Recent zbMATH articles in MSC 14Dhttps://zbmath.org/atom/cc/14D2023-12-07T16:00:11.105023ZWerkzeugNondensity of integral points and variations of Hodge structureshttps://zbmath.org/1522.110672023-12-07T16:00:11.105023Z"Maculan, Marco"https://zbmath.org/authors/?q=ai:maculan.marcoSummary: In the early eighties, Faltings proved that every nonsingular projective curve of genus at least \(2\), defined over a number field \(K\) has only finitely many points with coordinates in \(K\) a statement often referred to as `Mordell's conjecture'. Recently, \textit{B. Lawrence} and \textit{A. Venkatesh} [Invent. Math. 221, No. 3, 893--999 (2020; Zbl 1455.11093)] have come up with a new method to show nondensity (for the Zariski topology) of integral points of an algebraic variety defined over a number field. Applied to curves, this technique leads to a new proof of Mordell's conjecture; applied to varieties parameterizing nonsingular hypersurfaces of the projective space (Lawrence-Venkatesh) or of an abelian variety (Lawrence-Sawin), it yields finiteness results out of scope of the previous methods.
For the entire collection see [Zbl 1522.00191].New constructions of nef classes on self-products of curveshttps://zbmath.org/1522.140132023-12-07T16:00:11.105023Z"Fulger, Mihai"https://zbmath.org/authors/?q=ai:fulger.mihai"Murayama, Takumi"https://zbmath.org/authors/?q=ai:murayama.takumiSummary: We study the nef cone of self-products of a curve. When the curve is very general of genus \(g>2\), we construct a nontrivial class of self-intersection 0 on the boundary of the nef cone. Up to symmetry, this is the only known nontrivial boundary example that exists for all \(g > 2\). When the curve is general, we identify nef classes that improve on known examples for arbitrary curves. We also consider self-products of more than two copies of the curve.Motivic Chern classes and K-theoretic stable envelopeshttps://zbmath.org/1522.140182023-12-07T16:00:11.105023Z"Fehér, László M."https://zbmath.org/authors/?q=ai:feher.laszlo-m"Rimányi, Richárd"https://zbmath.org/authors/?q=ai:rimanyi.richard"Weber, Andrzej"https://zbmath.org/authors/?q=ai:weber.andrzejCharacteristic classes of singular varieties are important tools of study in algebraic geometry and enumerative geometry. In this paper the authors study the notion of equivaraint motivic Chern classes, after defining this they compute the motivic Chern classes for certain kind of varieties. Consider a smooth algebraic variety equipped with an action of an algebraic group having finitely many orbits. The notion of equivariant motivic Chern class is studied in the \(K\)-theory of the ambient space. They present an axiomatic characterization of the equivariant motivic Chern class. The axiom system is inspired by the work of Okounkov. There is a notion of Okounkov's \(K\)-theoretic stable envelope. These are analogous to the motivic Chern classes defined in this paper, in the sense that there is an anlogy of the defining axiom system. This imply that the motivic equivaraint Chern class coincides with the notion of stable envelope after a certain identification.
Secondly the authors' understood the motivic Chern classes of matrix Schubert cells. In theorem 6.4, 7.4 the authors present formulas's for the equivariant motivic Chern classes of Schubert cells in partial flag varieties and those of matrix Schubert cells.
Finally the authors study the motivic Chern classes of orbits of \(\mathrm{Hom}(\mathbb C^k, \mathbb C^n)\) acted upon by \(\mathrm{GL}_k(\mathbb C)\times \mathrm{GL}_n(\mathbb C)\). This representation is called \(A_2\) quiver representation and after projectivization the orbit closures are called determinantal varieties. The authors prove formulas' for the motivic Chern classes of the orbits.
Reviewer: Kalyan Banerjee (Chennai)Arithmeticity (or not) of monodromyhttps://zbmath.org/1522.140192023-12-07T16:00:11.105023Z"Sarnak, Peter"https://zbmath.org/authors/?q=ai:sarnak.peter-cIn [Discrete Subgroups of Lie Groups Appl. Moduli, Pap. Bombay Colloq. 1973, 32--134 (1975; Zbl 0355.14003)], \textit{P. Griffiths} and \textit{W. Schmid} have asked the question whether monodromy groups of families of varieties acting on cohomology are arithmetic or not. In spite of known explicit examples, the problem is still open. In the case of families of Calabi-Yau three-folds, of the well-known 14 families, 3 are arithmetic, 7 are not arithmetic and 4 remain undecided.
The author asks the question: What is the geometric significance of being arithmetic?
He gives three comments on this problem from Peter Sarnak's e-mail correspondence.
For the entire collection see [Zbl 1437.11003].
