Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1119topological 4–manifolds. A new TQFT on 4–manifolds was discoveredin SW studies of electric–magnetic duality of supersymmetric gauge theory.As discussed before, Seiberg and Witten [Seiberg and Witten (1994a);Seiberg and Witten (1994b)] studied the electric–magnetic duality of N = 2supersymmetric SU(2) YM gauge theory, by using a version of Montonen–Olive duality and obtained exact solutions. According to this result, theexact low energy effective action can be determined by a certain ellipticcurve with a parameter u = 〈Tr(φ) 2 〉, where φ is a complex scalar fieldin the adjoint representation of the gauge group, describing the quantummoduli space. For large u, the theory is weakly–coupled and semi–classical,but at u = ±Λ 2 corresponding to strong coupling regime, where Λ is thedynamically generated mass scale, the elliptic curve becomes singular andthe situation of the theory changes drastically. At these singular points,magnetically charged particles become massless. Witten showed that atu = ±Λ 2 the TQFT was related to the moduli problem of counting the solutionof the (Abelian) ‘Seiberg–Witten monopole equations’ [Witten (1994)]and it gave a dual description for the SU(2) Donaldson theory.It turns out that in 3D a particular TQFT of Bogomol’nyi monopolescan be obtained from a dimensional reduction of Donaldson theory and thepartition function of this theory gives the so–called Casson invariant of3–manifolds [Atiyah and Jeffrey (1990)].Ohta [Ohta (1998)] discussed TQFTs associated with the 3D versionof both Abelian and non–Abelian SW–monopoles, by applying Batalin–Vilkovisky quantization procedure. In particular, Ohta constructed thetopological actions, topological observables and BRST transformation rules.In this subsection, mainly following [Ohta (1998)], we will discussTQFTs associated with both Abelian and non–Abelian SW–monopoles.We will use the following notation.Let X be a compact orientable Spin 4–manifold without boundary andg µν be its Riemannian metric tensor (with g = det g µν ). Here we usex µ as the local coordinates on X. γ µ are Dirac’s gamma matrices andσ µν = [γ µ , γ ν ]/2 with {γ µ , γ ν } = g µν . M is a Weyl fermion and M is acomplex conjugate of M. (We will suppress spinor indices.) The Lie algebrag is defined by [T a , T b ] = if abc T c , where T a is a generator normalizedas Tr(T a T b ) = δ ab . The symbol f abc is a structure constant of g and isantisymmetric in its indices. The Greek indices µ, ν, α etc run from 0 to 3.The Roman indices a, b, c, · · · are used for the Lie algebra indices runningfrom 1 to dim g, whereas i, j, k, · · · are the indices for space coordinates.Space–time indices are raised and lowered with g µν . The repeated indices