Quantisation. We have seen above that the classical equations of motion can be solved up to a single second order evolution. We have also constructed the conserved momentum and central charges in the (2, 0) algebra. In this section, rather than solve the second order classical evolution equation we instead wish to quantise the system using P− as the Hamiltonian. In particular we see that it can be written as ∫ − 2g2 P = 1 Tr d4x ∂ Ai∂− Ai — 2∂− AiDiA− + DiA− DiA− + g4DiXI DiXI . (7.40) The first term gives the kinetic energy and can be expressed in terms of the metric gαβ on instanton moduli space defined by Tr ∫ d4x δAiδAi = gαβδmαδmβ . (7.41) Here δAi = ∂Ai/∂mαδmα + Diδω, with δω is the gauge transformation required to preserve DiδAi = 0. Next we have a term that is linear in time derivatives: Tr ∫ d4x ∂−AiDiA− = Tr I ∂−Arw = Lαm˙ α . (7.42) where r is the radial normal direction to the sphere at infinity, m˙ α = ∂−mα and Lα is a vector field on the instanton moduli space. We note that it is proportional to w, i.e. it is determined by the vacuum expectation value of A−, and can be viewed as a background gauge field. α α β β The last two terms can be written as a boundary integral and contribute to the potential. Thus we find that the Hamiltonian is P− = 2g2 gαβ(m˙ — L )(m˙ — L ) + V , (7.43) r r where 2g2 αβ V = — g LαLβ + 1 Tr I g4XI D XI + A D A 2g2 For w = 0 this Hamiltonian has appeared before [111] and is known to admit 8 supersymmetries, which correspond to the Q− here. In particular it was shown that g2 αKβ where Kα is a tri-holomorphic Killing vector on the instanton moduli space which can be expressed purely in terms of the asymptotic values of XI and the ADHM data [111] . By construction the Hamiltonian is also invariant under 8 supersymmetries when w /= 0. The next step is to decide on a momentum conjugate to the moduli coordinates mα. The obvious choice is pα = gαβm˙ β . (7.46) An alternative quantisation could be pα = gαβ(m˙ β — Lβ) however since Lα depends on wa this quantisation would then differ in various sectors of the theory. It would be interesting to ob- tain a symplectic structure on the entire (2, 0) system that leads to this. Quantisation is now straightforward and we just consider wavefunctions Ψ(mα, x−) and define α ∂mα pˆ Ψ = —i , mˆ αΨ = mαΨ , (7.47) where a hat denotes the quantum operator. There is one issue that requires some discussion, namely the moduli space generically contains singularities where the instantons shrink to zero size. ...
Quantisation. In order to quantise the theory, it is convenient to express ζ in terms of Fourier modes. For the non-winding mode sector, we have ζ(t, φ) = √2π n∈ΣZ/{0} αn(t)e
