Abstract This paper explores the relationship between L -equivalence and D -equivalence for K3 surfaces and hyperkähler manifolds. Building on Efimov’s approach using Hodge theory, we prove that very general L -equivalent K3 surfaces are D -equivalent, leveraging the Derived Torelli Theorem for K3 surfaces. Our main technical contribution is that two distinct lattice structures on an integral, irreducible Hodge structure are related by a rational endomorphism of the Hodge structure. We partially extend our results to hyperkähler fourfolds and moduli spaces of sheaves on K3 surfaces.

A bstract The moduli space and generalised global symmetries of 3d $$ \mathcal{N}=5 $$ N = 5 superconformal field theories are investigated, with a focus on the orthosymplectic ABJ theories and their discrete gauging variants. We extend the known classification of $$ \mathcal{N}=5 $$ N = 5 moduli spaces as orbifolds ℍ 2 N / Γ, where Γ is a quaternionic reflection group, to theories incorporating Spin, O − , and Pin-type gauge groups. In these cases, we find that the moduli space is governed not by Γ itself, but by a ℤ 2 central extension thereof, for which we explicitly describe the generators. We provide a systematic method to construct the group Γ ′ governing the moduli space of a theory 𝒯 ′ obtained by gauging a ℤ 2 zero-form symmetry of an original theory 𝒯. This is achieved by identifying the specific generator that must be added to Γ. We compute the Hilbert series for these moduli spaces and verify them against the corresponding limits of the superconformal index, finding perfect agreement. We also discuss how ’t Hooft anomalies for the zero-form symmetries manifest in the superconformal index and the moduli space. Furthermore, we revisit the symmetry category of the 𝔰𝔬(2 N ) 2 k × 𝔲𝔰𝔭(2 N ) −k theories. Building on previous work that identified the symmetry category for all parities of N and k , we provide the explicit symmetry webs for the opposite parity D 8 case. We find that the details of these webs differ from the previously studied D 8 webs corresponding to the both even parity case. Finally, we analyse theories with unequal ranks, those containing the 𝔰𝔬(2 N + 1) gauge algebra, and the two SCFT variants based on the F (4) superalgebra.

The real locus of the moduli space of stable genus zero curves with marked points, \overline{{\mathcal{M}}}_{0,{n+1}}({\mathbb{R}}) , is known to be a smooth manifold and is the Eilenberg–MacLane spaces for the so-called pure cactus groups. We describe the operad formed by these spaces in terms of a homotopy quotient of an operad of associative algebras. Using this model, we identify various Hopf models for the algebraic operad of chains and homologies of \overline{{\mathcal{M}}}_{0,{n+1}}({\mathbb{R}}) . In particular, we show that the operad \overline{{\mathcal{M}}}_{0,{n+1}}({\mathbb{R}}) is not formal. As an application of these operadic constructions, we prove that for each n , the cohomology ring H^{\bullet}(\overline{{\mathcal{M}}}_{0,{n+1}}({\mathbb{R}});{\mathbb{Q}}) is a Koszul algebra, and that the manifold \overline{{\mathcal{M}}}_{0,{n+1}}({\mathbb{R}}) is not formal for n\geq 6 but is a rational K(\pi,1) -space. Additionally, we describe the Lie algebras associated with the lower central series filtration of the pure cactus groups.

Abstract We give an explicit algorithm to reduce the ramification order of any exponential factor of an irregular connection on , using the same types of basic operations as in the Katz–Deligne–Arinkin algorithm for rigid irregular connections. The exponential factor reached when the algorithm terminates is, up to admissible deformations, the unique factor with minimal ramification order in the orbit of the initial factor under successive applications of basic operations. Furthermore, we show that for every even integer , there is up to admissible deformations a finite number of nonsimplifiable exponential factors at infinity such that the corresponding elementary wild character variety has complex dimension , which conjecturally implies that there is a finite number of isomorphism classes of elementary wild character varieties in any dimension. These results can be viewed as saying that the set of all possible level data of exponential factors has the structure of a disjoint union of an infinite number of infinite rooted trees, each tree being associated to a given dimension and with a finite number of trees for each . In particular, in dimension 2 there is a unique tree, corresponding to the Painlevé I moduli space.

