However heinous it is to make a living selling pubic vermin over the internet, it is somehow even more despicable to take people’s money and then NOT send them pubic lice.

Did all of this exist before the Web? Or have people just gotten a whole lot weirder in the past 15 years?

Probably everyone else knew, but I was pleased to learn that Dmitry Podolsky has a new blog. Dmitry’s main focus is on cosmology (he was a student of Starobinski), but his blog runs the gamut of subjects, and he’s been churning out posts of very high quality. His latest is on the limits of validity of cosmological perturbation theory, a subject which has seen several interesting papers, since I last blogged about it.

Adam FalkowskiJester has a scathing review of a CERN seminar/recent paper by John Moffat. Moffat wants to avoid introducing a Higgs (or other new degrees of freedom) into the Standard Model, by having the theory become nonlocal at a scale of about a TeV (more precisely, at ${\Lambda }_W=541.189$ GeV (!)). Nonlocality is a sort of magic pixie dust that makes all of the obvious problems go away. The scattering amplitude for longitudinal W-bosons grows like $s$, violating the unitarity bound above a TeV or so? No problem: in Moffat’s nonlocal theory, the amplitude just vanishes for $s\gtrsim 1$ TeV. This, in turn, violates the Cerulus-Martin bound¹, $\mid A(s,\cos \theta )\mid \ge e^{-f(\theta )\sqrt{s}\log (s)}$? Don’t worry …

I suppose I could go on in this vein, but someone will doubtless come along and accuse me of bias. Suffice to say that introducing nonlocality in some willy-nilly fashion like this is bad mojo. And, even were it totally unfair, Jester’s account is wittier than mine.

So I decided to run a memory test:

% sudo memtester 2048 1
memtester version 4.0.8 (32-bit)
Copyright (C) 2007 Charles Cazabon.
Licensed under the GNU General Public License version 2 (only).

pagesize is 4096
pagesizemask is 0xfffffffffffff000
want 2048MB (2147483648 bytes)
got  2048MB (2147483648 bytes), trying mlock ...locked.
Loop 1/1:
 Stuck Address       : testing   1FAILURE: possible bad address line at offset 0x10ec6907.
Skipping to next test...
 Random Value        : ok
 Compare XOR         : ok
 Compare SUB         : ok
 Compare MUL         : ok
 Compare DIV         : ok
 Compare OR          : ok
 Compare AND         : ok
 Sequential Increment: ok
 Solid Bits          : testing   2FAILURE: 0x00000000 != 0x01000000 at offset 0x00ec6907.
 Block Sequential    : testing  14FAILURE: 0x0e0e0e0e != 0x0f0e0e0e at offset 0x00ec6907.
 Checkerboard        : testing   4FAILURE: 0xaaaaaaaa != 0xabaaaaaa at offset 0x00ec6907.
 Bit Spread          : testing  22FAILURE: 0xfebfffff != 0xffbfffff at offset 0x00ec6907.
 Bit Flip            : testing   4FAILURE: 0x00000001 != 0x01000001 at offset 0x00ec6907.
 Walking Ones        : testing  39FAILURE: 0xfeffffff != 0xffffffff at offset 0x00ec6907.
 Walking Zeroes      : testing  26FAILURE: 0x04000000 != 0x05000000 at offset 0x00ec6907.

Done.

and discovered that there’s a stuck bit on one of the memory modules.

Things should be better now (no more random crashes) with new RAM installed. One thing I discovered along the way was that the version of memtester that comes with Fink is incapable, at least on PPC hardware, of testing more than 2GB of RAM. (Use a number larger than 2048 in the above, and you get a slew of malloc errors.)

The thing to do is download the current source, apply this patch and type

% make
% sudo make install

This will build a Universal binary, which is 64bit-enabled on G5 (and Intel Core 2/Xeon) hardware.

That’s good, because — when all the dust settles — golem will have 6.5GB of RAM. Now, if only my own memory were similarly upgradable…

I was remiss (read: lazy, overworked, or whatever) in not writing, earlier, about Meade, Seiberg and Shih’s paper on gauge mediation. But Patrick Meade was visting this week, so perhaps I can make amends.

There is a huge literature on models of gauge mediated supersymmetry-breaking. And there are a variety of characteristic predictions that emerge from particular classes of models. What these guys do is provide a model-independent characterization of gauge-mediation and try to isolate what features are generic to all models versus those which are special to particular subclasses of models of gauge mediation.

Their characterization is very simple: in gauge mediation, there the MSSM sector and a hidden sector, $S$, in which supersymetry is dynamically broken. When the MSSM gauge couplings, ${\alpha }_r,r=\mathrm{1,2,3}$, are set to zero, the two sectors decouple and the MSSM sector is supersymmetric.

