Recent zbMATH articles in MSC 14Dhttps://zbmath.org/atom/cc/14D2024-03-13T18:33:02.981707ZWerkzeugBad representations and homotopy of character varietieshttps://zbmath.org/1528.140062024-03-13T18:33:02.981707Z"Guérin, Clément"https://zbmath.org/authors/?q=ai:guerin.clement"Lawton, Sean"https://zbmath.org/authors/?q=ai:lawton.sean"Ramras, Daniel"https://zbmath.org/authors/?q=ai:ramras.daniel-aGiven a connected reductive complex affine algebraic group \(G\) and a finitely generated group \(\Gamma\), the \(G\)-character variety of \(\Gamma\) is defined as the GIT quotient
\[
\mathcal{X}_{\Gamma}(G)=\Hom(\Gamma,G) /\!\!/ G,
\]
where \(G\) acts by conjugation. Character varieties play an important role in several areas, such as Representation Theory, Nonabelian Hodge Theory or Geometric Topology. In this paper, the authors focus on the case where \(\Gamma=F_r\), the free group of rank r, and compute the higher homotopy groups \(\pi_k(\mathcal{X}_{F_r}(G))\), for \(0\leq k \leq 4\), extending previous results [\textit{C. Florentino} et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 17, No. 1, 143--185 (2017; Zbl 1403.14011)]. They also prove that
\[
\pi_k(\mathcal{X}_{F_r}(G))\cong \pi_k(G)^r \times \pi_{k-1}(PG),
\]
in a certain range, where \(PG\) is the quotient of \(G\) by its center \(Z(G)\), for both classical \(\left( \text{type } A_n,B_n,C_n,D_n \right)\) and exceptional groups \(G\) \(\left(G_2,F_4, E_6,E_7,E_8 \right)\).
Results are based on a detailed analysis of the singular locus of the character variety, which leads to the study of bad and ugly representations, which are irreducible representations whose \(G\)-stabilizer is larger than \(Z(G)\) and representations that produce topological singularities of \(\mathcal{X}_{F_r}(G)\), respectively. Proofs rely on previous results of Richardson on the algebraic singularities of \(G^r\) [\textit{R. W. Richardson}, Duke Math. J. 57, No. 1, 1--35 (1988; Zbl 0685.20035)] and codimension bounds of the singular locus, built upon results of \textit{C. Guérin} [J. Group Theory 21, No. 5, 789--816 (2018; Zbl 1437.20027); Geom. Dedicata 195, 23--55 (2018; Zbl 1418.20005)]. Their analysis leads to the study of Borel-de Siebenthal subalgebras of the Lie algebra \(\mathfrak{g}\) of \(G\), which are also classified in the article and used to describe bad representations.
Reviewer: Javier Martínez-Martínez (Madrid)On the splitting principle of Beniamino Segrehttps://zbmath.org/1528.140142024-03-13T18:33:02.981707Z"Felisetti, Camilla"https://zbmath.org/authors/?q=ai:felisetti.camilla"Fontanari, Claudio"https://zbmath.org/authors/?q=ai:fontanari.claudioIn the paper under review, the authors prove a stronger version of the classical principle of connectedness saying that if the general member \(X_t\) of a flat family \(X \to T\) over an irreducible curve of finite type is connected, then \(X_t\) is connected for all \(t \in T\).
The authors present with the language of modern algebraic geometry a result about the irreducibility of fibers of a flat family of nodal curves lying on a smooth surface. The result was firstly claimed by \textit{B. Segre} [Ann. Mat. Pura Appl. (4) 17, 107--126 (1938; Zbl 0019.07901; JFM 64.0688.02)].
Reviewer: Paolo Lella (Trento)Moduli spaces of modules over even Clifford algebras and Prym varietieshttps://zbmath.org/1528.140152024-03-13T18:33:02.981707Z"Lee, Jia Choon"https://zbmath.org/authors/?q=ai:lee.jia-choonSummary: A conic fibration has an associated sheaf of even Clifford algebras on the base. In this paper, we study the relation between the moduli spaces of modules over the sheaf of even Clifford algebras and the Prym variety associated to the conic fibration. In particular, we construct a rational map from the moduli space of modules over the sheaf of even Clifford algebras to the special subvarieties in the Prym variety, and check that the rational map is birational in some cases. As an application, we get an explicit correspondence between instanton bundles of minimal charge on cubic threefolds and twisted Higgs bundles on curves.Generators for the cohomology ring of the moduli of 1-dimensional sheaves on \(\mathbb{P}^2\)https://zbmath.org/1528.140162024-03-13T18:33:02.981707Z"Pi, Weite"https://zbmath.org/authors/?q=ai:pi.weite"Shen, Junliang"https://zbmath.org/authors/?q=ai:shen.junliangSummary: We explore the structure of the cohomology ring of the moduli space of stable 1-dimensional sheaves on \(\mathbb{P}^2\) of any degree. We obtain a minimal set of tautological generators, which implies an optimal generation result for both the cohomology and the Chow ring of the moduli space. Our approach is through a geometric study of tautological relations.Moduli spaces of vector bundles on a curve and opershttps://zbmath.org/1528.140372024-03-13T18:33:02.981707Z"Biswas, Indranil"https://zbmath.org/authors/?q=ai:biswas.indranil"Hurtubise, Jacques"https://zbmath.org/authors/?q=ai:hurtubise.jacques-c"Roubtsov, Vladimir"https://zbmath.org/authors/?q=ai:roubtsov.vladimirLet \(X\) be a compact connected Riemann surface of genus \(g \ge 2\). Let \({\mathcal M}_X (r)\) be the moduli space of stable vector bundles of rank \(r \ge 1\) and degree zero over \(X\). Fix a theta characteristic on \(X\); that is, a line bundle \(\mathbb{L} \to X\) such that \(\mathbb{L}^{\otimes 2}\) is isomorphic to the canonical bundle \(K_X\).
