Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 47 (2014)

From the Berkovits formulation to the Witten formulationin open superstring field theory†
Y. Iimori,∗1 T. Noumi,∗2 Y. Okawa,∗3 and S. Torii∗2Gauge invariance plays a fundamental role in thecurrent formulation of covariant string ﬁeld theory. Inopen bosonic string ﬁeld theory,1) behind the gauge invariance is the algebraic structure called the A∞ structure,2,3) which is closely related to the covering of themoduli space of Riemann surfaces. In open superstring
ﬁeld theory, we therefore expect that the structure underlying its gauge invariance be a supersymmetric extension of the A∞ structure, which would be closelyrelated to the covering of the supermoduli space ofsuper-Riemann surfaces. However, there is very little
understanding of gauge invariance, and some of the
problems we are confronted with in open superstring
ﬁeld theory seem to be related to the lack of our understanding in this perspective.
For example, in the Witten formulation of open
superstring ﬁeld theory,4) the gauge symmetry has
proven to be singular because of the collision of picturechanging operators.5) There are related divergences intree-level amplitudes, which are also caused by the collision of picture-changing operators. It is possible thatthe source of these divergences is related to the singularcovering of the supermoduli space of super-Riemannsurfaces. At the moment, however, such understanding is negligible.
On the other hand, gauge transformation does not
suﬀer from any singularity in the Berkovits formulationof open superstring ﬁeld theory6) in the Neveu-Schwarz
sector. We do not, however, understand why it works
well in the context of the covering of the supermoduli space of super-Riemann surfaces. In the Berkovits
formulation, the action contains interaction vertices
higher than cubic. We know that the bosonic moduli space of Riemann surfaces is covered by Feynman
diagrams with cubic vertices alone, and the higherorder vertices do not contribute to the covering of thebosonic moduli space. Since gauge invariance requires
the higher-order vertices, it is expected that these vertices play a role in the covering of the supermoduli
space. At the moment, however, such understandingis missing.
In view of recent developments in the understanding of the supermoduli space,7–10) the exploration of
the relation between gauge invariance in open superstring ﬁeld theory and the covering of the supermoduli space of super-Riemann surfaces can be crucially
important for the profound question of whether open
†

∗1
∗2
∗3

superstring ﬁeld theory can be a consistent quantum
theory by itself. In this report, as a ﬁrst step towards
this direction, we address the question of how the divergences in the Witten formulation can be resolved in
the Berkovits formulation.
The Hilbert space of the string ﬁeld in the Berkovits
formulation is larger than that in the Witten formulation and, correspondingly, the gauge symmetry in theBerkovits formulation is larger than that in the Witten formulation. We perform partial gauge ﬁxing in
the Berkovits formulation to associate it with the Witten formulation. We introduce a one-parameter familyof judicious gauge choices labeled by λ, and the cubic interaction in the Berkovits formulation reduces tothat in the Witten formulation in the singular limit
λ → 0. We can think of the Berkovits formulation
which is partially gauge ﬁxed with ﬁnite λ as a regularization of the Witten formulation. We ﬁnd thatthe divergence in the four-point amplitude as λ → 0 iscanceled by the quartic interaction. We also ﬁnd thatthe divergence in the gauge variation of the action to
the second order in the coupling constant as λ → 0 is
resolved by incorporating the quartic interaction. Our
approach based on the one-parameter family of gauge
choices enables us to discuss the nature of these divergences in a concrete and well-deﬁned setting. Our
next step will be to translate the mechanism of canceling the divergences into the language of the covering ofthe supermoduli space of super-Riemann surfaces, andour ultimate goal is to reveal a supersymmetric extension of the A∞ structure underlying open superstring
ﬁeld theory.
References
1) E. Witten, Nucl. Phys. B268, 253 (1986).
2) J. D. Stasheﬀ, Trans. Amer. Math. Soc. 108, 275(1963).
3) J. D. Stasheﬀ, Trans. Amer. Math. Soc. 108, 293(1963).
4) E. Witten, Nucl. Phys. B276, 291 (1986).
5) C. Wendt, Nucl. Phys. B314, 209 (1989).
6) N. Berkovits, Nucl. Phys. B 450, 90 (1995) [Erratumibid. B 459, 439 (1996)].
7) E. Witten, arXiv:1209.2459.
8) E. Witten, arXiv:1209.5461.
9) E. Witten, arXiv:1304.2832.
10) R. Donagi and E. Witten, arXiv:1304.7798.

Condensed from the article in arXiv:1312.1677.
Department of Physics, Nagoya University
RIKEN Nishina Center
Institute of Physics, The University of Tokyo- 127 -