Reviewer: Vladimir P. Kostov (Nice)Irreducible components of the moduli space of rank 2 sheaves of odd determinant on projective spacehttps://zbmath.org/1522.140202023-12-07T16:00:11.105023Z"Almeida, Charles"https://zbmath.org/authors/?q=ai:almeida.charles"Jardim, Marcos"https://zbmath.org/authors/?q=ai:jardim.marcos"Tikhomirov, Alexander S."https://zbmath.org/authors/?q=ai:tikhomirov.alexander-sSummary: We describe new irreducible components of the moduli space of rank 2 semistable torsion free sheaves on the three-dimensional projective space whose generic point corresponds to non-locally free sheaves whose singular locus is either 0-dimensional or consists of a line plus disjoint points. In particular, we prove that the moduli spaces of semistable sheaves with Chern classes \((c_1, c_2, c_3) = (- 1, 2 n, 0)\) and \((c_1, c_2, c_3) = (0, n, 0)\) always contain at least one rational irreducible component. As an application, we prove that the number of such components grows as the second Chern class grows, and compute the exact number of irreducible components of the moduli spaces of rank 2 semistable torsion free sheaves with Chern classes \((c_1, c_2, c_3) = (- 1, 2, m)\) for non negative values for \(m\); all components turn out to be rational. Furthermore, we also prove that these moduli spaces are connected, showing that some of sheaves here considered are smoothable.Betti numbers of stable map spaces to Grassmannianshttps://zbmath.org/1522.140212023-12-07T16:00:11.105023Z"Bagnarol, Massimo"https://zbmath.org/authors/?q=ai:bagnarol.massimoSummary: Let \(\overline{M}_{0,n}(G(r,V), d)\) be the coarse moduli space of stable degree \(d\) maps from \(n\)-pointed genus 0 curves to a Grassmann variety \(G(r,V)\). We provide a recursive method for the computation of the Hodge numbers and the Betti numbers of \(\overline{M}_{0,n}(G(r,V), d)\) for all \(n\) and \(d\). Our method is a generalization of \textit{E. Getzler} and \textit{R. Pandharipande}'s work for maps to projective spaces [J. Algebr. Geom. 15, No. 4, 709--732 (2006; Zbl 1114.14032)].
{{\copyright} 2022 Wiley-VCH GmbH.}Rational pullbacks of toric foliationshttps://zbmath.org/1522.140222023-12-07T16:00:11.105023Z"Gargiulo Acea, Javier"https://zbmath.org/authors/?q=ai:gargiulo-acea.javier"Molinuevo, Ariel"https://zbmath.org/authors/?q=ai:molinuevo.ariel"Velazquez, Sebastián"https://zbmath.org/authors/?q=ai:velazquez.sebastianThis article deals with deformation and, more specifically, first order unfoldings of codimension-\(1\) foliations. An \textit{unfolding} of a foliation given by an integrable \(1\)-form \(\omega\in\Omega^1_X\) is a foliation on a product space \(X\times S\) given by a \(1\)-form \(\widetilde{\omega}\in\Omega^1_{X\times S}\) that restricts to \(\omega\) in the fiber over a point \(0\in S\), that is \(\widetilde{\omega}|_{X\times\{0\}} =\omega\). When \(S=\mathrm{Spec}(k[x]/(x)^2)\) we talk of a first order unfolding, and it determines a deformation of the foliation given by \(\omega\). Here the authors study the cases where \(X\) is a simplicial toric variety first and then the case where the ambient variety is \(\mathbb{P}^n\) and the foliation is the pull-back \(\varphi^*\alpha\) of a foliation \(\alpha\) on a simplicial toric surface \(X\) by a rational map \(\varphi:\mathbb{P}^n\to X\).
The deformations of foliations \(\varphi^*\alpha\) that comes from unfoldings are characterized as being of the form \(\varphi_\epsilon^*\alpha\) with \(\varphi_\epsilon^*\) a deformation of the rational map \(\varphi\).
The results are obtained from a careful study of the expression of \(\alpha\) and \(\varphi\) in homogeneous coordinates of \(\mathbb{P}^1\) and the toric surface \(X\).
Reviewer: Federico Quallbrunn (Buenos Aires)Rigid analytic \(p\)-adic Simpson correspondence for line bundleshttps://zbmath.org/1522.140272023-12-07T16:00:11.105023Z"Song, Ziyan"https://zbmath.org/authors/?q=ai:song.ziyanSummary: The \(p\)-adic Simpson correspondence due to \textit{G. Faltings} [Adv. Math. 198, No. 2, 847--862 (2005; Zbl 1102.14022)] is a \(p\)-adic analogue of non-abelian Hodge theory. The following is the main result of this article: The correspondence for line bundles can be enhanced to a rigid analytic morphism of moduli spaces under certain smallness conditions. In the complex setting, Simpson shows that there is a complex analytic morphism from the moduli space for the vector bundles with integrable connection to the moduli space of representations of a finitely generated group as algebraic varieties. We give a \(p\)-adic analogue of Simpson's result.Hodge classes on the moduli space of \(W(E_6)\)-covers and the geometry of \(\mathcal{A}_6\)https://zbmath.org/1522.140372023-12-07T16:00:11.105023Z"Alexeev, Valery"https://zbmath.org/authors/?q=ai:alexeev.valery-a"Donagi, Ron"https://zbmath.org/authors/?q=ai:donagi.ron-y"Farkas, Gavril"https://zbmath.org/authors/?q=ai:farkas.gavril"Izadi, Elham"https://zbmath.org/authors/?q=ai:izadi.elham"Ortega, Angela"https://zbmath.org/authors/?q=ai:ortega.angelaSummary: In previous work we showed that the Hurwitz space of \(W(E_6)\)-covers of the projective line branched over \(24\) points dominates via the Prym-Tyurin map the moduli space \(\mathcal{A}_6\) of principally polarized abelian \(6\)-folds. Here we determine the \(25\) Hodge classes on the Hurwitz space of \(W(E_6)\)-covers corresponding to the \(25\) irreducible representations of the Weyl group \(W(E_6)\). This result has direct implications to the intersection theory of the toroidal compactification \(\overline{\mathcal{A}}_6\). In the final part of the paper, we present an alternative, elementary proof of our uniformization result on \(\mathcal{A}_6\) via Prym-Yurin varieties of type \(W(E_6)\).The Hurwitz space Picard rank conjecture for \(d > g - 1\)https://zbmath.org/1522.140402023-12-07T16:00:11.105023Z"Mullane, Scott"https://zbmath.org/authors/?q=ai:mullane.scottLes espaces d'Hurwitz constituent un outil clef pour l'étude de l'espace de modules des surfaces de Riemann, tant du point de vue géométrique [\textit{B. Riemann}, J. Reine Angew. Math. 54, 115--155 (1857; Zbl 2750360)] que topologique [\textit{A. Clebsch}, Math. Ann. 6, 216--230 (1873; JFM 05.0285.03)]. Rappelons que ces espaces paramétrisent les revêtements de degré \(d\) de la sphère par des surfaces de Riemann de genre \(g\) ayant des profils de ramifications fixés. Dans le cas où toutes les ramifications sont simples et dans des fibres distinctes, on note cet espace \(\mathcal{H}_{g,d}\). Malgré leur place centrale, de nombreuses questions, même dans les cas des profils les plus simples, restent ouvertes.
