Abstract
One of the simplest (1, 0) supersymmetric theories in six dimensions lives on the world volume of one M5 brane at a D type singularity ℂ2/Dk. The low energy theory is given by an SQCD theory with Sp(k − 4) gauge group, a precise number of 2k flavors which is anomaly free, and a scale which is set by the inverse gauge coupling. The Higgs branch at finite coupling {mathrm{mathscr{H}}}_f is a closure of a nilpotent orbit of D2k and develops many more flat directions as the inverse gauge coupling is set to zero (violating a standard lore that wrongly claims the Higgs branch remains classical). The quaternionic dimension grows by 29 for any k and the Higgs branch stops being a closure of a nilpotent orbit for k > 4, with an exception of k = 4 where it becomes overline{{ min}_{E_8}} , the closure of the minimal nilpotent orbit of E8, thus having a rare phenomenon of flavor symmetry enhancement in six dimensions. Geometrically, the natural inclusion of {mathrm{mathscr{H}}}_fsubset {mathrm{mathscr{H}}}_{infty } fits into the Brieskorn Slodowy theory of transverse slices, and the transverse slice is computed to be overline{{ min}_{E_8}} for any k > 3. This is identified with the well known small E8 instanton transition where 1 tensor multiplet is traded with 29 hypermultiplets, thus giving a physical interpretation to the geometric theory. By the analogy with the classical case, we call this the Kraft Procesi transition.
Highlights
A much less studied phenomenon, which certainly deserves full attention, is the phenomenon of the small instanton transition, which was first pointed out in [8], where the quantities nV and 29nT +nH remain fixed while the numbers nH and nT change values
A 3d Coulomb branch allows for this evaluation and we find that the new massless states at the tensionless string limit transform in the spinor representation of the global symmetry, while all other massless states are composites of these new states, together with the states that already generate the Higgs branch at finite coupling — mesons present in the IR theory and transform in the adjoint representation of the global symmetry
It should be noted that while results on Kraft Procesi (KP) transitions are mostly known for Hasse diagrams of nilpotent orbits, the results of this paper show a nice extension to the case where the bigger moduli space is not a closure of a nilpotent orbit
Summary
The finite coupling Higgs branch Hf of this theory, using the F and D term equations, is given by the set of all 4k × 4k antisymmetric matrices M with complex entries that square to 0 and has rank at most 2k − 8: Hf = M4k×4k|M + M T = 0, M 2 = 0, r(M ) ≤ 2k − 8 This gives an algebraic description of the closure of the nilpotent orbit of SO(4k) of height , corresponding to the partition [22k−8, 116] of 4k.3. This quiver shows up in [30] in the study of Slodowy slices.): They are equal to the Higgs and Coulomb branch dimensions of USp(2k − 8) gauge theory with 2k flavours, respectively.
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