Boundary state from Ellwood invariants

Abstract

Boundary states are given by appropriate linear combinations of Ishibashi states.Starting from any open string field theory solution and assuming Ellwood conjecture we show that every coefficient of such a linear combination is given by an Ellwood invariant, computed in a slightly modified theory where it does not trivially vanish by the on-shell condition. Unlike the previous construction of Kiermaier, Okawa and Zwiebach, ours is linear in the string field, it is manifestly gauge invariant and it is also suitable for solutions known only numerically. The correct boundary state is readily reproduced in the case of known analytic solutions and, as an example, we compute the energy momentum tensor of the rolling tachyon from the generalized invariants of the corresponding solution. We also compute the energy density profile of Siegel-gauge multiple lump solutions and show that, as the level increases, it correctly approaches a sum of delta functions. This provides a gauge invariant way of computing the separations between the lower dimensional D-branes.

BCFT moduli space in level truncation

“…Already at level 2 we find a non universal branch in the tachyon potential which is connected to the perturbative vacuum and which is reasonably flat for (very) small tachyon VEV t. We look at the value of the marginal parameter λ S on this newly found tachyon branch and, at level 5, we find that it starts showing a maximum as a function of t. By improving the solutions up to L = 18 we confirm that λ S has indeed a maximum. We numerically relate the tachyon VEV t with the BCFT modulus λ B , by fitting the Ellwood invariants [25] against their expected value from BCFT, [21,26]. We find that the Ellwood invariants are remarkably close to the Ishisbashi states coefficients of the known BCFT boundary state, even when (at the reachable level) the equation of motion for the tachyon is quite far from being satisfied.…”

“…We choose the cos X marginal deformation to allow comparison with the most recent results from [21]. Given a numerical solution Ψ, its boundary state can be computed using the generalized Ellwood invariants discussed in [26]. The primary operators of a compact free boson at R = 1, which define the Ishibashi states, can be classified by the SU(2) symmetry, see for example [28].…”

“…The primary operators of a compact free boson at R = 1, which define the Ishibashi states, can be classified by the SU(2) symmetry, see for example [28]. An holomorphic operator with SU(2) spin (j, m) has conformal weight h = j 2 and left-handed momentum k = m. Leaving the details of the construction to [26], the coefficients of the first Virasoro Ishibashi states are explicitly computed from a given solution Ψ (together with a tachyon vacuum solution Ψ tv ) as…”

“…We leave the detailed numerical algorithms to other references (see [31] for conservation laws for cubic vertices, appendix of [30] for the evaluation of the action and for solving the equations of motion, [26] for Ellwood invariants.) In [33] some of the algorithms we used will be described in detail. With a C++ code running on parallel clusters we can reach level 18, where we have 34842 fields.…”

“…Fully estimating the errors of the L → ∞ extrapolations is not really possible. We easily compute a "statistical" error by the standard deviation of different extrapolations with varying parameters (typically the number of included levels, as in [26]). However there are important unknown "systematic" errors that come from the fact the some quantities can have different asymptotics at high levels or, as it often happens with Ellwood invariants, anomalous JHEP04(2016)057 behaviour coming from Padé-Borel approximation.…”