A quantum system will stay near its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough. This quantum adiabatic behavior is the basis of a new class of algorithms for quantum computing. We tested one such algorithm by applying it to randomly generated hard instances of an NP-complete problem. For the small examples that we could simulate, the quantum adiabatic algorithm worked well, providing evidence that quantum computers (if large ones can be built) may be able to outperform ordinary computers on hard sets of instances of NP-complete problems.
Many N = (2, 2) two-dimensional nonlinear sigma models with Calabi-Yau target spaces admit ultraviolet descriptions as N = (2, 2) gauge theories (gauged linear sigma models). We conjecture that the two-sphere partition function of such ultraviolet gauge theories -recently computed via localization by Benini et al. and Doroud et al. -yields the exact Kähler potential on the quantum Kähler moduli space for Calabi-Yau threefold target spaces. In particular, this allows one to compute the genus zero Gromov-Witten invariants for any such Calabi-Yau threefold without the use of mirror symmetry. More generally, when the infrared superconformal fixed point is used to compactify string theory, this provides a direct method to compute the spacetime Kähler potential of certain moduli (e.g., vector multiplet moduli in type IIA), exactly in α ′ . We compute these quantities for the quintic and for Rødland's Pfaffian Calabi-Yau threefold and find agreement with existing results in the literature. We then apply our methods to a codimension four determinantal Calabi-Yau threefold in P 7 , recently given a nonabelian gauge theory description by the present authors, for which no mirror Calabi-Yau is currently known. We derive predictions for its Gromov-Witten invariants and verify that our predictions satisfy nontrivial geometric checks.1 The work of Böhm [24, 25] provides a promising proposal, but we have been unable to implement it well enough to produce a mirror for this example.2 Local special Kähler manifolds are also often called projective special Kähler manifolds and are distinct from special Kähler manifolds -see, e.g., [32].3 This geometric structure gives rise to a Hodge filtration F 3 ⊂ F 2 ⊂ F 1 ⊂ F 0 of weight 3, with F 3 ≃ L, F 2 ≃ V, F 1 ≃ L ⊥ , and F 0 ≃ V C , where L ⊥ is the subspace of V C perpendicular to L with respect to the symplectic pairing ·, · -see, e.g., [33,34].The last expression involves the periods of Ω,B J Ω(ξ) , I, J = 0, . . . , h 2,1 , (2.6) with respect to a canonical symplectic basis (A I , B J ) of H 3 (Y ξ , Z) satisfying A I , B J = δ I J , A I , A J = B I , B J = 0 . (2.7) 2.3 The quantum Kähler moduli space M KählerThe main player of this note is the quantum-corrected Kähler moduli space M Kähler of a Calabi-Yau threefold Y , which is defined as the corresponding space of chiral-antichiral and antichiralchiral moduli of the underlying SCFT. It is also a local special Kähler manifold, parametrizing 15 We thank C. Vafa for pointing this out to us. 16 Note that in spacetime, some of these fields become quaternionic and are subject to further gs corrections.
We construct worldsheet descriptions of heterotic flux vacua as the IR limits of N = 2 gauge theories. Spacetime torsion is incorporated via a 2d Green-Schwarz mechanism in which a doublet of axions cancels a oneloop gauge anomaly. Manifest (0, 2) supersymmetry and the compactness of the gauge theory instanton moduli space suggest that these models, which include Fu-Yau models, are stable against worldsheet instantons, implying that they, like Calabi-Yaus, may be smoothly extended to solutions of the exact beta functions. Since Fu-Yau compactifications are dual to KST-type flux compactifications, this provides a microscopic description of these IIB RR-flux vacua.
The two-dimensional supersymmetric gauged linear sigma model (GLSM) with abelian gauge groups and matter fields has provided many insights into string theory on Calabi-Yau manifolds of a certain type: complete intersections in toric varieties. In this paper, we consider two GLSM constructions with nonabelian gauge groups and charged matter whose infrared CFTs correspond to string propagation on determinantal Calabi-Yau varieties, furnishing another broad class of Calabi-Yau geometries in addition to complete intersections. We show that these two models -which we refer to as the PAX and the PAXY model -are dual descriptions of the same low-energy physics. Using GLSM techniques, we determine the quantum Kähler moduli space of these varieties and find no disagreement with existing results in the literature.
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