17.3 Superstring Field Theory
353
To be more concrete, given a PCO insertion X (y 1 ), the variation of its location
inserts a factor −∂ξ(y 1 ). Integrating this term between two components labelled by
i and j —keeping everything else fixed—leads to a factor ξ(y
(j )
1 ) − ξ(y
(j )
1 ).
When several PCOs are involved, it is not sufficient to integrate the vertical
segment along a path where only one PCO varies at a time. Indeed, because a hole
is left in the process of the vertical integration. Additional segments must be added
and integrated over. This avoids the spurious poles, and one can show that it yields a
well-defined amplitude. Moreover, it agrees with the large Hilbert space approach.
Finally, it remains to address the question of the Feynman diagrams construction.
In this case, every graph obtained by plumbing fixture inherits its PCO locations
from the lower-dimensional surfaces, and there is no control on the resulting
distribution. It can be shown that no spurious singularity is generated in the
gluing process if the lower-dimensional graphs have no spurious poles. Hence, it
is sufficient to ensure that the fundamental graphs have no spurious poles.
17.3 Superstring Field Theory
The construction of super-SFT has proceeded along different directions (for reviews,
see [2,7,24]). There are two main strategies for constructing the superstring vertices:
1. brute-force construction: build the vertices recursively from amplitude factorization;
2. dress the bosonic products with superconformal ghosts.
While the second approach is simpler and preferred for explicit construction, the
first allows to derive the general structure as was done for the bosonic string. There
are two main strategies for dressing the vertices:
1. Munich construction (homotopy algebra bootstrap): Use the L ∞ and A ∞
structures to derive the superstring vertices from the bosonic vertices (small
Hilbert space).
2. Berkovits’ construction (WZW action): Generalize Witten’s cubic bosonic open
SFT (NS/R in large/small Hilbert space).
As indicated in parenthesis, a super-SFT can be written in the small or large
Hilbert space (or a combination). The different approaches have been shown to be
equivalent at the classical level.
The main difficulty in building a super-SFT is to properly describe the Ramond
sector. This can be done following two different approaches:
• constraining the Ramond string field;
• using an auxiliary string field.
353
To be more concrete, given a PCO insertion X (y 1 ), the variation of its location
inserts a factor −∂ξ(y 1 ). Integrating this term between two components labelled by
i and j —keeping everything else fixed—leads to a factor ξ(y
(j )
1 ) − ξ(y
(j )
1 ).
When several PCOs are involved, it is not sufficient to integrate the vertical
segment along a path where only one PCO varies at a time. Indeed, because a hole
is left in the process of the vertical integration. Additional segments must be added
and integrated over. This avoids the spurious poles, and one can show that it yields a
well-defined amplitude. Moreover, it agrees with the large Hilbert space approach.
Finally, it remains to address the question of the Feynman diagrams construction.
In this case, every graph obtained by plumbing fixture inherits its PCO locations
from the lower-dimensional surfaces, and there is no control on the resulting
distribution. It can be shown that no spurious singularity is generated in the
gluing process if the lower-dimensional graphs have no spurious poles. Hence, it
is sufficient to ensure that the fundamental graphs have no spurious poles.
17.3 Superstring Field Theory
The construction of super-SFT has proceeded along different directions (for reviews,
see [2,7,24]). There are two main strategies for constructing the superstring vertices:
1. brute-force construction: build the vertices recursively from amplitude factorization;
2. dress the bosonic products with superconformal ghosts.
While the second approach is simpler and preferred for explicit construction, the
first allows to derive the general structure as was done for the bosonic string. There
are two main strategies for dressing the vertices:
1. Munich construction (homotopy algebra bootstrap): Use the L ∞ and A ∞
structures to derive the superstring vertices from the bosonic vertices (small
Hilbert space).
2. Berkovits’ construction (WZW action): Generalize Witten’s cubic bosonic open
SFT (NS/R in large/small Hilbert space).
As indicated in parenthesis, a super-SFT can be written in the small or large
Hilbert space (or a combination). The different approaches have been shown to be
equivalent at the classical level.
The main difficulty in building a super-SFT is to properly describe the Ramond
sector. This can be done following two different approaches:
• constraining the Ramond string field;
• using an auxiliary string field.