Duality cascades and parallelotopes

Abstract

Duality cascades are a series of duality transformations in field theories, which can be realized as the Hanany-Witten transitions in brane configurations on a circle. In the setup of the ABJM theory and its generalizations, from the physical requirement that duality cascades always end and the final destination depends only on the initial brane configuration, we propose that the fundamental domain of supersymmetric brane configurations in duality cascades can tile the whole parameter space of relative ranks by translations, hence is a parallelotope. We provide our arguments for the proposal.

40 bilinear relations of q-Painlevé VI from $$ \mathcal{N} $$ = 4 super Chern-Simons theory

“…See figure 5. The results match the picture of duality cascades indicated in [61,62], which shows that the more distant from the origin the relative JHEP08(2023)191 ranks are, the higher overall rank N a non-vanishing partition function requires. Namely, we have found that the results obtained from bilinear relations of q-Painlevé equations are also consistent with duality cascades.…”

“…As in [61,62], under duality cascades the relative ranks (M 0 , M 1 , M 3 ) are shifted translationally with the overall rank N decreasing and finally we arrive at the fundamental domain (2.12) with the lowest overall rank N . It is, however, subtle when the relative ranks are located on the boundaries of the fundamental domain since relative ranks on one boundary are shifted to those on the opposite one and both should be treated on an equal footing.…”

“…With the help of a computer we can evaluate all terms of the lowest order for various rank differences. Firstly, it is interesting to observe that the expression is only non-vanishing in the fundamental domain proposed in [61,62], where duality cascades in three dimensions are shown to terminate uniquely. In particular for k = 1, 2, the exact expressions of the grand partition function at the lowest order are simple enough, which allows us to discover the bilinear relations (1.1) heuristically.…”

“…Note that not all values of ranks (M 0 , M 1 , M 3 ) are allowed when the overall rank N is vanishing. Indeed, in [61,62] it was found that the brane configurations are subject to breakdown of supersymmetries due to negative ranks appearing in the Hanany-Witten transitions unless…”

“…Duality cascades for three-dimensional super Chern-Simons theories were discussed in [68,69]. In [61,62] duality cascades were defined as the following process.…”