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1. 2d $$ \mathcal{N} $$ = (0, 1) gauge theories and Spin(7) orientifolds

   https://doi.org/10.1007/JHEP03(2022)150  

   Franco, Sebastián; Mininno, Alessandro; Uranga, Ángel M.; Yu, Xingyang (March 2022, Journal of High Energy Physics)

   A bstract We initiate the geometric engineering of 2d $$ \mathcal{N} $$ N = (0 , 1) gauge theories on D1-branes probing singularities. To do so, we introduce a new class of backgrounds obtained as quotients of Calabi-Yau 4-folds by a combination of an anti-holomorphic involution leading to a Spin(7) cone and worldsheet parity. We refer to such constructions as Spin(7) orientifolds . Spin(7) orientifolds explicitly realize the perspective on 2d $$ \mathcal{N} $$ N = (0 , 1) theories as real slices of $$ \mathcal{N} $$ N = (0 , 2) ones. Remarkably, this projection is geometrically realized as Joyce’s construction of Spin(7) manifolds via quotients of Calabi-Yau 4-folds by anti-holomorphic involutions. We illustrate this construction in numerous examples with both orbifold and non-orbifold parent singularities, discuss the role of the choice of vector structure in the orientifold quotient, and study partial resolutions. 

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2. Twisted characters and holomorphic symmetries

   https://doi.org/10.1007/s11005-020-01319-4  

   Saberi, Ingmar; Williams, Brian R. (October 2020, Letters in Mathematical Physics)

   null (Ed.)

   Abstract We consider holomorphic twists of arbitrary supersymmetric theories in four dimensions. Working in the BV formalism, we rederive classical results characterizing the holomorphic twist of chiral and vector supermultiplets, computing the twist explicitly as a family over the space of nilpotent supercharges in minimal supersymmetry. The BV formalism allows one to work with or without auxiliary fields, according to preference; for chiral superfields, we show that the result of the twist is an identical BV theory, the holomorphic $$\beta \gamma $$ β γ system with superpotential, independent of whether or not auxiliary fields are included. We compute the character of local operators in this holomorphic theory, demonstrating agreement of the free local operators with the usual index of free fields. The local operators with superpotential are computed via a spectral sequence and are shown to agree with functions on a formal mapping space into the derived critical locus of the superpotential. We consider the holomorphic theory on various geometries, including Hopf manifolds and products of arbitrary pairs of Riemann surfaces, and offer some general remarks on dimensional reductions of holomorphic theories along the $$(n-1)$$ ( n - 1 ) -sphere to topological quantum mechanics. We also study an infinite-dimensional enhancement of the flavor symmetry in this example, to a recently studied central extension of the derived holomorphic functions with values in the original Lie algebra, that generalizes the familiar Kac–Moody enhancement in two-dimensional chiral theories. 

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3. Holomorphic isometric maps from the complex unit ball to reducible bounded symmetric domains

   https://doi.org/10.1515/crelle-2022-0029  

   Xiao, Ming (June 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract The first part of the paper studies the boundary behavior of holomorphic isometric mappings F = ( F 1 , … , F m ) {F=(F_{1},\dots,F_{m})} from the complex unit ball 𝔹 n {\mathbb{B}^{n}} , n ≥ 2 {n\geq 2} , to a bounded symmetric domain Ω = Ω 1 × ⋯ × Ω m {\Omega=\Omega_{1}\times\cdots\times\Omega_{m}} up to constant conformal factors, where Ω i ′ {\Omega_{i}^{\prime}} s are irreducible factors of Ω. We prove every non-constant component F i {F_{i}} must map generic boundary points of 𝔹 n {\mathbb{B}^{n}} to the boundary of Ω i {\Omega_{i}} . In the second part of the paper, we establish a rigidity result for local holomorphic isometric maps from the unit ball to aproduct of unit balls and Lie balls. 

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4. Arbitrarily slow decay in the Möbius disjointness conjecture

   https://doi.org/10.1017/etds.2022.61  

   ALGOM, AMIR; WANG, ZHIREN (September 2023, Ergodic Theory and Dynamical Systems)

   Abstract Sarnak’s Möbius disjointness conjecture asserts that for any zero entropy dynamical system $(X,T)$ , $$({1}/{N})\! \sum _{n=1}^{N}\! f(T^{n} x) \mu (n)= o(1)$$ for every $$f\in \mathcal {C}(X)$$ and every $$x\in X$$ . We construct examples showing that this $o(1)$ can go to zero arbitrarily slowly. In fact, our methods yield a more general result, where in lieu of $$\mu (n)$$ , one can put any bounded sequence $$a_{n}$$ such that the Cesàro mean of the corresponding sequence of absolute values does not tend to zero. Moreover, in our construction, the choice of x depends on the sequence $$a_{n}$$ but $(X,T)$ does not. 

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5. Cosets of Free Field Algebras via Arc Spaces

   https://doi.org/10.1093/imrn/rnac367  

   Linshaw, Andrew_R; Song, Bailin (January 2023, International Mathematics Research Notices)

   Abstract Using the invariant theory of arc spaces, we find minimal strong generating sets for certain cosets of affine vertex algebras inside free field algebras that are related to classical Howe duality. These results have several applications. First, for any vertex algebra $${{\mathcal {V}}}$$, we have a surjective homomorphism of differential algebras $$\mathbb {C}[J_{\infty }(X_{{{\mathcal {V}}}})] \rightarrow \text {gr}^{F}({{\mathcal {V}}})$$; equivalently, the singular support of $${{\mathcal {V}}}$$ is a closed subscheme of the arc space of the associated scheme $$X_{{{\mathcal {V}}}}$$. We give many new examples of classically free vertex algebras (i.e., this map is an isomorphism), including $$L_{k}({{\mathfrak {s}}}{{\mathfrak {p}}}_{2n})$$ for all positive integers $$n$$ and $$k$$. We also give new examples where the kernel of this map is nontrivial but is finitely generated as a differential ideal. Next, we prove a coset realization of the subregular $${{\mathcal {W}}}$$-algebra of $${{\mathfrak {s}}}{{\mathfrak {l}}}_{n}$$ at a critical level that was previously conjectured by Creutzig, Gao, and the 1st author. Finally, we give some new level-rank dualities involving affine vertex superalgebras. 

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