An Intro. to Modern Japanese [Book 1 - Grammar] by R. Bowring, et. al.,

By R. Bowring, et. al.,

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R, ± B) = S W (x ). 63) , with s, = ±l. The sum over basic classes of X fact orizes into a sum over t he Xi and a sum over t he Si . 61) [34]. 61) was derived by means of th e u-plane int egral. 6. 11), whose coefficients are polynomials in t he local observables of t he to pological field t heory. In t he case of SU(2), t his is rat her explicit from the t heory of ellipt ic functions. [34]' it tu rns out that an elegant and powerful recursive meth od to perform t his expa nsion in th e case of SU( N) - up t o arbitrary orde r in th e tim es- , follows from the t heory of hyp erelliptic Kleinian funct ions [43, 44, 45, 46].

Manifest calculat ions with t he Riema nn surface allow one to analyze t he monodromy prop ert ies of du al variabl es when moving in t he space of the ord er parameters . For inst ance, in the simplest SU(2) case, on th e u-pl an e of t he single order 4WC define t he symb ols f and res with ad dit iona l fact ors (211'i)- 1 so that T his explains the appea rance of 211'i facto rs in t he Riemann ident ities and in per turbativ e pr epo t ential s in sectio ns 4 and 5. Accordingly, t he th et a-fun cti ons are periodic wit h period 211'i , and since periods of th e J acob i transformation (i =' t di», belong to 211'i (Il + Til )· 40 parameter there are three singular points, and th e magnetic and electr ic vari ables mix when encircli ng t hese points.

Rr' Thus, the problem effectively reduces to the determination of one holomorphic function . If one man age to fix its behaviour nearby singularities, the function can be un ambi guously restored . e. to the per turbati ve limit is und er control (sin ce the theory is asy mptotically frcc). All ot her singularit ies are tr eated with t he use of du ality and of th e non-renorm alizat ion t heorems for the central cha rges of t he SUSY algebra. A combinat ion of t hese two ideas allows one to pr edict t he spec trum of t he st ab le BP S states which become massless in t he deep nonperturbative region and are in charge of all ot her singular ities.

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