Scope of mirror symmetry as a gauge symmetry: the Giveon-Witten Calabi-Yau conjecture and the classification of self-mirror N=2 group models

Giveon and Witten (Phys. Lett. B 332 (1994) 44) proved that mirror duality for $N=2$ supersymmetric string backgrounds whose target is a group manifold is a gauge symmetry: the mirror transformation, which flips the sign of one chirality's $N=2$ $U(1)$ current, is a symmetry of the given $N=2$ structure precisely when it is realized as a Weyl rotation of the chiral group symmetry, making the model self-mirror and letting mirror act on the $N=2$ moduli space as a gauge transformation. They verified the criterion for $SU(2)\times U(1)$, $SU(3)$ and the $A_{2n}$ series, showed it fails for $SU(2)\times SU(2)$, and conjectured - as the stated main motivation of their work - that mirror symmetry for $c=9$ Calabi-Yau compactifications is a gauge symmetry in the same sense: a mirror pair should arise at the boundary of a moduli space of $N=2$ backgrounds with sigma-model target-space dimension $d>6$, flowing from a self-mirror point of enhanced symmetry along exactly marginal (current-current type) deformation lines, with orbifold identifications and GSO twists. The objective is to determine whether the Giveon-Witten Calabi-Yau gauge-symmetry conjecture holds - proving or refuting it together with its constructive content - and to complete the classification of mirror-as-gauge on $N=2$ groups and cosets.

Background

In string theory a two-dimensional conformal field theory (CFT) describes string propagation in a target spacetime, and target-space dualities are equivalences between geometrically distinct backgrounds sharing the same worldsheet CFT. The prototype is T-duality, $R\to 1/R$, for a string on a circle of radius $R$. T-duality can be understood as a gauge symmetry: at the self-dual radius the worldsheet current algebra is enlarged to $SU(2)_L\times SU(2)_R$, and the duality acts at that enhanced-symmetry point as a Weyl reflection - an element of the Weyl group, the finite group generated by reflections in the roots of a compact Lie group - so away from the point it survives only as a residual discrete gauge transformation acting on the moduli. Giveon and Kiritsis extended this picture to lines of exactly marginal current-current deformations (exactly marginal: perturbations that preserve conformal invariance to all orders in perturbation theory; current-current: deformations by products of chiral affine currents) of Wess-Zumino-Witten (WZW) models - two-dimensional CFTs whose target is a group manifold $G$ at level $k$ and whose currents generate an affine Lie algebra (the central extension of the loop algebra of the Lie algebra of $G$) - showing that the resulting axial-vector duality is a broken gauge symmetry relating coset backgrounds (models obtained from a WZW model by gauging a subgroup and restricting to invariant fields) of different topology. Mirror symmetry is another target-space duality, specific to $N=2$ supersymmetric worldsheet theories. The $N=2$ superconformal algebra is an extended chiral algebra containing, besides the stress tensor, two supercurrents and a $U(1)$ current $J$; sigma models with $(2,2)$ worldsheet supersymmetry on Calabi-Yau manifolds (Kahler manifolds with vanishing first Chern class) are the basic geometric examples, with internal central charge $c=9$ giving critical type II strings in four flat dimensions. A mirror transformation is an isomorphism of $N=2$ CFTs acting on the chiral $U(1)$ currents as $J\to -J$ in one chirality while fixing the other; geometrically it exchanges the complex-structure and Kahler moduli of Calabi-Yau targets. Giveon and Witten proved that on group-manifold backgrounds mirror duality is itself a gauge symmetry. Any even-dimensional group admits an $N=2$ affine structure: one chooses a complex structure $h$ on the Lie algebra, equivalently a decomposition of the complexified Lie algebra into conjugate closed subalgebras $T=T_+\oplus T_-$. The mirror transformation acts on the left-moving complex structure as $h\to -h$, and the key criterion is that mirror is a symmetry of the given $N=2$ model if and only if it is realized by a Weyl rotation $m$ in the left-moving group $G_L$; the model is then self-mirror, and on the $N=2$ moduli space spanned by exactly marginal current-current deformations of the Cartan currents (the commuting generators of the chiral algebra) the mirror acts as the linear map on deformation parameters induced by $m$. They verified the criterion for $SU(2)\times U(1)$ (the mirror is a $\pi$-rotation of $SU(2)_L$), for $SU(3)$, and for the $A_{2n}$ series, and showed it fails for $SU(2)\times SU(2)$, where the needed Weyl element does not exist and mirror maps the chosen $N=2$ structure to a different one. The paper's central conjecture (its Section 8, described by the authors as the main motivation of the work) concerns Calabi-Yau compactifications. It is not known how to connect the two members of a Calabi-Yau mirror pair by marginal deformations of their $c=9$ CFTs alone. By analogy with the two-dimensional black hole, where the axially gauged coset (the cigar) and the vectorially gauged coset (the trumpet) cannot be connected within the coset CFT but are exchanged by mirror-as-gauge inside the moduli space of $N=2$ backgrounds with four-dimensional target connected to $SL(2)\times U(1)$, Giveon and Witten conjectured that a mirror pair of Calabi-Yau backgrounds (times a non-compact space) could appear at the boundary of the moduli space of $N=2$ backgrounds with sigma-model target-space dimension $d>6$ (here $d$ is the real target-space dimension of the worldsheet sigma model; the Calabi-Yau $c=9$ sigma model itself has $d=6$), that this moduli space may contain a self-mirror point with enhanced symmetry group $G$, and that mirror symmetry is a gauge transformation in $G$ there. As evidence they exhibited a tensor product of an $SU(2)/U(1)$ minimal-model coset ($N=2$ minimal models being the simplest rational $N=2$ superconformal theories) with four $SU(2)$ WZW factors at total central charge $c=15$, projected by a GSO (Gliozzi-Scherk-Olive) twist (an orbifold projection by $\exp(2\pi i J_0)$ using the $N=2$ $U(1)$ current, required for spacetime supersymmetry in fermionic string constructions): deforming the Cartan torus, one boundary gives the Calabi-Yau sigma model associated with a Gepner model (a tensor product of $N=2$ minimal models that at special levels is equivalent to a Calabi-Yau hypersurface in projective space), and the other boundary gives its mirror through $\mathbb{Z}_k$ orbifold identifications (quotients by the discrete symmetry $\mathbb{Z}_k$, whose consistent worldsheet description includes twisted sectors). Whether such a gauge-symmetry realization exists beyond such special backgrounds - in particular for generic Calabi-Yau mirror pairs - remains unproven, and the group-manifold case has never been systematically classified.

