Abstract
We show that it is possible to construct a well-defined effective field theory incorporating string winding modes without using strong constraint in double field theory. We show that X-ray (Radon) transform on a torus is well-suited for describing weakly constrained double fields, and any weakly constrained fields are represented as a sum of strongly constrained fields. Using inverse X-ray transform we define a novel binary operation which is compatible with the level matching constraint. Based on this formalism, we construct a consistent gauge transform and gauge invariant action without using strong constraint. We then discuss the relation of our result to the closed string field theory. Our construction suggests that there exists an effective field theory description for massless sector of closed string field theory on a torus in an associative truncation.
Highlights
We show that it is possible to construct a well-defined effective field theory incorporating string winding modes without using strong constraint in double field theory
As we have shown in the previous section, a weakly constrained field f (XI ) is represented by summing over all possible X-ray images fΠ(zi), which are strongly constrained fields defined on closed d-dimensional null planes D0(XI, Π)
We start by using the previous formulation, which is manifestly compatible with level matching constraint, to construct a consistent gauge transformation and corresponding gauge invariant action without using the strong constraint
Summary
Before considering the X-ray transform, we will review some basic facts about d-dimensional closed plane on a doubled torus T 2d [14,15,16]. A closed d-dimensional plane D(XI , Π) on a T 2d passing through a point XI ∈ T 2d is parametrized as. For the periodicity of the coordinates XI , the range of the parameters ti are given by 0 to 1. For later use we present a parametrization of a d-dimensional closed null plane: D0(XI , Π) = {XI + tiΠiI |0 ≤ ti < 1 and Π ∈ Pd0} ,. Where the Pd0 is a subset of the Pd whose row vectors of Π ∈ Pd0 are null and mutually orthogonal. If two slicing matrices Π′ and Π are related by P SL(d, Z) rotation, they parametrize the same d-plane because the a ∈ P SL(d, Z) can be absorbed into the parameter ti by redefining t′i = tjaji
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