Conformal bootstrap and Mirror symmetry of states in Gepner models

We consider two explicit constructions of states in the orbifolds of a product of Minimal $N=(2,2)$ models which are based on twisting by spectral flow, mutual locality and operator algebra requirement. It is shown that these two constructions lead to the Berglund-Hubsh-Krawitz dual orbifold groups which define mirror pairs of isomorphic models. Then we generalize our construction for the orbifolds of Gepner models of superstring compactification and explicitly build IIA/IIB mirror map of the space of states of the superstrings using light-cone gauge.

💡 Research Summary

The paper investigates the construction of states in orbifolds of products of N = (2, 2) minimal conformal field theories (CFTs) and extends the analysis to Gepner models, ultimately providing an explicit IIA↔IIB mirror map in light‑cone gauge. The authors begin by reviewing the N = (2, 2) super‑Virasoro algebra of a single minimal model M_k, introducing the spectral flow automorphisms U_t (t ∈ ½ + ℤ) that shift the U(1) charge and conformal weight. Using the free‑boson representation of the U(1) current, they write U_t = exp(i t c/3 φ) and show how the spectral flow generates all NS and R primary states via the operators V_{l,t} = (U G^{-1/2})^t Φ_{c,l}. The periodicity U_{k+2}≈1 is highlighted.

Next, they consider a tensor product of r minimal models, M̃_k = ∏{i=1}^r M{k_i}, whose total central charge is 9, making it suitable as the internal sector of a superstring compactification. The full discrete symmetry group is G_tot = ∏i ℤ{k_i+2}. Within this group they define an “admissible” subgroup G_adm by imposing two conditions: (i) the vector (1,1,…,1) belongs to G_adm, and (ii) the weighted sum Σ_i w_i k_i+2 is an integer. This second condition guarantees that the holomorphic and anti‑holomorphic spin‑3/2 currents (the (3,0) and (0,3) forms on the Calabi‑Yau three‑fold) are mutually local and therefore part of the operator algebra.

The construction of orbifold states proceeds in three steps. First, for each element w∈G_adm they introduce twisted NS fields of the form Ψ_{NS}^{~l,~t,~w}(z, \bar z) = V_{~l,~t+~w}(z) \bar V_{~l,~t}(\bar z), where V_{~l,~t+~w} is built from the spectral‑flow operators shifted by the twist vector. Second, they enforce mutual locality between any two twisted sectors, which yields the integer condition X_i w_i^{(1)}(q_i^{(2)} − w_i^{(2)}) + w_i^{(2)}(q_i^{(1)} − w_i^{(1)}) ∈ ℤ. This equation is shown to be precisely the defining relation of the Berglund‑Hübsch‑Krawitz (BHK) dual group G*_adm. Consequently, G_adm and G*_adm form a dual pair: the twisted sectors of the original orbifold are labeled by G_adm, while the set of mutually local fields is labeled by G*adm. The third step generates the R‑sector fields by acting with the half‑unit spectral flow U{1/2} on the NS fields.

Having established that the NS and R fields close under OPEs and satisfy all conformal bootstrap axioms, the authors argue that the two seemingly different constructions (spectral‑flow‑based versus mutual‑locality‑based) are in fact equivalent, both leading to the same BHK dual groups. This provides a rigorous CFT‑level proof that the orbifold theories related by the BHK duality are isomorphic as N = (2, 2) superconformal field theories, thereby realizing mirror symmetry at the level of the full operator algebra.

The paper then extends the framework to Gepner models, which are tensor products of minimal models further orbifolded by a GSO‑type projection to achieve integral U(1) charge. Working in light‑cone gauge, the authors write down the IIA and IIB spacetime supersymmetry generators and derive the generalized GSO conditions. By expressing these conditions in terms of the admissible and dual groups, they demonstrate that the GSO constraints are automatically mirror‑symmetric: the same set of equations governs both IIA and IIB spectra when the twist vectors are exchanged via the BHK duality. Consequently, every physical state in the IIA theory (including NS‑NS, NS‑R, R‑NS, and R‑R sectors) maps bijectively to a counterpart in the IIB theory, with matching conformal dimensions, U(1) charges, and spacetime quantum numbers. The construction explicitly exhibits the holomorphic (3,0) and anti‑holomorphic (0,3) currents on both sides, confirming that the Calabi‑Yau three‑form is preserved under the mirror map.

In the concluding section the authors summarize that (i) the spectral‑flow‑twist method and the mutual‑locality method both lead to the same BHK dual groups, (ii) this equivalence follows directly from conformal bootstrap requirements, and (iii) the resulting mirror symmetry extends naturally to the full superstring theory, providing an explicit IIA↔IIB state‑by‑state correspondence. They suggest future work on non‑A‑type minimal models, more general toric hypersurfaces, and applications to D‑brane spectra and modular forms.