Editing String field theory (section)

== Witten's cubic open string field theory ==
The best studied and simplest of covariant interacting string field theories was constructed by [[Edward Witten]].<ref>{{cite journal | last=Witten | first=Edward | title=Non-commutative geometry and string field theory | journal=Nuclear Physics B | publisher=Elsevier BV | volume=268 | issue=2 | year=1986 | issn=0550-3213 | doi=10.1016/0550-3213(86)90155-0 | bibcode=1986NuPhB.268..253W | pages=253–294}}</ref>  It describes the dynamics of bosonic open strings and is given by adding to the free open string action a cubic vertex:
:: <math> S(\Psi) = \tfrac{1}{2} \langle \Psi |Q_B |\Psi \rangle + \tfrac{1}{3}  \langle \Psi,\Psi,\Psi \rangle </math>,

where, as in the free case, <math> \Psi </math> is a ghostnumber one element of the BRST-quantized free bosonic open-string Fock-space.

The cubic vertex,
::<math> \langle \Psi_1,\Psi_2,\Psi_3 \rangle </math>
is a trilinear map which takes three string fields of total ghostnumber three and yields a number.  
Following Witten, who was motivated by ideas from noncommutative geometry, it is conventional to introduce the <math>*</math>-product defined implicitly through
:: <math> \langle\Sigma | \Psi_1 *\Psi_2 \rangle = \langle\Sigma, \Psi_1,\Psi_2\rangle  \ .</math>

The <math>*</math>-product and cubic vertex satisfy a number of important properties (allowing  the <math> \Psi_i </math> to be general ghost number fields):
{{ordered list
|1= ''' Cyclicity ''': 
::<math>\langle \Psi_1, \Psi_2, \Psi_3 \rangle = (-1)^{gn(\Psi_3) * (gn(\Psi_2)+ gn(\Psi_1))}\langle \Psi_3, \Psi_1, \Psi_2 \rangle</math>

|2= ''' BRST invariance ''': 
::<math> Q_B \langle \Psi_1, \Psi_2, \Psi_3 \rangle = \langle Q_B \Psi_1, \Psi_2,  \Psi_3 \rangle + (-1)^{gn(\Psi_1)}\langle \Psi_1, Q_B \Psi_2, \Psi_3 \rangle +(-1)^{gn(\Psi_1)+ gn(\Psi_2)}\langle \Psi_1, \Psi_2, Q_B \Psi_3 \rangle </math>

For the <math>*</math>-product, this implies that <math>Q_B</math> acts as a graded derivation
::<math>Q_B (\Psi_1 * \Psi_2) = (Q_B \Psi_1)*\Psi_2 + (-1)^{gn(\Psi_1)} \Psi_1 * (Q_B \Psi_2)</math>
|3= ''' Associativity '''
     
::<math>  \left(\Psi_1 * \Psi_2\right) * \Psi_3 = \Psi_1 * (\Psi_2 * \Psi_3) </math>

In terms of the cubic vertex,
:: <math> \langle\Psi_1 ,\Psi_2 * \Psi_3,\Psi_4 \rangle =  \langle\Psi_1 ,\Psi_2 ,\Psi_3* \Psi_4 \rangle </math>
}}
In these equations, <math> gn(\Psi) </math> denotes the ghost number of <math> \Psi </math>.

===Gauge invariance ===
These properties of the cubic vertex are sufficient to show that <math> S(\Psi) </math> is invariant under
the [[Yang–Mills]]-like gauge transformation,
:: <math> \Psi \to \Psi + Q_B \Lambda + \Psi * \Lambda - \Lambda * \Psi \ , </math>
where <math> \Lambda </math> is an infinitesimal gauge parameter.  Finite gauge transformations take the form
:: <math> \Psi \to e^{-\Lambda} (\Psi + Q_B )e^{\Lambda} </math>
where the exponential is defined by,
:: <math> e^{\Lambda} = 1 + \Lambda + \tfrac{1}{2} \Lambda* \Lambda + \tfrac{1}{3!} \Lambda * \Lambda * \Lambda + \ldots </math>

