Editing String field theory (section)

==Covariant closed string field theory ==
Covariant closed string field theories are considerably more complicated than their open string cousins.  Even if one wants to construct a string field theory which only reproduces ''tree-level'' interactions between closed strings, the classical action must contain an ''infinite'' number of vertices <ref>{{cite journal | last1=Sonoda | first1=Hidenori | last2=Zwiebach | first2=Barton | title=Covariant closed string theory cannot be cubic | journal=Nuclear Physics B | publisher=Elsevier BV | volume=336 | issue=2 | year=1990 | issn=0550-3213 | doi=10.1016/0550-3213(90)90108-p | bibcode=1990NuPhB.336..185S | pages=185–221}}</ref> consisting of string polyhedra.<ref>{{cite journal | last1=Saadi | first1=Maha | last2=Zwiebach | first2=Barton | title=Closed string field theory from polyhedra | journal=Annals of Physics | publisher=Elsevier BV | volume=192 | issue=1 | year=1989 | issn=0003-4916 | doi=10.1016/0003-4916(89)90126-7 | bibcode=1989AnPhy.192..213S | pages=213–227}}</ref><ref>{{cite journal | last1=Kugo | first1=Taichiro | last2=Suehiro | first2=Kazuhiro | title=Nonpolynomial closed string field theory: Action and its gauge invariance | journal=Nuclear Physics B | publisher=Elsevier BV | volume=337 | issue=2 | year=1990 | issn=0550-3213 | doi=10.1016/0550-3213(90)90277-k | bibcode=1990NuPhB.337..434K | pages=434–466}}</ref>

If one demands that on-shell scattering diagrams be reproduced to all orders in the string coupling, one must also include additional vertices arising from higher genus (and hence higher order in <math> \hbar </math>) as well.  In general, a manifestly BV invariant, quantizable action takes the form<ref>{{cite journal | last=Zwiebach | first=Barton | title=Closed string field theory: Quantum action and the Batalin-Vilkovisky master equation | journal=Nuclear Physics B | volume=390 | issue=1 | year=1993 | issn=0550-3213 | doi=10.1016/0550-3213(93)90388-6 | pages=33–152|arxiv=hep-th/9206084| bibcode=1993NuPhB.390...33Z | s2cid=119509701 }}</ref> 
:: <math> S(\Psi) = \hbar \sum_{g \ge 0} (\hbar g_c)^{g-1} \sum_{n \ge 0} \frac{1}{n!} \{\Psi^n \}_g </math>
where <math> \{ \Psi^n \}_g </math> denotes an <math>n</math>th order vertex arising from a genus <math> g </math> surface and <math> g_c </math> is the closed string coupling.  The structure of the vertices is in principle determined by a minimal area prescription,<ref>{{cite journal | last=Zwiebach | first=Barton | title=Quantum Closed Strings from Minimal Area | journal=Modern Physics Letters A | publisher=World Scientific Pub Co Pte Lt | volume=05 | issue=32 | date=1990-12-30 | issn=0217-7323 | doi=10.1142/s0217732390003218 | bibcode=1990MPLA....5.2753Z | pages=2753–2762}}</ref> although, even for the polyhedral vertices, explicit computations have only been performed to quintic order.<ref>{{cite journal | last=Moeller | first=Nicolas | title=Closed bosonic string field theory at quintic order: five-tachyon contact term and dilaton theorem | journal=Journal of High Energy Physics | volume=2007 | issue=3 | date=2007-03-12 | issn=1029-8479 | doi=10.1088/1126-6708/2007/03/043 | pages=043|arxiv=hep-th/0609209| bibcode=2007JHEP...03..043M | s2cid=11634790 }}</ref><ref>{{cite journal | last=Moeller | first=Nicolas | title=Closed bosonic string field theory at quintic order II: marginal deformations and effective potential | journal=Journal of High Energy Physics | volume=2007 | issue=9 | date=2007-09-26 | issn=1029-8479 | doi=10.1088/1126-6708/2007/09/118 | pages=118|arxiv=0705.2102| bibcode=2007JHEP...09..118M | s2cid=16383969 }}</ref>