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Abstract: Abstract We give a complete classification of twists of supersymmetric Yang–Mills theories in dimensions \(2\le n \le 10\) . We formulate supersymmetric Yang–Mills theory classically using the BV formalism, and then we construct an action of the supersymmetry algebra using the language of \(L_\infty \) algebras. For each orbit in the space of square-zero supercharges in the supersymmetry algebra, under the action of the spin group and the group of R-symmetries, we give a description of the corresponding twisted theory. These twists can be described in terms of mixed holomorphic-topological versions of Chern–Simons and BF theory. PubDate: 2022-08-08

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Abstract: Abstract In this paper we prove a formula relating the equivariant Euler characteristic of K-theoretic stable envelopes to an object known as the index vertex for the cotangent bundle of the full flag variety. Our formula demonstrates that the index vertex is the power series expansion of a rational function. This result is a consequence of the 3d mirror self-symmetry of the variety considered here. In general, one expects an analogous result to hold for any two varieties related by 3d mirror symmetry. PubDate: 2022-07-11

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Abstract: Abstract We study generalized Deligne categories and related tensor envelopes for the universal two-dimensional cobordism theories described by rational functions, recently defined by Sazdanovic and one of the authors. PubDate: 2022-07-08

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Abstract: Abstract For a semisimple Lie algebra defined over a discrete valuation ring with field of fractions K, we prove that any primitive ideal with rational central character in the affinoid enveloping algebra, \(\widehat{U({\mathfrak {g}})_{K}}\) , is the annihilator of an affinoid highest weight module. In the case \(n>0\) , we characterise all the primitive ideals in the affinoid algebra \(\widehat{U(\mathfrak {{g}})_{n,K}}\) . PubDate: 2022-07-07

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Abstract: Abstract We use techniques from Gromov–Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane arrangement, our invariants coincide with virtual fundamental classes used to define the logarithmic Gromov–Witten theory of wonderful models of arrangement complements, for any logarithmic structure supported on the wonderful boundary. When the boundary is empty, this implies that the quantum cohomology ring of a hyperplane arrangement’s wonderful model is a combinatorial invariant, i.e., it depends only on the matroid. When the boundary divisor is maximal, we use toric intersection theory to convert the virtual fundamental class into a balanced weighted fan in a vector space, having the expected dimension. We explain how the associated Gromov–Witten theory is completely encoded by intersections with this weighted fan. We include a number of questions whose positive answers would lead to a well-defined Gromov–Witten theory of non-realizable matroids. PubDate: 2022-06-18

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Abstract: Abstract We establish the existence of an involution on tabloids that is analogous to Schützenberger’s evacuation map on standard Young tableaux. We find that the number of its fixed points is given by evaluating a certain Green’s polynomial at \(q = -1\) , and satisfies a “domino-like” recurrence relation. PubDate: 2022-06-14

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Abstract: Abstract In this paper, we construct various simple vertex superalgebras which are extensions of affine vertex algebras, by using abelian cocycle twists of representation categories of quantum groups. This solves the Creutzig and Gaiotto conjectures (Creutzig and Gaiotto in Comm Math Phys 379:785–845, 2020, Conjecture 1.1 and 1.4) in the case of type ABC. If the twist is trivial, the resulting algebras correspond to chiral differential operators in the chiral case, and to WZW models in the non-chiral case. PubDate: 2022-06-14

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Abstract: Abstract The stated skein algebra of a punctured bordered surface (or equivalently, a marked surface) is a generalization of the well-known Kauffman bracket skein algebra of unmarked surfaces and can be considered as an extension of the quantum special linear group \({\mathcal {O}}_{q^2}(SL_2)\) from a bigon to general surfaces. We show that the stated skein algebra of a punctured bordered surface with non-empty boundary can be embedded into quantum tori in two different ways. The first embedding can be considered as a quantization of the map expressing the trace of a closed curve in terms of the shear coordinates of the enhanced Teichmüller space, and is a lift of Bonahon-Wong’s quantum trace map. The second embedding can be considered as a quantization of the map expressing the trace of a closed curve in terms of the lambda length coordinates of the decorated Teichmüller space, and is an extension of Muller’s quantum trace map. We explain the relation between the two quantum trace maps. We also show that the quantum cluster algebra of Muller is equal to a reduced version of the stated skein algebra. As applications we show that the stated skein algebra is an orderly finitely generated Noetherian domain and calculate its Gelfand-Kirillov dimension. PubDate: 2022-05-27 DOI: 10.1007/s00029-022-00781-3