Par exemple, on sait que \(\mathcal{H}_{g,d}\) possède un groupe de Picard (rationnel) trivial pour \(d\leq5\) et \(d> 2g-2\). La conjecture est que le groupe de Picard est trivial \(\mathcal{H}_{g,d}\) pour toutes les valeurs de \(g\) et \(d\). Le résultat principal de cet article est que \(\mathcal{H}_{g,d}\) est uniréglé de groupe de Picard trivial pour \(d>g-1\). L'idée principale de la preuve consiste à identifier ces espaces avec des ouverts denses de strates de différentielles du second type (dans le même esprit que [\textit{S. Mullane}, Ann. Inst. Fourier 72, No. 4, 1379--1416 (2022; Zbl 1502.14021)]).
Reviewer: Quentin Gendron (Ciudad de México)Moduli spaces of vector bundles on a nodal curvehttps://zbmath.org/1522.140482023-12-07T16:00:11.105023Z"Hu, Wenyao"https://zbmath.org/authors/?q=ai:hu.wenyao"Sun, Xiaotao"https://zbmath.org/authors/?q=ai:sun.xiaotaoSummary: Constructions of moduli spaces are given for more general cases and the semistability of parabolic sheaves is modified so that it is equivalent to the GIT-semistability. We formulate and discuss a conjectural vanishing problem for moduli spaces of semistable parabolic sheaves on a nodal curve.
For the entire collection see [Zbl 1508.20002].3-dimensional mirror symmetryhttps://zbmath.org/1522.140522023-12-07T16:00:11.105023Z"Webster, Ben"https://zbmath.org/authors/?q=ai:webster.ben"Yoo, Philsang"https://zbmath.org/authors/?q=ai:yoo.philsangFrom the text: After I gave a talk at the Institute for Advanced Study in 2008, Sergei Gukov pointed out to me that
physicists already knew that these secret passages should exist based on a known duality: 3-dimensional mirror
symmetry. As explained above, this definitely did not resolve all of our questions; to this day, an explanation
of several of the observations we had made remains elusive. More generally, this duality was poorly understood
by physicists at the time (and many questions remain), but at least it provided an explanation of why such a passage
should exist and a basis to search for it. In the 15 years since that conversation, enormous progress has been made on the connections between mathematics and 3-dimensional QFT. The purpose of this article is to give a short explanation of this progress and some of the QFT behind it for mathematicians. It is, of necessity, painfully incomplete, but we hope that it will be a useful guide for mathematicians of all ages to learn more.Disconnected moduli spaces of stable bundles on surfaceshttps://zbmath.org/1522.140592023-12-07T16:00:11.105023Z"Coskun, Izzet"https://zbmath.org/authors/?q=ai:coskun.izzet"Huizenga, Jack"https://zbmath.org/authors/?q=ai:huizenga.jack"Kopper, John"https://zbmath.org/authors/?q=ai:kopper.johnGiven a projective surface \(X\subset\mathbb{P}^n\), the moduli spaces \(M_X(2;c_1,c_2)\) of stable rank 2 bundles \(\mathcal{E}\) with Chern classes \(c_i:=c_i(\mathcal{E})\) tend to be well-behaved for large enough \(c_2\). In particular, they tend to be irreducible and reduced. On the other hand, a different behaviour can be expected for small \(c_2\).
In this paper, the authors deal with these issues constructing projective surfaces \(X\) of general type and Picard number one for which \(M_X(2;c_1,c_2)\) has a large number of components. More concretely, their examples consist on complete intersection \(X=\cap_{i=1}^{n-2}X_i\) of \(n-2\) hypersurfaces \(X_i\) in \(\mathbb{P}^n\) of degrees \(3\leq d_1\leq\dots\leq d_{n-2}\) such that \(D_1\) contains linear subspaces of codimension \(2\). It is proved that every connected component of the Fano scheme \(F(D_1)\) parameterizing such linear spaces corresponds, by means of a version of Serre correspondence, to a distinct connected component of \(M_X(2,H,(d_1-1)\prod_{i=2}^{n-2}d_i)\). Then, examples of hypersurfaces \(D_1\subset\mathbb{P}^n\) such that \(F(D_1)\) has many components are found.
