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Register a new userInvolutions, obstructions and mirror symmetry
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Authors
Jake P. Solomon
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Consider a Maslov zero Lagrangian submanifold diffeomorphic to a Lie group on which an anti-symplectic involution acts by the inverse map of the group. We show that the Fukaya $A_infty$ endomorphism algebra of such a Lagrangian is quasi-isomorphic to its de Rham cohomology tensored with the Novikov field. In particular, it is unobstructed, formal, and its Floer and de Rham cohomologies coincide. Our result implies that the smooth fibers of a large class of singular Lagrangian fibrations are unobstructed and their Floer and de Rham cohomologies coincide. This is a step in the SYZ and family Floer cohomology approaches to mirror symmetry. More generally, our result continues to hold if the Lagrangian has cohomology the free graded algebra on a graded vector space $V$ concentrated in odd degree, and the anti-symplectic involution acts on the cohomology of the Lagrangian by the induced map of negative the identity on $V.$ It suffices for the Maslov class to vanish modulo $4.$
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We study SYZ mirror symmetry in the context of non-Kaehler Calabi-Yau manifolds. In particular, we study the six-dimensional Type II supersymmetric $SU(3)$ systems with Ramond-Ramond fluxes, and generalize them to higher dimensions. We show that Fourier-Mukai transform provides the mirror map between these Type IIA and Type IIB supersymmetric systems in the semi-flat setting. This is concretely exhibited by nilmanifolds.
This is an expository article on the A-side of Kontsevichs Homological Mirror Symmetry conjecture. We give first a self-contained study of $A_infty$-categories and their homological algebra, and later restrict to Fukaya categories, with particular emphasis on the basics of the underlying Floer theory, and the geometric features therein. Finally, we place the theory in the context of mirror symmetry, building towards its main predictions.
The main result of the present paper concerns finiteness properties of Floer theoretic invariants on affine log Calabi-Yau varieties $X$. Namely, we show that: (a) the degree zero symplectic cohomology $SH^0(X)$ is finitely generated and is a filtered deformation of a certain algebra defined combinatorially in terms of a compactifying divisor $mathbf{D}.$ (b) For any Lagrangian branes $L_0, L_1$, the wrapped Floer groups $WF^*(L_0,L_1)$ are finitely generated modules over $SH^0(X).$ We then describe applications of this result to mirror symmetry, the first of which is an ``automatic generation criterion for the wrapped Fukaya category $mathcal{W}(X)$. We also show that, in the case where $X$ is maximally degenerate and admits a ``homological section, $mathcal{W}(X)$ gives a categorical crepant resolution of the potentially singular variety $operatorname{Spec}(SH^0(X))$. This provides a link between the intrinsic mirror symmetry program of Gross and Siebert and the categorical birational geometry program initiated by Bondal-Orlov and Kuznetsov.
Following a question of K. Hori at K. Fukayas 60th birthday conference, we relate the recently established WDVV-type relations for real Gromov-Witten invariants to topological recursion relations in a real setting. We also describe precisely the connections between the relations themselves previously observed by A. Alcolado.
Motivated by Nekrasovs instanton counting, we discuss a method for calculating equivariant volumes of non-compact quotients in symplectic and hyper-Kahler geometry by means of the Jeffrey-Kirwan residue-formula of non-abelian localization. In order to overcome the non-compactness, we use varying symplectic cuts to reduce the problem to a compact setting, and study what happens in the limit that recovers the original problem. We implement this method for the ADHM construction of the moduli spaces of framed Yang-Mills instantons on $R^{4}$ and rederive the formulas for the equivariant volumes obtained earlier by Nekrasov-Shadchin, expressing these volumes as iterated residues of a single rational function.
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