The coupling between the two sectors is described by gauging an $\mathrm{SU}(3)\times \mathrm{SU}(2)\times U(1)$ subgroup of the global symmetry group of $S$. The corresponding conserved current(s), $j_{\mu }$, is part of supermultiplet, a real linear linear multiplet, $$\begin{array}{c}{D}^2𝒥={\overline{D}}^2𝒥=0\\ 𝒥=J+i\theta j-i\overline{\theta }\overline{j}-\theta {\sigma }^{\mu }\overline{\theta }{j}_{\mu }+\frac{1}{2}\theta \theta \overline{\theta }{\overline{\sigma }}^{\mu }{\partial }_{\mu }j-\frac{1}{2}\overline{\theta }\overline{\theta }\theta {\sigma }^{\mu }{\partial }_{\mu }\overline{j}-\frac{1}{4}\theta \theta \overline{\theta }\overline{\theta }\square J\end{array}$$

The physics of gauge mediation is governed by the two-point functions

Here, $M$ is a mass scale characterizing physics in the hidden sector, and $\Lambda$ is a UV cutoff, to regulate the short-distance singularity in the 2-point function.

For the case of an abelian symmetry, there’s also the possibility of a 1-point function, $$\langle J(p)\rangle ={(2\pi )}^4{\delta }^{(4)}(p)\zeta$$ But, for various phenomenological reasons, it’s best to assume that the hidden sector has a $𝒥\to -𝒥$, which enforces $\zeta =0$.

In the supersymmetric limit,

Since supersymmetry is restored in the large-$p$ limit, this means that

for some constant, $c$, independent of the spin, and that $B$ is UV-finite. The divergent part of $C_1$ gives the contribution of the hidden sector to the $\beta$-function of the gauge coupling. Specifically, the shift in the 1-loop $\beta$-function coefficient from integrating out the hidden sector is $$b_{\text{high}}=b_{\text{low}}-16{\pi }^{2}c$$ In a model which preserves coupling unification, this means that all of the $c^{(r)},r=\mathrm{1,2,3}$ are equal.

The contribution to the gaugino masses comes from a tree-level insertion of the $\langle j_{\alpha }{j}_{\beta }\rangle$ two point function, represented by the magenta blob: $$\begin{array}{cc} Tree Graph, with insertion of B & \Rightarrow m^{(r)}=g_r^{2}M{B}^{(r)}(0)\end{array}$$

The contributions to the sfermion masses come from one-loop diagrams with an insertion of the two-point functions (1), $C_s^{(r)}$, represented by the red ($s=0$), yellow ($s=1/2$) and blue ($s=1$) blobs, respectively: $$\begin{array}{c}\begin{array}{ccccc} Loop Graph, with insertion of C_0 & +& Loop Graph, with insertion of C_{1/2} & +& \\ Loop Graph, with insertion of C_1 & +& Another Loop Graph, with insertion of C_1 & & \Rightarrow \end{array}\\ m_f^2=\sum _{r=1}^3{g}_r^4{c}_2(f;r)A^{(r)}\end{array}$$ where $c_2(f;r)$ is the quadratic Casimir in representation $f$ and

Note that the integrand in (4) is UV-finite, and vanishes in the supersymmetric limit, as a consequence of (2),(3). The sfermion masses obey two sum rules $$\begin{array}{c}\mathrm{Tr}Y{m}^2=0\\ \mathrm{Tr}(B-L)m^2=0\end{array}$$ or, more concretely, $$\begin{array}{c}{m}_Q^2-2m_U^2+m_D^2-m_L^2+m_E^2=0\\ 2m_Q^2-m_U^2-m_D^2-2m_L^2+m_E^2=0\end{array}$$ but are otherwise arbitrary.

And, since the $B^{(r)}$ have no a-priori relation to the $C_s^{(r)}$, there is, in this formalism, no prediction for gaugino mass unification.

Almost all concrete models, including those that come from String Theory, have more structure. In particular, supersymmetry is often broken at a scale much lower than $M$. That is, there’s a small parameter, $F/M^2$. And, at least at leading order in $F/M^2$, the supersymmetry-violating bits of the $C_s$ and $B$ seem to be related. This is what gives things like gaugino mass unification, and the prediction that the NLSP is a Bino or a stau.

They are, according to Patrick, hard at work trying to incorporate this feature into their analysis.

I committed an update to the Instiki BZR repository, and then did a

bzr log -v

which yielded an ominous

⋮
KnitCorrupt: Knit <bzrlib.knit._KnitAccess object at 0x24b6170> corrupt:
While reading {distler@golem.ph.utexas.edu-20080108060135-7ujf0nen62ge328t}
got IOError(CRC check failed 3032481332 2792320114)

Yikes!