Reviewer: Joan Pons-Llopis (Maó)Representations with minimal support for quantized Gieseker varietieshttps://zbmath.org/1522.160142023-12-07T16:00:11.105023Z"Etingof, Pavel"https://zbmath.org/authors/?q=ai:etingof.pavel-i"Krylov, Vasily"https://zbmath.org/authors/?q=ai:krylov.vasily"Losev, Ivan"https://zbmath.org/authors/?q=ai:losev.ivan-v"Simental, José"https://zbmath.org/authors/?q=ai:simental.jose-eSummary: We study the minimally supported representations of quantizations of Gieseker moduli spaces. We relate them to \(\mathrm{SL}_n\)-equivariant D-modules on the nilpotent cone of \(\mathfrak{sl}_n\) and to minimally supported representations of type A rational Cherednik algebras. Our main result is character formulas for minimally supported representations of quantized Gieseker moduli spaces.The proportion of derangements characterizes the symmetric and alternating groupshttps://zbmath.org/1522.200212023-12-07T16:00:11.105023Z"Poonen, Bjorn"https://zbmath.org/authors/?q=ai:poonen.bjorn"Slavov, Kaloyan"https://zbmath.org/authors/?q=ai:slavov.kaloyanPermutations with no fixed points are called derangements. The paper under review shows that the proportion of derangements characterizes the symmetric groups among permutation groups. In fact, one can even consider cosets of permutation groups. More precisely, the authors prove that if \(G\) is a subgroup of \(S_n\) (with \(n\geq 1\)) and there is a coset \(C\) of \(G\) in \(S_n\) such that
\[
\frac{|\{\sigma \in C : \sigma\text{ is a derangement}\}|}{|C|} = \frac{|\{\sigma \in S_n : \sigma\text{ is a derangement}\}|}{|S_n|},
\]
then necessarily \(C = G = S_n\) (Theorem 1.1). They also show that by a similar argument, the same statement holds when \(S_n\) is replaced by \(A_n\), provided that \(n\geq 7\) is imposed (Theorem. 5.1). Theorem 1.1 was originally motivated by a question about monodromy groups and the application is explained in Section 1.2 (we omit the details here).
Reviewer: Cindy Tsang (Tōkyō)Local types of \((\Gamma, G)\)-bundles and parahoric group schemeshttps://zbmath.org/1522.202012023-12-07T16:00:11.105023Z"Damiolini, Chiara"https://zbmath.org/authors/?q=ai:damiolini.chiara"Hong, Jiuzu"https://zbmath.org/authors/?q=ai:hong.jiuzuSummary: Let \(G\) be a simple algebraic group over an algebraically closed field \(k\). Let \(\Gamma\) be a finite group acting on \(G\). We classify and compute the local types of \((\Gamma , G)\)-bundles on a smooth projective \(\Gamma \)-curve in terms of the first nonabelian group cohomology of the stabilizer groups at the tamely ramified points with coefficients in \(G\). When \(\text{char}(k)=0\), we prove that any generically simply connected parahoric Bruhat-Tits group scheme can arise from a \((\Gamma ,G_{\mathrm{ad}})\)-bundle. We also prove a local version of this theorem, that is, parahoric group schemes over the formal disc arise from constant group schemes via tamely ramified coverings.
{{\copyright} 2023 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}The Hodge filtration on complements of complex subspace arrangements and integral representations of holomorphic functionshttps://zbmath.org/1522.320262023-12-07T16:00:11.105023Z"Eliyashev, Yury V."https://zbmath.org/authors/?q=ai:eliyashev.yury-vSummary: We compute the Hodge filtration on cohomology groups of complements of complex subspace arrangements. By means of this result we construct integral representations of holomorphic functions such that kernels of these representations have singularities on subspace arrangements.Open problems on structure of positively curved projective varietieshttps://zbmath.org/1522.320572023-12-07T16:00:11.105023Z"Matsumura, Shin-ichi"https://zbmath.org/authors/?q=ai:matsumura.shin-ichiSummary: We provide supplements and open problems related to structure theorems for maximal rationally connected fibrations of certain positively curved projective varieties, including smooth projective varieties with semi-positive holomorphic sectional curvature, pseudo-effective tangent bundle, and nef anti-canonical divisor.Harmonic morphisms and their Milnor fibrationshttps://zbmath.org/1522.320672023-12-07T16:00:11.105023Z"Ribeiro, M. F."https://zbmath.org/authors/?q=ai:ribeiro.maico-f"Araújo dos Santos, R. N."https://zbmath.org/authors/?q=ai:araujo-dos-santos.raimundo-nonato"Dreibelbis, D."https://zbmath.org/authors/?q=ai:dreibelbis.daniel"Griffin, M."https://zbmath.org/authors/?q=ai:griffin.mark|griffin.matt|griffin.malcolm-p|griffin.michael-j|griffin.maryclareSummary: In this paper, we study the relationships between harmonic morphisms and fibered structures in the local setting.On the relationships between Hopf fibrations and Cartan hypersurfaces in sphereshttps://zbmath.org/1522.530502023-12-07T16:00:11.105023Z"Hashimoto, Hideya"https://zbmath.org/authors/?q=ai:hashimoto.hideyaSummary: The famous theorem due to Hurwitz stated that the normed (division) algebra is isomorphic to one of the following four algebras; the field \(\mathbb{R}\) of real numbers, the field \(\mathbb{C}\) of complex numbers, the algebra \(\mathbb{H}\) of quaternions, and the non-associative algebra \(\mathbb{O}\) of octonions. By using these algebraic structures, we can construct the Hopf fibrations, and Cartan hypersurfaces in a sphere. The purpose of this paper is to give the relationship between these Hopf fibrations and Cartan hypersurfaces.