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Abstract: Abstract We show that any toric Kähler cone with smooth compact cross-section admits a family of Calabi–Yau cone metrics with conical singularities along its toric divisors. The family is parametrized by the Reeb cone and the angles are given explicitly in terms of the Reeb vector field. The result is optimal, in the sense that any toric Calabi–Yau cone metric with conical singularities along the toric divisor (and smooth elsewhere) belongs to this family. We also provide examples and interpret our results in terms of Sasaki–Einstein metrics. PubDate: 2022-05-04 DOI: 10.1007/s00029-022-00778-y

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Abstract: Abstract We formulate the geometric P=W conjecture for singular character varieties. We establish it for compact Riemann surfaces of genus one, and obtain partial results in arbitrary genus. To this end, we employ non-Archimedean, birational and degeneration techniques to study the topology of the dual boundary complex of character varieties. We also clarify the relation between the geometric and the cohomological P=W conjectures. PubDate: 2022-05-04 DOI: 10.1007/s00029-022-00776-0

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Abstract: Abstract In this paper we will present a homological model for Coloured Jones Polynomials. For each colour \(N \in {\mathbb {N}}\) , we will describe the invariant \(J_N(L,q)\) as a graded intersection pairing of certain homology classes in a covering of a configuration space on the punctured disc. This construction is based on the Lawrence representation and a result due to Kohno that relates quantum representations and homological representations of the braid groups. PubDate: 2022-04-30 DOI: 10.1007/s00029-022-00772-4

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Abstract: Abstract For a Koszul Artin-Schelter regular algebra (also called twisted Calabi-Yau algebra), we show that it has a “twisted" bi-symplectic structure, which may be viewed as a noncommutative and twisted analog of the shifted symplectic structure introduced by Pantev, Toën, Vaquié and Vezzosi. This structure gives a quasi-isomorphism between the tangent complex and the twisted cotangent complex of the algebra, and may be viewed as a DG enhancement of Van den Bergh’s noncommutative Poincaré duality; it also induces a twisted symplectic structure on its derived representation schemes. PubDate: 2022-04-28 DOI: 10.1007/s00029-022-00774-2

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Abstract: Abstract For every one-relator monoid \(M = \langle A \mid u=v \rangle \) with \(u, v \in A^*\) we construct a contractible M-CW complex and use it to build a projective resolution of the trivial module which is finitely generated in all dimensions. This proves that all one-relator monoids are of type \(\mathrm{FP}_{\infty }\) , answering positively a problem posed by Kobayashi in 2000. We also apply our results to classify the one-relator monoids of cohomological dimension at most 2, and to describe the relation module, in the sense of Ivanov, of a torsion-free one-relator monoid presentation as an explicitly given principal left ideal of the monoid ring. In addition, we prove the topological analogues of these results by showing that all one-relator monoids satisfy the topological finiteness property \(\mathrm{F}_\infty \) , and classifying the one-relator monoids with geometric dimension at most 2. These results give a natural monoid analogue of Lyndon’s Identity Theorem for one-relator groups. PubDate: 2022-04-27 DOI: 10.1007/s00029-022-00773-3

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Abstract: Abstract This article introduces a new method to construct volume-filling symplectic embeddings of 4-dimensional ellipsoids by employing polytope mutations in toric and almost toric varieties. The construction uniformly recovers the full sequences for the Fibonacci Staircase of McDuff–Schlenk, the Pell Staircase of Frenkel–Müller and the Cristofaro-Gardiner–Kleinman Staircase, and adds new infinite sequences of ellipsoid embeddings. In addition, we initiate the study of symplectic-tropical curves for almost toric fibrations and emphasize the connection to quiver combinatorics. PubDate: 2022-04-27 DOI: 10.1007/s00029-022-00765-3

