Dbrane Phase Transitions and Monodromy in K-theory

Majumder and Sen have given an explicit construction of a first order phase transition in a non–supersymmetric system of Dbranes that occurs when the B field is varied. We show that the description of this transition in terms of K-theory involves a bundle of K groups of non–commutative algebras over the Kähler cone with nontrivial monodromy. Thus the study of monodromy in K groups associated with quantized algebras can be used to predict the phase structure of systems of (non–supersymmetric) Dbranes. A major development in string theory over the past couple of years has been an increasing understanding of non–supersymmetric Dbranes and their dynamics, initiated largely by Sen [1]. The relevance of K-theory for the classification of Dbrane charges was pointed out by Witten [2] following earlier work of Minasian and Moore [3]. Reviews of these developments are [4–6]. The motivation for the present work was an intriguing observation made by Majumder and Sen [7]. Using an exact conformal field theory description [8] of non-supersymmetric Dbranes on a K3 surface, they found three different phases in the region on the moduli space where the surface looks like an orbifold X = T /ZZ2. The phase diagram in Fig. 1 represents different states (labeled as I, II and III) of two D-branes 1 wrapped over exceptional 2cycles S1 and S2 in X, which correspond to fixed points in the orbifold limit. In phase I the two Dbranes recombine into a single Dbrane wrapped over a non–supersymmetric cycle with the homology class [S] = [S1] + [S2]. This happens when the minimal radius of T 4 becomes less than the difference of the B-field fluxes through S1 and S2, ζ < R with the appropriate normalization. From the analysis of the tachyon potential [7] V (α) ∝ ( 1 4 (Rc −R) cos(απ) + ζ cos( 1 2 απ) )