Representations of KBc Algebra for Generating String Field Theory Solutions

String field theory (SFT) is a non-perturbative formulation of string theory in which the dynamics of relativistic strings is reformulated in the language of quantum field theory. In string theory, D-branes (short for “Dirichlet membranes”) are a class of extended objects upon which open strings can end with Dirichlet boundary conditions. However, in SFT, D-branes can be expressed as classical solutions to the equation of motion.

Researchers have previously constructed a solution representing the tachyon vacuum (a state with no D-branes at all). This solution requires an operator set (K,B,c) that satisfies a KBc algebra.

In this study, we present a general method for obtaining various solutions in SFT based on a solution that can be constructed with (K,B,c). To check this formalism, we have shown that this method reproduces known solutions that have been constructed previously by other methods.

Our methodology relies on the fact that a solution consisting of (K,B,c) remains a solution for another set (K',B',c') that satisfies the same KBc algebra as the original set. Notably, the new (K',B',c') set may consist of matter operators as well as the original (K,B,c) set.

In order to construct a new (K',B',c') set like this, we first constructed two operations in the original (K,B,c) space. These corresponded to the interior product and the Lie derivative. We defined these operations to satisfy the KBc algebra as proper operations in the (K,B,c) space. We then obtained the new (K',B',c') set through successive application of the Lie derivatives that were specified by a one-parameter family of “tangent vectors.”

Next, we applied our method to generate two known solutions representing a single D-brane with matter operators by starting from a single D-brane solution without matter operators. Thus, we identified a new (K',B',c') set and the one-parameter family of tangent vectors associated with each of the two D-brane solutions with matter operators.

While we have primarily reproduced simpler solutions that are already known, we believe that our formalism is capable of generating unknown solutions in SFT that are physically interesting. Moreover, the fact that the operations in the space of (K,B,c) are the same as those in the theory of differential forms could provide a deeper understanding of the mathematical structure of SFT along with a unified description of classical D-brane solutions.