The McKay correspondence for dihedral groups: The moduli space and the tautological bundles

This is the first of two papers on the global topology of the space S u b ( G ) Sub(G) of all closed subgroups of G = P S L 2 ( R ) G=PSL_2(\mathbb {R}) , equipped with the Chabauty topology. In this paper, we study the spaces of lattices and elementary subgroups of G G , and prove a continuity result for conformal grafting of (possibly infinite type) vectored orbifolds that will be useful in both papers. More specifically, we first identify the homotopy type of the space of elementary subgroups of G G , following Baik–Clavier. Then for a fixed finite type hyperbolizable 2 2 -orbifold S S , we show that the space S u b S ( G ) Sub_S(G) of all lattices Γ > G \Gamma > G with Γ ∖ H 2 ≅ S \Gamma \backslash \mathbb {H}^2 \cong S is a fiber orbibundle over the moduli space M ( S ) \mathcal M(S) . We describe the closure S u b S ( G ) ¯ \overline {Sub_S(G)} in S u b ( G ) Sub(G) and show that ∂ S u b S ( G ) \partial Sub_S(G) has a neighborhood deformation retract within S u b S ( G ) ¯ \overline {Sub_S(G)} . When S S is not one of finitely many low complexity orbifolds, we show that S u b S ( G ) ¯ \overline {Sub_S(G)} is simply connected. In the simplest exceptional case, when S S is a sphere with three total cusps and cone points, we show that S u b S ( G ) ¯ \overline {Sub_S(G)} is a (usually nontrivial) lens space. Finally, we show that when ( X i , v i ) → ( X ∞ , v ∞ ) (X_i,v_i) \to (X_\infty ,v_\infty ) is a (possibly infinite type) smoothly converging sequence of vectored hyperbolic 2 2 -orbifolds, and we graft in Euclidean annuli along suitable collections of simple closed curves in the X i X_i , then after uniformization, the resulting vectored hyperbolic 2 2 -orbifolds converge smoothly to the expected limit. As part of the proof, we give a new lower bound on the hyperbolic distance between points in a grafted orbifold in terms of their original distance.

A bstract The species cutoff is a moduli-dependent quantity signaling the onset of quantum gravitational phenomena, whose form can be oftentimes determined from higher-derivative and higher-curvature corrections within low-energy gravitational EFTs. In this work, we point out that these Wilson coefficients are eigenfunctions of an appropriate second-order elliptic operator defined over moduli space in theories with more than four supercharges. This was already known to be the case for the leading $$ {\mathcal{R}}^4 $$ R 4 -correction to the two-derivative (bosonic) action of maximal supergravity in d ≤ 10. Here, we reconsider this fact from the Swampland point of view and show how, in d = 10, 9, 8, solving a Laplace equation imposes non-trivial restrictions on the species hull vectors. We further argue that this property is also satisfied in settings with less supersymmetry. In particular, we focus on the $$ {\mathcal{R}}^4 $$ R 4 -operator in minimal supergravity theories in d = 10, 9, and on the leading $$ {\mathcal{R}}^2 $$ R 2 -term in setups with 8 supercharges in d = 6, 5, 4. Finally, we provide a symmetry-based criterion for determining when the relevant elliptic operator should be the Laplacian. A bottom-up rationale for this constraint remains to be fully understood, and we conclude by outlining some compelling possibilities.

Abstract For $r\geq 3$ and $g= \frac {r(r+1)}{2}$ , we study the Prym-Brill-Noether variety $V^r(C,\eta )$ associated to Prym curves $[C,\eta ]$ . The locus $\mathcal {R}_g^r$ in $\mathcal {R}_g$ parametrizing Prym curves $(C, \eta )$ with nonempty $V^r(C,\eta )$ is a divisor. We compute some key coefficients of the class $[\overline {\mathcal {R}}_g^r]$ in $\mathrm {Pic}_{\mathbb {Q}}(\overline {\mathcal {R}}_g)$ . Furthermore, we examine a strongly Brill-Noether divisor in $\overline {\mathcal {M}}_{g-1,2}$ : we show its irreducibility and compute some of its coefficients in $\mathrm {Pic}_{\mathbb {Q}}(\overline {\mathcal {M}}_{g-1,2})$ . As a consequence of our results, the moduli space $\mathcal {R}_{14,2}$ is of general type.

