52
3 String Theory
Of course, one may wonder how Fig. 1.1 modifies in the world of IBL ∞ -algebras.
This would lead us to properads, their cobar construction and representations. We
will, however, not pursue this here.
Concerning the scattering amplitudes, i.e. the induced loop homotopy structure
on the Q-cohomology, this works very much the same way as already described
in Chap. 1, cf. (2.35). Although, in formula (3.30), we gave only the result for
(cyclic) L ∞ -algebras, the modification of that formula to loop homotopy algebras is
straightforward. Note that (3.30) is the formula from the homological perturbation
lemma. It expresses the change of the trivial differential on functions on the Qcohomology H induced by the perturbation of the BRST operator Q by the map δ
introduced there. Now, the loop homotopy version leads to a similar formula, e.g. L
in δ is replaced by ¯
hΔ+L, where we think of the BV operator Δ as an operation with
zero inputs and two outputs and where L comprises also all higher genus operations.
As in Remark 2.10, it is enough for our purposes to think about morphisms of loop
homotopy algebras as of (nonlinear) maps between the underlying vector spaces
compatible with the symplectic structures and intertwining between the respective
full BV operators.
Further Reading
There are many good textbooks for the world-sheet formulation of string theory
which is relevant, in particular, for the calculation of scattering amplitudes. Standard
textbooks are
• M. B. Green, J. H. Schwarz and E. Witten, “Superstring theory. Vol. 1 and 2”,
Cambridge Monographs On Mathematical Physics, ( 1987),
• J. Polchinski, “String theory. Vol. 1 and 2”, Cambridge University Press (2005),
• R. Blumenhagen, D. Luest and S. Theisen, “Basic concepts of string Theory”,
Springer Verlag (2013).
For early work on BRST invariant string field theory, see
• W. Siegel and B. Zwiebach, “Gauge string fields”, Nucl.Phys. B263 (1986) 105–
128, and
• C. B. Thorn, “String field theory”, Phys. Rept. 175, 1–101 (1989).
For the covariant formulation of interacting quantum closed string field theory, see
• B. Zwiebach, “Closed string field theory: Quantum action and the B-V master
equation”, Nucl.Phys. B390, (1993) 33–152, and
• B. Zwiebach, “Oriented open - closed string theory revisited”, Annals Phys. 267,
(1998) 193–248.
3 String Theory
Of course, one may wonder how Fig. 1.1 modifies in the world of IBL ∞ -algebras.
This would lead us to properads, their cobar construction and representations. We
will, however, not pursue this here.
Concerning the scattering amplitudes, i.e. the induced loop homotopy structure
on the Q-cohomology, this works very much the same way as already described
in Chap. 1, cf. (2.35). Although, in formula (3.30), we gave only the result for
(cyclic) L ∞ -algebras, the modification of that formula to loop homotopy algebras is
straightforward. Note that (3.30) is the formula from the homological perturbation
lemma. It expresses the change of the trivial differential on functions on the Qcohomology H induced by the perturbation of the BRST operator Q by the map δ
introduced there. Now, the loop homotopy version leads to a similar formula, e.g. L
in δ is replaced by ¯
hΔ+L, where we think of the BV operator Δ as an operation with
zero inputs and two outputs and where L comprises also all higher genus operations.
As in Remark 2.10, it is enough for our purposes to think about morphisms of loop
homotopy algebras as of (nonlinear) maps between the underlying vector spaces
compatible with the symplectic structures and intertwining between the respective
full BV operators.
Further Reading
There are many good textbooks for the world-sheet formulation of string theory
which is relevant, in particular, for the calculation of scattering amplitudes. Standard
textbooks are
• M. B. Green, J. H. Schwarz and E. Witten, “Superstring theory. Vol. 1 and 2”,
Cambridge Monographs On Mathematical Physics, ( 1987),
• J. Polchinski, “String theory. Vol. 1 and 2”, Cambridge University Press (2005),
• R. Blumenhagen, D. Luest and S. Theisen, “Basic concepts of string Theory”,
Springer Verlag (2013).
For early work on BRST invariant string field theory, see
• W. Siegel and B. Zwiebach, “Gauge string fields”, Nucl.Phys. B263 (1986) 105–
128, and
• C. B. Thorn, “String field theory”, Phys. Rept. 175, 1–101 (1989).
For the covariant formulation of interacting quantum closed string field theory, see
• B. Zwiebach, “Closed string field theory: Quantum action and the B-V master
equation”, Nucl.Phys. B390, (1993) 33–152, and
• B. Zwiebach, “Oriented open - closed string theory revisited”, Annals Phys. 267,
(1998) 193–248.