Seiberg–Witten theory
Seiberg–Witten theory
Seiberg–Witten curves
In the original derivation by Nathan Seiberg and Edward Witten, they extensively used holomorphy and electric-magnetic duality to constrain the prepotential, namely the metric of the moduli space of vacua.
Consider the example with gauge group SU(n). The classical potential is
| **(1)** |
![V(x) = \frac{1}{g^2} \operatorname{Tr} [\phi , \bar{\phi} ]^2 \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfcc5eeb72a25f8b026cd66005490299617cb705)
This must vanish on the moduli space, so vacuum expectation value of φ can be gauge rotated into Cartan subalgebra, so it is a traceless diagonal complex matrix.
In terms of this prepotential the Lagrangian can be written in the form:
| **(3)** |
![\frac{1}{4\pi} \operatorname{Im} \Bigl[ \int d^4 \theta \frac{dF}{dA} \bar{A} + \int d^2 \theta \frac{1}{2} \frac{d^2 F}{dA^2} W_\alpha W^\alpha \Bigr] \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb23a0b33f085127decb4294c718d2cf9ec756f)
| **(4)** |
The first term is a perturbative loop calculation and the second is the instanton part where k labels fixed instanton numbers.
From this we can get the mass of the BPS particles.
| **(5)** |
| **(6)** |
One way to interpret this is that these variables a and its dual can be expressed as periods of a meromorphic differential on a Riemann surface called the Seiberg–Witten curve.
Relation to integrable systems
The special Kähler geometry on the moduli space of vacua in Seiberg–Witten theory can be identified with the geometry of the base of complex completely integrable system. The total phase of this complex completely integrable system can be identified with the moduli space of vacua of the 4d theory compactified on a circle. See Hitchin system.
Seiberg–Witten prepotential via instanton counting
| **(10)** |
holds.