Tautological relations and integrable systems

Inside the moduli space of curves of genus three with one marked point, we consider the locus of hyperelliptic curves with a marked Weierstrass point, and the locus of non-hyperelliptic curves with a marked hyperflex. These loci have codimension two. We compute the classes of their closures in the moduli space of stable curves of genus three with one marked point. Similarly, we compute the class of the closure of the locus of curves of genus four with an even theta characteristic vanishing with order three at a certain point. These loci naturally arise in the study of minimal strata of Abelian differentials.

It was proved recently that the correlation functions of a semisimple cohomological field theory satisfy the so-called local Eynard–Orantin topological recursion. We prove that in the settings of singularity theory, the local Eynard–Orantin recursion is equivalent to N copies of Virasoro constraints for the total ancestor potential. The latter follow easily from some known properties of the period integrals in singularity theory. Our approach generalizes easily to an arbitrary semisimple cohomological field theory, which yields a simple proof of the local Eynard–Orantin recursion for an arbitrary semisimple cohomological field theory.

We define and study the stack $\\mathcal{U}^{ns,a}_{g,g}$ of (possibly singular) projective curves of arithmetic genus $g$ with $g$ smooth marked points forming an ample nonspecial divisor. We define an explicit closed embedding of a natural $\\mathbb{G}_{m}^{g}$ -torsor $\\widetilde{\\mathcal{U}}^{ns,a}_{g,g}$ over $\\mathcal{U}^{ns,a}_{g,g}$ into an affine space, and we give explicit equations of the universal curve (away from characteristics $2$ and $3$ ). This construction can be viewed as a generalization of the Weierstrass cubic and the $j$ -invariant of an elliptic curve to the case $g\\gt 1$ . Our main result is that in characteristics different from $2$ and $3$ the moduli space $\\widetilde{\\mathcal{U}}^{ns,a}_{g,g}$ is isomorphic to the moduli space of minimal $A_{\\infty}$ -structures on a certain finite-dimensional graded associative algebra $E_{g}$ (introduced by Fisette and Polishchuk).

A tree-based algorithm for the integration of monomials in the Chow ring of the moduli space of stable marked curves of genus zero

We compute the effective cone for the moduli space of stable rational curves with at most six marked points.

The moduli space of stable rational curves with marked points has two distinguished families of maps: the forgetful maps, given by forgetting some of the markings, and the Kapranov maps, given by complete linear series of $\psi $-classes. The collection of all these maps embeds the moduli space into a product of projective spaces. We call the multidegrees of this embedding “Kapranov degrees,” which include as special cases the work of Witten, Silversmith, Gallet–Grasegger–Schicho, Castravet–Tevelev, Postnikov, Cavalieri–Gillespie–Monin, and Gillespie–Griffin–Levinson. We establish, in terms of a combinatorial matching condition, upper bounds for Kapranov degrees and a characterization of their positivity. The positivity characterization answers a question of Silversmith and gives a new proof of Laman’s theorem characterizing generically rigid graphs in the plane. We achieve this by proving a recursive formula for Kapranov degrees and by using tools from the theory of error correcting codes.

We develop techniques for studying fundamental groups and integral singular\nhomology of symmetric Delta-complexes, and apply these techniques to study\nmoduli spaces of stable tropical curves of unit volume, with and without marked\npoints. As one application, we show that Delta_g and Delta_{g,n} are simply\nconnected, for positive g. We also show that Delta_3 is homotopy equivalent to\nthe 5-sphere, and that Delta_4 has 3-torsion in H_5.\n

Anomalous finite-temperature transport has recently been observed in numerical studies of various integrable models in one dimension; these models share the feature of being invariant under a continuous non-Abelian global symmetry. This work offers a comprehensive group-theoretic account of this elusive phenomenon. For an integrable quantum model with local interactions, invariant under a global non-Abelian simple Lie group G, we find that finite-temperature transport of Noether charges associated with symmetry G in thermal states that are invariant under G is universally superdiffusive and characterized by the dynamical exponent z=3/2. This conclusion holds regardless of the Lie algebra symmetry, local degrees of freedom (on-site representations), Lorentz invariance, or particular realization of microscopic interactions: We accordingly dub it “superuniversal.” The anomalous transport behavior is attributed to long-lived giant quasiparticles dressed by thermal fluctuations. We provide an algebraic viewpoint on the corresponding dressing transformation and elucidate formal connections to fusion identities amongst the quantum-group characters. We identify giant quasiparticles with nonlinear soliton modes of classical field theories that describe low-energy excitations above ferromagnetic vacua. Our analysis of these field theories also provides a complete classification of the low-energy (i.e., Goldstone-mode) spectra of quantum isotropic ferromagnetic chains.Received 24 October 2020Revised 4 May 2021Accepted 19 May 2021DOI:https://doi.org/10.1103/PhysRevX.11.031023Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasNonequilibrium statistical mechanicsQuantum statistical mechanicsQuantum transportTechniquesIntegrable systemsStatistical PhysicsGeneral Physics

