image of $q$ under $f$ with multiplicity of $m_1$. Then I guess one will say that $\nu_f(p_1) = m_1$. Now is it obvious that any "small" perturbation of $q$ can only "split" $p_1$ into $m_1$ points each with $\nu_f = 1$? That nothing else can happen? For "large" enough perturbation to $q$ isn't it possible for many of its pre-images to "join up" and have larger ramifications than initially?
consider this set, $p \in X' \vert f^{-1} (p)$ has all points with $\nu_f(p)=1$ (called "simple points"?). Is this set open and dense in $X'$?
Finally a curiosity - is there a "simple" way to see the Riemann-Hurwitz formula without using the Poincare-Hopf formula?

$Aut(\mathbb{CP}^n)$ [..especially $n=1$ and $n=2$..]

2011-09-14T20:38:17Z

I am confused and curious about the meaning of the $Aut(\mathbb{CP}^n)$.

Is what is called the "linear automorphism group" of $\mathbb{CP}^n$ the same as $Aut(\mathbb{CP}^n)$? It somehow seems to me to be very non-trivial if they are the same things.
I see the statement that $Aut(\mathbb{CP}^1) = { z \mapsto \frac{az+b}{cz+d} , ad-bc \neq 0 }$ How am I supposed to interprete this statement? If $z$ is the homogeneous coordinate then its not clear to me that this map is well defined on a projective space. How does one prove this?
Is there a similar way to write down $Aut(\mathbb{CP}^2)$?
One wants to show that any two irreducible conic sections in $\mathbb{CP}^2$ are "projectively equivalent". I would like to know how this is shown. Does this mean that there exists an element of $Aut(\mathbb{CP}^2)$ that transforms one to the other? Is there a way to write down a general expression for an irreducible conic in $\mathbb{CP}^2$?

Some questions on Nicolai Reshetikhin's lectures on quantization of gauge theories.

2011-05-15T19:17:25Z

This in reference to this fascinating lecture by Nicolai Reshetikhin-

http://staff.science.uva.nl/~nresheti/Holb-Quant-Gauge.pdf

Given what is said on page 13 in section 4.1 its not clear to me why the ``partition function" should be a vector in the same space of states which is being assigned to the boundary of the space-time manifold.

I am confused to see the claim that $Z(M) \in H(\partial M)$.I would have thought that the partition function is a linear function on the space of states which takes in a boundary configuration and gives back a number.

I would like to know what is meant by saying (on the same page) that "..(this vector space of states) may depend on the extra structure at the boundary (it can be a vector bundle over the moduli space of such structures).."
How does the above relate to the claim on page 47 that for Chern-Simons theory the "..space of states assigned to the boundary is the space of holomorphic sections of the geometric quantization line bundle over the moduli space of flat connections in a trivial principal G-bundle over the boundary (provided we made a choice of complex structure)..."

I would like to know what the above means and I would be happy to get back some further references about this...especially what is a "geometric quantization bundle"?

Is the above somehow purely an effect of quantization? Thinking classically intuitively i would have thought that the space of states assigned to the boundary is the space of gauge equivalence classes of flat connections on the manifold which can be extended to a flat connection on the whole space-time. Is the above wrong?

I can't see where the structures of a vector bundle and may be even its sections over the above moduli space seem to be getting involved.

Is there a general argument to see that the space of states attached to the boundary is always a symplectic manifold and that the subspace of that will ever be picked up by solving the Euler-lagrangian equations in the interior will be some Lagrangian submanifold of it?

In these lectures some comment seems to be made about how the gauge invariance may complicate the above scenario.I would like to know more about that.

At the start of the lecture the author seems to demand that the space-times belong to a category of manifolds where the morphism is that of cobordism. I would like to know what was the intuition behind making this choice. Why not some other simpler morphism?

Does working in this category of cobordisms somehow help in justifying the functoriality demand on the partition function with respect to gluing of manifolds?

