Ivancevic_Applied-Diff-Geom

Introduction 35mediate the forces. The Lagrangian of each set of mediating bosons isinvariant under a transformation called a gauge transformation, so thesemediating bosons are referred to as gauge bosons. There are twelve different‘flavours’ of fermions in the Standard Model. The proton, neutron areThe real novelty of (2 + 1)D is that, instead of considering this ‘reduced’ form ofMaxwell theory, we can also define a completely different type of gauge theory: a Chern–Simons gauge theory. It satisfies the usual criteria for a sensible gauge theory: it isLorentz invariant, gauge invariant, and local. The Chern–Simons Lagrangian is (see,e.g., [Dunne (1999)])L CS = κ 2 ɛµνρ A µ∂ νA ρ − A µJ µ . (1.3)Two things are important about this Chern–Simons Lagrangian. First, it does not lookgauge invariant, because it involves the gauge field A µ itself, rather than just the (manifestlygauge invariant) field strength F µν. Nevertheless, under a gauge transformation,the Chern–Simons Lagrangian changes by a total space–time derivativeδL CS = κ 2 ∂µ (λ ɛµνρ ∂ νA ρ) . (1.4)Therefore, if we can neglect boundary terms then the corresponding Chern–Simons action,ZS CS = d 3 x L CS ,is gauge invariant. This is reflected in the fact that the classical Euler–Lagrange equationsκ2 ɛµνρ F νρ = J µ , or equivalently F µν = 1 κ ɛµνρJ ρ , (1.5)are clearly gauge invariant. Note that the Bianchi identity, ɛ µνρ ∂ µF νρ = 0, is compatiblewith the current conservation: ∂ µJ µ = 0, which follows from the NoetherTheorem. A second important feature of the Chern–Simons Lagrangian (1.3) is thatit is first–order in space–time derivatives. This makes the canonical structure of thesetheories significantly different from that of Maxwell theory. A related property is thatthe Chern–Simons Lagrangian is particular to (2 + 1)D, in the sense that we cannotwrite down such a term in (3 + 1)D – the indices simply do not match up. Actually, it ispossible to write down a ‘Chern–Simons theory’ in any odd space–time dimension (forexample, the Chern–Simons Lagrangian in 5D space–time is L = ɛ µνρστ A µ∂ νA ρ∂ σA τ ),but it is only in (2 + 1)D that the Lagrangian is quadratic in the gauge field.Recently, increasingly popular has become Seiberg–Witten gauge theory. It refers toa set of calculations that determine the low–energy physics, namely the moduli spaceand the masses of electrically and magnetically charged supersymmetric particles asa function of the moduli space. This is possible and nontrivial in gauge theory withN = 2 extended supersymmetry, by combining the fact that various parameters of theLagrangian are holomorphic functions (a consequence of supersymmetry) and the knownbehavior of the theory in the classical limit. The extended supersymmetry is supersymmetrywhose infinitesimal generators Q α i carry not only a spinor index α, but also anadditional index i = 1, 2... The more extended supersymmetry is, the more it constrainsphysical observables and parameters. Only the minimal (un–extended) supersymmetryis a realistic conjecture for particle physics, but extended supersymmetry is very importantfor analysis of mathematical properties of quantum field theory and superstringtheory.