We study the birational geometry of the moduli spaces of hyperelliptic curves with marked points. We show that these moduli spaces have non- Q \mathbb {Q} -factorial singularities. We complete the Kodaira classification by proving that these spaces have Kodaira dimension 4 g + 3 4g+3 when the number of markings is 4 g + 6 4g+6 and are of general type when the number of markings is n ≥ 4 g + 7 n\geq 4g+7 . Similarly, we consider the natural finite cover given by ordering the Weierstrass points. In this case, we provide a full Kodaira classification showing that the Kodaira dimension is negative when n ≤ 3 n\leq 3 , one when n = 4 n=4 , and of general type when n ≥ 5 n\geq 5 . For this, we carry out a singularity analysis of ordered and unordered pointed Hurwitz spaces. We show that the ordered space has canonical singularities and the unordered space has noncanonical singularities. We describe all noncanonical points and show that pluricanonical forms defined on the full regular locus extend to any resolution. Further, we provide a full classification of the structure of the pseudo-effective cone of Cartier divisors for the moduli space of hyperelliptic curves with marked points. We show the cone is nonpolyhedral when the number of markings is at least two and polyhedral in the remaining cases.

The tautological ring of the moduli space of stable curves has been studied extensively in the last decades. We present a SageMath implementation of many core features of this ring. This includes lists of generators and their products, intersection numbers and verification of tautological relations. Maps between tautological rings induced by functoriality, that is pushforwards and pullbacks under gluing and forgetful maps, are implemented. Furthermore, many interesting cycle classes, such as the double ramification cycles, strata of k-differentials and hyperelliptic or bielliptic cycles are available. In this paper we show how to apply the package, including concrete example computations.

We have seen that the equivariant localization formalism is an excellent, conceptual geometric arena for studies of supersymmetric and topological field theories, and more generally of (quantum) integrability. Given that the Hamiltonians in an integrable hierarchy are functionals of action variables alone [106], the equivariant localization formalism might yield a geometric characterization of quantum integrability, and perhaps some deeper connection between quantum-integrable bosonic theories and supersymmetric quantum field theories. This is particularly interesting from the point of view of examining corrections to the localization formulas, which in the last Chapter we have seen reflect global properties of the theory. This would be of particular interest to analyse more closely, as it could then lead to a unified description of localization in the symplectic loop space, the supersymmetric loop space and in topological quantum field theory.KeywordsLoop SpaceEquivariant CohomologySupersymmetric Quantum MechanicLocalization FormulaEquivariant LocalizationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the context of spin foam models for quantum gravity, group field theories are a useful tool allowing on the one hand a non-perturbative formulation of the partition function and on the other hand admitting an interpretation as generalized matrix models. Focusing on 2d group field theories, we review their explicit relation to matrix models and show their link to a class of non-commutative field theories invariant under a quantum-deformed 3d Poincaré symmetry. This provides a simple relation between matrix models and non-commutative geometry. Moreover, we review the derivation of effective 2d group field theories with non-trivial propagators from Boulatov's group field theory for 3d quantum gravity. Besides the fact that this gives a simple and direct derivation of non-commutative field theories for the matter dynamics coupled to (3d) quantum gravity, these effective field theories can be expressed as multi-matrix models with a non-trivial coupling between matrices of different sizes. It should be interesting to analyze this new class of theories, both from the point of view of matrix models as integrable systems and for the study of non-commutative field theories.

Quantum [formula omitted] Drinfeld–Sokolov hierarchy and quantum singularity theory

Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno–Drinfeld Lie algebra $\Xmathfrak {t}_{\hspace *{.3pt}n}$. We show that Gaudin subalgebras form a variety isomorphic to the moduli space $\bar M_{0,n+1}$ of stable curves of genus zero with n+1 marked points. In particular, this gives an embedding of $\bar M_{0,n+1}$ in a Grassmannian of (n−1)-planes in an n(n−1)/2-dimensional space. We show that the sheaf of Gaudin subalgebras over $\bar M_{0,n+1}$ is isomorphic to a sheaf of twisted first-order differential operators. For each representation of the Kohno–Drinfeld Lie algebra with fixed central character, we obtain a sheaf of commutative algebras whose spectrum is a coisotropic subscheme of a twisted version of the logarithmic cotangent bundle of $\bar M_{0,n+1}$.

We compare the construction of 2D integrable models through two gauge field theories. The first one is the 4D Chern–Simons (4D-CS) theory proposed by Costello and Yamazaki. The second one is the 2D generalization of the Hitchin integrable systems constructed by means of affine Higgs bundles (AHB). We illustrate the latter approach by considering 1 + 1 field versions of integrable systems including the Calogero–Moser field theory, the Landau–Lifshitz model and the field theory generalization of the elliptic Gaudin model.