For any \(n \ge 1\), the authors show (in several steps) how to associate naturally to each \(E \in {\mathcal M}_X (r)\) a connection \({\mathcal D} (E)\) on the bundle \(J^{n-1} \left( \mathbb{L}^{\otimes (1-n)} \right)\) of jets of order \(n-1\) of \(\mathbb{L}^{\otimes (1 - n)}\). As \(\mathbb{L}^{\otimes 2} \cong K_X\), one has \(\det J^{n-1} \left( \mathbb{L}^{\otimes (1-n)} \right) \cong {\mathcal O}_X\). The authors show that \({\mathcal D} (E)\) induces the trivial connection on \(\det J^{n-1} \left( \mathbb{L}^{\otimes (1-n)} \right)\). Then by \textit{A. Beilinson} and \textit{V. Drinfeld} [``Opers'', Preprint, \url{arXiv:math/0501398}], the connection \({\mathcal D} (E)\) defines an \(\mathrm{SL}(n)\)-oper over \(X\).
The above association defines an algebraic morphism \(\tilde{\Psi} \colon {\mathcal M}_X ( r ) \to \mathrm{Op}_X ( n )\), where \(\mathrm{Op}_X (n)\) is the moduli space of \(\mathrm{SL}(n)\)-opers on \(X\) (see [loc. cit.]). The authors observe moreover that \(\tilde{\Psi}\) factorises via the involution \({\mathcal I}\) of \({\mathcal M}_X ( r )\) given by \(F \mapsto F^*\).
Now let \(\xi \to X\) be a line bundle such that \(\xi^{\otimes 2}\) is trivial. The involution \({\mathcal I}\) restricts to an involution of the sublocus \({\mathcal M}_X ( r, \xi )\) of \({\mathcal M}_X ( r )\) consisting of bundles of determinant \(\xi\), and so \(\tilde{\Psi}\) gives rise to an algebraic morphism \(\Psi \colon {\mathcal M}_X ( r , \xi ) / {\mathcal I} \to \mathrm{Op}_X ( n )\).
The authors go on to observe that for \(n = r\), there is a coincidence of dimensions
\[
\dim {\mathcal M}_X ( r , \xi ) / {\mathcal I} \ = \ ( r^2 - 1 ) ( g - 1 ) \ = \ \dim \mathrm{Op}_X ( r ) .
\]
They conclude by posing the question of how close \(\Psi\) is to being injective or surjective.
Reviewer: George H. Hitching (Oslo)On the \(P = W\) conjecture for \(\mathrm{SL}_n\)https://zbmath.org/1528.140382024-03-13T18:33:02.981707Z"de Cataldo, Mark Andrea"https://zbmath.org/authors/?q=ai:de-cataldo.mark-andrea-a"Maulik, Davesh"https://zbmath.org/authors/?q=ai:maulik.davesh"Shen, Junliang"https://zbmath.org/authors/?q=ai:shen.junliangFor a smooth projective complex curve \(C\), non abelian Hodge theory provides a real analytic isomorphism
\[
\mathcal{M}^{Dol}_{n,d}(\mathrm{GL}(n,\mathbb{C}))\cong \mathcal{M}^{B}_{n,d}(\mathrm{GL}(n,\mathbb{C}))
\]
where
\begin{itemize}
\item \(\mathcal{M}^{Dol}_{n,d}(\mathrm{GL}(n,\mathbb{C})\) is the Dolbeault moduli spaces of \(\mathrm{GL}(n,\mathbb{C})\) \textit{Higgs bundles}, namely pairs \((E,\phi)\) where \(E\) is a holomorphic vector bundle on \(C\) of rank \(n\) and degree \(d\) and \(\phi\in H^0(End(E)\otimes K_C)\);
\item \(\mathcal{M}^{B}_{n,d}(\mathrm{GL}(n,\mathbb{C}))\) is the character variety of \(d\)-twisted representations of \(\pi_1(C)\) into \(\mathrm{GL}(n,\mathbb{C})\).
\end{itemize}
These moduli spaces are complex algebraic varieties and are smooth when \(n\) and \(d\) are coprime; however since the above isomoprhism is highly trascendental it does not preserve the complex structures on these moduli spaces, nor the hodge theoretic properties on their cohomology groups.