Quotient space of $\mathbb{C}^5$ under the action of $SL(2,\mathbb{C})$

2010-12-15T21:56:32Z

One sees that given the $SL(2,\mathbb{C})$ action on $\mathbb{C}^5$, thought of as the space of polynomials of the form,

$$a_0 x^4 + 4a_1 x^3 y + 6a_2x^2y^2 + 4a_3xy^3 + a_4 y^4$$

the ring of invariants is generated by the following functions,

$$g_2(a) = a_0a_4 - 4a_1 a_3 + 3a_2^2$$

and

$$g_3(a) = a_0a_2a_4 - a_0a_3^2 - a_1^2a_4 + 2a_1a_2a_3 - a_2^3$$

But if these $g_2$ and $g_3$ satisfy the discriminant $=0$ condition then there are inequivalent $SL(2,\mathbb{C})$ polynomials which map to the same $(g_2,g_3)$ point.

But if I look at say Theorem 5.9 in the book by Mukai then I get to see that the closure equivalent classes of orbits of the action of a linearly reductive group like $SL(2,\mathbb{C})$ on $\mathbb{C}^5$ are in bijective correspondence to the the points of $\mathbb{C}^5//SL(2,\mathbb{C})$ (which is defined as the spectrum of the invariant polynomials under $SL(2,\mathbb{C})$)

Also look at the theorem at the end of page 11 of this paper.

In the above paper "//" is defined as identifying points in the affine variety if one lies in the closure of the orbit through the other.

Are these two notions of "//" equivalent? If yes, how?
In the light of the above two theorems, can one say that the $SU(2)$ invariant polynomials among binary homogeneous quartics are in bijection with those closure equivalent classes of orbits of $SU(2)^{\mathbb{C}} = SL(2,\mathbb{C})$ which are labeled by the pairs of invariants $(g_2,g_3)$ such that $g_2^3 - 27 g_3^2 \neq 0$ ?
Hence if I am interested in only closure equivalent orbits can I just forget those pairs of values of the invariants which lie on the discriminant $0$ curve?
Conversely given a $(g_2,g_3)$ for which the above discriminant condition is satisfied can one write down the family of $SU(2)$ invariant polynomials explicitly?
I would anyway like to know how to distinguish the orbits corresponding to the discriminant $0$ condition.

In light of the various extremely helpful and references that have come up, I realize that there there is a notion of a "discriminant" for homogeneous polynomials of degree $d$ in $n$ variables. (call this space of polynomials as $P(n,d)$) This discriminant is in some sense a "homogeneous invariant" and for the $n=2$ case in which I am interested in, it generates the subalgebra of the coordinate ring over of these polynomials which is invariant under this group action (call that $\mathbb{C}[P(n=2,d=4)]^{SL(2,\mathbb{C})}$). (this is lucky!)

I guess the discriminant in this case has to be a polynomial in polynomials of homogeneous degree $4$ in $2$ variables. The above I guess implies that $\mathbb{C}[P(n=2,d=4)]^{SL(2,\mathbb{C})}$ is generated by just one such polynomial in polynomials.

I would like to know how is a ``discriminant" defined for such polynomials. (searching and asking around I am only getting definitions for the single variable case)

I want to know if knowing the generator of $\mathbb{C}[P(n=2,d=4)]^{SL(2,\mathbb{C})}$ tells me about the initial objective of knowing $P(n=2,d=4)\text{ }mod\text{ }SL(2,\mathbb{C})$

I wonder if this unique generator of the invariant subalgebra is related to the null-cone that was pointed out by Bart in his comment.

Also I would like to be pointed out if there is any mistake in what I said above!

I had recently tried asking a similar question here. But I think I could not precisely convey what I was looking for. Let me here try to give a specific situation that I need to understand coming from certain other considerations in Superconformal Quantum Field Theories.

I can think of $\mathbb{C}^5$ as being the space of all homogeneous degree $4$ polynomials in $2$ variables. On this space $SL(2,\mathbb{C})$ has the standard action.

I want to know what is the most explicit (or the best!) way to describe the quotient space thus obtained. I want to understand how do the orbits look like.

[EDIT: I was initially asking if there exists fixed subspaces etc but then from the comments I realized that I was missing the elementary fact that it is an irreducible representation! Hence nothing like this can exist.]

