Modular Group

The modular group is PSL(2,Z) which naturally acts on the upper half plane/the Poincare unit disk.

The standard generators are S and T. S sends z to -1/z and T sends z to z+1. The relations are S^2=I and (ST)^3=I. Calling U=ST we see it is generated by S and U which have no relations between them. Therefore the group is the free product of S's and U's.
Now for something totally different. Braids on 3 strands. Define a=s1*s2*s1 and b=s1*s2 where si is take the ith strand over the i+1st. Using the braid relations we get a^2=b^3. This element is in the center. Just try taking the commutator with s1 and s2. Quotienting by this cyclic subgroup gives a group with a and b with relations a^2=I and b^3=I. Precisely the same one we had before. We also see there is nothing else in the center. So B3 mod center = PSL(2,Z).

The PSL(2,R) can be easily identified by looking at the defining representation of these groups and then centrally extending both top and bottom.

So we know it acts on points of the upper half plane. Therefore it can act on functions on that space too. Just replace all the z's with gz for some group element g.

Some of these functions are particularly recurrent like the eta function which rears it head all the gorram time. Raising it to the 24th power (for the 24 transverse directions to the bosonic string) and calling that delta gives.



Replace that 12 with arbitrary k and you get general modular forms. If you chose k=0 you get nothing but the constants. There aren't any other functions that are invariant under the group. You have to pay the cost of that (cz+d) factor, but at least it's not that costly.

Some k's have modular forms some don't. They must be even natural numbers and at least 4. In fact even cooler is the fact that you only need two modular forms and you can get the others by polynomials. These two are the first two of the Eisenstein series.

You get a lot of identities with these See here

The key way to work with these functions is to Fourier transform from the tau variable to q



This turns them into Taylor series in q which converge because tau being in the upper half plane means q will be inside the unit disk. The numbers that appear as coefficients are spectacular.

The ones in the Eisenstein series give the summatory function which adds up powers of the divisors of n.



The ones for the inverse eta function after you get rid of that 24th root of q is the partition function. The partition function is cool. Partitions label representations of SU(N) and the symmetric group. So this function counts representations with a fixed number of boxes in the Young Tableaux.