In this paper, we study Spin*(8)-Higgs bundles over compact Riemann surfaces, extending the work of Bradlow, García-Prada, and Gothen on SO*(8). The group Spin*(8) is exceptional among classical real forms, as its complexification Spin(8,C) admits triality, an outer automorphism of order 3, but triality does not preserve the real form Spin*(8). We establish the Toledo bound |τ|≤4(g−1) for semistable Spin*(8)-Higgs bundles and characterize maximal bundles through rigidity theorems. We prove that the moduli space of maximal bundles fibers over the SO*(8) moduli space with discrete fibers parametrized by spin structures, and has a dimension of 15(g−1), one less than expected. Using Morse theory, we establish connectedness of moduli spaces for τ=0 and maximal |τ|. Via the non-abelian Hodge correspondence, our results yield connectedness theorems for character varieties of surface group representations into Spin*(8). We analyze how triality determines the decomposition of the isotropy representation despite not acting on the real form.

A bstract We study infinite distance limits in the complex structure moduli space of Type IIB compactifications on Calabi-Yau threefolds, in light of the Emergent String Conjecture. We focus on the so-called type II limits, which, based on the asymptotic behaviour of the physical couplings in the low-energy effective theory, are candidates for emergent string limits. However, due to the absence of Type IIB branes of suitable dimensionality, the emergence of a unique critical string accompanied by a tower of Kaluza-Klein states has so far remained elusive. By considering a broad class of type II b limits, corresponding to so-called Tyurin degenerations, and studying the asymptotic behaviour of four-dimensional EFT strings in this geometry, we argue that the worldsheet theory of the latter describes a unique critical heterotic string on T 2 × K3 with a gauge bundle whose rank depends on b . In addition, we establish the presence of an infinite tower of BPS particles arising from wrapped D3-branes by identifying a suitable set of special Lagrangian 3-cycles in the geometry. The associated BPS invariants are conjectured to be counted by generalisations of modular forms. As a consistency check, we further show that in special cases mirror symmetry identifies the EFT strings with the well-understood emergent string limits in the Kähler moduli space of Type IIA compactifications on K3-fibred Calabi-Yau threefolds. Finally, we discuss the implications of the Emergent String Conjecture for type II limits which do not correspond to Tyurin degenerations, and predict new constraints on the possible geometries of type II degenerations which resemble those arising in the Kulikov classification of degenerations of K3 surfaces.

We give a general formula for generators of the NL cone on an orthogonal modular variety. This is the cone of effective divisors linearly equivalent to an effective linear combination of irreducible components of Noether–Lefschetz divisors. We apply this to describe, in terms of minimal generators, the NL cone of various moduli spaces of geometric origin such as those of polarized K3 surfaces, cubic fourfolds, and hyperkähler manifolds. Additionally, we establish uniruledness for many moduli spaces of primitively polarized hyperkähler manifolds of {textrm{OG6}} and {textrm{Kum}}_n-type. Finally, in analogy with the case of K3 surfaces of degree 2, we show that any family of polarized {textrm{Kum}}_2-type hyperkähler manifolds with divisibility 2 and polarization degree 2 over a projective base is isotrivial.

In this memoir, we study the counterpart of Grothendieck’s projectivization construction in the context of derived algebraic geometry. Our main results are as follows: First, we define the derived projectivization of a connective complex, study its fundamental properties such as finiteness properties and functorial behaviors, and provide explicit descriptions of their relative cotangent complexes. We then focus on the derived projectivizations of complexes of perfect-amplitude contained in [ 0 , 1 ] [0, 1] . In this case, we prove a generalized Serre’s theorem, a derived version of Beilinson’s relations, and establish semiorthogonal decompositions for their derived categories. Finally, we show that many moduli problems fit into the framework of derived projectivizations, such as moduli spaces that arise in Hecke correspondences. We apply our results to these situations.

A bstract N = 2, 4 and 8 supersymmetric string theories in four dimensional flat space-time have moduli space of vacua. We argue that starting from a theory where the moduli approach a particular moduli space point A at infinity, we can construct a classical solution that contains an arbitrarily large space-time region where the moduli take values corresponding to any other moduli space point B of our choice to any desired accuracy. Therefore the observables of a theory with a given set of asymptotic values of the moduli will have complete information on the observables for any other asymptotic values of the moduli. Also it is physically impossible for any experiment, performed over a finite time, to determine the asymptotic values of the moduli. We point out the difference between asymptotically flat space-time and asymptotically AdS space-time in this regard and discuss the possible implication of these results for holographic duals of string theories in flat space-time. For N=2 supersymmetric theories, A and B could correspond to compactifications on topologically distinct Calabi-Yau manifolds related by flop or conifold transitions.