Suppose from now on that \(n\) and \(d\) are coprime, so that the moduli spaces are nonsingular. The \(P=W\) predicts that, under the natural isomorphism
\[
H^*(\mathcal{M}^{Dol}_{n,d}(\mathrm{GL}(n,\mathbb{C})),\mathbb{Q})\cong H^*(\mathcal{M}^{B}_{n,d}(\mathrm{GL}(n,\mathbb{C}),\mathbb{Q})
\]
the weight filtration arising from the mixed Hodge structure on \(H^*(\mathcal{M}^{B}_{n,d}(\mathrm{GL}(n,\mathbb{C})),\mathbb{Q})\) corresponds to another filtration, i.e. the \textit{Perverse filtration} on \(H^*(\mathcal{M}^{Dol}_{n,d}(\mathrm{GL}(n,\mathbb{C})),\mathbb{Q})\). This latter filtration is constructed starting by a proper map \(h:\mathcal{M}^{Dol}_{n,d}(\mathrm{GL}(n,\mathbb{C}))\rightarrow \bigoplus_i H^0(C,K_C^{\otimes i}) \), called the Hitchin fibration, which sends any pair \((E,\phi)\) into the characteristic polynomial of \(\phi\).
One can consider an analogous picture for any reductive group \(G\), in particular the cases \(G=\mathrm{SL}(n,\mathbb{C}), \mathrm{PGL}(n,\mathbb{C})\). It is well known that the \(P=W\) conjecture for \(\mathrm{PGL}(n,\mathbb{C})\) is equivalent to that for \(\mathrm{GL}(n,\mathbb{C})\) and that the \(\mathrm{SL}(n,\mathbb{C})\) case implies \(\mathrm{GL}(n,\mathbb{C})\).
The present paper deals with the \(P=W\) conjecture for \(\mathrm{SL}(n,\mathbb{C})\). The main result asserts that for \(n=p\) prime then the \(P=W\) conjecture for \(\mathrm{GL}(p,\mathbb{C})\) is equivalent to that for \(\mathrm{SL}(p,\mathbb{C})\). The proof relies on the computation of the perverse filtration and the weight filtration for the variant cohomology associated with the \(\mathrm{SL}(p,\mathbb{C})\) Dolbeault moduli space and the \(\mathrm{SL}(p,\mathbb{C})\)-twisted character variety, relying on Gröchenig-Wyss-Ziegler's recent proof of the topological mirror conjecture by Hausel-Thaddeus.
Reviewer: Camilla Felisetti (Modena)Compact moduli of elliptic K3 surfaceshttps://zbmath.org/1528.140412024-03-13T18:33:02.981707Z"Ascher, Kenneth"https://zbmath.org/authors/?q=ai:ascher.kenneth"Bejleri, Dori"https://zbmath.org/authors/?q=ai:bejleri.doriThe authors study compactifications of moduli spaces of elliptic \(K3\) surfaces. These moduli spaces are constructed via the framework of KSBA stable pairs, using the earlier work [\textit{K. Ascher} and \textit{D. Bejleri}, Proc. Lond. Math. Soc. (3) 122, No. 5, 617--677 (2021; Zbl 1464.14037)], where they constructed compactifications of moduli spaces of log canonical models of Weierstrass elliptic surface pairs.
There are other compactifications of moduli spaces of \(K3\) surfaces (e.g. GIT or the Baily-Borel compactification) but, unlike the spaces constructed via the KSBA framework, such compactifications lack an intrinsic modular interpretation at the boundary. The authors construct explicit maps between their KSBA spaces and GIT and Satake-Baily-Borel (SBB) compactifications to endow their boundaries with a modular meaning.
The main results of the paper are as follows. If \(C\) is a smooth curve and \(f:X \rightarrow C\) is an elliptic \(K3\) surface with a section \(S\), the authors define a KSBA compactification \(\overline{\mathcal{W}}(\mathcal{A})\) of the moduli space of pairs \((f:X \rightarrow C, S+F_{\mathcal{A}})\), where
\[
F_{\mathcal{A}} = \sum a_i F_i
\]
is a \(\mathbb{Q}\)-divisor defined for \(a_i \in \mathbb{Q} \cap [0,1]^{24}\), where \(\mathcal{A} = (a_1, \ldots, a_{24})\) and \(F_i\) are the 24 nodal fibres of \(f\). For \(\mathcal{A} = (a, \ldots, a)\), they let \(\overline{\mathcal{W}}(a)\) denote \(\overline{\mathcal{W}}(\mathcal{A})\) and define \(\overline{\mathcal{W}}_{\sigma}(a)\) as the quotient of \(\overline{\mathcal{W}}(a)\) by the symmetric group \(S_{24}\) acting on the fibres \(F_i\).
If \(\mathcal{W}\) is the moduli space of elliptic \(K3\) surfaces with a section, constructed as a locally symmetric variety with coarse space \(W\) and \(\overline{W}^*\) is the Baily-Borel compactification of \(W\), the authors prove that:
``(Theorems 6.13, 6.15 and 6.14 and Figure 1) The proper Deligne-Mumford stacks \(\overline{\mathcal{W}}(a)\) for \(a \in \mathbb{Q} \cap [0,1]\) give modular compactifications of \(\mathcal{W}\). There is an explicit classification of the broken elliptic \(K3\) surfaces parametrized by \(\overline{\mathcal{W}}_{\sigma}(\epsilon)\), and an explicit morphism from the coarse space \(\overline{W}_{\sigma}(\epsilon)\) to \(\overline{W}^*\), the SBB compactification of \(W\). Furthermore, the surfaces parametrized by \(\overline{\mathcal{W}}_{\sigma}(\epsilon)\) satisfy \(H^1(X, \mathcal{O}_X)=0\) and \(\omega_X \cong \mathcal{O}_X\).''
