Question

How does one determine the genus of the ultraviolet curve from Seiberg-Witten curve?

Answer 1

In Gaiotto's construction of 4d N=2 theories, one starts with 6d (2,0) theory and compactify it on the Riemann surface, which is called the ultraviolet curve of the N=2 theory. In the case of SU(2) ( A1 theories ) with only regular punctures on Riemann surface, one can literally read off the structure of the ultraviolet curve from the quiver diagram.  When one considers asymptotically free theories or Argyres-Douglas theories, then the only way we could construct the ultraviolet curve is from the known Seiberg-Witten curve and the Seiberg-Witten differential. For instance, in the paper http://arxiv.org/pdf/1112.1691.pdf the maximally superconformal point of SU(n−1) theory with 2 flavors has the curve  y~2=xn−2+c1xn−3+⋯+cn−2+cn−1x+C2x2 and the SW differential λ=y~dx has poles at 0 of degree 2, and at ∞ with degree n+2. But how do we know the ultraviolet curve has genus 0, i.e., it is a sphere?

Answer 2

We study the quantum Seiberg-Witten (SW) curves for (A 1, G)-type Argyres-Douglas (AD) theory by taking the scaling limit of the quantum SW curve of N = 2 gauge theory.