The .bzr directory is a labyrinth of plain text and gzip-compressed files. Evidently one of the latter was corrupted. But which?

find . -name '*.kndx' -print0 \
 | xargs -0 grep -l 'distler@golem.ph.utexas.edu-20080108060135-7ujf0nen62ge328t' \
 | sed -e 's/\.kndx/.knit/' | xargs zcat >/dev/null

failed on the file .bzr/repository/revisions.knit. Replace the whole file from backups? Not ideal, since I’d just made a commit. Further experimentation revealed that

bzr log -v -r1..204

and

bzr log -v -r206..231

worked fine. But anything including revision 205 yielded the above CRC error.

How to fix it?

The first thing to understand is the the .knit file is a concatenation of gzipped files. Looking in .bzr/repository/revisions.kndx, I found the line

distler@golem.ph.utexas.edu-20080108060135-7ujf0nen62ge328t fulltext 94495 460 230 :

This says that the gzip file corresponding to the revision in question is 460 bytes long, and starts at offset 94495 from the beginning of the file.

So I split the file into 3 pieces, corresponding to revision 1–204, the troublesome revision 205, and revisions 206-231. Now to find a replacement for the bad piece. If I had been at the office, I would have tried retrieving something from backup tapes. As it was, it was more convenient to poke around in the BZR repository on my home machine. I extracted the corresponding file. Oddly, it was 466 bytes long. Ungzipping it, it looked like

version distler@golem.ph.utexas.edu-20080108060135-7ujf0nen62ge328t 11 eead9ea844a899fc58fd9b9d15a2dbe20226549a
<revision committer="Jacques Distler &lt;distler@golem.ph.utexas.edu&gt;" format="5" inventory_sha1="be93437c54bcbb9ff6a3c73ffe9a50a835513ae5" revision_id="distler@golem.ph.utexas.edu-20080108060135-7ujf0nen62ge328t" timestamp="1199772095.276" timezone="-21600">
<message>Update to latest HTML5lib, Add Maruku testdir
Sync with the latest html5lib.
Having the Maruku unit tests on-hand may be useful for debugging; so let's include them.
</message>
<parents>
<revision_ref revision_id="distler@golem.ph.utexas.edu-20080103212703-037sbbvkyntk6mqs" />
</parents>
<properties><property name="branch-nick">svn</property>
</properties>
</revision>
end distler@golem.ph.utexas.edu-20080108060135-7ujf0nen62ge328t

which looked just fine. So I cat‘ed the three files together, shifted the offsets of each of the subsequent lines in .bzr/repository/revisions.kndx by 6 bytes and changed the length of this one from “460” to “466”. Then I moved the new revisions.knit and revisions.kndx files into position, and crossed my fingers.

Problem solved!

But, surely, surely there must be a better way.

As anyone who’s been following this weblog is doubtless bored to tears to hear, serving XHTML is a complicated, finicky, business, requiring jumping through elaborate hoops to ensure well-formedness. It would be so much easier to serve this content as text/html, and rely on the liberal parsing of the HTML parser. Hence it’s very cool that future browsers will support precisely that. As far as I can tell, the only change that would be required, here, is to send the SVG unprefixed. But, since the prefixing is done programmatically (to keep the MTValidate plugin happy), this would be a very easy change. Instiki already emits unprefixed SVG.

I say “as far as I can tell,” because there are no implementations of this days-old addition to the Specification to test against. Eventually, there will be, though I wonder how MathPlayer would handle a change to text/html. There would be a grim irony if IE+MathPlayer became the only browser which needed to be sent application/xhtml+xml.

I look forward to the day when this blog becomes a dinosaur. And, now that MathML is part of the the HTML Spec, I look forward to some more browser implementations.

With a few minor tweaks, this worked quite satisfactorily for over a decade, until the arXivs introduced their new identifier format. The change in format made it rather tricky to craft something that would work seamlessly with both old- and new-style identifiers. And SPIRES’s lackadaisical implementation (lumping everything into the eprint field) didn’t help.

I no longer knew who to contact at SPIRES, and folks at the arXivs didn’t seem too interested in taking up the issue. So things languished … for a year.

But, then, last week, the arXiv Admins contacted me, and the ball started rolling. After some back-and forth discussions, Travis Brooks implemented the new scheme at SPIRES. So, just in time for the 1^st anniversary of the new arXiv identifier scheme, there’s a new version of utphys.bst.

