Editing String field theory (section)

== Light-cone string field theory ==
Light-cone string field theories were introduced by [[Stanley Mandelstam]]<ref>{{cite journal | last=Mandelstam | first=S. | title=Interacting-string picture of dual-resonance models | journal=Nuclear Physics B | publisher=Elsevier BV | volume=64 | year=1973 | issn=0550-3213 | doi=10.1016/0550-3213(73)90622-6 | bibcode=1973NuPhB..64..205M | pages=205–235}}</ref><ref>{{cite journal | last=Mandelstam | first=S. | title=Interacting-string picture of the Neveu-Schwarz-Ramond model | journal=Nuclear Physics B | publisher=Elsevier BV | volume=69 | issue=1 | year=1974 | issn=0550-3213 | doi=10.1016/0550-3213(74)90127-8 | bibcode=1974NuPhB..69...77M | pages=77–106| s2cid=120638932 | url=https://escholarship.org/uc/item/73m8j1dz }}</ref> and developed by Mandelstam, [[Michael Green (physicist)|Michael Green]], [[John Henry Schwarz|John Schwarz]] and Lars Brink.<ref>{{cite journal | last1=Green | first1=Michael B. | last2=Schwarz | first2=John H. | title=Supersymmetric dual string theory: (II). Vertices and trees| journal=Nuclear Physics B | publisher=Elsevier BV | volume=198 | issue=2 | year=1982 | issn=0550-3213 | doi=10.1016/0550-3213(82)90556-9 | bibcode=1982NuPhB.198..252G | pages=252–268}}</ref><ref>{{cite journal | last1=Green | first1=Michael B. | last2=Schwarz | first2=John H. | title=Superstring interactions | journal=Nuclear Physics B | publisher=Elsevier BV | volume=218 | issue=1 | year=1983 | issn=0550-3213 | doi=10.1016/0550-3213(83)90475-3 | bibcode=1983NuPhB.218...43G | pages=43–88}}</ref><ref>{{cite journal | last1=Green | first1=Michael B. | last2=Schwarz | first2=John H. | last3=Brink | first3=Lars | title=Superfield theory of type (II) superstrings | journal=Nuclear Physics B | publisher=Elsevier BV | volume=219 | issue=2 | year=1983 | issn=0550-3213 | doi=10.1016/0550-3213(83)90651-x | bibcode=1983NuPhB.219..437G | pages=437–478}}</ref><ref>{{cite journal | last1=Green | first1=Michael B. | last2=Schwarz | first2=John H. | title=Superstring field theory | journal=Nuclear Physics B | publisher=Elsevier BV | volume=243 | issue=3 | year=1984 | issn=0550-3213 | doi=10.1016/0550-3213(84)90488-7 | bibcode=1984NuPhB.243..475G | pages=475–536}}</ref><ref>{{cite journal | last=Mandelstam | first=Stanley | title=Interacting-String Picture of the Fermionic String | journal=Progress of Theoretical Physics Supplement | publisher=Oxford University Press (OUP) | volume=86 | year=1986 | issn=0375-9687 | doi=10.1143/ptps.86.163 | bibcode=1986PThPS..86..163M | pages=163–170|doi-access=free}}</ref>  An explicit description of the second-quantization of the light-cone string was given by [[Michio Kaku]] and [[Keiji Kikkawa]].<ref>{{cite journal | last1=Kaku | first1=Michio | last2=Kikkawa | first2=K. | title=Field theory of relativistic strings. I. Trees | journal=Physical Review D | publisher=American Physical Society (APS) | volume=10 | issue=4 | date=1974-08-15 | issn=0556-2821 | doi=10.1103/physrevd.10.1110 | bibcode=1974PhRvD..10.1110K | pages=1110–1133}}</ref><ref>{{cite journal | last1=Kaku | first1=Michio | last2=Kikkawa | first2=K. | title=Field theory of relativistic strings. II. Loops and Pomerons | journal=Physical Review D | publisher=American Physical Society (APS) | volume=10 | issue=6 | date=1974-09-15 | issn=0556-2821 | doi=10.1103/physrevd.10.1823 | bibcode=1974PhRvD..10.1823K | pages=1823–1843}}</ref>

Light-cone string field theories were the first string field theories to be constructed and are based on the simplicity of string scattering in light-cone gauge.  For example, in the [[bosonic string theory|bosonic closed string]] case, the worldsheet scattering diagrams naturally take a Feynman diagram-like form, being built from two ingredients, a [[propagator]],

::[[Image:Light Cone String Propagator.svg]]

and two vertices for splitting and joining strings, which can be used to glue three propagators together,

::[[Image:Closed String Light Cone Vertex.svg]]

These vertices and propagators produce a single cover of the moduli space of <math>n</math>-point closed string scattering amplitudes so no higher order vertices are required.<ref>{{cite journal | last1=D'Hoker | first1=Eric | last2=Giddings | first2=Steven B. | title=Unitarity of the closed bosonic Polyakov string | journal=Nuclear Physics B | publisher=Elsevier BV | volume=291 | year=1987 | issn=0550-3213 | doi=10.1016/0550-3213(87)90466-4 | bibcode=1987NuPhB.291...90D | pages=90–112}}</ref> Similar vertices exist for the open string.

When one considers light-cone quantized ''superstrings'', the discussion is more subtle as divergences can arise when the light-cone vertices collide.<ref>{{cite journal | last1=Greensite | first1=J. | last2=Klinkhamer | first2=F.R. | title=New interactions for superstrings | journal=Nuclear Physics B | publisher=Elsevier BV | volume=281 | issue=1–2 | year=1987 | issn=0550-3213 | doi=10.1016/0550-3213(87)90256-2 | bibcode=1987NuPhB.281..269G | pages=269–288}}</ref>  To produce a consistent theory, it is necessary to introduce higher order vertices, called contact terms, to cancel the divergences.

Light-cone string field theories have the disadvantage that they break manifest [[Lorentz invariance]].  However, in backgrounds with [[light-like]] [[Killing vector field|Killing vectors]], they can considerably simplify the quantization of the string action.  Moreover, until the advent of the Berkovits string<ref>{{cite journal | last=Berkovits | first=Nathan | title=Super-Poincare covariant quantization of the superstring | journal=Journal of High Energy Physics | volume=2000 | issue=4 | date=2000-04-15 | issn=1029-8479 | doi=10.1088/1126-6708/2000/04/018 | pages=018|arxiv=hep-th/0001035| bibcode=2000JHEP...04..018B |doi-access=free}}</ref> it was the only known method for quantizing strings in the presence of [[Ramond–Ramond field]]s.  In recent research, light-cone string field theory played an important role in understanding strings in pp-wave backgrounds.<ref>M. Spradlin and A.  Volovich, "Light-cone string field theory in a plane wave", Lectures given at ICTP Spring School on Superstring Theory and Related Topics, Trieste, Italy, 31 Mar – 8 Apr (2003) hep-th/0310033.</ref>