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1. On an Equivalence of Divisors on $\overline {\text {M}}_{0,n}$ from Gromov-Witten Theory and Conformal Blocks

   https://doi.org/10.1007/s00031-022-09752-6  

   Chen, L. ; Gibney, A. ; Heller, L. ; Kalashnikov, E. ; Larson, H. ; Xu, W. ( August 2022 , Transformation Groups)

   We consider a conjecture that identifies two types of base point free divisors on$\overline {\text {M}}_{0,n}$${\overline{M}}_{0,n}$. The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated with simple Lie algebras in type A. Here we reduce this conjecture on$\overline {\text {M}}_{0,n}$${\overline{M}}_{0,n}$to the same statement forn= 4. A reinterpretation leads to a proof of the conjecture on$\overline {\text {M}}_{0,n}$${\overline{M}}_{0,n}$for a large class, and we give sufficient conditions for the non-vanishing of these divisors.

    

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2. Equivariant quantum differential equation and qKZ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-Theorem

   Giordano Cotti, Alexander Varchenko ( January 2021 , Proceedings of symposia in pure mathematics)

   Novikov, Krichever (Ed.)

   In [TV19a] the equivariant quantum differential equation (qDE) for a projective space was considered and a compatible system of difference qKZ equations was introduced; the space of solutions to the joint system of the qDE and qKZ equations was identified with the space of the equivariant K-theory algebra of the projective space; Stokes bases in the space of solutions were identified with exceptional bases in the equivariant K-theory algebra. This paper is a continuation of [TV19a]. We describe the relation between solutions to the joint system of the qDE and qKZ equations and the topological-enumerative solution to the qDE only, defined as a generating function of equivariant descendant Gromov-Witten invariants. The relation is in terms of the equivariant graded Chern character on the equivariant K-theory algebra, the equivariant Gamma class of the projective space, and the equivariant first Chern class of the tangent bundle of the projective space. We consider a Stokes basis, the associated exceptional basis in the equivariant K-theory algebra, and the associated Stokes matrix. We show that the Stokes matrix equals the Gram matrix of the equivariant Grothendieck-Euler-Poincaré pairing wrt to the basis, which is the left dual to the associated exceptional basis. We identify the Stokes bases in the space of solutions with explicit full exceptional collections in the equivariant derived category of coherent sheaves on the projective space, where the elements of those exceptional collections are just line bundles on the projective space and exterior powers of the tangent bundle of the projective space. These statements are equivariant analogs of results of G. Cotti, B. Dubrovin, D. Guzzetti, and S. Galkin, V. Golyshev, H. Iritani. 

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3. Wall-crossing in genus-zero hybrid theory

   https://doi.org/10.1515/advgeom-2021-0010  

   Clader, Emily ; Ross, Dustin ( June 2021 , Advances in Geometry)

   null (Ed.)

   Abstract The hybrid model is the Landau–Ginzburg-type theory that is expected, via the Landau–Ginzburg/ Calabi–Yau correspondence, to match the Gromov–Witten theory of a complete intersection in weighted projective space. We prove a wall-crossing formula exhibiting the dependence of the genus-zero hybrid model on its stability parameter, generalizing the work of [21] for quantum singularity theory and paralleling the work of Ciocan-Fontanine–Kim [7] for quasimaps. This completes the proof of the genus-zero Landau– Ginzburg/Calabi–Yau correspondence for complete intersections of hypersurfaces of the same degree, as well as the proof of the all-genus hybrid wall-crossing [11]. 

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4. Decomposition of degenerate Gromov–Witten invariants

   https://doi.org/10.1112/S0010437X20007393  

   Abramovich, Dan ; Chen, Qile ; Gross, Mark ; Siebert, Bernd ( October 2020 , Compositio Mathematica)

   null (Ed.)

   We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family $X \longrightarrow B$ with singular fibre over $b_0\in B$ yields a family $\mathscr {M}(X/B,\beta ) \longrightarrow B$ of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over $b_0$ in terms of rigid tropical maps to the tropicalization of $X/B$ . This generalizes one aspect of known results in the case that the fibre $X_{b_0}$ is a normal crossings union of two divisors. We exhibit our formulas in explicit examples. 

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5. Categorical Primitive Forms and Gromov–Witten Invariants of An Singularities

   https://doi.org/10.1093/imrn/rnz315  

   Căldăraru, Andrei ; Li, Si ; Tu, Junwu ( December 2019 , International Mathematics Research Notices)

   We introduce a categorical analogue of Saito’s notion of primitive forms. For the category $\textsf{MF}(\frac{1}{n+1}x^{n+1})$ of matrix factorizations of $\frac{1}{n+1}x^{n+1}$, we prove that there exists a unique, up to non-zero constant, categorical primitive form. The corresponding genus zero categorical Gromov–Witten invariants of $\textsf{MF}(\frac{1}{n+1}x^{n+1})$ are shown to match with the invariants defined through unfolding of singularities of $\frac{1}{n+1}x^{n+1}$.

    

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