Editing String field theory (section)

== Supersymmetric covariant open string field theories ==
There are two main constructions of [[supersymmetric]] extensions of Witten's cubic open string field theory.  The first is very similar in form to its bosonic cousin and is known as ''modified cubic superstring field theory''.  The second, due to [[Nathan Berkovits]] is very different and is based on a [[Wess–Zumino–Witten model|WZW]]-type action.

===Modified cubic superstring field theory===
The first consistent extension of Witten's bosonic open string field theory to the RNS string was constructed by Christian Preitschopf, [[Charles Thorn]] and Scott Yost and independently by Irina Aref'eva, P. B. Medvedev and A. P. Zubarev.<ref>{{cite journal | last1=Preitschopf | first1=Christian R. | last2=Thorn | first2=Charles B. | last3=Yost | first3=Scott | title=Superstring field theory | journal=Nuclear Physics B | publisher=Elsevier BV | volume=337 | issue=2 | year=1990 | issn=0550-3213 | doi=10.1016/0550-3213(90)90276-j | bibcode=1990NuPhB.337..363P | pages=363–433| osti=7241635 | url=https://www.osti.gov/biblio/7241635 }}</ref><ref>{{cite journal | last1=Aref'eva | first1=I.Ya. | last2=Medvedev | first2=P.B. | last3=Zubarev | first3=A.P. | title=New representation for string field solves the consistency problem for open superstring field theory | journal=Nuclear Physics B | publisher=Elsevier BV | volume=341 | issue=2 | year=1990 | issn=0550-3213 | doi=10.1016/0550-3213(90)90189-k | bibcode=1990NuPhB.341..464A | pages=464–498}}</ref>  The NS string field is taken to be a ghostnumber one picture zero string field in the small Hilbert space (i.e. <math> \eta_0 |\Psi\rangle = 0 </math>). The action takes a very similar form to bosonic action,
:: <math> S(\Psi) = \tfrac{1}{2} \langle \Psi |Y(i) Y(-i) Q_B |\Psi \rangle
  +\tfrac{1}{3} \langle \Psi | Y(i) Y(-i) |\Psi * \Psi\rangle  \ ,</math>
where, 
:: <math> Y(z) = -\partial \xi e^{-2 \phi} c(z) </math>
is the inverse picture changing operator. The suggested <math> -\tfrac{1}{2} </math> picture number extension of this theory to the Ramond sector might be problematic.

This action has been shown to reproduce tree-level amplitudes and has a tachyon vacuum solution with the correct energy.<ref>{{cite journal | last=Erler | first=Theodore | title=Tachyon vacuum in cubic superstring field theory | journal=Journal of High Energy Physics | volume=2008 | issue=1 | date=2008-01-07 | issn=1029-8479 | doi=10.1088/1126-6708/2008/01/013 | pages=013|doi-access=free|arxiv=0707.4591| bibcode=2008JHEP...01..013E }}</ref> The one subtlety in the action is the insertion of picture changing operators at the midpoint, which imply that the linearized equations of motion take the form
:: <math> Y(i)Y(-i) Q_B \Psi = 0 \left.\right. \ .</math>
Because <math> Y(i) Y(-i) </math> has a non-trivial kernel, there are potentially extra solutions that are not in the cohomology of <math> Q_B </math>.<ref>N. Berkovits, "Review of open superstring field theory", hep-th/0105230</ref>  However, such solutions would have operator insertions near the midpoint and would be potentially singular, and importance of this problem remains unclear.

===Berkovits superstring field theory===
A very different supersymmetric action for the open string was constructed by Nathan Berkovits.  It takes the form<ref>{{cite journal | last=Berkovits | first=Nathan | title=Super-Poincaré invariant superstring field theory | journal=Nuclear Physics B | publisher=Elsevier BV | volume=450 | issue=1–2 | year=1995 | issn=0550-3213 | doi=10.1016/0550-3213(95)00259-u | pages=90–102|arxiv=hep-th/9503099| bibcode=1995NuPhB.450...90B | s2cid=14495743 }}</ref>
:: <math>
  S = \tfrac{1}{2} \langle e^{-\Phi} Q_B e^{\Phi} | e^{-\Phi} \eta_0 e^{\Phi} \rangle
 - \tfrac{1}{2} \int_0^1 dt\langle e^{ -\hat{\Phi}} \partial_t e^{\hat{\Phi}}|\{e^{-\hat{\Phi}} Q_B
e^{\hat{\Phi}} , e^{-\hat{\Phi}} \eta_0 e^{\hat{\Phi}} \} \rangle
</math>
where all of the products are performed using the <math>*</math>-product including the anticommutator <math> \{,\} </math>, and <math>\hat{\Phi}(t) </math> is any string field such that <math> \hat{\Phi}(0)  = 0</math> and <math> \hat{\Phi}(1)  = \Phi</math>.  The string field <math> \Phi </math> is taken to be in the NS sector of the large Hilbert space, i.e. ''including'' the zero mode of <math> \xi </math>.  It is not known how to incorporate the R sector, although some preliminary ideas exist.<ref>{{cite journal | last=Michishita | first=Yoji | title=A Covariant Action with a Constraint and Feynman Rules for Fermions in Open Superstring Field Theory | journal=Journal of High Energy Physics | volume=2005 | issue=1 | date=2005-01-07 | issn=1029-8479 | doi=10.1088/1126-6708/2005/01/012 | pages=012|arxiv=hep-th/0412215| bibcode=2005JHEP...01..012M |doi-access=free}}</ref>