I tried something naive. I wrote down the most general element of $SL(2,\mathbb{C})$ using its canonical polar decomposition and then acted it on the most general homogeneous degree $4$ polynomial in $2$ variables and tried to see how the coefficients change. Unfortunately the equations are very complicated and I didn't see any hope of me being able to solve them to find the fixed points.

Apart from this specific example I would also like to know of references to simpler examples than this where a similar question is asked and answered.

Counting and summing over solutions of a Diophantine equation

2011-04-11T19:39:06Z

Say I have a Diophantine equation of the form $a_1 x_1 + a_2 x_2 + ... + a_m x_m = n$ such that the $a_is$ are all co-prime to each other. And I also have a function say $f$ which depends only on the $x_i's$ (and will be evaluated on solutions of the equations)

Is there a general method or simple examples of summing over the values of $f$ evaluated on the non-negative integral solutions of the equation?
Is there a way to count the number of non-negative integral solutions of such Diophantine equations? (...I am aware that it is trivially doable in some special cases like when all the $a_is$ are equal to $1$ or when $a_i = i$ and $m=n$...)

Maurer-Cartan form

2010-08-03T17:26:53Z

I suppose given a Lie Group (G) and its corresponding Lie Algebra (g) every element in its dual defines a Maurer-Cartan form on the whole Lie Group?

Let $\omega \in g^*$ be a Maurer-Cartan form and let $X$ and $Y$ be two elements of $g$ then in what sense are $\omega(X)$ and $\omega(Y)$ "constant functions" on G ? (such that we can write $X\omega(Y)=Y\omega(X) =0$)

Assuming the above one can immediately write the Maurer-Cartan equation, $d\omega(X,Y)= -\frac{1}{2}\omega([X,Y])$

Thinking of $\omega$ as Lie Algebra valued 1-form on the Lie Group and using the fact from linear algebra that $V^* \otimes W = Hom(V,W)$ one can write them as $\omega = \sum _i \omega_i \otimes B_i$ where $\omega_i$ are ordinary 1-forms on G and B_i are a basis on g. (should there be some arbitrary coefficients in front of every term in the above sum?)

Say $c^i_{jk}$ are the structure constants of the Lie Algebra then I do not understand how the Maurer-Cartan equations can be recast as,

$$d\omega_i = -\frac{1}{2}\sum_{j,k} c^i_{jk} \omega_j \wedge \omega_k$$

which apparently can be further recast as the equation,

$$d\omega = -\frac{1}{2} [\omega,\omega]$$

I would be happy if someone can explain how the above two forms of the Maurer-Cartan equation can be obtained knowing the first form which is more familiar form to me.

Also finding the structure constants of a Lie Algebra is not so hard for at least the common ones. Knowing that one fully "knows" the Maurer-Cartan Equation. Now is there any sense in which one can "solve" this to find out the Maurer-Cartan forms? (I would guess a basis might be obtainable)

Witten Index, letter partition function and superconformal representations.

2011-01-30T11:07:00Z

Except in a few papers I have seen so little written about this that I am not sure I can even frame this question properly.

I would like to know of expository references and explanations on the concept of "single/multi trace letter partition function" and how it connects to Witten Index and superconformal field theory.

I haven't been able to find any reference which explains the concept of letter partition function and how techniques from representation theory get used to calculate them. (especially in the context of superconformal representations)

For example one can see between page 15 and 30 of this paper to see some usages of this.

As said above this technology comes up often in the context of superconformal group representations. I would be happy see references which give explanations about them.

In superconformal representations one often lists out "long" and "short" representations labelled by the "primaries" and then one calculates the Witten Index of them. (which apparently always vanishes for the long ones) To give an example of a case where Witten Index is calculated,

So for ${\cal N} = 2$ superconformal algebra in $2+1$ dimensions the symmetry group is $SO(3,2)\times SO(2)$ and possibly the primary states of this algebra are labelled by the tuple $(\Delta, j,h)$ where $\Delta$ is the scaling dimension and $j$ is the spin and $h$ is its $R$ charge (or whatever it means to call it the $R$ charge highest weight)

I would like to know what are the precise eigenvalue equations used to do the above labeling.

