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 {displaystyle N=2} supersymmetric gauge theory—namely the metric of the moduli space of vacua. V ( x ) = 1 g 2 Tr ⁡ [ ϕ , ϕ ¯ ] 2 {displaystyle V(x)={frac {1}{g^{2}}}operatorname {Tr} ^{2},}     (1) 1 4 π Im ⁡ [ ∫ d 4 θ d F d A A ¯ + ∫ d 2 θ 1 2 d 2 F d A 2 W α W α ] {displaystyle {frac {1}{4pi }}operatorname {Im} {Bigl },}     (3) F = i 2 π A 2 ln ⁡ A 2 Λ 2 + ∑ k = 1 ∞ F k Λ 4 k A 4 k A 2 {displaystyle F={frac {i}{2pi }}{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) M ≈ | n a + m a D | {displaystyle Mapprox |na+ma_{D}|,}     (5) a D = d F d a {displaystyle a_{D}={frac {dF}{da}},}     (6) Z ( a ; ε 1 , ε 2 , Λ ) = exp ⁡ ( − 1 ε 1 ε 2 ( F ( a ; Λ ) + O ( ε 1 , ε 2 ) ) {displaystyle Z(a;varepsilon _{1},varepsilon _{2},Lambda )=exp left(-{frac {1}{varepsilon _{1}varepsilon _{2}}}({mathcal {F}}(a;Lambda )+{mathcal {O}}(varepsilon _{1},varepsilon _{2}) ight),}     (10) 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 {displaystyle N=2} supersymmetric gauge theory—namely the metric of the moduli space of vacua. 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 {displaystyle 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 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.