User jimeree - MathOverflow

Seiberg-Witten curve for product SU(2)^N gauge theories

2012-09-29T11:10:00Z

In equation 2.10 of this article, the author gives the Seiberg-Witten curve for a $U(N)$ gauge theory with $L<2N$ massive flavours with masses given by $m_i$ as:

$y^{2}=\left\langle \mathrm{det}\left(z\mathbb{I}-\Phi\right)\right\rangle ^{2}-4\Lambda^{2N-L}\prod_{i=1}^{L}\left(z+m_{i}\right)$

Where $\Lambda$ is the cutoff scale of the theory, and $\Phi$ is the adjoint scalar in the vector multiplet.

Does anyone know what the analogous equation for a product $SU(2)^N$ gauge theory is? There is a discussion of such Seiberg-Witten curves in e.g. this article by Gaiotto, but as far as I can see no explicit equation like the one above.

Are congruence subgroups of the modular group finitely presented?

2012-08-22T13:55:21Z

Are the congruence subgroups of the modular group $\Gamma\equiv\mathrm{PSL}\left(2,\mathbb{Z}\right)$ (e.g. $\Gamma\left(n\right)$, $\Gamma_{0}\left(n\right)$, $\Gamma_{1}\left(n\right)$ etc.) finitely presented? If so, is there a proof of this? Assuming they are finitely presented, are the presentations of e.g. the principal congruence subgroups $\Gamma\left(n\right)$ (for small $n$) documented anywhere? I know they could be computed using the Reidemeister-Schreier process, but it would be nice to have some independent confirmation of the presentations.

Many thanks!

Simplifying presentations of modular subgroups

2012-08-26T17:33:43Z

I've been using the Reidemeister-Schreier process (detailed in e.g. Holt et al. - Handbook of Computational Group Theory) to find the presentations of various modular subgroups. For example, this process tells us that the presentation of the principal congruence subgroup $\Gamma\left(4\right)$ is (before simplification):

$\left\langle \left[a..y\right]|a,d,e,f,g,ho,p,q,kl,s,t,vx,b,c,i,j,m,n,wr,yu\right\rangle$

I want to simplify this expression using Tietze transformations, again as described in Holt. However, doing so seems to give some worrying results. The first simplification is to remove all the "trivial" generators with monadic relators (e.g. $a,d,e,\ldots$). The second is to remove one of the two generators in each of the "double" relators, since the one will simply be the inverse of the other (i.e. delete one of $h$ or $o$, delete one of $k$ or $l$, etc.). However, doing these simplifications seems to leave the following presentation:

$\left\langle h,k,w,x,y|\right\rangle$

And this doesn't look like any good kind of presentation at all! If someone's able to tell me where I'm going wrong here, that would be hugely appreciated.

Thanks!

Finding mappings between j-invariants for Calabi-Yau threefolds

2012-08-16T13:56:25Z

Suppose we have an elliptic curve of the form:

$y^{2}=x^{3}-f\left(z_{1},z_{2}\right)x-g\left(z_{1},z_{2}\right)$

This describes a Calabi-Yau threefold as an elliptic fibration over $\mathbb{P}^{1}\times\mathbb{P}^{1}$, where $\left(z_{1},z_{2}\right)$ are the coordinates of the $\mathbb{P}^{1}\times\mathbb{P}^{1}$; $f$ is a homogenous polynomial of degree 8 in each of the $z_i$; and $g$ is a homogenous polynomial of degree 12 in each of the $z_i$. The $j$-invariant should be:

$j=\frac{4f^{3}}{4f^{3}-27g^{2}}$

Now suppose that this $j$-invariant is also expressed in terms of $t\left(z_{1},z_{2}\right)$. Specifically, I am looking at the Index 36 $j$-invariants on page 5 here: http://mysite.science.uottawa.ca/asebbar/publi/mcse.pdf.

I want to find a substitution of the form:

$t=\frac{P\left(z_{1},z_{2}\right)}{Q\left(z_{1},z_{2}\right)}$

Where $P$ and $Q$ are polynomials of some appropriate degree, such that we reproduce the form the $j\left(f,g\right)$ above. Is there any general procedure for doing this? Or is it just frustrating guesswork?

Many thanks!

Identifying Subgroups of the Modular Group via Permutation Representations on Cosets

2012-08-06T20:40:15Z

Suppose we have a known group G and an unknown subgroup H. The permutation representation of G on the cosets of H gives a permutation group C, which is known. Is it possible to identify the generators of H using this information? If so, how? If not, what further information is needed?

(For background, my G is the modular group PSL(2,Z); H is a not-necessarily-congruence subgroup of G. C is sometimes called a "cartographic group").

Many thanks!

Comment by Jimeree

2012-08-26T20:36:18Z

Yes, you're right. I don't know why I thought the presentation couldn't be right. Anyway, thanks!