W allC rossing,D iscrete A ttractor Flow and B orcherds A lgebra ?

A bstract. The appearance ofa generalized (or Borcherds{) Kac{M oody algebra in the spectrum ofBPS dyonsin N = 4, d = 4 string theory iselucidated. From the low-energy supergravity analysis,we identify itsrootlattice asthe lattice ofthe T-duality invariants ofthe dyonic charges,the sym m etry group ofthe root system as the extended S-duality group PGL(2;Z) ofthe theory,and the walls ofW eylcham bers as the walls ofm arginal stability fortherelevanttwo-centered solutions.Thisleadstoan interpretation fortheW eyl group asthegroup ofwall-crossing,orthegroup ofdiscreteattractorflows.Furtherm orewe proposean equivalencebetween a \second-quantized m ultiplicity" ofa charge-and m odulidependenthighestweightvectorand thedyon degeneracy,and show thatthe wall-crossing form ulafollowingfrom ourproposalagreeswith thewall-crossingform ulaobtained from the supergravity analysis. Thiscan be thoughtofasproviding a m icroscopicderivation ofthe wall-crossing form ula ofthistheory.