On exceptional instanton strings
HJE
On exceptional instanton strings
Michele Del Zotto 0 1 3
Guglielmo Lockhart 0 1 2 3
0 Science Park 904 , 1098 XH Amsterdam , The Netherlands
1 17 Oxford Street, Cambridge, MA 02138 , U.S.A
2 Institute for Theoretical Physics, University of Amsterdam
3 Je erson Physical Laboratory, Harvard University
According to a recent classi cation of 6d (1; 0) theories within F-theory there are only six pure" 6d gauge theories which have a UV superconformal xed point. The corresponding gauge groups are SU(3); SO(8); F4; E6; E7, and E8. These exceptional models have BPS strings which are also instantons for the corresponding gauge groups. For G simply-laced, we determine the 2d N = (0; 4) worldsheet theories of such BPS instanton strings by a simple geometric engineering argument. These are given by a twisted S2 compacti cation of the 4d N = 2 theories of type H2; D4; E6; E7 and E8 (and their higher rank generalizations), where the 6d instanton number is mapped to the rank of the corresponding 4d SCFT. This determines their anomaly polynomials and, via topological strings, establishes an interesting relation among the corresponding T 2 and the Hilbert series for moduli spaces of G instantons. Such relations allow to bootstrap the corresponding elliptic genera by modularity. As an example of such procedure, the elliptic genera for a single instanton string are determined. The same method also xes the elliptic genus for case of one F4 instanton. These results unveil a rather surprising relation with the Schur index of the corresponding 4d N = 2 models. ArXiv ePrint: 1609.00310
Nonperturbative E ects; Supersymmetric Gauge Theory; F-Theory; Topo-
-
logical Strings
1 Introduction
2
3 Instanton strings and HgG
theories
(k)
A lightning review of He G(k) models The -twisted He G(k) models and 6d instanton strings
Some generalities about the 2d (0; 4) he(Gk) SCFTs
Topological strings and elliptic genera
6d BPS strings and topological strings
Elliptic genera and Hilbert series
Modular bootstrap of the elliptic genera
From anomaly four-form to modular transformation
Constraining one-string elliptic genera with modularity
Elliptic genus of one SU(3) string
Elliptic genus of one SO(8) string
Elliptic genera of exceptional instanton strings
6.5.1
6.5.2
6.5.3
6.5.4
G = F4
G = E6
G = E7
G = E8
7
Relation with the Schur index of HG(1)
7.1
7.2
The case G = SU(3)
Generalization to other G
A Explicit expressions for the elliptic genera
A.1 Explicit form of the numerator terms
A.2 Tables of coe cients
Introduction
Recently, many new results have been obtained in the context of 6d (1; 0) theories [1{25];
nonetheless, many of their properties remain rather mysterious. A distinctive feature of
these theories is that among their excitations they have self-dual BPS strings preserving
2d (0; 4) supersymmetry on their worldsheet (see e.g. [26]). The 2d (0; 4) theories on the
worldsheets of the BPS strings give an interesting perspective on the physics of the 6d
(1; 0) models [27{43]. Often, such 2d worldsheet theories can be determined using brane
engineerings in IIA or IIB superstrings [44{47]; however, these perturbative brane
engineerings are less helpful in the case of 6d (1,0) systems with exceptional gauge groups, a fact
which is related to the absence of an ADHM construction for exceptional instanton moduli
spaces [48{51].1 On the other hand, it is well-known that systems with exceptional gauge
symmetries are ubiquitous in the landscape of 6d (1; 0) models realized within F-theory [53],
which rely upon the gauge symmetries of non-perturbative seven-brane stacks [54{57]. The
main aim of this paper is to begin lling this gap, shedding some light on the 2d (0; 4)
sigma models with target space the exceptional instanton moduli spaces.
The rank of a 6d SCFT is de ned to be the dimension of its tensor branch, i.e. the
number of independent abelian tensor
elds. Each tensor eld is paired up with a BPS
string which sources it. As our aim is to characterize the exceptional instanton strings, we
prefer to avoid the complications arising from bound states of strings of di erent types, and
we choose to work with rank one theories. The list of 6d (1; 0) rank one theories realized
within F-theory can be found in section 6.1 of [8]. It is rather interesting to remark that
there are only six \pure" gauge theories of rank one which can be completed to SCFTs. The
corresponding gauge groups are SU(3); SO(8); F4; E6; E7 and E8, while the Dirac pairing
of the corresponding strings is n = 3; 4; 5; 6; 8; 12.
One of the most intriguing features of the 6d (1; 0) theories which arise in F-theory is
that some gauge groups are \non-Higgsable" [58, 59], which is the case for the exceptional
models above. These models arise, for instance, in the context of the Heterotic E8
E8
superstring compacti ed on K3 with instanton numbers (12
n; 12 + n) for the two E8
factors. Whenever n 6= 0, the Heterotic string has a strong coupling singularity [26 (...truncated)