Quantum McKay Correspondence Research Articles

Antisymmetric Characters and Fourier Duality

Inspired by the quantum McKay correspondence, we consider the classical ADE Lie theory as a quantum theory over $$\mathfrak {sl}_2$$ . We introduce anti-symmetric characters for representations of quantum groups and investigate the Fourier duality to study the spectral theory. In the ADE Lie theory, there is a correspondence between the eigenvalues of the Coxeter element and the eigenvalues of the adjacency matrix. We formalize related notions and prove such a correspondence for representations of Verlinde algebras of quantum groups: this includes generalized Dynkin diagrams over any simple Lie algebra $$\mathfrak {g}$$ at any level $$k$$ . This answers a recent comment of Terry Gannon on an old question posed by Victor Kac in 1994.

Quantum McKay correspondence for disc invariants of toric Calabi-Yau 3-orbifolds

We announce a result on quantum McKay correspondence for disc invariants of outer legs in toric Calabi-Yau 3-orbifolds, and illustrate our method in a special example $[\mathbb C^3 /\mathbb Z_5 (1, 1, 3)]$.

Gauged Linear Sigma Model for Disc Invariants

In this paper, we find the extended charge vectors for disc invariants of outer branes in a toric Calabi–Yau 3-orbifold, and use these extended charge vectors to establish a quantum McKay correspondence for disc invariants of branes intersecting effective outer legs, and give an enumerative explanation to the integrality of open–closed mirror map in terms of Ooguri–Vafa invariants.

Quantum McKay correspondence and global dimensions for fusion and module-categories associated with Lie groups

Global dimensions for fusion categories Ak(G) defined by a pair (G,k), where G is a Lie group and k a positive integer, are expressed in terms of Lie quantum superfactorial functions. The global dimension is defined as the square sum of quantum dimensions of simple objects, for the category of integrable modules over an affine Lie algebra at some level. The same quantities can also be defined from the theory of quantum groups at roots of unity or from conformal field theory WZW models. Similar results are also presented for those associated module-categories that can be obtained via conformal embeddings (they are “quantum subgroups” of a particular kind). As a side result, we express the classical (or quantum) Weyl denominator of simple Lie groups in terms of classical (or quantum) factorials calculated for the exponents of the group. Some calculations use the correspondence existing between periodic quivers for simply-laced Lie groups and fusion rules for module-categories associated with Ak(SU(2)).

STRING FIELD THEORY FROM QUANTUM GRAVITY

Recent work on neutrino oscillations suggests that the three generations of fermions in the standard model are related by representations of the finite group A(4), the group of symmetries of the tetrahedron. Motivated by this, we explore models which extend the EPRL model for quantum gravity by coupling it to a bosonic quantum field of representations of A(4). This coupling is possible because the representation category of A(4) is a module category over the representation categories used to construct the EPRL model. The vertex operators which interchange vacua in the resulting quantum field theory reproduce the bosons and fermions of the standard model, up to issues of symmetry breaking which we do not resolve. We are led to the hypothesis that physical particles in nature represent vacuum changing operators on a sea of invisible excitations which are only observable in the A(4) representation labels which govern the horizontal symmetry revealed in neutrino oscillations. The quantum field theory of the A(4) representations is just the dual model on the extended lattice of the Lie group E6, as explained by the quantum McKay correspondence of Frenkel, Jing and Wang. The coupled model can be thought of as string field theory, but propagating on a discretized quantum spacetime rather than a classical manifold.

The quantum McKay correspondence for singularities of type D

We prove the quantum McKay correspondence formulae conjectured by J. Bryan and A. Gholampour for the type D (binary) polyhedral groups in SU(2) and SO(3). We use the method of induction by the WDVV equation and from the normal subgroups by J. Bryan and A. Gholampour, and the polynomiality technique developed in this article. We are also based on the validity of the corresponding conjecture for type A groups, which is proved by T. Coates, A. Corti, H. Iritani, and H.-H. Tseng.

The quantum McKay correspondence for polyhedral singularities

Let G be a polyhedral group, namely a finite subgroup of SO(3). Nakamura’s G-Hilbert scheme provides a preferred Calabi-Yau resolution Y of the polyhedral singularity ℂ3/G. The classical McKay correspondence describes the classical geometry of Y in terms of the representation theory of G. In this paper we describe the quantum geometry of Y in terms of R, an ADE root system associated to G. Namely, we give an explicit formula for the Gromov-Witten partition function of Y as a product over the positive roots of R. In terms of counts of BPS states (Gopakumar-Vafa invariants), our result can be stated as a correspondence: each positive root of R corresponds to one half of a genus zero BPS state. As an application, we use the Crepant Resolution Conjecture to provide a full prediction for the orbifold Gromov-Witten invariants of [ℂ3/G].

Fractional two-branes, toric orbifolds and the quantum McKay correspondence

We systematically study and obtain the large-volume analogues of fractional two-branes on resolutions of orbifolds C^3/Z_n. We study a generalisation of the McKay correspondence proposed in hep-th/0504164 called the quantum McKay correspondence by constructing duals to the fractional two-branes. Details are explicitly worked out for two examples -- the crepant resolutions of C^3/Z_3 and C^3/Z_5.

A quantum McKay correspondence for fractional 2p-branes on LG orbifolds

We study fractional 2p-branes and their intersection numbers in non-compact orbifolds as well the continuation of these objects in Kahler moduli space to coherent sheaves in the corresponding smooth non-compact Calabi-Yau manifolds. We show that the restriction of these objects to compact Calabi-Yau hypersurfaces gives the new fractional branes in LG orbifolds constructed by Ashok et. al. in hep-th/0401135. We thus demonstrate the equivalence of the B-type branes corresponding to linear boundary conditions in LG orbifolds, originally constructed in hep-th/9907131, to a subset of those constructed in LG orbifolds using boundary fermions and matrix factorization of the world-sheet superpotential. The relationship between the coherent sheaves corresponding to the fractional two-branes leads to a generalization of the McKay correspondence that we call the quantum McKay correspondence due to a close parallel with the construction of branes on non-supersymmetric orbifolds. We also provide evidence that the boundary states associated to these branes in a conformal field theory description corresponds to a sub-class of the boundary states associated to the permutation branes in the Gepner model associated with the LG orbifold.

Localized Tachyons and the Quantum McKay Correspondence

The condensation of closed string tachyons localized at the fixed point of a d/Γ orbifold can be studied in the framework of renormalization group flow in a gauged linear sigma model. The evolution of the Higgs branch along the flow describes a resolution of singularities via the process of tachyon condensation. The study of the fate of D-branes in this process has lead to a notion of a ``quantum McKay correspondence.'' This is a hypothetical correspondence between fractional branes in an orbifold singularity in the ultraviolet with the Coulomb and Higgs branch branes in the infrared. In this paper we present some nontrivial evidence for this correspondence in the case 2/n by relating the intersection form of fractional branes to that of ``Higgs branch branes,'' the latter being branes which wrap nontrivial cycles in the resolved space.