The equations of motion take the form
::<math> \eta_0 \left(e^{-\Phi} Q_B e^{\Phi} \right) = 0 .</math>

The action is invariant under the gauge transformation
:: <math> e^{\Phi} \to e^{Q_B \Lambda} e^{\Phi} e^{\eta_0 \Lambda'} .</math>

The principal advantage of this action is that it free from any insertions of picture-changing operators.  It has been shown to reproduce correctly tree level amplitudes<ref>{{cite journal | last1=Berkovits | first1=Nathan | last2=Echevarria | first2=Carlos Tello | title=Four-point amplitude from open superstring field theory | journal=Physics Letters B | publisher=Elsevier BV | volume=478 | issue=1–3 | year=2000 | issn=0370-2693 | doi=10.1016/s0370-2693(00)00246-x | pages=343–350|arxiv=hep-th/9912120| bibcode=2000PhLB..478..343B | s2cid=17003177 }}</ref> and has been found, numerically, to have a tachyon vacuum with appropriate energy.<ref>{{cite journal | last=Berkovits | first=Nathan | title=The tachyon potential in open Neveu-Schwarz string field theory | journal=Journal of High Energy Physics | volume=2000 | issue=4 | date=2000-04-19 | issn=1029-8479 | doi=10.1088/1126-6708/2000/04/022 | pages=022|doi-access=free|arxiv=hep-th/0001084| bibcode=2000JHEP...04..022B }}</ref><ref>{{cite journal | last1=Berkovits | first1=Nathan | last2=Sen | first2=Ashoke | last3=Zwiebach | first3=Barton | title=Tachyon condensation in superstring field theory | journal=Nuclear Physics B | volume=587 | issue=1–3 | year=2000 | issn=0550-3213 | doi=10.1016/s0550-3213(00)00501-0 | pages=147–178| arxiv=hep-th/0002211 | bibcode=2000NuPhB.587..147B | s2cid=11853254 }}</ref>  The known analytic solutions to the classical equations of motion include the tachyon vacuum<ref>{{cite journal | last=Erler | first=Theodore | title=Analytic solution for tachyon condensation in Berkovits' open superstring field theory | journal=Journal of High Energy Physics | volume=2013 | issue=11 | year=2013 | issn=1029-8479 | doi=10.1007/jhep11(2013)007 | page=7|arxiv=1308.4400| bibcode=2013JHEP...11..007E | s2cid=119114830 }}</ref> and marginal deformations.

===Other formulations of covariant open superstring field theory===
A formulation of superstring field theory using the non-minimal pure-spinor variables was introduced by Berkovits.<ref>{{cite journal | last=Berkovits | first=Nathan | title=Pure spinor formalism as an N= 2 topological string | journal=Journal of High Energy Physics | volume=2005 | issue=10 | date=2005-10-27 | issn=1029-8479 | doi=10.1088/1126-6708/2005/10/089 | pages=089|doi-access=free|arxiv=hep-th/0509120| bibcode=2005JHEP...10..089B }}</ref> The action is cubic and includes a midpoint insertion whose kernel is trivial. As always within the pure-spinor formulation, the Ramond sector can be easily treated. However, it is not known how to incorporate the GSO- sectors into the formalism.

In an attempt to resolve the allegedly problematic midpoint insertion of the modified cubic theory, Berkovits and Siegel proposed a superstring field theory based on a non-minimal extension of the RNS string,<ref>{{cite journal | last1=Berkovits | first1=Nathan | last2=Siegel | first2=Warren | title=Regularizing cubic open Neveu-Schwarz string field theory | journal=Journal of High Energy Physics | volume=2009 | issue=11 | date=2009-11-05 | issn=1029-8479 | doi=10.1088/1126-6708/2009/11/021 | pages=021|arxiv=0901.3386| bibcode=2009JHEP...11..021B | s2cid=16824165 }}</ref> which uses a midpoint insertion with no kernel. It is not clear if such insertions are in any way better than midpoint insertions with non-trivial kernels.