A strong coupling test of S-duality

An Approach to Script N = 4 ADE Gauge Theory onK3

“…In (2.5), τ is the gauge coupling constant including theta angle, χ(X) is Euler number of X and r is the rank of G. Introduction of q − 1 24 factor is required by the modular property like the case of η(τ ) function. For SU (N )/Z N , this conjecture is nothing but Vafa-Witten's [21]. With this result, Vafa and Witten conjectured the behavior of the partition functions under the action of SL(2, Z) on τ .…”

“…The duality of this theory is believed to be the one proposed by Montonen and Olive. On the basis of this thought, the study of N = 4 SU (N ) gauge theory has enjoyed many progresses with the aid of the S-duality conjecture: the sharpened version of Montonen-Olive duality [21,11,13]. On the other hand, there has been almost no attempt to determine the partition function of D, E gauge theory in spite of the fact that the S duality conjecture for these cases has already been established [22,16].…”

“…A special class of K3 surfaces can be constructed from blowing up sixteen singularities of the quotient space T 4 /Z 2 = S 0 . Therefore, the partition function on K3 should be given as the product of the partition function of S 0 and the blow-up formula [24,21,9]. This attempt was successful not only for SU (2) but also for SU (N ).…”

“…The verification of this attempt mainly depends on the following two facts. The first one is that the SU (N ) partition function on K3 can be described by a Hecke transformation of order N of 1/η(τ ) 24 [21,13,26,11]. The second one is that 1/η(τ /N ) can be written as the form θ 2 A N−1 (τ )/η(τ ) N , where θ 2 A N−1 (τ ) is a kind of A N −1 theta function [7].…”

“…Then we take SL(2, Z) orbit of this function (precisely speaking, we have to include the transformation τ → τ + 1/m in some cases). After that, we take linear combination of all the building blocks so obtained, whose coefficients are determined to satisfy the S-duality conjecture, as was done in VafaWitten 's original work [21]. The result is nothing but the order N Hecke transformation of 1/η(τ ) 24 .…”