Geometric classification of 4d N=2 SCFTs
A bstract The classification of 4d N = 2 SCFTs boils down to the classification of conical special geometries with closed Reeb orbits (CSG). Under mild assumptions, one shows that the underlying complex space of a CSG is (birational to) an affine cone over a simply-connected ℚ-factorial log -Fano va...
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| Published in: | The journal of high energy physics 2018-07, Vol.2018 (7), p.1-108 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | eng |
| Subjects: | |
| Online Access: | Get full text |
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| Summary: | A
bstract
The classification of 4d
N
=
2
SCFTs boils down to the classification of conical special geometries with closed Reeb orbits (CSG). Under mild assumptions, one shows that the underlying complex space of a CSG is (birational to) an affine cone over a simply-connected ℚ-factorial
log
-Fano variety with Hodge numbers
h
p,q
=
δ
p,q
. With some plausible restrictions, this means that the Coulomb branch chiral ring
is a graded polynomial ring generated by global holomorphic functions
u
i
of dimension Δ
i
. The coarse-grained classification of the CSG consists in listing the (finitely many) dimension
k
-tuples {Δ
1
,
Δ
2
,
⋯
,
Δ
k
} which are realized as Coulomb branch dimensions of some rank-
k
CSG: this is the problem we address in this paper. Our sheaf-theoretical analysis leads to an Universal Dimension Formula for the possible {Δ
1
,
⋯
,
Δ
k
}’s. For Lagrangian SCFTs the Universal Formula reduces to the fundamental theorem of Springer Theory.
The number
N
(
k
) of dimensions allowed in rank
k
is given by a certain sum of the Erdös-Bateman Number-Theoretic function (sequence A070243 in OEIS) so that for large
k
N
k
=
2
ζ
2
ζ
3
ζ
6
k
2
+
o
k
2
.
In the special case
k
= 2 our dimension formula reproduces a recent result by Argyres et al.
Class Field Theory implies a subtlety: certain dimension
k
-tuples {Δ
1
,
⋯
,
Δ
k
} are consistent only if supplemented by additional selection rules on the electro-magnetic charges, that is, for a SCFT with these Coulomb dimensions not all charges/fluxes consistent with Dirac quantization are permitted.
Since the arguments tend to be abstract, we illustrate the various aspects with several concrete examples and perform a number of explicit checks. We include detailed tables of dimensions for the first few
k
’s. |
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| ISSN: | 1029-8479 |