Tree-level S-matrix of superstring field theory with homotopy algebra structure

Abstract

We show that the tree-level S-matrices of the superstring field theories based on the homotopy-algebra structure agree with those obtained in the first-quantized formulation. The proof is given in detail for the heterotic string field theory. The extensions to the type II and open superstring field theories are straightforward.

Type II superstring field theory revisited

“…Using HPT, we can show that the tree-level S-matrix derived from (super)string field theory agrees with that calculated using the first-quantization method [22][23][24]. We show in this section that the new prescription proposed in the previous section simplifies this proof for the type II superstring field theory.…”

“…which can be shown, similar to the case in [24], using the alternative form of the (extended) L ∞relations (3.49) and the fact that generic intermediate states are off-shell. Next, differentiating eq.…”

“…14) is given by [D , D ′ ] 11 = n,r,r m,s,s [ D n | (2r,2r) , D ′ m | (2s,2s) ] | (2(r+s),2(r+s)) , (A.1a) [D , D ′ ] 21 = n,r,r m,s,s[ D n | (2r,2r) , D ′ m | (2s,2s) ] | (2(r+s−1),2(r+s)) , (A.1b) [D , D ′ ] 12 = n,r,r m,s,s [ D n | (2r,2r) , D ′ m | (2s,2s) ] | (2(r+s),2(r+s−1)) , (A.1c) [D , D ′ ] 22 = n,r,r m,s,s [ D n | (2r,2r) , D m | (2s,2s) ] | (2(r+s−1),2(r+s−1)) , (A.1d) for coderivations D = n,r,r D n | (2r,2r) andD ′ = m,s,s D ′ m | (2s,2s) . An alternative expressions obtained by projecting the intermediate state[24],[ D , D ′ ] 11 = n D n π (0,0) 1 D ′ ∧ I n−1 − (−) |D||D ′ | D ′ n π (0,0) ∧ I n−1 , (A.2a) [ D , D ′ ] 21 = n ′ ∧ I n−1 − (−) |D||D ′ | D ′ n π ′ ∧ I n−1 − (−) |D||D ′ | D ′ n π ′ ∧ I n−1 − (−) |D||D ′ | D ′ n π ∧ I n−1 , (A.2d)are also useful. Here, |D| (|D ′ |) is equal to 0 or 1 when the degree of coderivation D (D ′ ) is even or odd, respectively.…”

Open-closed homotopy algebra in superstring field theory

“…In the superstring field theories, the homotopy algebra structure is more important. Since it seems inevitable to avoid associativity anomaly [12], the A ∞ structure becomes essential to determine the gauge-invariant action in the open superstring field theory [13][14][15]. The L ∞ structure again plays the role of guiding principle to determine the action with appropriate picture numbers in the heterotic and type II superstring field theories [16][17][18][19][20].…”