They also consider the space \(\overline{\mathcal{K}}_{\epsilon}\), obtained as a compactification of the moduli space of pairs \((f:X \rightarrow \mathbb{P}^1, S + \epsilon F)\) where \(F\) is a single nodal fibre, and the associated coarse moduli space \(\overline{K}_{\epsilon}\). They prove that:
``(Theorems 8.1 and 8.2, and Figure 1) The compact moduli space \(\overline{\mathcal{K}}_{\epsilon}\) parametrizes irreducible semi-log canonical Weierstrass elliptic \(K3\) surfaces satisfying \(H^1(X, \mathcal{O}_X) = 0 \) and \(\omega_X \cong \mathcal{O}_X\). Moreover, there is an explicit generically finite morphism from the coarse space \(\overline{K}_{\epsilon}\) to \(\overline{W}^*\).''
The final moduli space they consider is \(\overline{\mathcal{F}}_{\epsilon}\), defined as for \(\overline{\mathcal{K}}(\epsilon)\), but where \(F\) is any fibre of \(f\) (i.e. not necessarily singular) with weight \(\epsilon \ll 1\). They prove that:
``(Theorem 8.8 and Figure 1) There exists a smooth proper Deligne-Mumford stack \(\overline{\mathcal{F}}_{\epsilon}\) parametrizing semi-log canonical elliptic \(K3\) surfaces with a single marked fiber. Its coarse space is isomorphic to an explicit GIT quotient \(\widetilde{W}^G\) of Weierstrass \(K3\) surfaces and a chosen fiber. Furthermore, the surfaces parametrized by \(\overline{\mathcal{F}}_{\epsilon}\) satisfy \(H^1(X, \mathcal{O}_X) = 0 \) and \(\omega_X \cong \mathcal{O}_X\).''
Much of the paper is dedicated to the results of \S5 and \S6, in which a detailed analysis of the surfaces parametrized by \(\overline{\mathcal{W}}(a)\) is performed as \(a\) varies. This is via an explicit description of wall crossings using MMP techniques. In \S7, the authors enumerate the boundary strata of \(\overline{\mathcal{W}}\), in the spirit of Kulikov models.
Stable pairs compactification of the space of elliptic \(K3\) surfaces were also constructed by Brunyate using Kulikov models: these results offer a new approach.
Reviewer: Matthew Dawes (Cardiff)Slope inequalities for KSB-stable and K-stable familieshttps://zbmath.org/1528.140422024-03-13T18:33:02.981707Z"Codogni, Giulio"https://zbmath.org/authors/?q=ai:codogni.giulio"Tasin, Luca"https://zbmath.org/authors/?q=ai:tasin.luca"Viviani, Filippo"https://zbmath.org/authors/?q=ai:viviani.filippoThe paper is devoted to generalize the classical slope inequality for fibrations of surfaces to curves due independently to Xiao Gang and Cornalba-Harris.
The result is a series of inequalities for higher dimensional fibrations to curves, under different assumptions. Roughly speaking they organize their inequalities in three groups, one for families of KSB-stable (canonically polarized) pairs, one for families of K-stable (log Fano) pairs, and one for arbitrarily polarized families.
Reviewer: Roberto Pignatelli (Trento)Isotriviality of smooth families of varieties of general typehttps://zbmath.org/1528.140432024-03-13T18:33:02.981707Z"Wei, Chuanhao"https://zbmath.org/authors/?q=ai:wei.chuanhao"Wu, Lei"https://zbmath.org/authors/?q=ai:wu.leiSummary: In this paper, we proved that a log smooth family of log general type klt pairs with a special (in the sense of Campana) quasi-projective base is isotrivial. As a consequence, we proved the generalized Kebekus-Kovács conjecture [the authors, Int. Math. Res. Not. 2023, No. 1, 708--743 (2023; Zbl 1511.14020)], Conjecture 1.1, for smooth families of general type varieties as well as log smooth families of log canonical pairs of log general type, assuming the existence of relative good minimal models.Nearby cycle sheaves for symmetric pairshttps://zbmath.org/1528.220112024-03-13T18:33:02.981707Z"Grinberg, Mikhail"https://zbmath.org/authors/?q=ai:grinberg.mikhail"Vilonen, Kari"https://zbmath.org/authors/?q=ai:vilonen.kari"Xue, Ting"https://zbmath.org/authors/?q=ai:xue.tingThe authors present a nearby cycle sheaf construction in the development of the theory of character sheaves in the context of symmetric spaces. They allow equivariant local systems as coefficients, and this construction, and its variant, produce all character sheaves up to parabolic induction in the setting of classical symmetric spaces.
