Script N = 4 SYM matrix integrals for almost all simple gauge groups (exceptE₇andE₈)

Abstract

In this paper the partition function of N = 4 D = 0 super Yang-Mills matrix theory with arbitrary simple gauge group is discussed. We explicitly computed its value for all classical groups of rank r 11 and for the exceptional groups G 2 , F 4 and E 6 . In the case of classical groups of arbitrary rank we conjecture general formulas for the B r , C r and D r series in addition to the known result for the A r series. Also, the relevant boundary term contributing to the Witten index of the corresponding supersymmetric quantum mechanics has been explicitly computed as a simple function of rank for the orthogonal and symplectic groups SO(2N + 1), Sp(2N ), SO(2N ).

Twisted partition functions and H-saddles

“…The latter set of numbers agree with older analytical results by Staudacher [12] and by Pestun [13] as well as with the Monte Carlo estimates by Krauth and Staudacher [20] within the latter's error bars. Interestingly, Z G is consistently smaller than I G bulk whenever the two disagree.…”

“…Thankfully, this has been worked out in the past for SYMQ. A particularly powerful version is via a 0d localization which leaves only rank-many contour integrals as [11][12][13]…”

“…With these, the identity (4.31) simplifies to These identities are checked affirmatively by Table 1, and their analog for N = 8 SYMQ. For classical groups of general ranks, a conjectural formula is available for the numerical limit of Z for N = 4, 8 [13], while its 1d counterpart Ω's has been also computed [2,16]. The comparison of these two sets of numbers via the above identities is given in the appendix.…”

“…The formula (A.2) has been further evaluated for classical groups as [13], Ω Sp(r) = Ω SO(2r+1) = 1 2 2r r! On the other hand, the same reference offered conjectural formulae for the matrix integral counterpart.…”

D-particles on orientifolds and rational invariants

“…(4.2) 4.1 O(2N ) First, we turn to O(2N ) for 2N ≥ 4. For Ω O − (2N ) N =4,8,16 , we made an explicit JK-residue evaluation as in the previous section. The insertion of P can be represented by a Z 2 holonomy along the Euclidean time circle, diag2N ×2N (1, 1, .…”