4d $\mathcal{N}=2$ SCFT and singularity theory Part III: Rigid singularity

Abstract

We classify three fold isolated quotient Gorenstein singularity C 3 /G. These singularities are rigid, i.e. there is no non-trivial deformation, and we conjecture that they define 4d N = 2 SCFTs which do not have a Coulomb branch. 1(3) We can also turn on expectation value of operator E r,(0,0) : u r = E r,(0,0) .A central question of understanding 4d N = 2 SCFT is to understand the low energy physics for general deformations parameterized by S = (λ, m, u r ). The low energy physics is best captured by the Seiberg-Witten geometry [SW]. Usually Seiberg-Witten geometry is described by a family of Rieman surfaces fibered over space S, and it is conjectured in [XY] that more general Coulomb branch geometry can be captured by the miniversal deformation of certain kind of three fold singularity X [GLS]. Roughly speaking, a deformation is a flat morphism π : Y → S, with π −1 (0) isomorphic to the singularity X, and a mini-versal deformation essentially captures all the deformations. Here S is identified with the parameter space (λ, m, u r ) of our (generalized) Coulomb branch.Therefore the study of 4d N = 2 SCFT and its Coulomb branch solution are reduced to the study of singularity X and its mini-versal deformation. We have classified such X which can be described by complete intersection [XY, YY1, CX], and the physical aspects of these 4d N = 2 SCFTs are studied in [XY1, XY2, XY3, XYY]. All the complete intersection examples studied in [XY, YY1, CX] have non-trivial mini-versal deformation and therefore non-trivial Coulomb branch.The purpose of this note is to study non-complete intersection rational Gorenstein singularities. An interesting class of such singularities are quotient singularity C 3 /G with G a finite subgroup of SL(3). One of main results of this paper is the classification of the three dimensional isolated Gorenstein quotient singularity.We then would like to study mini-versal deformation of these singularities, and a surprising theorem by Schlessinger [S] shows that all such singularities are rigid, i.e. they have no non-trivial deformation 1 . Therefore the corresponding 4d theory has no Coulomb branch 2 . We call such theories rigid N = 2 theories. It would be very interesting to study more properties of these theories.2. Three-fold singularity and 4d N = 2 SCFT Let's discuss more about the interpretation of N = 2 SCFT defined using three fold rational Gorenstein singularity (they are also called canonical singularity [R]). There are two special ways of smoothing a singularity: crepant resolution [R] and mini-versal deformation [GLS]. For the singularities we are interested, we have following facts:1 See [V] for example of rigid compact Calabi-Yau manifolds. 2 Free hypermultiplets do have a Coulomb branch as we can turn on mass deformation.

Coulomb and Higgs branches from canonical singularities. Part 0

“…Our evidence in 4d is somewhat more limited, but points towards the same conclusion, that these rank-0 4d N = 2 theories, engineered at toric singularities [46], can be free hypermultiplets or discrete gauging thereof.…”

“…Similarly, the IIB engineering predicts the existence of infinite families of rank-zero 4d N = 2 theories, which may or may not be interacting, and correspond to singularities X whose complex-structure deformation X does not give rise to any dynamical U(1) vector multiplet in 4d (so that r = 0). This happens, in particular, for all isolated toric singularities; this point was first noted by [46] for toric orbifolds. The same singularities engineer well-studied (and conventional, higher rank) 5d SCFTs in M-theory.…”

“…(The 4d geometric engineering in IIB has also studied in a lot detail over the years, see e.g. [8,[42][43][44][45][46][47].) This gives us a complementary understanding of magnetic quivers, and of JHEP02(2021)003 their generalizations, directly in the geometric-engineering picture.…”

IIB flux non-commutativity and the global structure of field theories

“…Along similar lines, we could also consider compactifications on singular Calabi-Yau theefolds in the context of geometric engineering (starting with [105][106][107][108], and more recently [109][110][111][112][113][114][115][116]). The global structure of any theory that can be engineered in terms of a singular threefold or fourfold with isolated singularities can in principle be obtained via an extension of the methods described here.…”

Higher form symmetries of Argyres-Douglas theories

“…2 See [8][9][10][11][12][13][14][15][16] for more examples of this kind, as well as [17,18] for interesting generalizations beyond the class of hypersurface singularities. 3 The definition of defect group we use in this paper is [22]…”