Now consider a primary labelled by $(\Delta, j,h)$ such that it is in the long representation and hence $\Delta >j+\vert h\vert +1$. Then I see people listing something called the “conformal content” of this representation labelled by the above state. For the above case the conformal content apparently consists of the following states, $(\Delta, j,h)$, $(\Delta+0.5, j\pm 0.5,h\pm 1)$, $(\Delta + 1 , j,h \pm 2)$, $(\Delta +1 , j+1,h)$, twice $(\Delta + 1, j,h)$, $(\Delta + 1, j-1,h)$, $(\Delta + 1.5, j\pm 0.5,h \pm 1)$ and $(\Delta + 2, j,h)$

I would like to know what exactly is the definition of “conformal content” and how are lists like the above computed. The Witten Index of the above is supposed to be $0$ and I guess it was supposed to be obvious without explicitly enumerating the labels.

Similar lists can be constructed for various kinds of short representations like those labelled by $(j+h+1,j,h)$ ($j, h \neq 0$), by $(j+1,j,0)$, by $(h,0,h)$, by $(0.5,0,\pm 0.5)$, by $(h+1,0,h)$ and $(1,0,0)$. Its not completely clear to me a priori as to why some of these states had to be taken out separately from the general case, but I guess if I am explained the above queries I would be able to understand the complete construction.

Answer by Anirbit for Completeness of the future null infinity in defining a black hole

2011-01-12T16:08:40Z

At least about the definition of "complete future null infinity" I found some answers on the 8th page of these lectures by Klainerman.

I would be glad to hear of some explanations about how the two definitions given on that page relate to each other and how are they related to the notion explained by Tim in the comments to his answer. (Tim is calling the future null infinity to be complete if the whole manifold is geodesically complete)

Also these seem closely related to the idea of calling a hypersurface as being "generated by complete null geodesics". I would like to know what this means and why this is often used as a condition for the event horizon to satisfy.

Christodoulou's paper on naked singularities in inhomogeneous dust collapse

2010-09-05T17:56:01Z

I have been studying of late about formation of naked singularities in certain collapse scenarios in Einstein's theory. It seems to me that the canonical paper to read about how such a formation is established is the 1984 paper by Christodoulou in Communications in Mathematical Physics. ( http://www.ams.org/mathscinet-getitem?mr=742192 )

I was wondering if there is a reference which gives a more modern rewriting of the proof in that paper which say sort of highlights the generic technique of the proof which the reader can take away from there for other scenarios.

Somehow even the most recent books on Einstein's theory like the otherwise brilliant book by Choquet-Bruhat also doesn't dwell on techniques of testing in a collapse scenario whether the curvature singularity is naked or not.

I haven't seen till now any generic method or algorithm for testing this.

It would be great if someone can give me some references/explanations along these lines.

Action of $SL(2,\mathbb{C})$ on representations of $SU(2)$

2010-12-13T06:26:16Z

I want to precisely understand in what sense is (if it is!) $SL(2,\mathbb{C})$ the "complexified" version of $SU(2)$?

Can I think of it like choosing a natural matrix basis of the real three dimensional Lie algebra of $SU(2)$ (say the Pauli matrices) and looking at the vector space they would span over $\mathbb{C}$ (i.e look at the vector space of matrices spanned by linear sums of Pauli matrices with complex coefficients) and then exponentiate it down?

Is there a natural action of $SL(2,\mathbb{C})$ on the $5$ dimensional irreducible representation of $SU(2)$? If yes then how does one best understand the quotient space and precisely in what way does this action respect the representation.

(I would be interested about the general picture if there is any behind such actions, I chose the above particular example since it is most relevant to my current pursuits.)

Some questions about causal structure of space-time.

2010-11-23T12:13:51Z

Let $(\hat{M},\hat{g})$ be the conformal compactification of a space-time $(M,g)$. Let $I^+$ be the conformal null infinity and $J^{-}(I^+)$ be its causal past. Then the spacetime will be called "asymptotically strongly predictable" if there exists a subset $\hat{V}$ of $\hat{M}$ such that $(\hat{V},\hat{g})$ is globally hyperbolic and contains the closure in $\hat{M}$ of $M \cap J^{-}(I^+)$. Define as a "black hole" the region $B=M /\ J^{-}(I^+)$.
1. I think from the above it should imply that there cannot be a causal curve which starts in the region $B$ and enters $M \cap \hat{V}$. Is this correct and if yes then how does one prove it?