Gaudin models and moduli space of flower curves

Abstract The classical Losev–Manin space is a toric compactification of the moduli space of $n$ points in the affine line modulo translation and scaling. Motivated by this, we study its higher-dimensional toric counterparts, which compactify the moduli space of $n$ distinct labeled points in affine space modulo translation and scaling. We show that these moduli spaces are a fibration over a product of projective spaces—with fibers isomorphic to the Losev–Manin space—and that they are isomorphic to the normalization of a Chow quotient. Moreover, we present a criterion to decide whether the blow-up of a toric variety along the closure of a subtorus is a Mori dream space. As an application, we demonstrate that a related generalization of the moduli space of pointed rational curves constructed by Chen, Gibney, and Krashen is not a Mori dream space when the number of points is at least nine, regardless of the dimension.

A bstract Six dimensional $$ \mathcal{N} $$ N = (1, 0) supergravity features BPS strings whose properties encode highly nontrivial information about the parent 6d theory. We focus on a distinguished set of theories whose string charge lattice is one-dimensional. In geometric theories, the generator of the lattice arises from a D3 brane wrapping the hyperplane class in ℙ 2 . This hyperplane string is expected to remain stable even when one ventures beyond the geometric regime where it becomes challenging to verify which candidate 6d theories belong to the swampland. We identify five 6d models which from the perspective of the hyperplane string deviate the most from being geometric. For these theories we are able to provide an exact description of the left-moving sector of the hyperplane string worldsheet in terms of a rational conformal field theory and provide evidence for their consistency. In one instance, using RCFT methods we are able to determine the elliptic genus and find that in the unflavored limit it matches with the elliptic genus of geometric models. We argue that the non-geometric model is connected to geometric ones via a sequence of Higgsing transitions. These results lead us to formulate a proposal relating the quantum corrected moduli space of the hyperplane string CFT with a region of the landscape of 6d $$ \mathcal{N} $$ N = (1, 0) quantum gravity.

Toric varieties provide a rich class of examples in algebraic geometry that benefit from deep and fruitful interactions with combinatorics. This workshop highlighted recent interactions between toric geometry and mirror symmetry, matroids, deformation theory and moduli spaces, and non-commutative geometry, as well as some exciting new developments within toric geometry itself.

We investigate multi-mode GKP (Gottesman–Kitaev–Preskill) quantum error-correcting codes from a geometric perspective. First, we construct their moduli space as a quotient of groups and exhibit it as a fiber bundle over the moduli space of symplectically integral lattices. We then establish the Gottesman–Zhang conjecture for logical GKP Clifford operations, showing that all such gates arise from parallel transport with respect to a flat connection on this space. Specifically, non-trivial Clifford operations correspond to topologically non-contractible paths on the space of GKP codes, while logical identity operations correspond to contractible paths.

Free Maxwell theory on general four-manifolds may, under certain conditions on the background geometry, exhibit holomorphic factorization in its partition function. We show that when this occurs, new discrete symmetries emerge at orbifold points of the conformal manifold. These symmetries, which act only on a sublattice of flux configurations, are not associated with standard dualities, yet they may carry ’t Hooft anomalies, potentially causing the partition function to vanish even in the absence of apparent pathologies. We further explore their non-invertible extensions and argue that their anomalies can account for zeros of the partition function at smooth points in the moduli space.

Abstract We use freely acting asymmetric orbifolds of type IIB string theory to construct a class of theories in four dimensions with eight supercharges. Their low energy effective field theories resemble STU models, but have different duality groups: the orbifold’s free action reduce the duality groups to congruence subgroups of the modular group. The fundamental domain is consequently larger and contains new interesting points at infinite distance on the real axis bounding the upper half plane. We verify that the distance conjectures hold in the non-geometric compactification of string theory studied here. In particular, we find points at infinite distance in moduli space at which the theory decompactifies to a different orbifold construction.