Long multiplet bootstrap

Abstract

Applications of the bootstrap program to superconformal field theories promise unique new insights into their landscape and could even lead to the discovery of new models. Most existing results of the superconformal bootstrap were obtained form correlation functions of very special fields in short (BPS) representations of the superconformal algebra. Our main goal is to initiate a superconformal bootstrap for long multiplets, one that exploits all constraints from superprimaries and their descendants. To this end, we work out the Casimir equations for four-point correlators of long multiplets of the two-dimensional global N = 2 superconformal algebra. After constructing the full set of conformal blocks we discuss two different applications. The first one concerns two-dimensional (2,0) theories. The numerical bootstrap analysis we perform serves a twofold purpose, as a feasibility study of our long multiplet bootstrap and also as an exploration of (2,0) theories. A second line of applications is directed towards four-dimensional N = 3 SCFTs. In this context, our results imply a new bound c 13 24 for the central charge of such models, which we argue cannot be saturated by an interacting SCFT.

Bootstrapping N = 3 $$ \mathcal{N}=3 $$ superconformal theories

“…Note that both (2.25) and (2.27) diverge at c = 13 24 , which corresponds to the analytic central charge bound obtained in [75], following from the fact that the norm of one of the candidateB [3,3] operators is going to zero. We note that the chiral algebra in 2.2 was constructed with a generic central charge in mind and care was not given to possible null states arising at specific values of c. It is not clear that the solution we have is consistent for c = 13 24 , as null states are expected to decouple.…”

“…In particular there is an analytic lower bound for interacting N 2 SCFTs of c 11 30 ≈ 0.37 [47]. Furthermore it can be shown, by considering the N = 3 stress tensor four-point function in the chiral algebra, that any interacting N 3 SCFT must obey c 13 24 ≈ 0.54 [75]. These two bounds will be depicted as vertical dashed lines in all the numerical results.…”

“…The red line and dots corresponds to the solution of N = 4 SYM theories. The two dashed lines correspond to the minimum central charges for an interacting N = 2 [47] and N = 3 SCFTs [75] section 2.3 we can construct candidate operators, in the chiral algebra, to correspond to â B [3,3] . In this case there is only one candidate, and if one assumes it to be in fact aB [3,3] we find λB [3,3] 2 = 22 85 , 5) which is shown as a green dot in figure 3.…”

Conformal manifolds: ODEs from OPEs

“…This is not the case for long multiplets with nilpotent superconformal invariants. In these superconformal blocks, the coefficients of bosonic blocks that appear are theory dependent [41][42][43]. Even before these exercises are done, it is clear that (2.7) will not allow us to see individual cancellation within a given block.…”

$N=3$ four dimensional field theories

“…For an M-theoretic construction of the theories of exceptional type, see [11]. We should also mention that a complementary approach using bootstrap techniques has been recently initiated in [12,13]. Furthermore, the N = 3 theories described here fit nicely in the classification of rank-one N = 2 SCFTs of [14,15,16,17,18].…”