Special arithmetic of flavor

Abstract

Abstract:We revisit the classification of rank-1 4d N = 2 QFTs in the spirit of Diophantine Geometry, viewing their special geometries as elliptic curves over the chiral ring (a Dedekind domain). The Kodaira-Néron model maps the space of non-trivial rank-1 special geometries to the well-known moduli of pairs (E, F ∞ ) where E is a relatively minimal, rational elliptic surface with section, and F ∞ a fiber with additive reduction. Requiring enough Seiberg-Witten differentials yields a condition on (E, F ∞ ) equivalent to the "safely irrelevant conjecture". The Mordell-Weil group of E (with the Néron-Tate pairing) contains a canonical root system arising from (−1)-curves in special position in the Néron-Severi group. This canonical system is identified with the roots of the flavor group F: the allowed flavor groups are then read from the Oguiso-Shioda table of Mordell-Weil groups. Discrete gaugings correspond to base changes. Our results are consistent with previous work by Argyres et al.

Homological classification of 4d $$ \mathcal{N} $$ = 2 QFT. Rank-1 revisited

“…This has a physical meaning: the quiver (3.15) with k = 5 belongs to the mutation class of SU (2) SQCD with N f = 5, a QFT affected by Landau poles, so not defined as a QFT in its own right. However, as discussed on page 12, SU (2) SQCD with N f = 5 makes sense as a low-energy effective theory up to some cut-off (Ringel's 'curious fact', see footnote 17). The presence of Landau poles obviously spoils the CY property; a part for that, ∆ = 2 and F = SO(10) are the correct answers for this formal QFT.…”

“…The relation with the geometry of the rational elliptic surfaces may be described directly, without reference to the physical considerations of [17], as we are going to show.…”

“…so, in a sense, the derived category D b coh Y is obtained by gluing the two fractional Calabi-Yau categories of complementary dimensions. Geometrically, this correspond to the operation in [17] of gluing together the special geometries of two complementary rank-1 SCFT to get a compact geometry Y , easier to study than the open geometry of a single SCFT.…”

“…The clue comes from the classification of base-changes between rational elliptic surfaces satisfying the geometric requirements called "UV and SW completeness" obtained in [17] using the tables of [78]. All Z k gaugings 33 of a rank-1 SCFT which preserve N = 2 susy correspond to base-changes of elliptic surfaces (seen as elliptic curves over the field C(z) of rational functions) but the inverse statement is false.…”

“…Argyres et al have completed their program in rank-1, listing all N = 2 SCFTs with a one-dimensional Coulomb branch [3][4][5][6][7]. Their rank-1 classification was subsequently reinterpreted [17] in terms of the Mordell-Weil groups [18] of rational elliptic surfaces (with section) [19] having at least one additive Kodaira singular fiber F ∞ and at most one fiber of the three types {II, III, IV } not dual to affine A D E Dynkin graphs. In this approach, the rank-1 classifications is read from the Oguiso-Shioda table of Mordell-Weil groups [18,20].…”