Musings

One of the annoying misfeatures of itex2MML, heretofore, was that it used a rather nonstandard syntax,

\array{ ... } or
\array{ \arrayopts{...} ... }

to create matrices and aligned equations. I’m not sure of the historical origin of this peculiarly un-LaTeX-like syntax but, for some time now, I’ve been meaning to add support for the standard AMSLaTeX constructs.

With itex2MML 1.1, we now support

\begin{env}
   ...
\end{env}

where env is any one of the following

matrix

a matrix, $$\begin{matrix} a & b \\ c & d \end{matrix}\, .$$

pmatrix

a matrix, enclosed in parentheses, $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}\, .$$

bmatrix

a matrix, enclosed in square brackets, $$\begin{bmatrix} a & b \\ c & d \end{bmatrix}\, .$$

Bmatrix

a matrix, enclosed in brace brackets, $$\begin{Bmatrix} a & b \\ c & d \end{Bmatrix}\, .$$

vmatrix

a matrix, enclosed in vertical bars, $$\begin{vmatrix} a & b \\ c & d \end{vmatrix}\, .$$

Vmatrix

a matrix, enclosed in double vertical bars, $$\begin{Vmatrix} a & b \\ c & d \end{Vmatrix}\, .$$

smallmatrix

a small matrix, suitable for use in inline equations, $\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}$.

cases

cases construct, $$|x|=\begin{cases}x&\text{for}\, x\geq 0,\\ -x& \text{for}\, x\lt 0.\end{cases}$$

aligned

produces aligned equations (or sub-blocks thereof).

So

$$
\begin{aligned}
   A =& \begin{pmatrix}a & b \\ c & d \end{pmatrix} \\
   A^{-1} =& \frac{1}{\det A}  \begin{pmatrix}d & -b \\ -c & a \end{pmatrix} \\
     =& \frac{1}{a d - b c}  \begin{pmatrix}d & -b \\ -c & a \end{pmatrix}
\end{aligned}
$$

produces $$\begin{aligned} A =& \begin{pmatrix}a & b \\ c & d \end{pmatrix} \\ A^{-1} =& \frac{1}{\det A} \begin{pmatrix}d & -b \\ -c & a \end{pmatrix} \\ =& \frac{1}{a d - b c} \begin{pmatrix}d & -b \\ -c & a \end{pmatrix} \end{aligned}$$

Another AMSLaTeX construct I’ve been pining for is \substack{...}, as in

$$
   A = \sum_{\substack{m,n \in \mathbb{Z} \\ m \geq n }} a_{m,n}
$$

which now produces $$A = \sum_{\substack{m,n \in \mathbb{Z} \\ m \geq n }} a_{m,n}$$

If you’re viewing this in Mozilla/Firefox, you’ll note that the vertical spacing in all these matrices (and in \substack{...}) is screwed-up because of this bug.

For sheer lack of time, I don’t see myself implementing the myriad of other AMSLaTeX constructs anytime soon. But if there are particular ones you’ve been pining for, let me know.

As usual, my distribution comes with precompiled binaries for MacOSX and Linux and a plugin for MovableType.

In the process of reviewing the features of itex2MML, I made a slew of bugfixes/improvements. So it’s time to release itex2MML 1.1.2 with the following improvements.

• \backslash is now mapped correctly.
• \Box was added as a synonym for \square.
• Several characters were changed from <mo>s to <mi>s.
• Wide and regular-width accents now work as expected (well, except for this bug with \overline).
• Corrected support for \pmod and add support for \mod.
• Status-line messages in \toggle were a NOOP. Dropped.
• Fixed the \href command so that it will play nice with the W3C Validator, and hence will work here¹. If you downloaded itex2MML in the past few hours, download it again, to get the fixed-up version (sorry 'bout that!).

Previously, I wrote about AdS/CFT computations of the jet-quenching parameter. A related quantity is the energy loss of a heavy quark moving through the quark-gluon plasma. Herzog, Karch, Kovtun, Kozcaz and Yaffe have a beautiful recent paper treating this latter problem.

Ben Trott probably has other, more high-flying things to think about than a bug in a (probably long-forgotten) Perl Module. But Austin Frank recently uncovered an obscure bug in Crypt::OpenPGP, which we use here to verify PGP-signed comments.

Most people, these days, have a DSA primary key (used for signing) and an El-Gamal subkey (used for encryption). Austin has an RSA primary key and DSA (signing) and El-Gamal (encryption) subkeys. Nothing wrong with that and, as far as I can tell, GnuPG handles such keys just fine. But Crypt::OpenPGP seems to barf on any message signed by Austin, leading me to suspect a bug in its handling of keys of this sort.