(In specific examples what happens is that such a curve cannot remain always causal if it has to do such a journey. But I guess this not a good way of thinking since the restriction is purely topological)

Under the above conditions when does it also follow that $(M \cap \hat{V},\hat{g})$ and/or $(M\cap \hat{V},g)$ is globally hyperbolic?
What is the meaning of "trace of a terminally indecomposable past set" ? Is there any special significance if a compact set happens to be the trace of a terminally indecomposable past set of a maximal future directed time-like curve? (Reading around I get the impression that such compact sets somehow specify physically reasonable initial conditions)

Answer by Anirbit for Sasaki but not Einstein

2010-11-19T15:26:21Z

I am not very sure of everything here but I wonder if the example given in Appendix A on Page 21 of this paper by one of my professors meets your criteria.

Relationship between apparent, event and Cauchy horizons

2010-11-18T14:53:15Z

I would use the definition of an event horizon as being the boundary of the past of the future null infinity of a space-time, future/past Cauchy horizon of a closed achronal surface as the boundary of its future/past domain of dependence and apparent horizon as the outermost trapped surfaces.

I would like to know a reference for or a proof of the following two concepts,

For static/stationary space-times, the event horizon must equal the apparent horizon.
For static space-times, the event horizon is where the static Killing field becomes null.

In the maximally extended Reissner-Nordstrom black-hole space-time the inner horizon is a Cauchy horizon for the "t=0" space-like surfaces. ( Was there a way to see the above without doing the extension?)

I can't prove it but I think the outer horizon of a Reissner-Nordstrom black-hole space-time is not a Cauchy horizon for any closed achronal surface.

I would like to know what is the most precise statement one can make about the relationship between Cauchy horizons and event horizons.
Definition of a black hole as in Yvonne's book is the complement of the past of the set covered by the null geodesics which have an infinite future canonical parameter.

This definition doesn't seem to guarantee global hyperbolicity for either the outside or inside of a black hole and neither does it even demand time-orientability of the space-time nor does this want the space-time to have a "regular" Penrose compactification.

She needs to put in an extra definition of calling a space-time to be ``asymptotically strongly predictable" if the complement of the closure of the black hole region is globally hyperbolic.

Does the above criteria get automatically implied if one uses the definition of black hole as the complement of the past of the future null infinity for those space-times which have a regular Penrose compactifiaction?

Hence my question as to in how general a situation can one guarantee that the space-time in the complement of the black-hole region is globally hyperbolic?

What is the most precise connection known between existence of a black-hole and global hyperbolicity of its interior and exterior?

(Somehow I can't number the questions as 1,2,3,4 and the software insists on calling it 1,2 and again 1,2!)

Comment by Anirbit

2013-05-22T09:48:52Z

@Igor Khavkine You know of a reference with Feynman parameters with 3 factors as here? Its hard to find such examples in text-books. (..also looking at the 1964 Gelfand-Shillof edition I couldn't locate the identities you used the last time..)

Comment by Anirbit

2013-05-14T18:13:09Z

@Jeff Harvey Looking at their equation A.2 it seems that they have twisted the SU(2) R-symmetry into the flavour symmetry in some curious way. Even if I take on faith that a N=3 hyper splits into 2 N=2 chirals in conjugate representations of the gauge group the equation A.2 doesn't seem to follow automatically. I think their A.2 is very crucial since thats the complete specification of the flavour symmetry - it would be great if you can help understand its derivation

Comment by Anirbit

2013-05-14T18:12:09Z

@Jeff Harvey I have also been confused about this very difficult paper. Thanks for clarifying the notation! So the "a" indices on the fields ϕ are the gauge indices corresponding to the representation Ri that has been specified. And i,j go from 1..Nf corresponding to how many flavours they have? So they are never talking of any flavour representation here? So the USp(2Nf) falvour representation specified in the Appendix on page 31 comes from what? But in the very next line they seem to claim that the N=3 fields are in the fundamental of USp(2Nf) - how do they arrive at that?

