Abstract
Abstract We present an analytic construction of multi-brane solutions with any integer brane number in cubic open string field theory (CSFT) on the basis of the ${K\!Bc}$ algebra. Our solution is given in the pure-gauge form $\Psi=U{Q_\textrm{B}} U^{-1}$ by a unitary string field $U$, which we choose to satisfy two requirements. First, the energy density of the solution should reproduce that of the $(N+1)$-branes. Second, the equations of motion (EOM) of the solution should hold against the solution itself. In spite of the pure-gauge form of $\Psi$, these two conditions are non-trivial ones due to the singularity at $K=0$. For the $(N+1)$-brane solution, our $U$ is specified by $[N/2]$ independent real parameters $\alpha_k$. For the 2-brane ($N=1$), the solution is unique and reproduces the known one. We find that $\alpha_k$ satisfying the two conditions indeed exist as far as we have tested for various integer values of $N\ (=2, 3, 4, 5, \ldots)$. Our multi-brane solutions consisting only of the elements of the ${K\!Bc}$ algebra have the problem that the EOM is not satisfied against the Fock states and therefore are not complete ones. However, our construction should be an important step toward understanding the topological nature of CSFT, which has similarities to the Chern–Simons theory in three dimensions.
Highlights
Since Schnabl’s construction [1] of an analytic solution for tachyon condensation in cubic open string field theory (CSFT), there have appeared lots of studies on the analytic construction of solutions representing multiple D25-branes within the framework of the KBc algebra [2].1 Among them, the construction presented in [4] by using the boundary condition changing operators, in addition to the elements of the KBc algebra, may be a satisfactory one
In [8], we proposed that the 3-brane solution with N = 2 and T = 0 can be constructed in the form (1.2) by making use of the singularities both at K = 0 and K = ∞, and taking, for example, G(K) = (1 + K)2/K
We have presented an analytic expression of the multi-brane solutions of CSFT for arbitrary brane numbers
Summary
In this Appendix, we derive eq (3.28) for Sm1,m2,m3 (3.24) by using the (s, z)-integration formula for the Bcccc-correlators [5, 7]. For hQ (E.5), calculating the sum of residues at z = −1 and −1 ± (2πi/s) by using the formula (C.11) and carrying out the s-integration, we obtain. This leads to the expression of hQ given by (4.12) and (4.13). (±) θ(Q ≤ −2) 1F1(2 + Q, 3; ±z) − θ(Q ≥ 1) 1F1(1 − Q, 3; ±z)
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