The following (optional) fields are recognized:

archive

A Base-URL (defaults to “http://arxiv.org/abs”, if absent).

eprint

The eprint identifier. For an old-style eprint, this would be something like

eprint = "hep-th/9605023"

For a new-style eprint, this would be something like

eprint = "0707.3168"

primaryClass

The primary classification of new-style eprints; should be omitted for old-style eprints.

archivePrefix

The “archive prefix,” usually, the string “arXiv”.

Here are some examples.

• Old-style arXiv identifier (the previous behaviour):

  eprint = "hep-th/9605023",

  produces

  \href{http://arxiv.org/abs/hep-th/9605023}{{\tt hep-th/9605023}}

• New-style arXiv identifier:

  archivePrefix = "arXiv",
  eprint = "0707.3168",
  primaryClass = "hep-th",

  produces

  \href{http://arxiv.org/abs/0707.3168}{{\tt arXiv:0707.3168 [hep-th]}}

• Old-style arXiv identifier, with a prefix:

  archivePrefix = "arXiv",
  eprint = "hep-th/9605023",

  produces

  \href{http://arxiv.org/abs/hep-th/9605023}{{\tt arXiv:hep-th/9605023}}

• A different eprint archive

  
  archive = "http://cogprints.org",
  eprint = "5542",
  archivePrefix = "Cogprints",

  produces

  \href{http://cogprints.org/5542}{{\tt Cogprints:5542}} 

• Yet another eprint archive

  
  archive = "http://www.ncbi.nlm.nih.gov/pubmed",
  eprint = "2277438",
  archivePrefix = "PMID",

  produces

  \href{http://www.ncbi.nlm.nih.gov/pubmed/2277438}{{\tt PMID:2277438}} 

Enjoy version 2.0. Sorry it took this long to get it out the door.

@Misc{Distler:Lisi1,
   author = "Distler, Jacques",
   title = "A Little Group Theory",
   howpublished ="weblog entry",
   month = "November",
   year = "2007",
   url = "http://golem.ph.utexas.edu/~distler/blog/archives/001505.html",
   note = "See also \cite{Distler:Lisi2}."
}

@Misc{Distler:Lisi2,
   author = "Distler, Jacques",
   title = "A Little More Group Theory",
   howpublished ="weblog entry",
   month = "December",
   year = "2007",
   url = "http://golem.ph.utexas.edu/~distler/blog/archives/001532.html",
   note = "Followup to \cite{Distler:Lisi1}."
}

Of course, no one has the slightest interest in citing those two particular posts, but you get the idea …

Enjoy version 2.1.

Update (4/7/2008): DOI Support

Niklas Beisert suggested adding DOI support. So, in version 2.2, a bibtex entry, like

@Article{Distler:2006if,
     author    = "Distler, Jacques and Grinstein, Benjamin and Porto, Rafael
                  A. and Rothstein, Ira Z.",
     title     = "Falsifying Models of New Physics via {WW} Scattering",
     journal   = "Phys. Rev. Lett.",
     volume    = "98",
     year      = "2007",
     pages     = "041601",
     eprint    = "hep-ph/0604255",
     doi       = "10.1103/PhysRevLett.98.041601",
     SLACcitation  = "%%CITATION = HEP-PH/0604255;%%"
}

turns the journal-reference into a clickable link to the online-journal version of the paper. SPIRES outputs a doi field for all published papers for which a DOI identifier is available.

The low-energy theory on a (stack of) D2-brane(s) is a maximally-supersymmetric gauge theory in 2+1 dimensions. The Yang-Mill multiplet has a gauge field and 7 real scalars in the adjoint representation. At least if you are out on the Coulomb branch, where the gauge symmetry is Higgsed down to the Cartan, you can dualize the gauge fields to another scalar, which is circle-valued.

The M2-brane is obtained as the strong-coupling limit of this theory. The radius of the circles (one for each element of the Cartan subalgebra) go to infinity, and the $\mathrm{SO}(7)$ R-symmetry is promoted to $\mathrm{SO}(8)$. This strong-coupling limit is superconformal, but the above description is effective only for the free theory, where the M2-branes are separated (away from the origin, the moduli space looks like ${ℝ}^{8n}/S_n$). The theory of coincident M2-branes is an interacting SCFT which, so far, does not have a Lagrangian description. But there’s no theorem that rules out a Lagrangian description, so there may just be one.

Bagger and Lambert recently proposed a very interesting maximally supersymmetric interacting 2+1D Lagrangian field theory which, at least classically, seems to be superconformal. It does not arise as the dimensional reduction of some higher dimensional theory, and so it was missed in previous attempts at tackling this problem.

