On the homological mirror symmetry conjecture for pairs of pants
By Nick Sheridan. 2010
81 citations recorded by the source corpus.
The n-dimensional pair of pants is defined to be the complement of n+2 generic hyperplanes in CP^n. We construct an immersed Lagrangian sphere in the pair of pants and compute its endomorphism A_{\infty} algebra in the Fukaya category. On the level of cohomology, it is an exterior algebra with n+2 generators. It is not formal, and we compute certain higher products in order to determine it up to quasi-isomorphism. This allows us to give some evidence for the homological mirror symmetry conjecture: the pair of pants is conjectured to be mirror to the Landau-Ginzburg model (C^{n+2},W), where W = z_1 ... z_{n+2}. We show that the endomorphism A_{\infty} algebra of our Lagrangian is quasi-isomorphic to the endomorphism dg algebra of the structure sheaf of the origin in the mirror. This implies similar results for finite covers of the pair of pants, in particular for certain affine Fermat hypersurfaces.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/718ce413a7b24effebb4c485ca22c7b6dfc96dbb
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Citations in the corpus
- Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, methodology): “ingly the inhomogeneous terms vary over the punctured discs in a more complicated way. Other approaches use singular cochains [18], or Morse-Bott techniques similar to the one mentioned in Remark 3.2 [25, 10, 8, 7, 19, 31, 36]. In each of these approaches, one can ensure (based on our exactness assu”
- Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P^1_{a,b,c} (2013, methodology): “ned from the counting of holomorphic pearly trees (using the combination of Morse ows and J-holomorphic polygons). This construction was sketched in [Sei11] and details has been given in Section 4 of [She11]. Such a construction uses modulus- and domain-dependent perturbations to achieve transversal”
- Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, unknown): “The main result of [47] was the zeroth-order part of Proposition 6.”
- Generating the Fukaya categories of Hamiltonian 𝐺-manifolds (2015, background): “li split-generates then K is not displaceable from any Lagrangian submanifold L with HF(L,L) 6= 0. Finding a split-generator is also the first step in many modern proofs of homological mirror symmetry [6, 51, 52, 54, 55, 56]. The aim of this paperis todeterminewhen a free Lagrangian orbit of a Hamilt”
- Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces (2012, background): “These results provide a geometric foundation for mirror constructions that have appeared in the recent literature [11, 28, 27, 40, 42, 1, 20].”