Spacetime Supersymmetry in Low‐Dimensional Perturbative Heterotic Compactifications

Abstract

We study the constraints of spacetime supersymmetry for perturbative threeand two-dimensional Minkowski vacua of the critical heterotic string. Assuming a standard RNS construction of the spacetime supersymmetry generators and a compact unitary internal superconformal worldsheet theory, we describe the worldsheet structures associated to various spacetime supersymmetries. In three dimensions we show that there are no CFT surprises: each allowed spacetime supersymmetry is realized by a supergravity compactification. As a recent orbifold construction shows, in two dimensions there are more exotic possibilities, and we discuss how these fit into our analysis. 1 There is a classification [2][3][4] of the SCAs that only have currents with spins in {0, 1/2, 1, 3/2, 2} and a unique spin 2 current. All such SCAs have N ≤ 4, and those with N ≤ 3 are unique up to isomorphism; for N = 4 our notation A 4 c denotes the so-called "small" N=4 algebra. 2 The investigations of the worldsheet structure of Calabi-Yau compactifications and related theories [5,6] showed that A 2 9 SCA with integral R-charges was a sufficient condition for spacetime supersymmetry, and the work of [1] went on to show this to be necessary as well. 4 z 2 12 , J αβ 1 J γ δ 2 ∼ 1 z 2 12 δ αγ δ βδ − δ αδ δ βγFrom the last line it follows that the currents J α are chiral primary with respect to A 2 12 , while the J †α are their anti-chiral primary conjugates. Moreover, for any α the triplet {J α , J †α , J R } forms an su(2) 2 Kac-Moody algebra. As a result, we have the additional non-trivial OPEs

Non-duality in three dimensions

“…Moreover, N = 3 vacua cannot be obtained from any heterotic supergravity description. Recently, this result has also been extended to general perturbative heterotic string constructions [33]. The large amount of supersymmetry preserved suggests that we may have a hope of identifying non-perturbative heterotic ingredients of a possible dual to the M-theory constructions.…”

Generalized ADHM equations from marginal deformations in open superstring field theory

“…For compactifications to dimensions higher than four, it automatically follows that spacetime supersymmetry implies extended worldsheet superconformal symmetry, as one can always dimensionally reduce back to four dimensions. Results for compactifications to dimensions lower than four, which appeared only recently [33] (for the heterotic worldsheet), seem to suggest that the boundary worldsheet theory has a global N = 2 superconformal symmetry as long as the background conserves at least two spacetime supercharges with the same chirality in two non-compact dimensions (i.e. N = (2, 0) supersymmetry in 2d -we are going to discuss a concrete example of this minimal setting in subsection 5.3).…”

Spin(7)-manifolds as generalized connected sums and 3d $$ \mathcal{N}=1 $$ theories

“…In this paper we will not consider fluxes but focus on the geometric constructions. Of course it is interesting to consider this in the future and study the effects of these on supersymmetry breaking and potential obstructions to dualities, both M/F-duality [34], where in particular 4d Poincaré invariance could be broken, as well as obstructions to M-theory/weakly-coupled heterotic duality [35,36].…”