I never got around to blogging about Bagger and Lambert’s paper, but Bandres, Lipstein and Schwarz wrote a nice followup, which gives me an excuse to return to the subject.

Let me start with some notational preliminaries. The connected part of the Lorentz group in 2+1 dimensions is $\mathrm{SL}(2,ℝ)$, so we’ll use a Wess and Bagger-like notation for spinors¹. Spinor indices are raised and lowered using the 2-component $ϵ$-symbol, ${ϵ}_{\alpha \beta }=-{ϵ}_{\beta \alpha }$. The $\gamma$-matrices are ${\sigma }_{\alpha \beta }^{\mu }={\sigma }_{\beta \alpha }^{\mu }$, and are real. After raising an index using ${ϵ}^{\alpha \beta }$, we can write $({\gamma }^{\mu })^{\alpha }{}_{\beta }={ϵ}^{\alpha \delta }{\sigma }_{\delta \beta }^{\mu }$ in terms of Pauli matrices, with ${\gamma }^0=i{\sigma }_2$.

Also real are the $\mathrm{SO}(8)$ $\gamma$-matrices, which we can regard as a trilinear form $$\Gamma :V\otimes S\otimes C\to ℝ$$ where $V$, $S$, and $C$ are the three 8-dimensional real representations of $\mathrm{Spin}(8)$. Alternatively, letting “$I$” be an $\mathrm{SO}(8)$ vector index, we think of $${\Gamma }^I:C\to S$$ We’ll also need $$\begin{array}{rl}{\Gamma }^{IJ}& ={\tilde{\Gamma }}^{[I}{\Gamma }^{J]}:C\to C\\ {\Gamma }^{IJK}& ={\Gamma }^{[I}{\tilde{\Gamma }}^J{\Gamma }^{K]}:C\to S\end{array}$$ where ${\tilde{\Gamma }}^I={\left({\Gamma }^I\right)}^t:S\to C$.

What Bagger and Lambert do is introduce an auxiliary vector space, $W$ (really, a vector bundle, which we will take to be trivial). $W$ is endowed with a positive definite inner product, $$(\cdot ,\cdot ):{Sym}^2(W)\to ℝ$$ and a skew-symmetric quadrilinear form, $$f:{\wedge }^{4}W\to ℝ$$ Alternatively, using the metric, we can regard $f$ as a trilinear product

The fields of the model are $$\begin{array}{rl}\varphi & \text{a scalar field taking values in}W\otimes V\\ {\psi }_{\alpha }& \text{a spinor field taking values in}W\otimes C\\ A_{\mu }& \text{a}G\subset \mathrm{SO}(W)\text{gauge connection}\\ \end{array}$$

Now, $\mathrm{so}(W)\simeq {\wedge }^{2}W$ has the usual action on $W$,

But, because we have (1), we can contemplate an “exotic” action,

We demand that $f$ be invariant under this action, which amounts to requiring

Then the antisymmetry of $f$ ensures that the inner product, $(\cdot ,\cdot )$ is also invariant.

Unfortunately, (3) is not really satisfactory. The action of $\mathrm{so}(W)$ must satisfy the the Lie-algebra relations,

so it would make sense, for instance, to gauge that symmetry. If we try to impose that (3) satisfy (5), this would require

which does not hold in any of the known solutions.

The closest one comes is when $W$ is 4-dimensional. The quadrilinear form, $f$, is just the 4-index $ϵ$-symbol, and ${\wedge }^{2}W$ is decomposable into self-dual and anti-self-dual forms $${\wedge }^{2}W={\wedge }_{+}^{2}W\oplus {\wedge }_{-}^{2}W$$ with respect to this form. The action of ${\wedge }_{-}^{2}W$ on $W$ is the one induced from (3), while the action of ${\wedge }_{+}^{2}W$ on $W$ is the one induced from $$\alpha \wedge \beta :X\mapsto X-[\alpha ,\beta ,X]$$ These signs are made slightly more transparent by mapping ${ℝ}^4$ to the quaternions $$X=\frac{1}{2}\left(\begin{array}{cc}{X}_4+i{X}_3& i{X}_1-X_2\\ i{X}_1+X_2& X_4-i{X}_3\end{array}\right)$$ where we’ve represented the unit quaternions by $i$ times the Pauli matrices. In this notation, the inner product is $$\frac{1}{2}(X,Y)=\mathrm{Tr}(X^{†}Y)$$ The matrix representing $[X,Y,Z]$ is $$\begin{array}{rl}[X,Y,Z]\simeq & \frac{4}{3}(X{Y}^{†}Z+Y{Z}^{†}X+Z{X}^{†}Y\\ & -(Z{Y}^{†}X+Y{X}^{†}Z+X{Z}^{†}Y))\end{array}$$ In this notation, $\mathrm{SO}(4)\sim \mathrm{SU}(2{)}_L\times \mathrm{SU}(2{)}_R$ acts by conjugation $$X\mapsto g_{L}X{g}_R^{-1}$$ whereas (3) would have corresponded to $X\mapsto g_{L}X{g}_R$, which fails to satisfy the group law for $\mathrm{SU}(2{)}_R$.

