More Like this

1. A Landau–Ginzburg mirror theorem via matrix factorizations

   https://doi.org/10.1515/crelle-2022-0057  

   He, Weiqiang; Polishchuk, Alexander; Shen, Yefeng; Vaintrob, Arkady (October 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract For an invertible quasihomogeneouspolynomial 𝒘 {{\boldsymbol{w}}} we prove an all-genus mirror theoremrelating two cohomological field theories of Landau–Ginzburg type.On the B -side it is the Saito–Givental theory for a specificchoice of a primitive form. On the A -side, it is the matrix factorization CohFTfor the dual singularity 𝒘 T {{\boldsymbol{w}}^{T}} with the maximal diagonal symmetry group. 

   more » « less
2. Apéry extensions

   https://doi.org/10.1112/jlms.12825  

   Golyshev, Vasily; Kerr, Matt; Sasaki, Tokio (October 2023, Journal of the London Mathematical Society)

   Abstract TheApéry numbersof Fano varieties are asymptotic invariants of their quantum differential equations. In this paper, we initiate a program to exhibit these invariants as (mirror to) limiting extension classes of higher cycles on the associated Landau–Ginzburg (LG) models — and thus, in particular, as periods. We also construct anApéry motive, whose mixed Hodge structure is shown, as an application of the decomposition theorem, to contain the limiting extension classes in question. Using a new technical result on the inhomogeneous Picard–Fuchs equations satisfied by higher normal functions, we illustrate this proposal with detailed calculations for LG‐models mirror to several Fano threefolds. By describing the “elementary” Apéry numbers in terms of regulators of higher cycles (i.e., algebraic ‐theory/motivic cohomology classes), we obtain a satisfying explanation of their arithmetic properties. Indeed, in each case, the LG‐models are modular families of surfaces, and the distinction between multiples of and (or ) translates ultimately into one between algebraic and of the family. 

   more » « less
3. Proof of the elliptic expansion moonshine conjecture of Căldăraru, He, and Huang

   https://doi.org/10.1090/proc/16032  

   Hong, Letong; Mertens, Michael; Ono, Ken; Zhang, Shengtong (January 2022, Proceedings of the American Mathematical Society)

   Using predictions in mirror symmetry, Căldăraru, He, and Huang recently formulated a “Moonshine Conjecture at Landau-Ginzburg points” [arXiv:2107.12405, 2021] for Klein’s modular j j -function at j = 0 j=0 and j = 1728. j=1728. The conjecture asserts that the j j -function, when specialized at specific flat coordinates on the moduli spaces of versal deformations of the corresponding CM elliptic curves, yields simple rational functions. We prove this conjecture, and show that these rational functions arise from classical 2 F 1 _2F_1 -hypergeometric inversion formulae for the j j -function. 

   more » « less
4. More non-Abelian mirrors and some two-dimensional dualities

   https://doi.org/10.1142/S0217751X19501811  

   Gu, Wei; Parsian, Hadi; Sharpe, Eric (October 2019, International Journal of Modern Physics A)

   In this paper, we extend the non-Abelian mirror proposal of two of the authors from two-dimensional gauge theories with connected gauge groups to the case of [Formula: see text] gauge groups with discrete theta angles. We check our proposed extension by counting and comparing vacua in mirrors to the known dual two-dimensional [Formula: see text] gauge theories. The mirrors in question are Landau–Ginzburg orbifolds, and for mirrors to [Formula: see text] gauge theories, the critical loci of the mirror superpotential often intersect fixed-point loci, so that to count vacua, one must take into account the twisted sector contributions. This is a technical novelty relative to the mirrors of gauge theories with connected gauge groups, for which critical loci do not intersect fixed-point loci and so no orbifold twisted sector contributions are pertinent. The vacuum computations turn out to be a rather intricate test of the proposed mirrors, in particular as untwisted sector states in the mirror to one theory are often exchanged with twisted sector states in the mirror to the dual. In cases with nontrivial IR limits, we also check that the central charges computed from the Landau–Ginzburg mirrors match those expected for the IR SCFTs. 

   more » « less
5. Moment maps, nonlinear PDE and stability in mirror symmetry, I: geodesics.

   Collins, Tristan C.; Yau, Shing-Tung (January 2021, Annals of PDE)

   null (Ed.)

   In this paper, the first in a series, we study the deformed Hermitian-Yang-Mills (dHYM) equation from the variational point of view as an infinite dimensional GIT problem. The dHYM equation is mirror to the special Lagrangian equation, and our infinite dimensional GIT problem is mirror to Thomas' GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold H closely related to Solomon's space of positive Lagrangians. In the hypercritical phase case we prove the existence of smooth approximate geodesics, and weak geodesics with C1,α regularity. This is accomplished by proving sharp with respect to scale estimates for the Lagrangian phase operator on collapsing manifolds with boundary. As an application of our techniques we give a simplified proof of Chen's theorem on the existence of C1,α geodesics in the space of Kähler metrics. In two follow up papers, these results will be used to examine algebraic obstructions to the existence of solutions to dHYM and special Lagrangians in Landau-Ginzburg models. 

   more » « less