Quadratic differentials as stability conditions
The paper under this review is motivated by the physics papers of Gaiotto, Moore and Neitzke [\textit{D. Gaiotto} et al., Commun. Math. Phys. 299, No. 1, 163--224 (2010; Zbl 1225.81135); Adv. Math. 234, 239--403 (2013; Zbl 1358.81150)] in the references of this paper). Let us start this review with a rather long and rough explanation of what the physicists studies, since it is helpful to understand the setting and objects Bridgeland and Smith consider. The main object of the work of Gaiotto, Moore and Neitzke is the determination of BPS spectrum of the so-called ``class S theories'', which are certain 4-dimensional \(N=2\) supersymmetric gauge theories. The class S theories originate from compactification of M5-branes in the 6-dimensional theory on a punctured Riemann surface \(C\), and the theories are labelled by the punctured surface \(C\) and by the gauge group associated to the superconformal algebras. These theories are also related to the Hitchin system on \(C\) by the \(S^1\)- and \(C\)-compactification of the original 6-dimensional theory. Simply speaking, a Hitchin system on \(C\) encodes the Seiberg-Witten theory, which is the low energy effective theory of the 4-dimensional theory we started with. The Seiberg-Witten curve \(\Sigma\) is a branched cover of \(C\), and can be considered as \(\Sigma \subset T^* C\), and the canonical 1-form on \(T^* C\) recovers the Seiberg-Witten differential \(\lambda\) when restricted to \(\Sigma\). The paper [Zbl 1358.81150] described the BPS spectrum (in some examples of class S theories) in terms of Hitchin systems. If the gauge group labelling the theory is \(\mathrm{U}(2)\), then BPS states are associated to the curves on the surface \(C\) that minimize the length in terms of the metric induced by the coordinate \(w = \int_{z_0}^z \lambda\). BPS hypermultiplets are associated with curves that begin and end on branch points of \(\Sigma \to C\), and BPS vector-multiplets are associated with curves which are closed. The central charges of these BPS states are given by \(Z_\gamma = \oint \lambda\). In particular, the phase of the central charge is the angle between the straight line in the \(w\)-plane and the horizontal line \(\Im(w)=0\). The overall BPS spectrum corresponds to a triangulation of the surface \(C\), which they call a WKB triangulation, and BPS spectrum enjoy a wall-crossing behavior under changing triangulations associated to the choices of the Seiberg-Witten differentials. The paper [Zbl 1225.81135], which appeared before [Zbl 1358.81150], considers the \(S^1\)-compactification of 4-dimensional theories. Its low energy effective theory is given by 3-dimensional sigma model whose target space \(M\) should have a hyper-kaehler metric. The quantum corrections of this metric should depend only on the BPS spectrum of the original 4-dimensional theory, and the metric should enjoy a wall-crossing formula as a function on the moduli of 4-dimensional vacua. One of the results of [Zbl 1225.81135] is an explicit construction of these metrics, and the other one is that the wall-crossing coincides with the Kontsevich-Soibelman wall-crossing. In [Zbl 1358.81150] Gaiotto, Moore and Neitzke expected that there should be a category of quiver representations which underlies all the story explained above. The paper under review can be considered as a precise construction of mathematical theory of their expectation, and as a first step to understand the picture of [Zbl 1225.81135, Zbl 1358.81150]. Briefly speaking, Bridgeland and Smith identified the moduli spaces of certain meromorphic quadratic differentials on Riemann surfaces with spaces of stability conditions on 3-dimensional Calabi-Yau categories arising from Ginzburg algebra of quivers with potential associated to triangulated surfaces. Let us spell out the setting. A marked bordered surface \((\mathbb S, \mathbb M)\) means a pair of a compact oriented smooth surface \(\mathbb S\) (possibly with boundary) and a set of marked points \(\mathbb M \subset \mathbb S\). The marked points \(\mathbb P \subset \mathbb M\) not lying on the boundary of \(\mathbb S\) are called punctures. A quadratic differential on \((\mathbb S, \mathbb M)\) means a pair \((S,\phi)\) of a compact connected Riemann surface \(S\) and a meromorphic quadratic differential \(\phi\) with simple zeroes and arbitrary poles, where \(|\mathbb M|\) poles are of order \( \leq 2\), and the oriented real blow-ups at each pole of \(\phi\) of order \(\geq 3\) makes the resulting surface isomorphic to \(\mathbb S\) and identifies the poles of order \(\leq 2\) with \(\mathbb M\). The equivalence classes of quadratic differentials