Mixed Hodge structure and $\mathcal{N}=2$ Coulomb branch solution

Abstract

The Coulomb branch of four dimensional N = 2 theories can be solved by finding a Seiberg-Witten (SW) geometry and a SW differential. While lots of SW geometries are found, the extraction of low energy theory out of it is limited due to following reasons: (a) the difficulty of distinguishing electric-magnetic and flavor charges; (b) the difficulty of determining the low energy theory at singular point, (c) the lack of SW differential. We show that the mixed Hodge structure (MHS) can be used to fully solve the low energy physics of Coulomb branch at every vacuum. The MHS can be used to solve above three problems as follows: (a) The smooth fiber of SW geometry carries a Mixed Hodge Structure structure with two weights: one weight describes electric-magnetic charge and the other for the flavor charge; (b) for the singular fiber, there are three MHS which can used to determine the IR theory. (c) the MHS at ∞ is used to find the SW differential.

Snowmass White Paper on SCFTs

“…The methods which worked to compile the classification in rank-1 are of limited use for higher ranks. Thus for general answers a variety of approaches have been employed: (a) leveraging singularity theory to perform a systematic analysis of N = 2 SCFTs which can be engineered in type IIB on a Calabi-Yau threefold [230][231][232][233][234], (b) systematic study of the constraint of special Kähler geometry [235] constraining the set of allowed Coulomb branches [202,203,[236][237][238][239][240][241][242][243][244], and (c) systematic understanding of Higgs branches of N Q = 8 supercharge SCFTs as the Coulomb branch of 3D N = 4 SCFT magnetic quivers (as 3D mirror symmetry [245]) for Higgs branches of D > 3 SCFTs [140,145,[159][160][161][246][247][248][249][250][251]. These approaches are helpful given the limited effectiveness of bootstrap methods in this context [252].…”

The hitchhiker's guide to 4d $\mathcal{N}=2$ superconformal field theories

“…On the other hand, given a general SW geometry (which encodes the low energy EFT of a putative SCFT), it can be difficult to extract information of the corresponding SCFT. See[151] for a recent attempt using the mixed Hodge structure of the fiber of the SW geometry.…”

The Characteristic Dimension of Four-dimensional $\mathcal{N} = 2$ SCFTs

“…CBs of N = 2 SCFTs of rank r are stratified by singular loci supported in codimension r − p where the dynamics is that of a rank p SCFTs. The pattern of intersections of such singular loci give rise to a stratification of the CB [30,36,37]. Isotrivial SCFTs form a class which is necessarily closed upon stratification, meaning that all the strata are isotrivial theories themselves.…”