Noncommutative Field Theories and (Super)string Field Theories

Abstract

In this lecture notes we explain and discuss some ideas concerning noncommutative geometry in general, as well as noncommutative field theories and string field theories. We consider noncommutative quantum field theories emphasizing an issue of their renormalizability and the UV/IR mixing. Sen's conjectures on open string tachyon condensation and their application to the D-brane physics have led to wide investigations of the covariant string field theory proposed by Witten about 15 years ago. We review main ingredients of cubic (super)string field theories using various formulations: functional, operator, conformal and the half string formalisms. The main technical tools that are used to study conjectured D-brane decay into closed string vacuum through the tachyon condensation are presented. We describe also methods which are used to study the cubic open string field theory around the tachyon vacuum: construction of the sliver state, "comma" and matrix representations of vertices.

On the classical equivalence of superstring field theories

“…In this formalism there are no problems of collisions, since while the interaction vertex comes with a Y −2 factor, the propagator comes with a factor of its inverse, which is well defined. These operators are canceled in pairs and scattering amplitudes can be calculated and give (at least on-shell and for tree diagrams) the expected results [9]. The two different choices define two a-priori inequivalent theories.…”

“…In fact, a solution based on this vertex operator (in the theory with chiral Y −2 ) was found using lowest-level truncation in [30], where it was interpreted as a supersymmetry breaking solution. In AGM it was suggested that the correct resolution of these problems is to define the string field on the non-BPS D-brane using a gauge field with Erler's gauge field in the GSO(+) sector 2 and with 9) in the GSO(−) sector. It is immediate that the action is the same as in Erler's case.…”

Tachyon solution in a cubic Neveu-Schwarz string field theory

“…The unique (up to rescaling of fields) gauge invariant action unifying the GSO(+) and GSO(−) sectors is [8] (for notations see review [26])…”

The Higgs mechanism in non-commutative gauge theories

“…Several papers have explicitly shown that such theories as φ 4 and U(N) gauge theories, when formulated on a non-commutative space, are one loop renormalizable [5,6,7,8,9,10]. However, results in the literature [11] have shown that non-commutativity renders impossible the continuum renormalization of the spontaneously broken linear sigma model.…”