They consider a connected complex reductive group \(G\) and an involution \(\theta: G \to G\), giving rise to a symmetric pair \((G, K)\), with \(K = G^\theta\). They have a decomposition \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\) into \(+1\) and \(-1\) eigenspaces of the involution \(\theta\). Writing \(\mathcal{N}\) for the nilpotent cone in \(\mathfrak{g}\), let \(\mathcal{N}_{\mathfrak{p}} = \mathcal{N} \cap \mathfrak{p}\). Let \(\mathfrak{a} \subset \mathfrak{p}\) be a Cartan subspace, which is a maximal abelian subspace of \(\mathfrak{p}\) consisting of semisimple elements, and let \(W_{\mathfrak{a}} = N_K (\mathfrak{a}) / Z_K (\mathfrak{a})\) be the little Weyl group. Write \(\mathfrak{p}^{\text{rs}}\) for the set of regular semisimple elements of \(\mathfrak{p}\) and let \(\mathfrak{a}^{\text{rs}} = \mathfrak{a} \cap \mathfrak{p}^{\text{rs}}\). This gives the adjoint quotient map: \(f: \mathfrak{p} \to \mathfrak{p} /\!\!/ K \cong \mathfrak{a} / W_{\mathfrak{a}}\). Note that \(f^{-1} (0) = \mathcal{N}_{\mathfrak{p}}\). The fiber of this map at a point \(\bar{a}_0 \in \mathfrak{a}^{\text{rs}} / W_{\mathfrak{a}}\) is a regular semisimple \(K\)-orbit \(X_{\bar{a}_0}\), with equivariant fundamental group \(\pi_1^K (X_{\bar{a}_0}) = I := Z_K (\mathfrak{a}) / Z_K (\mathfrak{a})^0\).
The authors construct a nearby cycle sheaf \(P_\chi \in \text{Perv}_K (\mathcal{N}_{\mathfrak{p}})\), a \(K\)-equivariant perverse sheaf on the nilpotent cone \(\mathcal{N}_{\mathfrak{p}}\), for each character \(\chi \in \hat{I}\). They study the topological Fourier transform of \(P_\chi\), and show that this Fourier transform is an IC-extension of a \(K\)-equivariant local system on \(\mathfrak{p}^{\text{rs}}\) under a suitable identification of \(\mathfrak{p}\) and \(\mathfrak{p}^*\). The local systems on \(\mathfrak{p}^{\text{rs}}\) arising in this way are described in Theorem 3.6 (page 21), where Hecke algebras with parameters \(\pm 1\), attached to certain Coxeter subgroups of \(W_\mathfrak{a}\), enter the description. The construction of these Hecke algebras can be viewed as an instance of endoscopy. This nearby cycle sheaf construction can be regarded as a replacement for the Grothendieck--Springer resolution in classical Springer theory.
Reviewer: Mee Seong Im (Annapolis)The birational invariants of Lins Neto's foliationshttps://zbmath.org/1528.320482024-03-13T18:33:02.981707Z"Ling, Hao"https://zbmath.org/authors/?q=ai:ling.hao"Lu, Jun"https://zbmath.org/authors/?q=ai:lu.jun.1"Tan, Sheng-Li"https://zbmath.org/authors/?q=ai:tan.sheng-li\textit{A. Lins Neto} constructed in [Ann. Sci. Éc. Norm. Supér. (4) 35, No. 2, 231--266 (2002; Zbl 1130.34301)] families of singular holomorphic foliations in \(\mathbb{CP}^2\) which are counterexamples to two famous questions.
Poincaré's Problem: is it possible to decide if an algebraic differential equation is algebraically integrable?
Painlevé's Problem: is it possible to decide if an algebraic differential equation has a rational first integral of a given genus g?
In the present work, the authors determine the minimal models of these families of singular holomorphic foliations in \(\mathbb{CP}^2\), calculate their Chern numbers, Kodaira dimension, and numerical Kodaira dimension. The work proves that the slopes of Lins Neto's foliations are at least 6, and their limits are bigger than 7.
Reviewer: Jesus Muciño Raymundo (Morelia)Extended Goldman symplectic structure in Fock-Goncharov coordinateshttps://zbmath.org/1528.530762024-03-13T18:33:02.981707Z"Bertola, M."https://zbmath.org/authors/?q=ai:bertola.marco"Korotkin, D."https://zbmath.org/authors/?q=ai:korotkin.dmitrii-aAuthors' abstract: The goal of this paper is to express the extended Goldman symplectic structure ([\textit{W. M. Goldman}, Adv. Math. 54, 200--225 (1984; Zbl 0574.32032)]) on the SL(n) character variety of a punctured Riemann surface in terms of Fock-Goncharov coordinates([\textit{V. Fock} and \textit{A. Goncharov}, Publ. Math., Inst. Hautes Étud. 103, 1--211 (2006; Zbl 1099.14025)]). The associated symplectic form has integer coefficients expressed via the inverse of the Cartan matrix. The main technical tool is a canonical two-form associated to a flat graph connection. We discuss the relationship between the extension of the Goldman Poisson structure and the Poisson structure defined by Fock and Goncharov. We elucidate the role of the Rogers' dilogarithm as generating function of the symplectomorphism defined by a graph transformation.
We note the related work earlier than this one: [\textit{Z. Sun} et al., Geom. Funct. Anal. 30, No. 2, 588--692 (2020; Zbl 1528.32020)]. However, it is not at all clear what the exact relationships are at the moment.