Anyway, introducing the covariant derivative², $$D_{\mu }X={\partial }_{\mu }X+A_{\mu }^{L}X-X{A}_{\mu }^R$$ we can write the action as

where $k\in \mathbb{Z}$,

and

is the Chern-Simons action at level-1 for $\mathrm{SU}(2{)}_L$ and level-($-1$) for $\mathrm{SU}(2{)}_R$.

The supercharges are spacetime spinors in the $S$ of $\mathrm{Spin}(8)$, and the supersymmetry variations

While it’s probably unsurprising, Bandres et al verify that the rest of the $\mathrm{OSp}(8\mid 4)$ superconformal algebra holds at the classical level, as well. Perhaps a good challenge for our ERGE friends would be to check that (7) is superconformal at the quantum level.

Parity is implemented, in this theory, in a slightly nonstandard way: accompanying a reflection in one of the spatial coordinates, is an exchange of the two $\mathrm{SU}(2)$s and Hermitian conjugation on the matrices $\Phi$ and $\Psi$, $$\begin{array}{c}{A}^L\leftrightarrow A^R\\ {\Phi }^I\to {\Phi }^{I†}\\ \Psi \to {\gamma }^1{\Psi }^{†}\end{array}$$ A Majorana mass term, $\psi \psi$ is a pseudo-scalar in 2+1 dimensions which, as noted by Bandres et al accounts the requisite sign in the transformation under parity of the second line of (8).

Are there other realizations, where we demand that (3), (4), (5) hold for just some subalgebra, $𝔤\subset \mathrm{so}(W)$? An obvious guess would be to replace $W=ℍ$ by $W=ℍ(n)$, the space of $n\times n$ matrices of quaternions. Bandres et al looked for other, more nontrivial, examples, but didn’t find any.

Update: Moduli Space

Van Raamsdonk points out that the moduli space — the space of zeroes of the scalar potential in (8), modulo $\mathrm{SO}(4)$ gauge transformations — of the Bagger-Lambert model is $({ℝ}^8\times {ℝ}^8)/\mathrm{SO}(2)$. This is because a zero of the potential requires that all 8 of the ${\varphi }^I$ lie in a common 2-plane in $W$. Configurations which differ by a rotation within that 2-plane are gauge-equivalent, so the moduli space seems to be $V\otimes {ℝ}^2/\mathrm{SO}(2)$. Rotations in the orthogonal 2-plane comprise a residual unbroken $\mathrm{SO}(2)$ gauge symmetry, which does does not act on the moduli space.

This looks like a puzzle, because it’s hard to see how the bosonic spectrum (15 massless scalars and a nondynamical $U(1)$ gauge field) could be $N=8$ supersymmetric. Fortunately, Mukhi and Papageorgakis ride to the rescue. They show that, when one integrates out the massive modes, the $U(1)$ gauge field actually becomes dynamical³. A dynamical $U(1)$ gauge field can be dualized to a circle-valued scalar, so the full moduli space is $$\frac{{ℝ}^8\times {ℝ}^8}{\mathrm{SO}(2)}\times S^1$$ which is the desired answer for the moduli space of a pair of M2-branes. (If I were a little more energetic, I would attempt to show that this actually gets the discrete identifications right, and that the moduli space is $({ℝ}^8\times {ℝ}^8)/{\mathbb{Z}}_2$.)

If you’ve tried reading this blog in Safari, or on an iPhone, you know how great that would be.

Just don’t let someone cajole you into settling for a stylesheet.

Back in 2000, in my pre-blog era, I decided to compile a few statistics to test out these assertions. Now that 2008 has rolled around, I’ve updated my spreadsheet to include the George W Bush era, and answer a few of the objections to the previous, not-widely-circulated, version.

I decided to look at two items: budget deficits and real GDP growth. The historical data goes back to 1930. And I did the simplest possible thing: separate out the time-series into Republican and Democratic Administrations, computing the average annual budget deficit, and the average annual real GDP growth for each.

Now, there are several immediate objections you could raise.

