By Warren Siegel
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Extra info for Introduction to string field theory
For the ﬁelds described in sect. 1) where J AB for A = (+, −, α) (graded antisymmetric in its indices) are the generators of OSp(1,1|2), and we have set k = 0, so that the p+ factor is the Hilbert space metric. 2) implies that only the part of Φ at x− = xα = 0 can be gauge invariant. 1. OSp(1,1|2) 53 since p+ = 0 in light-cone formalisms. 5) where the J −+ and J αβ each can be freely moved to either side of the [J −α , J +β ]. 6) where φ now depends only on the usual spacetime coordinates xa , and for irreducible Poincar´e representations φ has indices which are the result of starting with an irreducible representation of OSp(D−1,1|2) in the massless case, or OSp(D,1|2) in the massive case, and then truncating to the Sp(2) singlets.
8), any state of the form ψ + Qλ is equivalent to ψ. The physical states are therefore said to belong to the “cohomology” of Q: those satisfying Qψ = 0 modulo gauge transformations δψ = Qλ. 6a) is. 3. , the conformal group in 1 anticommuting dimension). From the gauge generators G i , which act on only the physical variables, we can deﬁne IGL(1)-invariant generalizations which transform also C, as the adjoint representation: G i = Q, ∂ ∂C i = G i + C j f jik ∂ ∂C k . 13) The G’s are gauge-ﬁxed versions of the gauge generators G.
This cannot be done by straightforward truncation, since expansion of the ﬁeld in this anticommuting coordinate shows that one cannot eliminate consistently the ﬁelds in the c˜ sector. 6) which is a bracket with unusual statistics because of the anticommuting δ-function on the right-hand side. 4. 6) to disappear, enabling one to delete the x− argument from the ﬁelds and the corresponding δ-function to ﬁnd (using (c∂/∂c)c = c) [Φ† (1), Φ(2)]c = − 12 δ D (x2 − x1 )δ(c2 − c1 ) . 7) This is the covariant bracket.
Introduction to string field theory by Warren Siegel