Seiberg–Witten theory

In theoretical physics, Seiberg–Witten theory is a theory that determines an exact low-energy effective action (for massless degrees of freedom) of a N=2 supersymmetric gauge theory—namely the metric of the moduli space of vacua.

Seiberg–Witten curves

In general, effective Lagrangians of supersymmetric gauge theories are largely determined by their holomorphic properties and their behavior near the singularities. In particular, in gauge theory with N = 2 extended supersymmetry, the moduli space of vacua is a special Kähler manifold and its Kähler potential is constrained by above conditions.

In the original derivation by Seiberg and 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

$V(x) = \frac{1}{g^2} \operatorname{Tr} [\phi , \bar{\phi} ]^2 \,$

(1)

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.

Because the fields φ no longer have vanishing Vacuum expectation value. Because these are now heavy due to the Higgs effect, they should be integrated out in order to find the effective N=2 Abelian gauge theory. This can be expressed in terms of a single holomorphic function F.

In terms of this prepotential the Lagrangian can be written in the form:

$\frac{1}{4\pi} \operatorname{Im} \Bigl[ \int d^4 \theta \frac{dF}{dA} \bar{A} + \int d^2 \theta \frac{1}{2} \frac{d^2 F}{dA^2} W_\alpha W^\alpha \Bigr] \,$

(3)

$F = \frac{i}{2\pi} \mathcal{A}^2 \operatorname{\ln}\frac{\mathcal{A}^2}{\Lambda^2} + \sum_{k=1}^\infty F_k \frac{\Lambda^{4k}}{\mathcal{A}^{4k}} \mathcal{A}^2 \,$

(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.

$M \approx |na+ma_D| \,$

(5)

$a_D = \frac{dF}{da} \,$

(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 Kahler 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

Consider a super Yang-Mills theory in curved 6-dimensional background. After dimensional reduction on 2-torus, we obtain a 4d N = 2 super Yang-Mills theory with additional terms. Turning Wilson lines to compensate holonomies of fermions on the 2-torus, we get 4d N = 2 SYM in　Ω-background. Ω has 2 parameters, ε1,ε2, which go to 0 in the flat limit.

In Ω-background, we can integrate out all the non-zero modes, so the partition function (with the boundary condition φ → 0 at x → ∞) can be expressed as a sum of products and ratios of fermionic and bosonic determinants over instanton number. In the limit where ε1,ε2 approach 0, this sum is dominated by a unique saddle point. On the other hand, when ε1,ε2 approach 0,

$Z(a;\varepsilon _{{1}},\varepsilon _{{2}},\Lambda )=\exp(-{\frac {1}{\varepsilon _{{1}}\varepsilon _{{2}}}}({\mathcal {F}}(a;\Lambda )+O(\varepsilon _{{1}},\varepsilon _{{2}}))\,$

(10)

holds.

References

Jürgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag ISBN 3-540-42627-2 (See Section 7.2)

External links

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