336
16 Background Independence
The form of the equation
∂B 0,3 = V 0,3 − V
0,3
(16.33)
suggests to use homology theory. The interpretation of B 0,3 is that it is a space
interpolating between V 0,3 and V
0,3 . A preliminary step is to check that there is
no obstruction: since the LHS is already a boundary, one has ∂ 2 B 0,3 = 0 and one
should check that ∂(RHS) = 0 as well. It can be shown that it is indeed true. It
was proved in [10] that this equation admits a solution and that the equations for
higher g and n can all be solved. Hence, there exists a field redefinition and SFT is
background independent.
16.7 Suggested Readings
• Proof of the background independence under marginal deformations [10–12] (see
also [7–9] for earlier results laying foundations for the complete proof).
• L ∞ perspective [4, sec. 4] (see also [2, 3, sec. III.B].
• Connection on the space of CFTs [1, 5, 6, 8, 14].
References
1. M. Campbell, P. Nelson, E. Wong, Stress tensor perturbations in conformal field theory. Int. J.
Mod. Phys. A 06(27), 4909–4924 (1991). https://doi.org/10.1142/S0217751X9100232X
2. K. Muenster, I. Sachs, On homotopy algebras and quantum string field theory (2013). arXiv:
1303.3444
3. K. Muenster, I. Sachs, Quantum open-closed homotopy algebra and string field theory.
Commun. Math. Phys. 321(3), 769–801 (2013). https://doi.org/10.1007/s00220-012-1654-1.
arXiv: 1109.4101
4. K. Muenster, I. Sachs, Homotopy classification of bosonic string field theory. Commun. Math.
Phys. 330, 1227–1262 (2014). https://doi.org/10.1007/s00220-014-2027-8. arXiv: 1208.5626
5. K. Ranganathan, Nearby CFT’s in the operator formalism: the role of a connection. Nucl.
Phys. B 408(1), 180–206 (1993). https://doi.org/10.1016/0550-3213(93)90136-D. arXiv: hepth/9210090
6. K. Ranganathan, H. Sonoda, B. Zwiebach, Connections on the state-space over conformal field theories. Nucl. Phys. B 414(1–2), 405–460 (1994). https://doi.org/10.1016/05503213(94)90436-7. arXiv: hep-th/9304053
7. A. Sen, On the background independence of string field theory: II. Analysis of on-shell
S-matrix elements. Nucl. Phys. B 347(1), 270–318 (1990). https://doi.org/10.1016/05503213(90)90560-Z
8. A. Sen, On the background independence of string field theory. Nucl. Phys. B 345(2–3), 551–
583 (1990). https://doi.org/10.1016/0550-3213(90)90400-8
9. A. Sen, On the background independence of string field theory: III. Explicit field redefinitions.
Nucl. Phys. B 391(3), 550–590 (1993). https://doi.org/10.1016/0550-3213(93)90084-3. arXiv:
hep-th/9201041
10. A. Sen, Background independence of closed superstring field theory. J. High Energy Phys.
2018(2), 155 (2018). https://doi.org/10.1007/JHEP02(2018)155. arXiv: 1711.08468
16 Background Independence
The form of the equation
∂B 0,3 = V 0,3 − V
0,3
(16.33)
suggests to use homology theory. The interpretation of B 0,3 is that it is a space
interpolating between V 0,3 and V
0,3 . A preliminary step is to check that there is
no obstruction: since the LHS is already a boundary, one has ∂ 2 B 0,3 = 0 and one
should check that ∂(RHS) = 0 as well. It can be shown that it is indeed true. It
was proved in [10] that this equation admits a solution and that the equations for
higher g and n can all be solved. Hence, there exists a field redefinition and SFT is
background independent.
16.7 Suggested Readings
• Proof of the background independence under marginal deformations [10–12] (see
also [7–9] for earlier results laying foundations for the complete proof).
• L ∞ perspective [4, sec. 4] (see also [2, 3, sec. III.B].
• Connection on the space of CFTs [1, 5, 6, 8, 14].
References
1. M. Campbell, P. Nelson, E. Wong, Stress tensor perturbations in conformal field theory. Int. J.
Mod. Phys. A 06(27), 4909–4924 (1991). https://doi.org/10.1142/S0217751X9100232X
2. K. Muenster, I. Sachs, On homotopy algebras and quantum string field theory (2013). arXiv:
1303.3444
3. K. Muenster, I. Sachs, Quantum open-closed homotopy algebra and string field theory.
Commun. Math. Phys. 321(3), 769–801 (2013). https://doi.org/10.1007/s00220-012-1654-1.
arXiv: 1109.4101
4. K. Muenster, I. Sachs, Homotopy classification of bosonic string field theory. Commun. Math.
Phys. 330, 1227–1262 (2014). https://doi.org/10.1007/s00220-014-2027-8. arXiv: 1208.5626
5. K. Ranganathan, Nearby CFT’s in the operator formalism: the role of a connection. Nucl.
Phys. B 408(1), 180–206 (1993). https://doi.org/10.1016/0550-3213(93)90136-D. arXiv: hepth/9210090
6. K. Ranganathan, H. Sonoda, B. Zwiebach, Connections on the state-space over conformal field theories. Nucl. Phys. B 414(1–2), 405–460 (1994). https://doi.org/10.1016/05503213(94)90436-7. arXiv: hep-th/9304053
7. A. Sen, On the background independence of string field theory: II. Analysis of on-shell
S-matrix elements. Nucl. Phys. B 347(1), 270–318 (1990). https://doi.org/10.1016/05503213(90)90560-Z
8. A. Sen, On the background independence of string field theory. Nucl. Phys. B 345(2–3), 551–
583 (1990). https://doi.org/10.1016/0550-3213(90)90400-8
9. A. Sen, On the background independence of string field theory: III. Explicit field redefinitions.
Nucl. Phys. B 391(3), 550–590 (1993). https://doi.org/10.1016/0550-3213(93)90084-3. arXiv:
hep-th/9201041
10. A. Sen, Background independence of closed superstring field theory. J. High Energy Phys.
2018(2), 155 (2018). https://doi.org/10.1007/JHEP02(2018)155. arXiv: 1711.08468