For the entire collection see [Zbl 1508.53008].Graph gauge theory of mobile non-abelian anyons in a qubit stabilizer codehttps://zbmath.org/1522.810582023-12-07T16:00:11.105023Z"Lensky, Yuri D."https://zbmath.org/authors/?q=ai:lensky.yuri-d"Kechedzhi, Kostyantyn"https://zbmath.org/authors/?q=ai:kechedzhi.kostyantyn"Aleiner, Igor"https://zbmath.org/authors/?q=ai:aleiner.igor-l"Kim, Eun-Ah"https://zbmath.org/authors/?q=ai:kim.eun-ahSummary: Stabilizer braiding codes allow for non-local encoding and processing of quantum information. Deformations of stabilizer surface codes introduce new and non-trivial geometry, in particular leading to emergence of long sought after objects known as projective Ising non-abelian anyons. Braiding of such anyons is a key ingredient of topological quantum computation. We suggest a simple and systematic approach to construct effective unitary protocols for braiding, manipulation and readout of non-abelian anyons and preparation of their entangled states. We generalize the surface code to a more generic graph with vertices of degree 2, 3 and 4. Our approach is based on the mapping of the stabilizer code defined on such a graph onto a model of Majorana fermions charged with respect to two emergent gauge fields. One gauge field is akin to the physical magnetic field. The other one is responsible for emergence of the non-abelian anyonic statistics and has a purely geometric origin. This field arises from assigning certain rules of orientation on the graph known as the Kasteleyn orientation in the statistical theory of dimer coverings. Each 3-degree vertex on the graph carries the flux of this ``Kasteleyn'' field and hosts a non-abelian anyon. In our approach all the experimentally relevant operators are unambiguously fixed by locality, unitarity and gauge invariance. We illustrate the power of our method by making specific prescriptions for experiments verifying the non-abelian statistics.Towards a mathematical theory of the Madelung equations: Takabayasi's quantization condition, quantum quasi-irrotationality, weak formulations, and the Wallstrom phenomenonhttps://zbmath.org/1522.810792023-12-07T16:00:11.105023Z"Reddiger, Maik"https://zbmath.org/authors/?q=ai:reddiger.maik"Poirier, Bill"https://zbmath.org/authors/?q=ai:poirier.billSummary: Even though the Madelung equations are central to many `classical' approaches to the foundations of quantum mechanics such as Bohmian and stochastic mechanics, no coherent mathematical theory has been developed so far for this system of partial differential equations. Wallstrom prominently raised objections against the Madelung equations, aiming to show that no such theory exists in which the system is well-posed and in which the Schrödinger equation is recovered without the imposition of an additional `ad hoc quantization condition' -- like the one proposed by Takabayasi. The primary objective of our work is to clarify in which sense Wallstrom's objections are justified and in which sense they are not, with a view on the existing literature. We find that it may be possible to construct a mathematical theory of the Madelung equations which is satisfactory in the aforementioned sense, though more mathematical research is required. More specifically, this work makes five main contributions to the subject: First, we rigorously prove that Takabayasi's quantization condition holds for arbitrary \(C^1\)-wave functions. Nonetheless, we explain why there are serious doubts with regards to its applicability in the general theory of quantum mechanics. Second, we argue that the Madelung equations need to be understood in the sense of distributions. Accordingly, we review a weak formulation due to \textit{I. Gasser} and \textit{P. A. Markowich} [Asymptotic Anal. 14, No. 2, 97--116 (1997; Zbl 0877.76087)] and suggest a second one based on Nelson's equations. Third, we show that the common examples that motivate Takabayasi's condition do not satisfy one of the Madelung equations in the distributional sense, leading us to introduce the concept of `quantum quasi-irrotationality'. This terminology was inspired by a statement due to Schönberg. Fourth, we construct explicit `non-quantized' strong solutions to the Madelung equations in two dimensions, which were claimed to exist by Wallstrom, and provide an analysis thereof. Fifth, we demonstrate that Wallstrom's argument for non-uniqueness of solutions of the Madelung equations, termed the `Wallstrom phenomenon', is ultimately due to a failure of quantum mechanics to discern physically equivalent, yet mathematically inequivalent states -- an issue that finds its historic origins in the Pauli problem.SUSY localization for Coulomb branch operators in omega-deformed 3d \(\mathcal{N} = 4\) gauge theorieshttps://zbmath.org/1522.811142023-12-07T16:00:11.105023Z"Okuda, Takuya"https://zbmath.org/authors/?q=ai:okuda.takuya"Yoshida, Yutaka"https://zbmath.org/authors/?q=ai:yoshida.yutakaSummary: We perform SUSY localization for Coulomb branch operators of 3d \(\mathcal{N}=4\) gauge theories in \(\mathbb{R}^3\) with \(\Omega\)-deformation. Our study provides a path integral foundation to the so-called abelianization procedure that has been used to study the Coulomb branch. For the dressed monopole operators whose expectation values do not involve non-perturbative corrections, our computations reproduce the results of abelianization. For the expectation values of other operators and the correlation functions of multiple operators in \(U(N)\) gauge theories, we compute the non-perturbative corrections due to monopole bubbling using matrix models obtained by string theory (brane) construction. We relate the results of localization to algebraic structures discussed in the mathematical literature, and also uncover a similar relation for line operators in 4d \(\mathcal{N}=2\) gauge theories. For \(U(N)\) (quiver) gauge theories in 3d we demonstrate a direct correspondence between wall-crossing in matrix models and the ordering of operators.Example of 4-pt non-vacuum \(\mathcal{W}_3\) classical blockhttps://zbmath.org/1522.811252023-12-07T16:00:11.105023Z"Pavlov, Mikhail"https://zbmath.org/authors/?q=ai:pavlov.mikhail-sSummary: In this note we study a special case of the 4-pt non-vacuum classical block associated with the \(\mathcal{W}_3\) algebra. We formulate the monodromy problem for the block and derive monodromy equations within heavy-light approximation. Fixing the remaining functional arbitrariness using parameters of the 4-pt vacuum \(\mathcal{W}_3\) block, we compute the 4-pt non-vacuum \(\mathcal{W}_3\) block function.New quiver-like varieties and Lie superalgebrashttps://zbmath.org/1522.811262023-12-07T16:00:11.105023Z"Rimányi, R."https://zbmath.org/authors/?q=ai:rimanyi.richard"Rozansky, L."https://zbmath.org/authors/?q=ai:rozansky.levSummary: In order to extend the geometrization of Yangian \(R\)-matrices from Lie algebras \(\mathfrak{gl}(n)\) to superalgebras \(\mathfrak{gl}(M|N)\), we introduce new quiver-related varieties which are associated with representations of \(\mathfrak{gl}(M|N) \). In order to define them similarly to the Nakajima-Cherkis varieties, we reformulate the construction of the latter by replacing the Hamiltonian reduction with the intersection of generalized Lagrangian subvarieties in the cotangent bundles of Lie algebras sitting at the vertices of the quiver. The new varieties come from replacing some Lagrangian subvarieties with their Legendre transforms. We present superalgebra versions of stable envelopes for the new quiver-like varieties that generalize the cotangent bundle of a Grassmannian. We define superalgebra generalizations of the Tarasov-Varchenko weight functions, and show that they represent the super stable envelopes. Both super stable envelopes and super weight functions transform according to Yangian \(\check{R} \)-matrices of \(\mathfrak{gl}(M|N)\) with \(M+N=2\).A-branes, foliations and localizationhttps://zbmath.org/1522.813142023-12-07T16:00:11.105023Z"Banerjee, Sibasish"https://zbmath.org/authors/?q=ai:banerjee.sibasish"Longhi, Pietro"https://zbmath.org/authors/?q=ai:longhi.pietro"Romo, Mauricio"https://zbmath.org/authors/?q=ai:romo.mauricioSummary: This paper studies a notion of enumerative invariants for stable \(A\)-branes and discusses its relation to invariants defined by spectral and exponential networks. A natural definition of stable \(A\)-branes and their counts is provided by the string theoretic origin of the topological \(A\)-model. This is the Witten index of the supersymmetric quantum mechanics of a single \(D3\) brane supported on a special Lagrangian in a Calabi-Yau threefold. Geometrically, this is closely related to the Euler characteristic of the \(A\)-brane moduli space. Using the natural torus action on this moduli space, we reduce the computation of its Euler characteristic to a count of fixed points via equivariant localization. Studying the \(A\)-branes that correspond to fixed points, we make contact with definitions of spectral and exponential networks. We find agreement between the counts defined via the Witten index, and the BPS invariants defined by networks. By extension, our definition also matches with Donaldson-Thomas invariants of \(B\)-branes related by homological mirror symmetry.Divisor topologies of CICY 3-folds and their applications to phenomenologyhttps://zbmath.org/1522.813312023-12-07T16:00:11.105023Z"Carta, Federico"https://zbmath.org/authors/?q=ai:carta.federico"Mininno, Alessandro"https://zbmath.org/authors/?q=ai:mininno.alessandro"Shukla, Pramod"https://zbmath.org/authors/?q=ai:shukla.pramod-sSummary: In this article, we present a classification for the divisor topologies of the projective complete intersection Calabi-Yau (pCICY) 3-folds realized as hypersurfaces in the product of complex projective spaces. There are 7890 such pCICYs of which 7820 are favorable, and can be subsequently useful for phenomenological purposes. To our surprise we find that the whole pCICY database results in only 11 (so-called coordinate) divisors \((D)\) of distinct topology and we classify those surfaces with their possible deformations inside the pCICY 3-fold, which turn out to be satisfying \(1\leq h^{2, 0}(D)\leq 7\). We also present a classification of the so-called ample divisors for all the favorable pCICYs which can be useful for fixing all the (saxionic) Kähler moduli through a single non-perturbative term in the superpotential. We argue that this relatively unexplored pCICY dataset equipped with the necessary model building ingredients, can be used for a systematic search of physical vacua. To illustrate this for model building in the context of type IIB CY orientifold compactifications, we present moduli stabilization with some preliminary analysis of searching possible vacua in simple models, as a template to be adopted for analyzing models with a larger number of Kähler moduli.From quantum curves to topological string partition functionshttps://zbmath.org/1522.813362023-12-07T16:00:11.105023Z"Coman, Ioana"https://zbmath.org/authors/?q=ai:coman.ioana-alexandra"Pomoni, Elli"https://zbmath.org/authors/?q=ai:pomoni.elli"Teschner, Jörg"https://zbmath.org/authors/?q=ai:teschner.jorgSummary: This paper describes the reconstruction of the topological string partition function for certain local Calabi-Yau (CY) manifolds from the quantum curve, an ordinary differential equation obtained by quantising their defining equations. Quantum curves are characterised as solutions to a Riemann-Hilbert problem. The isomonodromic tau-functions associated to these Riemann-Hilbert problems admit a family of natural normalisations labelled by the chambers in the extended Kähler moduli space of the local CY under consideration. The corresponding isomonodromic tau-functions admit a series expansion of generalised theta series type from which one can extract the topological string partition functions for each chamber.Deformed WZW models and Hodge theory. I.https://zbmath.org/1522.815562023-12-07T16:00:11.105023Z"Grimm, Thomas W."https://zbmath.org/authors/?q=ai:grimm.thomas-w"Monnee, Jeroen"https://zbmath.org/authors/?q=ai:monnee.jeroenSummary: We investigate a relationship between a particular class of two-dimensional integrable non-linear \(\sigma\)-models and variations of Hodge structures. Concretely, our aim is to study the classical dynamics of the \(\lambda\)-deformed \(G/G\) model and show that a special class of solutions to its equations of motion precisely describes a one-parameter variation of Hodge structures. We find that this special class is obtained by identifying the group-valued field of the \(\sigma\)-model with the Weil operator of the Hodge structure. In this way, the study of strings on classifying spaces of Hodge structures suggests an interesting connection between the broad field of integrable models and the mathematical study of period mappings.The deformed Hermitian-Yang-Mills equation on the blowup of \(\mathbb{P}^n\)https://zbmath.org/1522.816332023-12-07T16:00:11.105023Z"Jacob, Adam"https://zbmath.org/authors/?q=ai:jacob.adam"Sheu, Norman"https://zbmath.org/authors/?q=ai:sheu.normanThis article concerns the so-called \textit{deformed Hermitian-Yang-Mills (dHYM) equation}, which can be formulated on a compact Kähler manifold \((X,J,\omega)\). A class \([\alpha]\in H^{1,1}(X,\mathbb R)\) is said to solve the dHYM equation with \textit{phase} \(e^{i\hat\theta}\in S^1\) if \([\alpha]\) admits a representative \(\beta\in [\alpha]\) such that \(\mathrm{Im}(e^{i\hat\theta}(\omega+i\beta)^n)=0\). Using the \(\partial\bar\partial\)-lemma, this equation is to be thought of as an elliptic nonlinear PDE for a real-valued function \(\phi\) on \(X\), which arises naturally in certain supersymmetric physical theories as well as in the setting of SYZ mirror symmetry.
