Non-orientable Boundary Superstring Field Theory with Tachyon Field

We use the BSFT method to study the non-orientable open string field theory (type I). The partition function on the Möbius strip is calculated. We find that, at the one-loop level, the divergence coming from planar graph and unoriented graph cancel each other as expected. 1 Introduction Tachyon condensation is an interesting issue in string theory [1, 2, 3]. To study it, Witten's Boundary String Field Theory (BSFT) [4]-[6] has proven to be a useful method. Tachyon condensation at tree level has been studied first by A. Gerasimov and S. Shatashvili [7] and D. Kutasov, M. Marino, G. Moore [8], then followed by many authors [9]-[18]. The BSFT method was also extended to the case of superstrings in [11], [19]-[23]. Recently, tachyon condensation at one-loop level has been discussed by many authors who have studied it by different methods and got similar results [24]-[37]. However, all of these considerations focus on type II superstring theories which are oriented string theories, i.e. including only annulus or cylinder diagrams at one-loop level. In this work, we will study type I superstring theory. In fact, with Chan-Paton charges, open-string theory in supersymmetric 10 dimensional space-time (type I) has to be non-orientable [38]. To study the non-orientable open-string theory, we need to include the Möbius strip at the one-loop level. It is well known that the gauge group in type I superstring theory has to be SO(32). There are several ways to verify this by anomaly or divergence cancellation. In [38, 44], the on-shell n strings amplitudes for different one-loop graphs are calculated. It was shown that the divergences, due to the integration over the modulus parameter, coming from planar and Möbius graphs are cancelled each other only for SO(32) gauge group. By using BSFT, we will show that, off-shell, this divergences cancellation also holds only for SO(32) gauge group. In the next section, we briefly recall the analysis of powers of the coupling constant in non-orientable string graphs. In the section 3, the partition function on the Möbius strip is calculated by the method of BSFT. In the section 4, we include Chan-Paton factors by considering an unstable Dp-brane in N D9-branes background, then show that the divergences due to the integration over the modulus parameter only cancel each other for N = 32, namely SO(32) gauge group. We summarize our work in the section 5. Boundary string theory is defined on all …