Reviewer: Suhyoung Choi (Daejeon)On character varieties of singular manifoldshttps://zbmath.org/1528.570252024-03-13T18:33:02.981707Z"González-Prieto, Ángel"https://zbmath.org/authors/?q=ai:gonzalez-prieto.angel"Logares, Marina"https://zbmath.org/authors/?q=ai:logares.marinaThe representation variety of \(X\), a topological space with finitely generated fundamental group, is the variety \(\mathfrak{X}_{G}(X)\) of its representations in a reductive algebraic group \(G\). Its GIT quotient by the natural action of \(G\) is the character variety \(\mathcal{R}_{G}(X)\). A parabolic structure on \(X\) is given by a finite collection of punctures with conjugacy classes of \(G\), called holonomies, assigned to them, and the parabolic character variety is defined analogously.
The paper aims to study the class of \(\mathcal{R}_{G}(X)\) in the Grothendieck ring of algebraic varieties, the so-called virtual class, for a generalization of topological manifolds that the authors termed nodefolds. Those are locally modeled on cones over closed manifolds and so can have nodal singularities. They come up in connection with mirror symmetry and computation of their invariants is complicated by the breakdown of the non-Adelian Hodge correspondence with the moduli of Higgs bundles because the standard arithmetic techniques cannot be used. The virtual class is the most general invariant preserving disjoint unions and products.
The first author initiated a TQFT approach to computing virtual classes of representation varieties over closed manifolds and without holonomies. In this paper, the approach is extended to nodefolds and parabolic structures. Specifically, the authors construct a lax monoidal symmetric functor from the category of nodefold bordisms (equipped with a parabolic structure, if any) with a finite number of base points to the category of modules over the ground ring. This functor turns out to be, naturally, a functor of \(2\)-categories. In particular, it allows them to relate virtual classes of representation varieties of 2D nodefolds to those of smooth surfaces via conic degenerations that collapse several points into nodes.
When, moreover, \(G=SL_r(k)\), the results are extended from representation varieties to character varieties using the first author's theory of pseudo-quotients. Those are weaker than GIT quotients but have the same virtual class and are compatible with convenient stratifications of \(\mathfrak{X}_{G}(X)\) with simpler action of \(G\) on the strata. The virtual class of \(\mathcal{R}_{G}(X)\) can then be assembled from those of the strata. In this case, the authors decompose \(\mathfrak{X}_{G}(X)\) into the loci of reducible and irreducible representations. On the latter, \(\mathfrak{X}_{G}(X)\) reduces to the orbit space, and on the former it is equivalent to a product of lower rank representations with \(G\) permuting those of the same rank. For \(G=SL_2(k)\) the closures of reducible orbits contain diagonal representations, so the authors' calculations lead to explicit formulas for the virtual class of \(\mathcal{R}_{SL_2(k)}(X)\). The results are extended to parabolic structures with holonomies of Jordan and semi-simple type.
Reviewer: Sergiy Koshkin (Houston)Schrödinger equation driven by the square of a Gaussian field: instanton analysis in the large amplification limithttps://zbmath.org/1528.811312024-03-13T18:33:02.981707Z"Mounaix, Philippe"https://zbmath.org/authors/?q=ai:mounaix.philippeSummary: We study the tail of \(p(U)\), the probability distribution of \(U = |\psi(0, L)|^2\), for \(\ln U \gg 1\), \(\psi(x, z)\) being the solution to \(\partial_z\psi - \frac{i}{2m}\nabla^2_\perp\psi = g|S|^2\psi\), where \(S(x, z)\) is a complex Gaussian random field, \(z\) and \(x\) respectively are the axial and transverse coordinates, with \(0 \leqslant z \leqslant L\), and both \(m \neq 0\) and \(g > 0\) are real parameters. We perform the first instanton analysis of the corresponding Martin-Siggia-Rose action, from which it is found that the realizations of \(S\) concentrate onto long filamentary instantons, as \(\ln U \to +\infty\). The tail of \(p(U)\) is deduced from the statistics of the instantons. The value of \(g\) above which \(\langle U \rangle\) diverges coincides with the one obtained by the completely different approach developed in [\textit{P. Mounaix} et al., Commun. Math. Phys. 264, No. 3, 741--758 (2006; Zbl 1111.35310)]. Numerical simulations clearly show a statistical bias of \(S\) towards the instanton for the largest sampled values of \(\ln U\). The high maxima -- or `hot spots' -- of \(|S(x, z)|^2\) for the biased realizations of \(S\) tend to cluster in the instanton region.Quantum sheaf cohomology on Grassmannianshttps://zbmath.org/1528.811582024-03-13T18:33:02.981707Z"Guo, Jirui"https://zbmath.org/authors/?q=ai:guo.jirui"Lu, Zhentao"https://zbmath.org/authors/?q=ai:lu.zhentao"Sharpe, Eric"https://zbmath.org/authors/?q=ai:sharpe.eric-rIn this article under reviewed, the authors describes and investigates quantum sheaf cohomolgy on Grassmanians with deformations of the tangent bundle. A ring structure is used to derive the main results. Quantum cohomology is an essential part in algebraic geometry and string theory. In particular, quantum sheaf cohomology is a generalization quantum cohomology. Many authors studied in details quantum sheaf cohomolgy on toric varieties. There are many stages to tackle and to deal with quantum sheaf cohomolgy. The first author of this article under reviewed \textit{J. Guo} has published an interesting and similar article in the title: [Commun. Math. Phys. 374, No. 2, 661--688 (2020; Zbl 1435.81130)]
A general description of the ring structure has been used and found from both physical prospective and mathematical prospectives. In fact, as was indicates in the article mentioned above, that one can study quantum sheaf cohomology by representing the theory with generators and relations.
