An Approach to Script N = 4 ADE Gauge Theory on<i>K</i>3

Jinzenji, Masao; Sasaki, Toru

doi:10.1088/1126-6708/2002/09/002

Cited by 5 publications

(48 citation statements)

References 31 publications

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“…In the next work, we tried to determine the N = 4 ADE partition function on K3 [8]. Using the generalization of (3.2) to ADE, we define the ADE blow-up formula:…”

Section: Our Previous Resultsmentioning

confidence: 99%

“…To second line in IIA side, we compactify T 2 ×ALE ADE and obtain N = 4 ADE on K3. To the second line in Hetero side, we compactify T 4 × T 2 and obtain (N = 4 U (1) on ALE ADE ) ⊗24 [8]. The origin of ⊗24 is explained by the compactification of K3 × T 2 in IIA side.…”

Section: Our Previous Resultsmentioning

confidence: 99%

“…, (4.7) 8) and request the modular property (4.5), then we have a = 1 4 and can determine the partition function…”

Section: Our Previous Resultsmentioning

confidence: 99%

“…Here we list up several explicit self-dualized D, E partition functions by using the results in [8]: …”

Section: Self-dualizing Our Original D E Partition Function On K3mentioning

confidence: 99%

“…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 S-duality conjecture for these cases has already been established [23,16]. Thus we tried to derive the D, E partition function on K3 in the previous article [8]. Fundamental strategy is the following: We speculated a piece of D, E partition function on K3, which was derived by using the denominator identity of affine Lie algebra for ADE group [9,13].…”

Section: Introductionmentioning

confidence: 99%

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GAP CONDITION AND SELF-DUALIZED ${\mathcal N}=4$ SUPER-YANG–MILLS THEORY FOR ADE GAUGE GROUP ON K3

Sasaki

2004

Mod. Phys. Lett. A

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“…In the next work, we tried to determine the N = 4 ADE partition function on K3 [8]. Using the generalization of (3.2) to ADE, we define the ADE blow-up formula:…”

Section: Our Previous Resultsmentioning

confidence: 99%

Section: Our Previous Resultsmentioning

confidence: 99%

“…, (4.7) 8) and request the modular property (4.5), then we have a = 1 4 and can determine the partition function…”

Section: Our Previous Resultsmentioning

confidence: 99%

“…Here we list up several explicit self-dualized D, E partition functions by using the results in [8]: …”

Section: Self-dualizing Our Original D E Partition Function On K3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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GAP CONDITION AND SELF-DUALIZED ${\mathcal N}=4$ SUPER-YANG–MILLS THEORY FOR ADE GAUGE GROUP ON K3

Sasaki

2004

Mod. Phys. Lett. A

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Hecke operator andS-duality of Script N = 4 super Yang-Mills for ADE gauge group onK3

Sasaki

2003

J. High Energy Phys.

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We determine the partition functions of N = 4 super Yang-Mills gauge theory for some ADE gauge groups on K3, under the assumption that they are holomorphic. Our partition functions satisfy the gap condition and Montonen-Olive duality at the same time, like the SU (N ) partition functions of Vafa and Witten. As a result we find a close relation between Hecke operator and S-duality of N = 4 super Yang-Mills for ADE gauge group on K3.

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S-duality in Vafa-Witten theory for non-simply laced gauge groups

2008

J. High Energy Phys.

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Vafa-Witten theory is a twisted N = 4 supersymmetric gauge theory whose partition functions are the generating functions of the Euler number of instanton moduli spaces. In this paper, we recall quantum gauge theory with discrete electric and magnetic fluxes and review the main results of Vafa-Witten theory when the gauge group is simply laced. Based on the transformations of theta functions and their appearance in the blow-up formulae, we propose explicit transformations of the partition functions under the Hecke group when the gauge group is non-simply laced. We provide various evidences and consistency checks. development in both topology and S-duality. Recently, there has been much interest in the third twist [24] for its relation to the geometric Langlands programme [20,9,8,34].In this paper, We revisit the Vafa-Witten theory with an emphasis on the roles of Langlands duals. While much progress has been made when the gauge group is simply laced, both in physics [25,22,16,17] and in mathematics [21,38,39,23,13], there has been almost no attempt in studying the Vafa-Witten theory in the non-simply laced case (see however [19]). The latter differs from the simply laced case in several crucial ways. First, as the gauge group is distinct from its dual, we need to consider simultaneously two sets of discrete fluxes and hence two sets of partition functions. Second, the Z 2 -duality of electricity and magnetism is not part of the modular group, but the Hecke group [10, 6, 1], which has different relations on the generators. Based on the transformations of theta functions and their appearance in the blow-up formulae, we propose explicit transformations of the partition functions with various discrete fluxes under the Hecke group. This would be the counterpart, when the gauge group is non-simply laced, of the sharpened S-duality conjecture of Vafa and Witten [31].The organisation of the paper is as follows. In Section 2, we review 't Hooft's discrete electric and magnetic fluxes [15] and their role in canonical and path integral quantisation. Given an arbitrary simple gauge group, we choose a subset of permitted discrete fluxes so that under S-duality, the discrete electric and magnetic fluxes are interchanged. In Section 3, we consider Vafa-Witten theory for simply laced gauge groups. The sharpened S-duality conjecture [31] specifies how the partition functions with various discrete fluxes transform under the modular group. From this and the above selection of discrete fluxes for arbitrary gauge groups, we deduce the usual Z 2 -duality which exchanges the gauge group and its Langlands dual. To compare with the non-simply laced case, we summarise the relevant mathematical results, especially the blow-up formulae [38,31,23,19], in which the universal factors contain theta functions constructed from the coroot lattice [19] and the Dedekind eta function. In Section 4, we study Vafa-Witten theory for non-simply laced gauge groups. There are two sets of partition functions which transform under the Hecke group. To find the...

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An Approach to Script N = 4 ADE Gauge Theory onK3

Abstract: We propose a recipe for determination of the partition function of N = 4 ADE gauge theory on K3 by generalizing our previous results of the SU (N ) case. The resulting partition function satisfies Montonen-Olive duality for ADE gauge group.

Cited by 5 publications

References 31 publications

GAP CONDITION AND SELF-DUALIZED ${\mathcal N}=4$ SUPER-YANG–MILLS THEORY FOR ADE GAUGE GROUP ON K3

GAP CONDITION AND SELF-DUALIZED ${\mathcal N}=4$ SUPER-YANG–MILLS THEORY FOR ADE GAUGE GROUP ON K3

Hecke operator andS-duality of Script N = 4 super Yang-Mills for ADE gauge group onK3

S-duality in Vafa-Witten theory for non-simply laced gauge groups

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