I’ve filed a bug report, but I’m not holding my breath.

It’s been brought to my attention that some people have recently been encountering an INTERNAL SERVER ERROR when attempting to comment here (or at the String Coffee Table).

The cause, alas, is my determination to be overly clever.

Several very interesting recent papers applying AdS/CFT techniques to study properties of the quark-gluon plasma, as seen at RHIC (see this post for some earlier applications of AdS/CFT to RHIC). I’ll talk about two here, and two in my next post.

The major Telecoms have been in the news a lot lately. And not in the most flattering of ways. “Is there a connection?” you ask. I didn’t see one. But Evan did.

In $4k+2$ dimensions, with Minkowski signature, The Hodge $*$-operator, $*^2=1$, when acting on $\Omega^{2k+1}$. One therefore has theories with $2k$-form gauge fields, whose $2k+1$-form field strength is (anti)self-dual. The classic examples are the chiral scalar in 2 dimensions and the 4-form gauge field of type IIB supergravity in 10 dimensions.

There isn’t a wholly satisfactory action principle for such fields, which make constructing the quantum theory somewhat less than straightforward. Recently, Moore and Belov came out with a beautiful paper on constructing an action principle for such fields and its connection to spin-Chern-Simons Theory in $4k+3$ dimensions.

One of the novelties (for me, at least), in writing a paper in a well-worn field like precision electroweak measurements, is that a lot of people have a prior stake in the subject. Our recent paper got a flood of very positive comments.

All very gratifying. But, at the end of last week, came one comment which made us realize that our bounds (from forward-scattering dispersion relations) could be considerably strengthened. So we spent most of the intervening week redoing our analysis, and rewriting our paper to incorporate the new, stronger bounds.

A while back, I wrote a post about the accessibility (or lack thereof) of mathematical text in PDF documents.

There is, apparently, a Working Group, trying to develop a standard for accessible PDF documents. Joe Clark decided that they needed some input from yours truly, and arranged for me to join one of their conference calls this week, and give some further input on their Wiki.

Someday, an accessible PDF document will be one which is properly tagged, and which, for each equation, contains an embedded MathML equivalent (in the same way that LaTeXit.app produces PDF files with embedded TeX source code). The MathML could then be used by assistive technology to produce either mathematical braille or voice.

Today, alas, pdftex doesn’t do tags (much less output embedded MathML), and the arXivs don’t use pdftex. So, like MathML itself, this is all a bit pie-in-the-sky.

Oh, wait …

A very pretty paper, today, by Gray, He, Jejjala and Nelson. They study the algebraic geometry of the space of supersymmetric vacua of $\mathcal{N}=1$ supersymmetric gauge theories.

The obvious (and well-known) statement is that the solution to the F-flatness conditions is an affine algebraic variety, $\mathcal{F}\subset \mathbb{C}^n$. The ring of functions on this variety is $$F = \mathbb{C}[\phi_1,\dots,\phi_n]/\langle\partial_i W\rangle$$ where $\langle\partial_i W\rangle$ denotes the ideal generated by the derivatives of the superpotential. Imposing D-flatness corresponds to the symplectic reduction, or GIT quotient $$\mathcal{F}//G \simeq \mathcal{F}/G_{\mathbb{C}}$$

Their application of interest is the MSSM. Needless to say, finding a Gröbner basis for this complicated ideal in $\mathbb{C}^{49}$ is a somewhat formidable computational task. So they end up looking at various simplified versions of the problem.

Setting all the colour-triplet fields to zero (that is, keeping only $H$,$\tilde{H}$, $\tilde{E}_i$, $L_i$), and turning on the most general quartic terms in the superpotential compatible with R-parity, they find that the vacuum manifold for this truncated theory is an affine cone over the Veronese surface (the degree-2 embedding of $\mathbb{P}^2$ in $\mathbb{P}^5$)¹.

It’s not at all clear what the significance of this observation is, though they do find other examples where the same Veronese structure arises, provided one is willing to tune the superpotential in various ways.

What would be really nice is a characterization of supersymmetry-breaking which allows one to carry over at least some of the techniques of algebraic geometry (which apply so beautifully to the supersymmetric case). Which reminds me that I ought to think some more about some of the obvious extensions of my old paper with Varadarajan, which explores similar questions, using similar techniques.