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Abstract: Abstract Let (X, o) be a complex normal surface singularity with rational homology sphere link and let \(\widetilde{X}\) be one of its good resolutions. Consider an effective cycle Z supported on the exceptional curve and the isomorphism classes \(\mathrm{Pic}(Z)\) of line bundles on Z. The set of possible values \(h^1(Z,\mathcal {L})\) for \(\mathcal {L}\in \mathrm{Pic}(Z)\) can be understood in terms of the dimensions of the images of the Abel maps, as subspaces of \(\mathrm{Pic}(Z)\) . In this note we present two algorithms, which provide these dimensions. Usually, the dimension of \(\mathrm{Pic}(Z)\) and of the dimension of the image of the Abel maps are not topological. However, we provide combinatorial formulae for them in terms of the resolution graph whenever the analytic structure on \(\widetilde{X}\) is generic. PubDate: 2022-04-27 DOI: 10.1007/s00029-022-00777-z

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Abstract: Abstract Given a connected reductive algebraic group G and a Borel subgroup \(B \subseteq G\) , we study B-normalized one-parameter additive group actions on affine spherical G-varieties. We establish basic properties of such actions and their weights and discuss many examples exhibiting various features. We propose a construction of such actions that generalizes the well-known construction of normalized one-parameter additive group actions on affine toric varieties. Using this construction, for every affine horospherical G-variety X we obtain a complete description of all G-normalized one-parameter additive group actions on X and show that the open G-orbit in X can be connected with every G-stable prime divisor via a suitable choice of a B-normalized one-parameter additive group action. Finally, when G is of semisimple rank 1, we obtain a complete description of all B-normalized one-parameter additive group actions on affine spherical G-varieties having an open orbit of a maximal torus \(T \subseteq B\) . PubDate: 2022-04-27 DOI: 10.1007/s00029-022-00775-1

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Abstract: Abstract We consider the Hodge filtration on the sheaf of meromorphic functions along free divisors for which the logarithmic comparison theorem holds. We describe the Hodge filtration steps as submodules of the order filtration on a cyclic presentation in terms of a special factor of the Bernstein–Sato polynomial of the divisor and we conjecture a bound for the generating level of the Hodge filtration. Finally, we develop an algorithm to compute Hodge ideals of such divisors and we apply it to some examples. PubDate: 2022-04-18 DOI: 10.1007/s00029-022-00767-1

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Abstract: Abstract We prove the height two case of a conjecture of Hovey and Strickland that provides a K(n)-local analogue of the Hopkins–Smith thick subcategory theorem. Our approach first reduces the general conjecture to a problem in arithmetic geometry posed by Chai. We then use the Gross–Hopkins period map to verify Chai’s Hope at height two and all primes. Along the way, we show that the graded commutative ring of completed cooperations for Morava E-theory is coherent, and that every finitely generated Morava module can be realized by a K(n)-local spectrum as long as \(2p-2>n^2+n\) . Finally, we deduce consequences of our results for descent of Balmer spectra. PubDate: 2022-04-10 DOI: 10.1007/s00029-022-00766-2

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Abstract: Abstract Motivated by the intersection theory of moduli spaces of curves, we introduce psi classes in matroid Chow rings and prove a number of properties that naturally generalize properties of psi classes in Chow rings of Losev-Manin spaces. We use these properties of matroid psi classes to give new proofs of (1) a Chow-theoretic interpretation for the coefficients of the reduced characteristic polynomials of matroids, (2) explicit formulas for the volume polynomials of matroids, and (3) Poincaré duality for matroid Chow rings. PubDate: 2022-04-01 DOI: 10.1007/s00029-022-00771-5

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Abstract: Abstract We study categorical primitive forms for Calabi–Yau \(A_\infty \) categories with semi-simple Hochschild cohomology. We classify these primitive forms in terms of certain grading operators on the Hochschild homology. We use this result to prove that, if the Fukaya category \({{\textsf {Fuk}}}(M)\) of a symplectic manifold M has semi-simple Hochschild cohomology, then its genus zero Gromov–Witten invariants may be recovered from the \(A_\infty \) -category \({{\textsf {Fuk}}}(M)\) together with the closed-open map. An immediate corollary of this is that in the semi-simple case, homological mirror symmetry implies enumerative mirror symmetry. PubDate: 2022-03-25 DOI: 10.1007/s00029-022-00769-z