Comment by Anirbit

2013-04-30T19:22:08Z

@Igor Khavkine Hmm..I did find a copy in my library..hope to read up the details! Thanks for the help!

Comment by Anirbit

2013-04-26T16:58:39Z

@Igor Khavkine You say that any $\nu_1$ and $\nu_2$ is allowed but isn't something uncomfortable happening when either or both are equal to $0$?

Comment by Anirbit

2013-04-22T15:33:24Z

@Igor Khavkine Ah..sorry..I guess the point is that the paper uses the notation of $k = \vert \vec{k} \vert$ and hence the flip of the sign doesn't matter and you get that effect by writing it as $\vec{k}.\vec{k}$. And I guess by thinking of $(\vec{k}+\vec{q})^2$ as $((-\vec{k}) - \vec{q})^2$ you are getting across the $(-1)^d$ issue that the OP had already mentioned. (...this is quite a queer integral I would say - very hard to find anywhere and softwares like Mathematica won't know either..)

Comment by Anirbit

2013-04-21T23:20:43Z

@Igor Khavkine My point about $k+q$ is this - if you see the linked reference then the answer is framed as "~k^$d - 2(\nu_1 + \nu_2)}$" In this form $k$ and $-k$ would give different answers. But you frame the answer as "$~(k^2)^(d/2 - (\nu_1+\nu_2))$" in which $k$ and $-k$ would give the same answers.

Comment by Anirbit

2013-04-21T20:44:10Z

@IgorKhavkine Thanks! And I wanted to clarify if anything here changes if in the denominator instead of $k-q$ it were $k+q$? Also is this applicable for any value of $\nu_1$ and $\nu_2$?

Comment by Anirbit

2013-04-18T18:51:43Z

And what do you denote as $S_{13}$? I am not getting your notation. And similarly for $g$? (..here the confusion stems from the fact that once there are these sources I can't anymore think of a separate equation for $f$ and $g$ with sources..)

Comment by Anirbit

2013-04-18T18:49:50Z

@Aaron Hoffman Are there a few "]" missing in your expression? So perturbatively if one is solving this expression then all of the $F(x')$ on your RHS be replaced by $f_{13}(x)$?

Comment by Anirbit

2013-04-18T16:36:57Z

@Aaron Hoffman Can you state what is the exact correct answer for $f$ and $g$ in terms of $G_f$, $G_g$, $f_{13}$ and $g_{13}$? (..i am hoping an integral representation..)

Comment by Anirbit

2013-04-16T21:35:02Z

@S.Carnahan As you can see in my comment above to Greg, I have explained why I parametrized using squares - to ensure negative definiteness.

Comment by Anirbit

2013-04-10T23:11:21Z

@Greg Martin Are you referring to the fact that I have written $N^2$ and $M^2$? That I did so that, $(p-1)/2 = - N^2$ is a negative definite integer and hence the second Gamma function has a pole. Similarly, $(5+p)/2 = -M^2$ is again a negative definite integer and hence the 3rd Gamma function has a pole. It would be great if you can help with the question.

Comment by Anirbit

2013-04-02T18:16:20Z

@S.Cranahan It is a very difficult question to really come up with a good definition of a CFT - roughly speaking its a system such that all the couplings are tuned to the zero of their respective beta-functions. But there can be a million more subtleties to this simple thinking like you can see these important recent works, arxiv.org/abs/1204.5221, arxiv.org/abs/1209.3424, arxiv.org/abs/1205.3994, arxiv.org/abs/1101.5385, arxiv.org/abs/hep-th/0006037

Comment by Anirbit

2013-04-01T16:56:13Z

@Anton Nazarov Aren't you needing to do a QFT of the action you have written? I mean - your currents need to be normal ordered or something? What is the fundamental Dirac bracket from which you are getting the $[J,J]$ commutation? (..also about your comment about comapactified Boson picture - thats part of my question - in closed Bosonic string theory compactified on a $(S^1)^{rank}$ isn't the affine algebra of the massless states restricted to be in the A, D, E?...And in that case what is Lagrangian?..)