Brane setups for the exceptional-series orbifold gauge theories: the $\mathbb{Z}_3$ generalization of the orientifold plane

The stepwise-projection analysis of Feng, Hanany, He and Prezas (JHEP 0201 (2002) 040) showed that the quiver gauge theory of D-branes at the orbifold singularity $\mathbb{C}^2/E_6$ ($E_6$ the binary tetrahedral subgroup of $SU(2)$) is obtained from the abelian $\mathbb{Z}_6$ orbifold theory (or, as an alternative parent, the binary dihedral $D_2$ orbifold theory) by a $\mathbb{Z}_3$ quotient acting on space-time and on the string world-sheet, in exactly the way the orientifold/ON-plane provides a $\mathbb{Z}_2$ quotient of the abelian $\mathbb{Z}_{2k}$ theory for the $D_k$ series. Whether an object of string theory — a generalized orientifold plane — exists that performs this order-3 (and, generically, an arbitrary finite symmetry $S$ governing the quiver node identifications and splittings) action, and thereby whether brane setups for the exceptional series $E_6, E_7, E_8$ and for generic non-abelian orbifold singularities can be constructed, is an open problem that later literature has neither solved nor refuted.

Background

Two equivalent routes exist for engineering the supersymmetric gauge theories living on D-branes — extended objects in string theory on which open strings can end. In the first, a stack of D3-branes (Dirichlet 3-branes) placed at an orbifold singularity $\mathbb{C}^k/\Gamma$, with $\Gamma$ a finite subgroup of $SU(k)$, produces a quiver gauge theory: a product of unitary gauge groups $U(n_i)$ with bifundamental matter encoded in a directed graph (the quiver) whose adjacency matrix is fixed by the representation theory of $\Gamma$ through the McKay correspondence (Douglas and Moore, arXiv:hep-th/9603167). In the second, the same theories are engineered by brane setups (Hanany and Witten, arXiv:hep-th/9611230): arrangements of D-branes stretched between NS5-branes (solitonic five-branes carrying NS-NS B-field charge), such as the elliptic model — D4-branes suspended between NS5-branes arranged on a circle — which is T-dual to the $\mathbb{C}^2/\mathbb{Z}_n$ orbifold. Since the two descriptions must agree, one expects a T-duality dictionary for arbitrary orbifolds, but around 2000 it existed only for special classes: the abelian singularities $\mathbb{C}^2/\mathbb{Z}_n$ (elliptic model), $\mathbb{C}^3/\mathbb{Z}_n\times\mathbb{Z}_m$ (brane box models, with branes suspended between two orthogonal arrays of NS5-branes), abelian $\mathbb{C}^4$ cases (the brane cube, with three such arrays), and — with orientifold planes — the binary dihedral $D_k$ series. An orientifold plane is the fixed locus of a string-theory quotient by the world-sheet parity involution $\Omega$ (possibly combined with signs such as $(-1)^{F_L}$, the left-moving world-sheet fermion number); it acts as a $\mathbb{Z}_2$ reflection on space-time and on the Chan-Paton labels (the multiplicity labels carried by open-string endpoints, which encode the gauge degrees of freedom), and the variant called the ON-plane supplies precisely the $\mathbb{Z}_2$ action needed to pass from the $\mathbb{Z}_{2k}$ to the $D_k$ theory (Kapustin, arXiv:hep-th/9806238). Muto's brane-box constructions for the non-abelian $\mathbb{C}^3/\Delta$-series orbifolds (arXiv:hep-th/9905230) already required an analogous but unfamiliar $\mathbb{Z}_3$ quotient of the abelian setup. The stepwise-projection algorithm of Feng, Hanany, He and Prezas (arXiv:hep-th/0012078) systematized this quotienting: every non-abelian orbifold quiver can be obtained from an abelian parent quiver by identifying and splitting nodes under a finite symmetry $S$ of the parent (justified by Frobenius' theory of induced representations, which builds the representations of a group out of those of a subgroup); the $D_k$ case reproduces the orientifold projection, while the $E_6$ case — $E_6$ denoting the binary tetrahedral group, the preimage in $SU(2)$ of the rotational symmetry group of the tetrahedron, alongside $E_7$ (binary octahedral) and $E_8$ (binary icosahedral) — requires a $\mathbb{Z}_3$ action on the space-time directions $X^{6789}$ of the brane configuration and on the string world-sheet. The mathematical quotient is fully controlled; what is missing is its physical implementation, a question the source paper leaves explicitly open.

Problem Statement

Within the Type IIA brane-setup (Hanany-Witten/elliptic-model) engineering of D-brane worldvolume gauge theories, the stepwise-projection analysis shows that the quiver theory of D-branes at $\mathbb{C}^2/E_6$ ($E_6$ the binary tetrahedral subgroup of $SU(2)$) is obtained from the abelian $\mathbb{Z}_6$ setup (or, as an alternative parent, the binary dihedral $D_2$ setup) by a $\mathbb{Z}_3$ action on the space-time directions $X^{6789}$ (the four directions, with $X^6$ the compact circle direction of the NS5-branes, in which the branes of the elliptic model are arranged) and on the string world-sheet, exactly as the ON-plane's $\mathbb{Z}_2$ action passes from the abelian $\mathbb{Z}_{2k}$ setup to the $D_k$ theories. Determine whether a consistent object of string theory — a generalization of the orientifold plane — exists that performs this order-3 action on space-time and world-sheet, including its action on open-string Chan-Paton data. If it exists, construct it and use it to build explicitly the brane setups for the exceptional-series theories $\mathbb{C}^2/E_6$, $\mathbb{C}^2/E_7$, $\mathbb{C}^2/E_8$ (with $E_6$ the minimal case), and carry the same mechanism to generic non-abelian orbifold singularities $\mathbb{C}^k/\Gamma$; the $\mathbb{C}^2/E_7$, $\mathbb{C}^2/E_8$ and generic $\mathbb{C}^k/\Gamma$ cases are extensions by the same mechanism. Brane setups for the same theories by alternative constructions are equally admissible, since the shared objective is completing the orbifold-to-brane-setup dictionary for non-abelian orbifold singularities. If no such object exists, establish the obstruction — which would, as the source notes, explain why the $E_6$ attempt has been prohibitively difficult. The construction's engineered worldvolume theory (gauge group, matter content, superpotential — the holomorphic function fixing the supersymmetric interactions — and amount of supersymmetry) must agree with the orbifold-projection computation for the same singularity, and its background must be internally consistent (world-sheet consistency, tadpole cancellation — the vanishing of the spurious bulk charges induced by the quotient — and anomaly cancellation).

References

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