String Field Theory - Witten's Cubic Open String Field Theory

Witten's Cubic Open String Field Theory

The best studied and simplest of covariant interacting string field theories was constructed by Edward Witten. It describes the dynamics of bosonic open strings and is given by adding to the free open string action a cubic vertex:

,

where, as in the free case, is a ghostnumber one element of the BRST-quantized free bosonic open-string Fock-space.

The cubic vertex,

is a triliniar 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 -product defined implicitly through

The -product and cubic vertex satisfy a number of important properties (allowing the to be general ghost number fields):

  1. Cyclicity :
  2. BRST invariance :
    For the -product, this implies that acts as a graded derivation
  3. Associativity
    In terms of the cubic vertex,

In these equations, denotes the ghost number of .

Read more about this topic:  String Field Theory

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