The article is well written. It gives crucial and vital methods to focus on physics derivations with excellent examples. These examples describe the correlation functions and quantum cohomollogy in certain specific situation. For the purpose to explain such examples, the authors consider only some special cases.
The article contains interesting sections. Without going in the technical details, this article under reviewed introduces a novel study and a significant approach to quantum sheaf cohomology on Grassmannians. In this article, the introduction provides sufficient background. Such introduction includes the relevant references. There are very good references in the end of the article. The methods of the article are clearly and adequately described. The research design is excellent. In fact, theory of Grassmannians is presented clearly. Non-abelain case is recalled in a good way. Then an interesting section on ring structures of quantum sheaf cohomolgy is presented clearly. This section of ring structure approach contains three subsection. The first is about Gauge invariant operators. Second one is about quantum sheaf cohomology ring with specialization to ordinary classical cohomolgy as well as specialization to quantum cohomology. Then a very excellent section for examples. The section of conclusion is presented. It support the results of the paper. In fact, this theory can be investigated more in the future either by physical approach or mathematical approach.
Reviewer: Ahmad Alghamdi (Makkah)Comments on contact terms and conformal manifolds in the AdS/CFT correspondencehttps://zbmath.org/1528.811992024-03-13T18:33:02.981707Z"Sakai, Tadakatsu"https://zbmath.org/authors/?q=ai:sakai.tadakatsu"Zenkai, Masashi"https://zbmath.org/authors/?q=ai:zenkai.masashiSummary: We study the contact terms that appear in the correlation functions of exactly marginal operators using the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. It is known that CFT with an exactly marginal deformation requires the existence of the contact terms with their coefficients having a geometrical interpretation in the context of conformal manifolds. We show that the AdS/CFT correspondence captures properly the mathematical structure of the correlation functions that is expected from the CFT analysis. For this purpose, we employ a holographic renormalization group to formulate a most general setup in the bulk for describing an exactly marginal deformation. The resultant bulk equations of motion are nonlinear and solved perturbatively to obtain the on-shell action. We compute three- and four-point functions of the exactly marginal operators using the GKP-Witten prescription, and show that they match the expected results precisely. The cut-off surface prescription in the bulk serves as a regularization scheme for conformal perturbation theory in the boundary CFT. As an application, we examine a double OPE limit of the four-point functions. The anomalous dimensions of double trace operators are written in terms of the geometrical data of a conformal manifold.TT deformations in general dimensionshttps://zbmath.org/1528.830042024-03-13T18:33:02.981707Z"Taylor, Marika"https://zbmath.org/authors/?q=ai:taylor.marikaSummary: It has recently been proposed that Zamoldchikov's \(T\bar{T}\) deformation of two-dimensional CFTs describes the holographic theory dual to \(\mathrm{AdS}_3\) at finite radius. In this note we use the Gauss-Codazzi form of the Einstein equations to derive a relationship in general dimensions between the trace of the quasi-local stress tensor and a specific quadratic combination of this stress tensor, on constant radius slices of AdS. We use this relation to propose a generalization of Zamoldchikov's \(T\bar{T}\) deformation to conformal field theories in general dimensions. This operator is quadratic in the stress tensor and retains many but not all of the features of \(T\bar{T}\). To describe gravity with gauge or scalar fields, the deforming operator needs to be modified to include appropriate terms involving the corresponding R currents and scalar operators and we can again use the Gauss-Codazzi form of the Einstein equations to deduce the forms of the deforming operators. We conclude by discussing the relation of the quadratic stress tensor deformation to the stress energy tensor trace constraint in holographic theories dual to vacuum Einstein gravity.Dominant energy condition and dissipative fluids in general relativityhttps://zbmath.org/1528.830422024-03-13T18:33:02.981707Z"Faraoni, Valerio"https://zbmath.org/authors/?q=ai:faraoni.valerio"Mokkedem, El Mokhtar Z. R."https://zbmath.org/authors/?q=ai:mokkedem.el-mokhtar-z-rSummary: Existing literature implements the Dominant Energy Condition for dissipative fluids in general relativity. It is pointed out that this condition fails to forbid superluminal flows, which is what it is ultimately supposed to do. Tilted perfect fluids, which formally have the stress-energy tensor of imperfect fluids, are discussed for comparison.Quantum modified gravity at low energy in the Ricci flow of quantum spacetimehttps://zbmath.org/1528.830512024-03-13T18:33:02.981707Z"Luo, M. J."https://zbmath.org/authors/?q=ai:luo.ming-jian|luo.meijin|luo.minjieSummary: Quantum treatment of physical reference frame leads to the Ricci flow of quantum spacetime, which is a quite rigid framework to quantum and renormalization effect of gravity. The theory has a low characteristic energy scale described by a unique constant: the critical density of the universe. At low energy long distance (cosmic or galactic) scale, the theory modifies Einstein's gravity which naturally gives rise to a cosmological constant as a counter term of the Ricci flow at leading order and an effective scale dependent Einstein-Hilbert action. In the weak and