• There are, to be sure, lots of exogenous factors which influence economic performance. In any given year, one can ascribe performance to something other than who occupies the White House. But that’s where the law of large number comes into play. If you average over many years, these exogenous factors should cancel out. The longer the historical baseline, the more likely it is that you’re seeing a real “inter-party” effect.
• That said, the Great Depression and World War II were truly singular events with a dramatic effect on these averages. In 1932, real GDP contracted 13%. Perhaps it’s unfair to blame the Great Depression on the Republicans. By the same token, real GDP contracted 11% in 1946, in the great post-War contraction. It would be equally unfair to blame that on the Democrats. On the deficit side, the cost of waging WWII was extraordinary. The On-Budget Deficit in 1943 was an eye-popping 30.8% of GDP. For both of these reason, you might not want to take the first two rows, in each of the tables below, too seriously.
• Less obvious, but equally salient, you probably should assign the performance during the first year of each Administration to the previous one. Arguably, the economic policies of the Administration only really begin to kick in its second year. I’ve presented the data both ways.

First, let’s look at the Deficit. The data comes from the OMB historical tables accompanying the FY 2008 Budget. I tallied the On-Budget Deficit, in constant (FY 2000) dollars, and as a percentage of GDP. The deflators, needed for the former, are only available for 1940-2007.

So, how did the fiscally-prudent Republicans do?

Setting aside the huge (as a percentage of GDP) outlays during WWII, the Democrats beat them like a drum. In any of the three postwar periods tabulated, the Republicans consistently show themselves to be the party of fiscal irresponsibility, racking up deficits which dwarf those of the Democrats, both in absolute terms and as a fraction of GDP.

Ascribing the first year of each Administration to the previous one changes the numbers only a little, tipping them ever-so-slightly further in favour of the Democrats.

But what about economic growth? Surely, the “party of business” is better at stimulating economic growth. This time, the statistics come from the Commerce Department’s Bureau of Economic Analysis. I tabulated real GDP growth during Republican and Democratic Administrations, and annualized the results.

Again, Democratic Adminstrations outstrip their Republican counterparts by 1%/year or more. It’s an astonishing, and astonishingly persistent difference.

But, hey, as my Republican friends like to point out, Bill Clinton really should get full blame for the 2001 recession. So let’s tabulate the same numbers, but ascribing the first year of each Administration’s performance to the previous Administration.

Only in the 1970-2007 period does the gap narrow much, but even there, the Democrats retain their lead.

My recommendation to John McCain: lay off the arguments about economic stewardship. You’re sure to lose that fight. Stick to your strongest argument: that Obama is a crypto-Muslim terrorist-lover, who hates America. That’s sure to win in November.

On average, families at the 95th percentile of the income distribution have experienced identical income growth under Democratic and Republican presidents, while those at the 20th percentile have experienced more than four times as much income growth under Democrats as they have under Republicans. These differences are attributable to partisan differences in unemployment (which has been 30 percent lower under Democratic presidents, on average) and GDP growth (which has been 30 percent higher under Democratic presidents, on average); both unemployment and GDP growth have much stronger effects on income growth at the bottom of the income distribution than at the top

Bottom line: unless your annual household income is greater that $174,000 (top 5%, in 2006), don’t even think about voting Republican.

Which means that Instiki gets more than its share of attention from those interested in the question of whether XHTML is suitable for the Web.

Philip Taylor has been tireless in poking holes in various peoples’ XHTML implementations. Recently, Philip found a pair of issues in Instiki. Both were quickly fixed, but they illustrate my general maxim that any instance of a well-formedness issue is very likely an XSS issue as well.

Of the two issues that Philip found, the more serious one had to do with the author IP Address displayed at the bottom of each wiki page, next to the author’s name. What could be dangerous about an IP Address?, you ask. Well, in this case, it’s generated using Rails’s request.remote_ip method. And that, in turn, uses the HTTP Client-Ip header, if one has been set.

Install, say, Firefox’s Modify Headers extension, and you can set the Client-Ip header to whatever the heck you want. As Philip ably demonstrated, this can make the targeted page ill-formed, but it can equally-well be used to inject an XSS attack.

Arguably, Rails itself should take care that this method returns an actual IP address, rather than arbitrary garbage, but it’s easy enough to fix at the application level.

require 'resolv'

def remote_ip
  ip = request.remote_ip
  logger.info(ip)
  ip.gsub!(Regexp.union(Resolv::IPv4::Regex, Resolv::IPv6::Regex), '\0') || 'bogus address'
end

Anyway, the bottom line is: if you’re using my branch of Instiki, please upgrade immediately to version 0.14pre(MML+).

If you’re using the main branch of Instiki, I have committed the requisite fixes to SVN Source Tree and contacted the maintainer (twice). Presumably, he will roll out a security update.