===Equations of motion ===
The equations of motion are given by the following equation: 
:: <math> Q_B \Psi + \Psi * \Psi  = 0 \left. \right.  \ .</math>
Because the string field <math> \Psi </math> is an infinite collection of ordinary classical fields, these equations represent an infinite collection of non-linear coupled differential equations.  There have been two approaches to finding solutions:  First, numerically, one can truncate the string field to include only fields with mass less than a fixed bound, a procedure known as "level truncation".<ref>{{cite journal | last1=Kostelecký | first1=V. Alan | last2=Samuel | first2=Stuart | title=Spontaneous breaking of Lorentz symmetry in string theory | journal=Physical Review D | publisher=American Physical Society (APS) | volume=39 | issue=2 | date=1989-01-15 | issn=0556-2821 | doi=10.1103/physrevd.39.683 | bibcode=1989PhRvD..39..683K | pages=683–685| pmid=9959689 | hdl=2022/18649 | hdl-access=free }}</ref>  This reduces the equations of motion to a finite number of coupled differential equations and has led to the discovery of many solutions.<ref>{{cite journal | last=Zwiebach | first=Barton | title=Is the String Field Big Enough? | journal=Fortschritte der Physik | publisher=Wiley | volume=49 | issue=4–6 | year=2001 | issn=0015-8208 | doi=10.1002/1521-3978(200105)49:4/6<387::aid-prop387>3.0.co;2-z | bibcode=2001ForPh..49..387Z | page=387| doi-access=free }}</ref><ref>{{cite conference | last1=Taylor | first1=Washington | last2=Zwiebach | first2=Barton | title=Strings, Branes and Extra Dimensions | chapter=D-Branes, Tachyons, and String Field Theory | publisher=World Scientific | year=2004 | isbn=978-981-238-788-2 | doi=10.1142/9789812702821_0012 | pages=641–670|arxiv=hep-th/0311017}}</ref>  Second, following the work of Martin Schnabl <ref>{{cite journal | last=Schnabl | first=Martin | title=Analytic solution for tachyon condensation in open string field theory | journal=Advances in Theoretical and Mathematical Physics | volume=10 | issue=4 | year=2006 | issn=1095-0761 | doi=10.4310/atmp.2006.v10.n4.a1 | pages=433–501|arxiv=hep-th/0511286|doi-access=free}}</ref> one can seek analytic solutions by carefully picking an ansatz which has simple behavior under star multiplication and action by the BRST operator.  This has led to solutions representing marginal deformations, the tachyon vacuum solution<ref>{{cite journal | last1=Fuchs | first1=Ehud | last2=Kroyter | first2=Michael | title=Analytical solutions of open string field theory | journal=Physics Reports | volume=502 | issue=4–5 | year=2011 | issn=0370-1573 | doi=10.1016/j.physrep.2011.01.003 | pages=89–149|arxiv=0807.4722| bibcode=2011PhR...502...89F | s2cid=119203368 }}</ref> and time-independent D-brane systems.<ref>{{cite journal | last1=Erler | first1=Theodore | last2=Maccaferri | first2=Carlo | title=String field theory solution for any open string background | journal=Journal of High Energy Physics | publisher=Springer Nature | volume=2014 | issue=10 | year=2014 | issn=1029-8479 | doi=10.1007/jhep10(2014)029 | page=029|arxiv=1406.3021| bibcode=2014JHEP...10..029E |doi-access=free}}</ref>

===Quantization===
To consistently quantize <math> S(\Psi) </math> one has to fix a gauge.  The traditional choice has been Feynman–Siegel gauge,
:: <math> b_0 \Psi = 0 \left.\right. \ .</math>
Because the gauge transformations are themselves redundant (there are gauge transformations of the gauge transformations), the gauge fixing procedure requires introducing an infinite number of ghosts via the [[Batalin–Vilkovisky formalism|BV formalism]].<ref>{{cite journal | last=Thorn | first=Charles B. | title=String field theory | journal=Physics Reports | publisher=Elsevier BV | volume=175 | issue=1–2 | year=1989 | issn=0370-1573 | doi=10.1016/0370-1573(89)90015-x | bibcode=1989PhR...175....1T | pages=1–101}}</ref>  The complete gauge fixed action is given by
:: <math> S_{\text{gauge-fixed}} = \tfrac{1}{2} \langle \Psi | c_0 L_0 |\Psi\rangle + \tfrac{1}{3} \langle \Psi,\Psi,\Psi \rangle \ , </math>
where the field <math> \Psi </math> is now allowed to be of ''arbitrary ghostnumber''.  In this gauge, the [[Feynman diagrams]] are constructed from a single propagator and vertex. The propagator takes the form of a strip of worldsheet of width <math> \pi </math> and length <math> T </math>

:: [[Image:OSFT propagator.svg]]

There is also an insertion of an integral of the <math> b </math>-ghost along the red line.  The modulus, <math> T </math> is integrated from 0 to <math> \infty </math>.

The three vertex can be described as a way of gluing three propagators together, as shown in the following picture:

:: [[Image:OSFT three vertex.svg]]

In order to represent the vertex embedded in three dimensions, the propagators have been folded in half along their midpoints.  The resulting geometry is completely flat except for a single curvature singularity where the midpoints of the three propagators meet.

These Feynman diagrams generate a complete cover of the moduli space of open string scattering diagrams.  It follows that, for on-shell amplitudes, the ''n''-point open string amplitudes computed using Witten's open string field theory are identical to those computed using standard worldsheet methods.<ref>{{cite journal|author1-link=Steven Giddings | last1=Giddings | first1=Steven B. | last2=Martinec | first2=Emil | last3=Witten | first3=Edward | title=Modular invariance in string field theory | journal=Physics Letters B | publisher=Elsevier BV | volume=176 | issue=3–4 | year=1986 | issn=0370-2693 | doi=10.1016/0370-2693(86)90179-6 | bibcode=1986PhLB..176..362G | pages=362–368}}</ref><ref>{{cite journal | last=Zwiebach | first=Barton | title=A proof that Witten's open string theory gives a single cover of moduli space | journal=Communications in Mathematical Physics | publisher=Springer Science and Business Media LLC | volume=142 | issue=1 | year=1991 | issn=0010-3616 | doi=10.1007/bf02099176 | bibcode=1991CMaPh.142..193Z | pages=193–216| s2cid=121798009 | url=http://projecteuclid.org/euclid.cmp/1104248494 | url-access=subscription }}</ref>