324
15 Closed String Field Theory
The gauge fixed 1PI effective action reads
S 1PI =
1
g 2
s
n≥0
g n
s
n!
V
1PI
n ((
n ) :=
1
2
| c
−
0 c
+
0 L
+
0 | +
1
g 2
s
n≥0
g n
s
n!
V
1PI
n ((
n ).
(15.58)
Here, the prime means again that the terms g = 0, n = 2 are excluded from
the definition of V 1PI
2 . The action has the same form as the classical gauge fixed
action (15.29), which is logical since it generates only tree-level Feynman graphs.
For this reason, the vertices V 1PI
n have exactly the same properties as the brackets
V 0,n . This fact can be used to write the 1PI gauge invariant action:
S 1PI =
1
2
| c
−
0 Q B | +
1
g 2
s
n≥0
g n
s
n!
V
1PI
n ((
n ),
(15.59)
which mirrors the classical gauge invariant action (15.36). Then, it is straightforward
to see that it enjoys the same gauge symmetry upon replacing the tree-level vertices
by the 1PI vertices. But, since this action incorporates all quantum corrections, this
also proves that the quantum theory is correctly invariant under a quantum gauge
symmetry.
Remark 15.2 The 1PI action (15.59) can also be directly constructed from the BV
action (15.47).
15.6 Suggested Readings
• Gauge fixed and classical gauge invariant closed SFT [17] (see also [7, 8]).
• BV closed SFT [17] (see also [15]).
• Construction of the open–closed BV SFT [18].
• 1PI SFT [3, 13, 14, sec. 4.1, 5.2].
References
1. A. Belopolsky, B. Zwiebach, Who changes the string coupling? Nucl. Phys. B 472(1–2), 109–
138 (1996). https://doi.org/10.1016/0550-3213(96)00203-9. arXiv: hep-th/9511077
2. O. Bergman, B. Zwiebach, The Dilaton theorem and closed string backgrounds. Nucl.
Phys. B 441(1–2), 76–118 (1995). https://doi.org/10.1016/0550-3213(95)00022-K. arXiv:
hep-th/9411047
3. C. de Lacroix, H. Erbin, S.P. Kashyap, A. Sen, M. Verma, Closed super-string field theory
and its applications. Int. J. Mod. Phys. A 32(28–29), 1730021 (2017). https://doi.org/10.1142/
S0217751X17300216. arXiv: 1703.06410
4. T. Erler, S. Konopka, I. Sachs, NS-NS sector of closed superstring field theory. J. High Energy
Phys. 2014(8) (2014). https://doi.org/10.1007/JHEP08(2014)158. arXiv: 1403.0940
5. O. Hohm, B. Zwiebach, L ∞ algebras and field theory. Fortschr. Phys. 65(3–4), 1700014
(2017). https://doi.org/10.1002/prop.201700014. arXiv: 1701.08824
15 Closed String Field Theory
The gauge fixed 1PI effective action reads
S 1PI =
1
g 2
s
n≥0
g n
s
n!
V
1PI
n ((
n ) :=
1
2
| c
−
0 c
+
0 L
+
0 | +
1
g 2
s
n≥0
g n
s
n!
V
1PI
n ((
n ).
(15.58)
Here, the prime means again that the terms g = 0, n = 2 are excluded from
the definition of V 1PI
2 . The action has the same form as the classical gauge fixed
action (15.29), which is logical since it generates only tree-level Feynman graphs.
For this reason, the vertices V 1PI
n have exactly the same properties as the brackets
V 0,n . This fact can be used to write the 1PI gauge invariant action:
S 1PI =
1
2
| c
−
0 Q B | +
1
g 2
s
n≥0
g n
s
n!
V
1PI
n ((
n ),
(15.59)
which mirrors the classical gauge invariant action (15.36). Then, it is straightforward
to see that it enjoys the same gauge symmetry upon replacing the tree-level vertices
by the 1PI vertices. But, since this action incorporates all quantum corrections, this
also proves that the quantum theory is correctly invariant under a quantum gauge
symmetry.
Remark 15.2 The 1PI action (15.59) can also be directly constructed from the BV
action (15.47).
15.6 Suggested Readings
• Gauge fixed and classical gauge invariant closed SFT [17] (see also [7, 8]).
• BV closed SFT [17] (see also [15]).
• Construction of the open–closed BV SFT [18].
• 1PI SFT [3, 13, 14, sec. 4.1, 5.2].
References
1. A. Belopolsky, B. Zwiebach, Who changes the string coupling? Nucl. Phys. B 472(1–2), 109–
138 (1996). https://doi.org/10.1016/0550-3213(96)00203-9. arXiv: hep-th/9511077
2. O. Bergman, B. Zwiebach, The Dilaton theorem and closed string backgrounds. Nucl.
Phys. B 441(1–2), 76–118 (1995). https://doi.org/10.1016/0550-3213(95)00022-K. arXiv:
hep-th/9411047
3. C. de Lacroix, H. Erbin, S.P. Kashyap, A. Sen, M. Verma, Closed super-string field theory
and its applications. Int. J. Mod. Phys. A 32(28–29), 1730021 (2017). https://doi.org/10.1142/
S0217751X17300216. arXiv: 1703.06410
4. T. Erler, S. Konopka, I. Sachs, NS-NS sector of closed superstring field theory. J. High Energy
Phys. 2014(8) (2014). https://doi.org/10.1007/JHEP08(2014)158. arXiv: 1403.0940
5. O. Hohm, B. Zwiebach, L ∞ algebras and field theory. Fortschr. Phys. 65(3–4), 1700014
(2017). https://doi.org/10.1002/prop.201700014. arXiv: 1701.08824