I’ve been reading Beasley, Heckman and Vafa’s recent 125 page opus, hoping to get through it before the promised Part II comes out.

F-theory is the fancy name for Type IIB string theory with 7-branes. If we compactify on $B$ (for compactifications down to 4 dimensions, we’re interested in $B$ a complex 3-fold), the 7-branes are wrapped on divisors in $B$. The complex IIB coupling, $\tau$, has monodromies as we circle those divisors and, viewing it as the modulus of an elliptic curve, we get the total space of an elliptically-fibered Calabi-Yau 4-fold, $X\to B$.

Except for the case where one has only D7-branes (and orientifold O7 planes), $\mathrm{Im}(\tau )$ cannot be taken to be uniformly large. So perturbative string theory techniques are not applicable. General configurations of 7-branes are hard to study, except in some special cases.

The interest, here, is to study a local model for a wrapped 7-brane, or perhaps a pair of 7-branes intersecting transversally, and study the local physics from the point of view of the twisted SYM theory living on the brane.

In the local model, $B$ is noncompact, and is the total space of the line bundle

where the normal bundle $N_{S\mid B}=K_S^p$, a power of the canonical bundle of the surface, $S$, on which the 7-brane is wrapped. Away from the zero section, we have an elliptic curve (with affine coordinates $x,y$) fibered over the base. Over the zero section, the curve degenerates, and the total space of $X\to S$ looks like an isolated ADE surface singularity fibered over $S$. Denoting the fiber coordinate of $B\to S$ as $z$, the total space of $X\to S$ is given as the locus $\{f(x,y,z)=0\}\subset V$, with

Now, I’m a little confused by the $A_n$ and $D_n$ cases. As written, these are not elliptically-fibered; away from $z=0$, the fiber of $X\to B$ looks to me like a smooth quadric. But the main focus of attention is on the exceptional cases where, indeed, we have a Weierstrass form for the equation of the elliptic fiber.

Now, the idea is that the local physics is captured by the 8D twisted SYM theory on $R^{\mathrm{3,1}}\times S$. The super Yang-Mills multiplet, in 8 dimensions, consists of a $G_S$-connection, a complex scalar in the adjoint representation, and some fermions, also in the adjoint. After twisting, the scalar becomes a 2-form $$\phi \in {\Omega }^{(\mathrm{2,0})}(S,\mathrm{ad}(P)),\overline{\phi }\in {\Omega }^{(\mathrm{0,2})}(S,\mathrm{ad}(P))$$ where $P\to S$ is a principal $G_S$-bundle. The left-handed fermions are $$\begin{array}{rl}{\eta }_{\alpha }& \in \Gamma (\mathrm{ad}(P))\\ {\psi }_{\alpha }& \in {\Omega }^{(\mathrm{0,1})}(S,\mathrm{ad}(P))\\ {\chi }_{\alpha }& \in {\Omega }^{(\mathrm{2,0})}(S,\mathrm{ad}(P))\end{array}$$ and their right-handed conjugates are $$\begin{array}{rl}{\overline{\eta }}_{\dot{\alpha }}& \in \Gamma (\mathrm{ad}(P))\\ {\overline{\psi }}_{\dot{\alpha }}& \in {\Omega }^{(\mathrm{1,0})}(S,\mathrm{ad}(P))\\ {\overline{\chi }}_{\dot{\alpha }}& \in {\Omega }^{(\mathrm{0,2})}(S,\mathrm{ad}(P))\end{array}$$ The conditions for a supersymmetric solution are

Here, $$F_S^{(\mathrm{2,0})}={\left(D_S^{(\mathrm{1,0})}\right)}^2,F_S^{(\mathrm{0,2})}={\left(D_S^{(\mathrm{0,1})}\right)}^2,F_S=\{D_S^{(\mathrm{1,0})},D_S^{(\mathrm{0,1})}\}$$ are, respectively, the $(\mathrm{2,0})$, $(\mathrm{0,2})$ and $(\mathrm{1,1})$ parts of the field strength on $S$, and $\omega$ is the Kähler form.

In the particular case¹ of $\phi =0$ (or, more generally, $[\phi ,\overline{\phi }]=0$), (2), these equations imply the Donaldson-Uhlenbeck-Yau equation, whose solution is an anti-self-dual connection, with field strength $F_S$, for some subgroup $H_S\subset G_S$. Correspondingly, there’s a reduction of the structure group of $P$ from $G_S$ to $H_S$, Denoting by ${\Gamma }_S$, the commutant of $H_S$ in $G_S$, we decompose $$\mathrm{ad}(P)\simeq {\oplus }_i(R_i\otimes {𝒯}_i)$$ where $R_i$ are irreps of