Correspondences in String Field Theory (The Importance of Being Noncommutative...)

Chapter 1 is an introduction to String Field Theory and its use to describe tachyon condensation. Recent reviews of the subject can be found in [24, 25]. Chapter 2 is a review of the works of Rastelli, Sen and Zwiebach that defined VSFT, [36, 37, 38). Chapter 4, a review of solitons in noncommutative field theory [61 J. The original part of this thesis is contained in chapters 3, 5 and 6 which refer to the three main results we obtained. The first one [78], chapter 3, concerns the definition of the multiplication operation in SFT, which is noncommutative. There are three different type of star products, one matter type and two ghosts. They differ in the Neumann coefficient which define the star product. We will show that such coefficients for the three stars are related to each other in a very simple way: a SL(2, R)-like map connects the matter ones with the so called reduced ghost ones; the same map but with an extra minus sign connects the reduceds with the twisted ghosts and, finally, the twisted ghost ones are equal up tp a minus sign to the matter ones. We enphasize the these two ghost star products are different although they give rise to the same solution of equation of motion in VSFT, source of confusion in the past. The second one [75, 76, 77], chapter 5, concerns the possibility to find solutions of VSFT if a B field is switched on, the differences between VSFT with or without the B field and the definition of a new infinite class of solutions that we called "Ancestors" because in the low energy limit they give rise to all so called GMS solitons, which we review in chapter 4. In particular we find that B field behaves as a natural regulator: in [43] it was shown that the geometry of the lower-dimensional lump states is singular at the string level because the midpoint of the string is confined on the brane and that is singular also at the lovv-energy level because in this limit you must introduce an ad hoe regulator by hand. In [76], starting from the lump solution ·with the B field, we showed that at high energy the string midpoint is no more confined on the brane and at low-energy the lump (representing a D-brane) becomes the simplest GMS soliton, using the Seiberg-Witten limit [6] that gives a noncommutative field theory from a string theory when a B field is turned on. This gave the inspiration to write down the Ancestors solutions. In particular, we pointed out a precise isomorphism which seem to be hidden between such solitonic solutions in VSFT and in noncommutative field theory. The third one [80, 81], chapter 6, concerns relations among the small "zoo" of projectors of the star algebra, which we review in chapter 2. They play an important role in the theory because they are solutions of matter equation of motion and/ or define the star algebra of string fields. It turns out that they can be rewritten in a general form involving a matrix U which, case by case, is nothing but the null matrix, the identity matrix, the twist matrix or even and odd powers of the fundamental matrix S which define the D25-brane, the so called "Sliver". In particular, we speculate the possibility to obtain such "general form" using a suitable resummation of the Ancestors. We can summarize saying that the first is a correspondence between the matter and the ghosts (note the plural) noncommutative structure, the second is a correspondence at the same time between B and not B regime and between SFT and noncommutative field theory and, finally, the third is a correspondence among the relevant actors playng in the game of star algebra. These are the correspondences we mean in the title of this thesis. Of course, it is crucial to have noncomrnutativity.