static gravity limit, the framework gives rise to a transition trend away from Newtonian gravity and similar to the MOdified Newtonian Dynamics (MOND) around the characteristic scale. When local curvature is large, Newtonian gravity is recovered. When local curvature is low enough to be comparable with the asymptotic background curvature corresponding to the characteristic energy scale, the transition trend produces the baryonic Tully-Fisher relation. For intermediate general curvature around the background curvature, the interpolating Lagrangian function yields a similar transition trend to the observed radial acceleration relation of galaxies. When the baryonic matter density is much lower than the critical density at the outskirt of a galaxy, there may be a universal ``acceleration floor'' corresponding to the acceleration expansion of the universe, which differs from MOND at its deep-MOND limit. The critical acceleration constant \(a_0\) introduced in MOND is related to the low characteristic energy scale of the theory. The cosmological constant gives a universal leading order contribution to \(a_0\) and the flow effect gives the next order scale dependent contribution, which equivalently induces the ``cold dark matter'' to the theory. \(a_0\) is consistent with galaxian data when the ``dark matter'' is about 5 times the baryonic matter.Stellar features of strange dark energy starshttps://zbmath.org/1528.830582024-03-13T18:33:02.981707Z"Salti, M."https://zbmath.org/authors/?q=ai:salti.mustafa|salti.mehmet"Aydogdu, O."https://zbmath.org/authors/?q=ai:aydogdu.oktay|aydogdu.omerSummary: Dark energy stars are theoretical objects that could potentially provide an explanation for the observable properties of black holes, including their event horizons and emission of the Hawking radiation. Because of this, they are considered to be an important area of research in modern theoretical physics and could help to answer fundamental questions about the Universe and the nature of dark energy. So, identifying and characterizing dark energy stars are a theoretically active and significant field in literature. In this context, for a theoretical model describing a new static symmetric singularity-free anisotropic dark energy star in view of the Rastall theory of gravity, which is an alternative to the general theory of relativity and is consistent with a wide range of astronomical observations, we focus our awareness in the present study on the modified field equations as well as the hydrostatic equilibrium expression and then discuss the corresponding solutions. It is shown that this new-type dark energy star meets all the necessary astronomical requirements, including stability and feasibility can potentially represent a compact cosmic structure.Gauge/frame invariant variables for the numerical relativity study of cosmological spacetimeshttps://zbmath.org/1528.831432024-03-13T18:33:02.981707Z"Ijjas, Anna"https://zbmath.org/authors/?q=ai:ijjas.anna(no abstract)A viable varying speed of light model in the RW metrichttps://zbmath.org/1528.831462024-03-13T18:33:02.981707Z"Lee, Seokcheon"https://zbmath.org/authors/?q=ai:lee.seokcheonSummary: The Robertson-Walker (RW) metric allows us to apply general relativity to model the behavior of the Universe as a whole (i.e., cosmology). We can properly interpret various cosmological observations, like the cosmological redshift, the Hubble parameter, geometrical distances, and so on, if we identify fundamental observers with individual galaxies. That is to say that the interpretation of observations of modern cosmology relies on the RW metric. The RW model satisfies the cosmological principle in which the 3-space always remains isotropic and homogeneous. One can derive the cosmological redshift relation from this condition. We show that it is still possible for us to obtain consistent results in a specific time-varying speed-of-light model without spoiling the success of the standard model. The validity of this model needs to be determined by observations.Codazzi tensors and their space-times and cotton gravityhttps://zbmath.org/1528.831482024-03-13T18:33:02.981707Z"Mantica, Carlo Alberto"https://zbmath.org/authors/?q=ai:mantica.carlo-alberto"Molinari, Luca Guido"https://zbmath.org/authors/?q=ai:molinari.luca-guidoSummary: We study the geometric properties of certain Codazzi tensors for their own sake, and for their appearance in the recent theory of Cotton gravity. We prove that a perfect-fluid tensor is Codazzi if and only if the metric is a generalized Stephani universe. A trace condition restricts it to a warped space-time, as proven by Merton and Derdziński. We also give necessary and sufficient conditions for a space-time to host a current-flow Codazzi tensor. In particular, we study the static and spherically symmetric cases, which include the Nariai and Bertotti-Robinson metrics. The latter are a special case of Yang Pure space-times, together with spatially flat FRW space-times with constant curvature scalar. We apply these results to the recent Cotton gravity by Harada. We show that the equation of Cotton gravity is Einstein's equation modified by the presence of a Codazzi tensor, which can be chosen freely and constrains the space-time where the theory is staged. In doing so, the tensor (chosen in forms appropriate for physics) implies the form of the Ricci tensor. The two tensors specify the energy-momentum tensor, which is the source in the equation of Cotton gravity for the metric implied by the Codazzi tensor. For example, we show that the Stephani, Nariai and Bertotti-Robinson space-times are characterized by a ``current flow'' Codazzi tensor. Because of it, they solve Cotton gravity with physically sensible energy-momentum tensors. Finally, we discuss Cotton gravity in constant curvature space-times.