on \((\mathbb S, \mathbb M)\) under holomorphic equivalence are parametrized by a complex orbifold \(\mathrm{Quad}(\mathbb S, \mathbb M)\) of dimension \(n := 6g-6+3p+\sum_i(k_i+3)\), where \(p := |\mathbb P|\) and \(k_i+2\) are the orders of poles of order \(\geq3\). A quadratic differential \((S, \phi)\) determines a double cover \(\pi:\widehat{S} \to S\), which is branched precisely at the zeroes and the simple poles of \(\phi\) and equipped with a meromorphic 1-form \(\psi\) satisfying \(\pi^*(\phi)=\psi^{\otimes 2}\). The hat-homology group of \((S,,\phi)\) is defined to be \[ \widehat{H}(\phi) := H_1({\widehat{S}}^\circ)^{-}, \] where \({\widehat{S}}^\circ \subset \widehat{S}\) denotes the complement of the poles of \(\psi\), and the superscript \(-\) means the anti-invariant part under the covering involution. \(\psi\) is holomorphic on \({\widehat{S}}^{\circ}\) and anti-invariant, We have a group homomorphism \[ Z_\phi: \widehat{H}(\phi) \longrightarrow \mathbb{C}, \quad\gamma \longmapsto \int_\gamma \psi. \] The hat-homology groups form a local system on a dense open subset of \(\mathrm{Quad}(\mathbb S, \mathbb M)\) consisting of differentials without simple poles. The local system extends to the one on the orbifold \[ \mathrm{Quad}H(\mathbb S,\mathbb M)=\mathrm{Quad}^{\pm}(\mathbb S, \mathbb M)/\mathbb Z_2^{\mathbb P}, \] where \(\mathrm{Quad}^{\pm}(\mathbb S, \mathbb M)\) denotes the orbifold of signed differentials, which are equipped with choices of signs of the residue of differentials at each puncture. The group \(\mathbb Z_2\) acts on each puncture, and \(\mathrm{Quad}H(\mathbb S,\mathbb M)\) is the quotient orbifold with respect to these groups. Next let us introduce a terminology on triangulations. An arc in \((\mathbb S, \mathbb M)\) is a smooth path \(\gamma\) in \((\mathbb S, \mathbb M)\) connecting points of \(\mathbb M\), whose interior lies in \(S \setminus (\mathbb M \cup \partial \mathbb S)\), which has no self-intersections in its interior and which are not homotopic to a single point, or to a path in \(\partial \mathbb S\) whose interior contains no points of \(\mathbb M\). An ideal triangulation \(T\) of \((\mathbb S, \mathbb M)\) is defined to be a maximal collection of equivalence classes of arcs under homotopy where we can find representatives whose interiors are pairwise disjoint. An equivalence class of arcs will be called an edge of \(T\). We can attach to an ideal triangulation \(T\) of \((\mathbb S, \mathbb M)\) a quiver \(Q(T)\) whose vertices are the edges of \(T\), and the number of arrows from the vertex \(e\) to \(f\) is set to be \(n(e,f) \in \mathbb Z_{\geq0}\) which is determined by a skew-symmetric form on the lattice \(\oplus_{e}\mathbb Z [e]\) (see \S8.4 for the detail). \textit{D. Labardini-Fragoso} [Proc. Lond. Math. Soc. (3) 98, No. 3, 797--839 (2009; Zbl 1241.16012)] showed that there is a potential \(W(T)\) attached to \(Q(T)\). In the case of non-degenerate ideal triangulation, where \(\partial \mathbb S = \empty\) and every vertex in \(T\) has valency at least \(3\), \(W(T)\) is given by \[ W(T) = \sum_f T(f) - \sum_p C(p), \] where \(T(f)\) is the clockwise 3-cycle attached to each face \(f\), and \(C(p)\) is the anti-clockwise cycle encircling each point \(p \in \mathbb M\). Then we can consider the derived category of the complete Ginzburg algebra of the quiver with potential \((Q(T),W(T))\), and let \(\mathcal D(T)\) be the full subcategory consisting of modules with finite-dimensional cohomology. It is a CY3 triangulated category and has a canonical \(\mathfrak{t}\)-structure, whose heart \(\mathcal A(T) \subset \mathcal D(T)\) is equivalent to the category of finite-dimensional modules for the completed Jacobi algebra of \((Q(T),W(T))\). By the work of \textit{B. Keller} and \textit{D. Yang} [Adv. Math. 226, No. 3, 2118--2168 (2011; Zbl 1272.13021)] and Labardini-Fragoso [Zbl 1241.16012], the category \(\mathcal D(T)\) is independent of the chosen triangulation, so that we may denote it by \(\mathcal D = \mathcal D(\mathbb S, \mathbb M)\). We freely use the terminology on the Bridgeland stability conditions [\textit{T. Bridgeland}, Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008); Proc. Symp. Pure Math. 80, 1--21 (2009; Zbl 1169.14303); Duke Math. J. 141, No. 2, 241--291 (2008; Zbl 1138.14022)]. Denote by \(\mathrm {Stab}(\mathcal D)\) the space of stability conditions on \(\mathcal D\), and by \(\mathrm {Stab}T(\mathcal D)\) the distinguished component containing the stability conditions whose heart is one of the form \(\mathcal A(T)\). Denote also by \({\mathrm{Aut}_{\triangle}}(\mathcal D)\) the subgroup of autoequivalences of \(\mathcal D\) which preserve \(\mathrm {Stab}T(\mathcal D)\). Finally set \({\mathrm{Aut}_{\triangle}^{\circ}}(\mathcal D)\) to be the quotient of \({\mathrm{Aut}_{\triangle}}(\mathcal D)\) by the subgroup of autoequivalences which act trivially on \(\mathrm {Stab}T(\mathcal D)\). Now we may state one of the main result of the paper (Theorem 1.2). For a marked bordered surface \((\mathbb S, \mathbb M)\) with \(g(\mathbb S)=0\) and \(|\mathbb M|\geq 9\) or \(g(\mathbb S)>0\) and \(|\mathbb M|\geq 6\), there is an isomorphism of complex orbifolds \[ \mathrm{Quad}H(\mathbb S, \mathbb M) \simeq \mathrm {Stab}T(\mathcal D)/{\mathrm{Aut}_{\triangle}^{\circ}}(\mathcal D). \] A key observation in the proof of this theorem is that a generic point of the space \(\mathrm{Quad}(\mathbb S, \mathbb M)\) determines an ideal triangulation of the surface \((\mathbb S, \mathbb M)\) well-defined up to the action of the mapping class group which goes back to the work of Gaiotto, Moore and Neitzke, as we mentioned at the first part of this review. In the course of the proof, the authors also demonstrate a correspondence between the finite-length trajectories of a quadratic differential and the stable objects of the corresponding stability condition, which is also included in the physics picture (BPS states are the physics replacement of stable objects). Let us also spell out the statement precisely. For a given \((S,\phi) \in \mathrm{Quad}(\mathbb S, \mathbb M)\), let \(\sigma \in \mathrm {Stab}(\mathcal D)\) be the corresponding stability condition under the above isomorphism. Assume that \(\sigma\) is generic in the sense of wall-chamber structure on \(\mathrm {Stab}(\mathcal D)\), and call \(\phi\) generic if the corresponding \(\sigma\) is generic. Denote by \(\mathcal M_\sigma(0)\) the moduli space of objects in \(\mathcal D\) that are stable in the stability condition \(\sigma\) and of phase \(0\). This space is isomorphic to a moduli space of stable representations of a finite-dimensional algebra, so that it is a quasi-projective scheme by the work of \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, No. 180, 515--530 (1994; Zbl 0837.16005)]. Now we may state Theorem 1.4 of the paper. If \(\phi\) is generic, then \(\mathcal M_\sigma(0)\) is smooth, and each of its connected components is either a point, or is isomorphic to the projective line \(\mathbb P^1\). Moreover, there are bijections between \(0\)-dimensional components of \(\mathcal M_\sigma(0)\) and non-closed saddle trajectories of \(\phi\), and \(1\)-dimensional components of \(\mathcal M_\sigma(0)\) and non-degenerate ring domains of \(\phi\). Let us explain the organization of the paper and comment on how-to-read. The paper is largely divided into four parts, the introduction (\S 1), quadratic differentials (\S\S2--6), stability conditions and the Ginzburg algebras (\S\S7--9) and the main proof (\S\S10--12). Since the number of the pages is over 100, the reviewer recommend the future readers to look first at the introduction and then go to \S12 where detailed examples are explained with explicit calculations. \S\S2--6 deals with the meromorphic quadratic differentials. The first half (mainly \S\S2--3) reviews basic notions and the standard reference here is \textit{K. Strebel}'s book [Quadratic differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Band 5. Berlin etc.: Springer-Verlag. (1984; Zbl 0547.30001)], although it is not necessary to read through the book since the material in the paper is almost complete for the understanding of the main result. The sections \S\S4--6 give a careful study of the moduli space of quadratic differentials. They also include several explicit proof of the may-be known propositions. The sections 7--9 are recollections of Bridgeland's stability conditions for triangulated categories, Ginzburg algebras attached to quivers with potentials and the 3-dimensional Calabi-Yau categories arising from these algebras. These sections are briefly written and the reviewer finds them very helpful, although they require some basic knowledge on the topics so that some readers might need to read references of unfamiliar topics, \S\S10-11 can be considered as the main body of this paper. \S10 describes a triangulation associated to a saddle-free differential, and the way it changes as one passes between neighbouring chambers. \S 11 contains the proofs of the main result. Let us close this long review with a short comment. Although the paper is very long, the material is presented carefully and the readers may find the text very readable. It also help us to understand the work of Gaiotto Moore and Neitzke. The reviewer believes it is worth taking much time to read the paper.
scientific article