Duality Cascades and Affine Weyl Groups

Abstract

Brane configurations in a circle allow subsequent applications of the Hanany-Witten transitions, which are known as duality cascades. By studying the process of duality cascades corresponding to quantum curves with symmetries of Weyl groups, we find a hidden structure of affine Weyl groups. Namely, the fundamental domain of duality cascades consisting of all the final destinations is characterized by the affine Weyl chamber and the duality cascades are realized as translations of the affine Weyl group, where the overall rank in the brane configuration associates to the grading operator of the affine algebra. The structure of the affine Weyl group guarantees the finiteness of the process and the uniqueness of the endpoint of the duality cascades. In addition to the original duality cascades, we can generalize to the cases with Fayet-Iliopoulos parameters. There we can utilize the Weyl group to analyze the fundamental domain similarly and find that the fundamental domain continues to be the affine Weyl chamber. We further interpret the Weyl group we impose as a "half" of the Hanany-Witten transition.

M2-branes and $\mathfrak{q}$-Painlevé equations

“…On the mathematical side, it provides new Fredholm determinant expressions for the τ -functions of q-Painlevé equations. Moreover, it is worth to observe that this viewpoint opens a new algebraic characterization of duality symmetries of supersymmetric gauge theories in three dimensions in terms of affine Weyl group associated to the q-difference equations [22,23,24,25]. Finally, we expect this line of thought to shed light on the classification problem of five dimensional superconformal field theories with eight supercharges.…”

“…), which is written as a (2N + M ) dimensional integral, can be further reduced to a N dimensional integral:Z k (N ; M 1 , M 2 , M, ζ 1 , ζ 2 ) = e iΘ k (M 1 ,M 2 ,M,ζ 1 ,ζ 2 )Z the grand partition function (3.2) can be written as[78] Ξ k (κ;M 1 , M 2 , M, ζ 1 , ζ 2 ) = ∞ N =0 κ N Z k (N ; M 1 , M 2 , M, ζ 1 , ζ 2 ) Z k (0; M 1 , M 2 , M, ζ 1 , ζ = 0 case When M = 0, the matrix model (B.22) simplifies to Z k (N ; M 1 , M 2 , 0, ζ 1 , ζ 2 ) =e iΘ k (M 1 ,M 2 ,0,ζ 1 ,ζ 2 ) Z m | ρ k (M 1 , M 2 , 0, ζ 1 , ζ 2 ) | µ n ] N ×N m,n , (B.24)whereρ k (M 1 , M 2 , 0, ζ 1 , ζ 2 ) = D VI 1 D VI 2 M =0 = S M 1 ( x) x) C M 2 ( x + 2πζ 2 ) x + 2πζ 2 ) . (B 25).…”

Duality Cascades and Parallelotopes

“…Then, duality cascades are realized by subsequent applications of the HW transitions in brane configurations on a circle with the reference changed when necessary [9]. Namely, given an arbitrary brane configuration with D3-branes compactified on a circle and 5-branes located perpendicularly to the D3-branes and tilted relatively to preserve supersymmetries, the working hypothesis of duality cascades can be formulated as follows.…”

“…Since duality cascades are realized by changing references, from the arguments of charge conservations (1.2), they can be interpreted as translations in the parameter space of relative ranks [8]. Then, the above questions can be reformulated geometrically as follows [9].…”

“…Especially, the brane configurations associated to the del Pezzo geometries of genus one enjoy large symmetries of exceptional Weyl groups. These group-theoretical structures have played central roles in previous studies of the matrix models [8,9,[33][34][35][36][37][38]. For those with interpretations of the del Pezzo geometries enjoying symmetries of Weyl groups, the above questions on duality cascades were answered by discovering the structure of affine Weyl groups [9].…”