The relationship between the existence of solutions to natural PDEs and algebraic stability conditions is one of the central themes of the past decades in complex (algebraic) geometry, and the main aim of the authors is to investigate this question for the dHYM equation in a simple setting, namely the case where \(X\) is the blow-up of \(\mathbb P^n\) at a point. The proofs of the main theorems rely strongly on the special structure afforded by this simplified setting, such as the simple second-degree cohomology of \(X\). To formulate the main results, we need to introduce some notation. For any irreducible, analytic subvariety \(V\subset X\), we can consider the number
\[
Z_{[\alpha],[\omega]}(V)= - \int_V e^{-i\omega+\alpha}
\]
where the exponential is interpreted as a formal power series and we take the convention that integration is done only over the term of this power series of degree \(\dim_{\mathbb R} V\).
Now fix \(X\) as the blowup of \(\mathbb P^n\) at a point, let \(\omega\) be any Kähler class and \(\alpha\) any real cohomology class of type \((1,1)\). Then the first theorem asserts the following: If \(Z_{[\alpha],[\omega]}\neq 0\) and there exists a sequence \((\epsilon_1,\dots,\epsilon_{n-1})\), \(\epsilon_i\in \pm 1\), such that for each \(k\)-dimensional subvariety \(V^k\subset X\) we have
\[
\mathrm{sign}\,\bigg(\mathrm{Im}\bigg(\frac{Z_{[\alpha],[\omega]}(V^k)}{Z_{[\alpha],[\omega]}(X)}\bigg)\bigg) = \epsilon_k,
\]
then \(\alpha\) admits a solution to the dHYM equation. This solution moreover satisfies the Calabi ansatz, meaning that it takes on a specific form on \(X\setminus (H\cup E)\cong \mathbb C^n \setminus\{0\}\), where \(H\) and \(E\) are the hyperplane and exceptional divisor, respectively.
While the above condition is sufficient, it is not clear that it is necessary. In fact, the authors prove that it is not, and formulate a pair of necessary and sufficient conditions for the existence of a solution to the dHYM equation. Defining the so-called average angle \(\hat\Theta_{V^k}\) as the argument of \(\int_{V^k}(\omega+i\alpha)^k\), these conditions are expressed in terms of (i) \(\hat\theta=\hat\Theta_X\) and (ii) the collection of all \(\hat\Theta_{V^{n-1}}\), where \(V^{n-1}\) is a divisor.
Reviewer: Daniel Thung (Amsterdam)Gluon scattering on self-dual radiative gauge fieldshttps://zbmath.org/1522.816752023-12-07T16:00:11.105023Z"Adamo, Tim"https://zbmath.org/authors/?q=ai:adamo.tim"Mason, Lionel"https://zbmath.org/authors/?q=ai:mason.lionel-j"Sharma, Atul"https://zbmath.org/authors/?q=ai:sharma.atul-kumar|sharma.atul-sSummary: We present all-multiplicity formulae, derived from first principles in the MHV sector and motivated by twistor string theory for general helicities, for the tree-level S-matrix of gluon scattering on self-dual radiative backgrounds. These backgrounds are chiral, asymptotically flat gauge fields characterised by their free radiative data, and their underlying integrability is captured by twistor theory. Tree-level gluon scattering scattering amplitudes are expressed as integrals over the moduli space of holomorphic maps from the Riemann sphere to twistor space, with the degree of the map related to the helicity configuration of the external gluons. In the MHV sector, our formula is derived from the Yang-Mills action; for general helicities the formulae are obtained using a background-coupled twistor string theory and pass several consistency tests. Unlike amplitudes on a trivial vacuum, there are residual integrals due to the functional freedom in the self-dual background, but for scattering of momentum eigenstates we are able to do many of these explicitly and even more is possible in the special case of plane wave backgrounds. In general, the number of these integrals is always less than expected from standard perturbation theory, but matches the number associated with space-time MHV rules in a self-dual background field, which we develop for self-dual plane waves.Primordial black holes and gravitational waves from non-canonical inflationhttps://zbmath.org/1522.830362023-12-07T16:00:11.105023Z"Papanikolaou, Theodoros"https://zbmath.org/authors/?q=ai:papanikolaou.theodoros"Lymperis, Andreas"https://zbmath.org/authors/?q=ai:lymperis.andreas"Lola, Smaragda"https://zbmath.org/authors/?q=ai:lola.smaragda"Saridakis, Emmanuel N."https://zbmath.org/authors/?q=ai:saridakis.emmanuel-n(no abstract)Stability of non-degenerate Ricci-type Palatini theorieshttps://zbmath.org/1522.830392023-12-07T16:00:11.105023Z"Annala, Jaakko"https://zbmath.org/authors/?q=ai:annala.jaakko"Räsänen, Syksy"https://zbmath.org/authors/?q=ai:rasanen.syksy(no abstract)Holographic entanglement entropy in \(T\overline{T}\)-deformed \(\mathrm{AdS}_3\)https://zbmath.org/1522.830432023-12-07T16:00:11.105023Z"He, Miao"https://zbmath.org/authors/?q=ai:he.miao"Sun, Yuan"https://zbmath.org/authors/?q=ai:sun.yuanSummary: In this work, we study the holographic entanglement entropy in \(\mathrm{AdS}_3\) gravity with the certain mixed boundary condition, which turns out to correspond to \(T\overline{T}\)-deformed 2D CFTs. By employing the Chern-Simons formalism and Wilson line technique, the holographic entanglement entropy in \(T\overline{T}\)-deformed BTZ black hole is obtained. We also get the same formula by calculating the RT surface. The holographic entanglement entropy agrees with the perturbation result derived from both \(T\overline{T}\)-deformed CFTs and cutoff \(\mathrm{AdS}_3\). Moreover, our result shows that the deformed entanglement entropy for large deformation parameter behaves like the entanglement entropy of CFT at zero temperature. We also consider the entanglement entropy of two intervals and study the effect of \(T\overline{T}\) deformation on phase transition.LVS de Sitter vacua are probably in the swamplandhttps://zbmath.org/1522.833372023-12-07T16:00:11.105023Z"Junghans, Daniel"https://zbmath.org/authors/?q=ai:junghans.danielSummary: We argue that dS vacua in the LARGE-volume scenario of type IIB string theory are vulnerable to various unsuppressed curvature, warping and \(g_s\) corrections. We work out in general how these corrections affect the moduli vevs, the vacuum energy and the moduli masses in the 4D EFT for the two Kähler moduli, the conifold modulus and a nilpotent superfield describing the anti-brane uplift. Our analysis reveals that the corrections are parametrically larger in the relevant expressions than one might have guessed from their suppression in the off-shell potential. Some corrections appear without any parametric suppression at all, which makes them particularly dangerous for candidate dS vacua. Other types of corrections can in principle be made small for appropriate parameter choices. However, we show in an explicit model that this is never possible for all corrections at the same time when the vacuum energy is positive. Some of the corrections we consider are also relevant for the stability of non-supersymmetric AdS vacua.Phantom scalar-tensor models and cosmological tensionshttps://zbmath.org/1522.834062023-12-07T16:00:11.105023Z"Ballardini, Mario"https://zbmath.org/authors/?q=ai:ballardini.mario"Ferrari, Angelo Giuseppe"https://zbmath.org/authors/?q=ai:ferrari.angelo-giuseppe"Finelli, Fabio"https://zbmath.org/authors/?q=ai:finelli.fabio(no abstract)Ghost and Laplacian instabilities in teleparallel Horndeski gravityhttps://zbmath.org/1522.834162023-12-07T16:00:11.105023Z"Capozziello, Salvatore"https://zbmath.org/authors/?q=ai:capozziello.salvatore"Caruana, Maria"https://zbmath.org/authors/?q=ai:caruana.maria"Levi Said, Jackson"https://zbmath.org/authors/?q=ai:said.jackson-levi"Sultana, Joseph"https://zbmath.org/authors/?q=ai:sultana.joseph(no abstract)Quantum cosmology, eternal inflation, and swampland conjectureshttps://zbmath.org/1522.834222023-12-07T16:00:11.105023Z"Fanaras, Georgios"https://zbmath.org/authors/?q=ai:fanaras.georgios"Vilenkin, Alexander"https://zbmath.org/authors/?q=ai:vilenkin.alexander(no abstract)Accelerating internal dimensions and nonzero positive cosmological constanthttps://zbmath.org/1522.834502023-12-07T16:00:11.105023Z"Park, Eun Kyung"https://zbmath.org/authors/?q=ai:park.eunkyung"Kwon, Pyung Seong"https://zbmath.org/authors/?q=ai:kwon.pyung-seongSummary: We present a new scenario for the moduli stabilization with a very small but nonzero positive cosmological constant \(\lambda\). In this scenario the complex structure moduli are still stabilized by the three-form fluxes as in the usual flux compactifications, but the Kähler modulus is not fixed by the KKLT scenario. In our case the scale factor (or the Kähler modulus) of the internal dimensions is basically allowed to change with time. But at the supergravity level it is fixed by a set of dynamical (plus constraint) equations defined on the 4D spacetime, not by the nonperturbative corrections of KKLT. Also at the supergravity level it is shown that \(\lambda\) is fine-tuned to zero, \(\lambda=0\), by the same set of 4D equations. This result changes once we admit \(\alpha'\)-corrections of the string theory. The fine-tuning \(\lambda=0\) changes into \(\lambda=\frac{2}{3}Q\), where \(Q\) is a constant representing quantum corrections of the brane and 6D action defined on the internal dimensions and its value is determined by the \(\alpha'\)-corrections. It is also shown that this nonzero \(\lambda\) must be positive and at the same time the internal dimensions must evolve with time almost at the same rate as the external dimensions in the case of nonzero \(\lambda\).