Problem Statement

Determine the scope of the statement that mirror duality is a gauge symmetry of string theory, in the sense established by Giveon and Witten for group-manifold backgrounds. The problem has two linked parts whose common objective is to chart exactly where the mirror-as-gauge picture holds. Part 1 (the Giveon-Witten Section-8 conjecture, Calabi-Yau case): establish or refute that mirror symmetry for $c=9$ Calabi-Yau compactifications is a gauge symmetry - that is, determine whether, for mirror pairs of Calabi-Yau backgrounds, there exists a moduli space of $N=2$ backgrounds with sigma-model target-space dimension $d>6$ containing a self-mirror point of enhanced symmetry $G$, such that (i) both members of the mirror pair arise at the boundary of this moduli space along exactly marginal, current-current-type deformation lines, possibly involving orbifold identifications and GSO twists, and (ii) the mirror involution ($J\to -J$ in one chirality, $J$ fixed in the other) is realized as a gauge transformation in the enhanced symmetry group at the self-mirror point. The source states this conjecture generally, with direct evidence only for Gepner-type backgrounds; its constructive content, across the class of mirror pairs the conjecture claims, consists in identifying the enhanced point, the deformation line, and the map between the sigma-model data (metric, $B$-field, fermionic couplings) along the line and the action of the mirror/gauge transformation, including the fermionic terms in current-current perturbations and the fate of orbifold twisted sectors at the boundaries. A refutation of the general conjecture requires an obstruction valid against all candidate enhanced points and marginal paths, not merely specific constructions. Part 2 (classification on groups and cosets): classify the $N=2$ superconformal structures $h$ on group manifolds $G$ - semi-simple and non-semi-simple - and on coset constructions such as the Kazama-Suzuki models (the standard $N=2$ superconformal coset constructions $G/(H\times U(1))$ of WZW models), into those for which the mirror transformation is a gauge symmetry, realizable by a Weyl rotation or group automorphism of the chiral symmetry, and those, like $SU(2)\times SU(2)$ with its standard structure, for which mirror necessarily maps the chosen $N=2$ structure to an inequivalent one. The known positive cases ($SU(2)\times U(1)$, $SU(3)$, the $A_{2n}$ series) and the single known negative case do not constitute a classification; the criterion of Giveon and Witten exists but has not been systematically evaluated, including for structures related by Poisson-Lie T-duality (a non-Abelian generalization of T-duality), where the mirror transformation is realized as a duality between two backgrounds rather than as the Weyl-rotation or group-automorphism gauge symmetry of a single model that the classification requires, so those structures still have to be evaluated against the criterion.

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