Abstract
We uncover a Kawai-Lewellen-Tye (KLT)-type factorization of closed string amplitudes into open string amplitudes for closed string states carrying winding and momentum in toroidal compactifications. The winding and momentum closed string quantum numbers map respectively to the integer and fractional winding quantum numbers of open strings ending on a D-brane array localized in the compactified directions. The closed string amplitudes factorize into products of open string scattering amplitudes with the open strings ending on a D-brane configuration determined by closed string data.
Highlights
An open string state can either carry momentum or winding along the circle, depending upon whether we impose Neumann or Dirichlet boundary conditions along the circle, but cannot carry both momentum and winding. The resolution of this conundrum is that, while the winding number wi of the i-th closed string can be identified with the winding number of the i-th open string obeying Dirichlet boundary conditions, the closed string momentum ni is encoded in the D-brane configuration where open string amplitudes are defined, and corresponds to the fractional winding number of an open string stretched between D-branes
We showed that the string amplitudes of winding closed strings factorize into quadratic products of amplitudes for open strings ending on an array of D-branes transverse to the compact directions
The winding number w of a closed string is mapped to an integer winding number of an open string wrapping around the compactified circle; while the momentum of a closed string is mapped to the fractional winding number that encodes how many D-branes the open string traverses, excluding the w times that the open string winds around the full circle
Summary
We consider string theory described by a sigma model that maps the worldsheet Σ to a ddimensional target space M. We denote the coordinates by XM = (Xμ, X), where M = 0, . A closed string that winds w times around the circle along the X-direction satisfies. The singlevaluedness of the operator exp(ipX) requires the quantization condition n p= ,. Where the counterclockwise oriented contour C encloses the vertex operator associated with the string. 2.1 Closed string tachyons and cocycles The closed string tachyon is described by the following vertex operator [17]: VC(z , z) = gc exp i 4 πα We have pRVC(z , z) = pRVC(z , z) This level matching condition implies that an asymptotic tachyonic state cannot have both nonzero Kaluza-Klein and winding numbers. Since both w1 and n2 are integers, we find that VC1(z1 , z1) and VC2(z2 , z2) commute
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