Let us now construct the holomorphic functions , satisfying the monodromies determined above. It is again done by using the Seiberg-Witten curve, which is given in this case by
(5.2.1) |
with auxiliary complex variables and , together with the Seiberg-Witten differential
(5.2.2) |
We dropped the subscript from to lighten the notation.
Again, we add a point and regard as a complex coordinate on the sphere . This is the ultraviolet curve. The variable is now a function on it, see Fig. 5.6. Note that is no longer a branch point; indeed, the local behavior of there is now
Note also that has a residue at . The curve is a two-sheeted cover of shown in Fig. 5.7.
We define cycles and as shown, and then the functions and are given by
(5.2.5) |
The proof goes exactly as in the pure case. The curve can be mapped to a parallelogram within a complex plane by , see Fig. 5.8. The poles with residues of are denoted explicitly in the figure. When a closed cycle on the torus winds the cycles times, cycles times, and the poles times, the integral of is then
(5.2.6) |
just as in the BPS mass formula (5.1.4).
Let us check that the curve correctly reproduces the running of the coupling in the weakly-coupled region. For simplicity, set , and assume . We put the cycle at . We easily find
(5.2.7) |
as before. As for the integral, two branch points are around and one branch point is around . The dominant contribution to the integral is then
(5.2.8) |
Then we find
(5.2.9) |
reproducing the running (5.1.6).
Let us next check that the curve correctly reproduces the singularity structure on the -plane. The branch points of the function can be determined by studying when the equation of , given in (5.2.1), has double roots. The equation for the branch points is given by
(5.2.10) |
The singularity in the -plane is caused by two of the branch points of colliding in the ultraviolet curve with the coordinate . This condition can be found by taking the discriminant of the equation of above, giving
(5.2.11) |
When , this equation simplifies to , giving the solutions
(5.2.12) |
for a constant , reproducing Fig. 5.3.
When , the equation (5.2.11) can be solved by making two separate approximations. Assuming is rather big, we can truncate the equation to just , finding a singularity at
(5.2.13) |
Next, assuming is rather small, we find giving
(5.2.14) |
Together, they reproduce Fig. 5.2. From this, we find that the effective pure theory in the region has the dynamical scale
(5.2.15) |
This agrees with what we saw in (5.1.11).
It is instructive to study another way to derive the singularity at from the curve. We would like to take the approximation . To facilitate to take the limit, we introduce in (5.2.1) and find
(5.2.16) |
Now the limit is easy to take: we just find
(5.2.17) |
Then it is clear that when , the equation can be factorized to
(5.2.18) |
therefore it represents two sheets intersecting at a point. When , two sheets are connected smoothly. The change is schematically shown in Fig. 5.9. We learned that the singularity at arises essentially from the structure in the curve.