(0,2) Mirror Symmetry on homogeneous Hopf surfaces

Álvarez-Cónsul, Luis; Hera, Andoni De Arriba de La; Garcia-Fernandez, Mario

doi:10.48550/arxiv.2012.01851

Cited by 5 publications

(11 citation statements)

References 39 publications

(77 reference statements)

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“…This implies that ( Ψ, ω) is of the form (2.18) and, since g is Vaisman, the fundamental group Γ of 𝑋 is as in (2.15). Using that Γ preserves (2.18) we necessarily have that (2.17) holds and furthermore 𝐻 ⊂ SU (2). Thus, we conclude that 𝑋 is quaternionic.…”

Section: Complex Surfaces and Aeppli Classesmentioning

confidence: 67%

“…Motivation for the twisted Hull-Strominger system (1.1) comes from generalized geometry [23] (see Subsection 4.1), supergravity (forthcoming paper by Garcia-Fernandez, Rubio, Shahbazi and Tipler), and mirror symmetry. Actually, the study of this system of equations has very recently led to the first non-Kähler examples of (0,2) mirror symmetry [2]. The Hull-Strominger system is recovered from (1.1) when the cohomology class of the Lee form [𝜃 𝜔 ] ∈ 𝐻 1 (𝑋, ℝ) vanishes (see Proposition 2.5).…”

Section: Summary Of Resultsmentioning

confidence: 99%

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Canonical metrics on holomorphic Courant algebroids

Garcia-Fernandez

Rubio

Shahbazi

et al. 2022

Proceedings of London Math Soc

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The solution of the Calabi Conjecture by Yau implies that every Kähler Calabi–Yau manifold X$X$ admits a metric with holonomy contained in SUfalse(nfalse)$\mathrm{SU}(n)$, and that these metrics are parameterized by the positive cone in H1,1(X,R)$H^{1,1}(X,\mathbb {R})$. In this work, we give evidence of an extension of Yau's theorem to non‐Kähler manifolds, where X$X$ is replaced by a compact complex manifold with vanishing first Chern class endowed with a holomorphic Courant algebroid Q$Q$ of Bott–Chern type. The equations that define our notion of best metric correspond to a mild generalization of the Hull–Strominger system, whereas the role of H1,1(X,R)$H^{1,1}(X,\mathbb {R})$ is played by an affine space of ‘Aeppli classes’ naturally associated to Q$Q$ via Bott–Chern secondary characteristic classes.

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Section: Complex Surfaces and Aeppli Classesmentioning

confidence: 67%

Section: Summary Of Resultsmentioning

confidence: 99%

Canonical metrics on holomorphic Courant algebroids

Garcia-Fernandez

Rubio

Shahbazi

et al. 2022

Proceedings of London Math Soc

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“…It would be interesting to reconcile these results with the complexified gauge transformations and moment maps in a GIT-type analysis of these solutions studied in [20,21]. There are also examples of mathematical interest, such as that related to the Hopf surface which do not have an obvious valid α -expansion, yet appear to be governed by a (0, 2) superconformal algebra [22]. Nonetheless, one could still try to construct the adjoint operator D † and compare with the space of deformations obtained.…”

Section: Discussionmentioning

confidence: 95%

“…Finding a suitable modification of this worldsheet theory and Hopf surface geometry so that we have a unitary theory flowing to a unitary (0, 2) conformal field theory would be very interesting. Recently, [22] conjectured some of the algebraic structures such a theory should have -it would be very interesting to try to turn this into a description of a unitary conformal field theory.…”

Section: B a Supergravity No-go Theoremmentioning

confidence: 99%

Heterotic Quantum Cohomology

McOrist¹,

Svanes²

2021

Preprint

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We reexamine the massless spectrum of a heterotic string vacuum at large radius and present two results. The first result is to construct a vector bundle Q and operator D whose kernel amounts to deformations solving 'F-term' type equations. This resolves a dilemma in previous works in which the spin connection is treated as an independent degree of freedom, something that is not the case in string theory. The second result is to utilise the moduli space metric, constructed in previous work, to define an adjoint operator D † . The kernel of D † amounts to deformations solving 'D-term' type equations. Put together, we show there is a vector bundle Q with a metric, a D operator and a gauge fixing (holomorphic gauge) in which the massless spectrum are harmonic representatives of D. This is remarkable as previous work indicated the Hodge decomposition of massless deformations were complicated and in particular not harmonic except at the standard embedding.

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“…Mirror symmetry for non-Kähler space has been investigated in some other settings. Most recently, in [5], Álvarez-Cónsul, de Arriba de la Hera, and Garcia-Fernandez show that certain homogeneous surfaces, including the Hopf surface, admit (0, 2) mirrors with isomorphic half-twisted models. In [17] Lau, Tseng, and Yau use a Fourier-Mukai transform to obtain a correspondence between different Ramond-Ramond flux terms for pairs of Type IIA and Type IIB supersymmetric systems, and in [23] Popovici shows that there is a classical mirror symmetry correspondence for the Iwasawa surface in the sense that there is an isomorphism between the Gauduchon cone (parametrizing the space of symplectic structures) and a space of complex deformations of the same surface.…”

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confidence: 99%

Homological mirror symmetry for elliptic Hopf surfaces

Ward¹

2021

Preprint

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We show homological mirror symmetry results relating coherent analytic sheaves on some complex elliptic surfaces and objects of certain Fukaya categories. We first define the notion of a non-algebraic Landau-Ginzburg model on R × S 1 3 and its associated Fukaya category, and show that non-Kähler surfaces obtained by performing two logarithmic transformations to the product of the projective plane and an elliptic curve have non-algebraic Landau-Ginzburg models as their mirror spaces; this class of surface includes the classical Hopf surface S 1 × S 3 and other elliptic primary and secondary Hopf surfaces. We also define localization maps from the Fukaya categories associated to the Landau-Ginzburg models to partially wrapped and fully wrapped categories. We show mirror symmetry results that relate the partially wrapped and fully wrapped categories to spaces of coherent analytic sheaves on open submanifolds of the compact complex surfaces in question, and we use these results to sketch a proof of a full HMS result. Contents 65 References 68 * r (L, L ′ ) → CF * r (ψ H (L) , L ′ ).

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(0,2) Mirror Symmetry on homogeneous Hopf surfaces

Cited by 5 publications

References 39 publications

Canonical metrics on holomorphic Courant algebroids

Canonical metrics on holomorphic Courant algebroids

Heterotic Quantum Cohomology

Homological mirror symmetry for elliptic Hopf surfaces

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