Researchers compute differential K-theory for moduli stacks.
problem Computing differential K-theory for moduli stacks of principal G-bundles.
method Using homotopy theory of presheaves of spaces and spectra, they formulate results in terms of invariant polynomials and representation rings.
result They successfully compute the connective differential K-theory and differential cohomology of moduli stacks.
The paper explores Higgs bundles and their moduli spaces on Riemann surfaces.
problem Developing the moduli theory of Higgs bundles and understanding their geometric properties.
method Establishing non-abelian Hodge correspondences and studying Hitchin fibration.
result Computing the Poincaré polynomial of rank 2 moduli space and verifying topological mirror symmetry.
Develops moduli theory for Calabi-Yau pairs, constructing a projective space.
problem Constructing a moduli space for Calabi-Yau pairs at the Calabi-Yau wall.
method Develops moduli theory, proving S-completeness and Θ-reductivity, constructing projective moduli space. result Constructs a projective moduli space for degenerate P2 pairs. Constructs moduli stacks of quiver bundles and applies to Higgs bundles.
problem Classifying morphisms of vector bundles over a fixed base.
method General method for constructing moduli stacks of diagrams of vector bundles indexed by a simplicial set.
result Recovery of Nakajima quiver varieties and alternate construction of moduli stacks of Higgs bundles.
Decouples moduli groups in heterotic string theory cohomology.
problem Decomposing moduli groups in heterotic string theory cohomology.
method Computing cohomology groups at the standard embedding and showing their direct sum decomposition.
result Decomposes moduli groups into a direct sum of cohomologies at the standard embedding.
New proof for discrete Morse theory using combinatorial construction.
problem Verifying the Morse differential in discrete Morse homology.
method Combinatorial construction of flowlines in discrete Morse theory.
result Morse differential squares to zero in discrete Morse homology.
We study the four-dimensional effective theory arising from ten-dimensional heterotic supergravity compactified on manifolds with torsion. In particular, given the heterotic superpotential appropriately corrected at O(α′) to account for the Green-Schwarz anomaly cancellation mechanism, we investigate proper…
We consider the extension of classical 2-dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We approach this using the theory of modular operads by introducing a new operad governing associative algebras with involution. This operad is Koszul an…
The paper proposes conjectures about moduli space rigidity.
problem Rigidity of moduli spaces in algebraic geometry.
method Group-theoretic, topological, and holomorphic approaches.
result Some conjectures have been proved.
Moduli spaces of doubly periodic monopoles, also called monopole walls or monowalls, are hyperkähler; thus, when four-dimensional, they are self-dual gravitational instantons. We find all monowalls with lowest number of moduli. Their moduli spaces can be identified, on the one hand, with Coulomb branches of five-dimens…
Constructs moduli stacks for quiver connections and extends non-Abelian Hodge theory.
problem Extending non-Abelian Hodge theory to moduli stacks of quiver connections.
method Formalizes and constructs moduli stacks of bundles with λ-connections over prestacks.
result Shows moduli stacks are algebraic and locally of finite presentation when base is smooth and projective.
This thesis is concerned with the application of operadic methods, particularly modular operads, to questions arising in the study of moduli spaces of surfaces as well as applications to the study of homotopy algebras and new constructions of 'quantum invariants' of manifolds inspired by ideas originating from physics.…
Study classifies moduli spaces of spin connections on 3D homogeneous spaces.
problem Classifying moduli spaces of spin connections on 3D homogeneous spaces.
method Analysis of the topology of moduli spaces, focusing on finite-dimensional topological manifolds with trivial homotopy groups.
result Moduli spaces are finite-dimensional topological manifolds with trivial homotopy groups, essential for consistent cosmological models.
These are notes on the theory of super Riemann surfaces and their moduli spaces, aiming to collect results that are useful for a better understanding of superstring perturbation theory in the RNS formalism.
Study knot singularities in Bogomolny equation solutions.
problem Understanding solutions with knot singularities.
method Analyzes the moduli space of solutions on R^3 with specific asymptotic conditions.
result Potential applications in low-dimensional topology and knot theory.
Universal geometry organizes heterotic moduli spaces.
problem Understanding the structure of heterotic compactifications.
method Fibering compactification data over moduli space and analyzing universal curvatures.
result Deformations are components of universal curvatures, incorporating α′2 corrections. We present ADHM-Nahm data for instantons on the Taub-NUT space and encode these data in terms of Bow Diagrams. We study the moduli spaces of the instantons and present these spaces as finite hyperkahler quotients. As an example, we find an explicit expression for the metric on the moduli space of one SU(2) instanton. W…
Constructs a space for stable holomorphic submersions over a fixed base.
problem Stability of holomorphic submersions over a compact Kaehler base.
method Geometric invariant theory combined with geometric PDEs.
result Moduli space is a Hausdorff complex space with a Weil-Petersson type Kaehler metric.
The goal of this paper is to develop some aspects of the deformation theory of piecewise flat structures on surfaces and use this theory to construct new geometric structures on the moduli space of Riemann surfaces.
The Bogomol'nyi-Prasad-Sommerfield (BPS) multi-wall solutions are constructed in supersymmetric U(N_C) gauge theories in five dimensions with N_F(>N_C) hypermultiplets in the fundamental representation. Exact solutions are obtained with full generic moduli for infinite gauge coupling and with partial moduli for finite …
The paper studies the local structure of a moduli space for a specific string theory system.
problem Investigating the local structure of the moduli space of solutions to the Hull--Strominger system.
method Using a vector bundle and studying the deformation complex associated with a differential operator $ar{D}$, establishing an isomorphism between cohomology groups.
result The moduli space has an expected dimension of zero.
Survey of methods for computing volumes of moduli spaces.
problem Computing volumes of moduli spaces for Riemann surfaces with different metrics.
method Combinatorial enumeration, intersection theory, recursion relations.
result Review of key results and methods in computing both Weil-Petersson and Masur-Veech volumes.
New Morse functions on curve moduli space via geodesics.
problem Understanding the moduli space of curves via geometric and combinatorial methods.
method Introducing Morse functions based on geodesic lengths and analyzing their critical points and indices.
result Found new explicit Morse functions on Mg,n, leading to a combinatorial cell decomposition. This thesis was motivated by a desire to understand the natural geometry of hyperbolic monopole moduli spaces. We take two approaches. Firstly we develop the twistor theory of singular hyperbolic monopoles and use it to study the geometry of their charge 1 moduli spaces. After this we introduce a new way to study the m…
Develops formal moduli theory for splitting complex supermanifolds.
problem Tackles the splitting problem of complex supermanifolds.
method Constructs a filtered dg Lie algebra to control splittings and transfers the theory to a minimal filtered L∞-model. result Recover classical obstruction classes as leading terms of Maurer-Cartan representatives and proves the existence of higher obstructions.
We apply gauge theory to study the space Fk(M) of smooth codimension-k framed foliations on a smooth manifold M. The quotient of Maurer-Cartan elements by the action of an infinite dimensional non-abelian gauge groupoid forms a moduli space, which contains Fk(M) as a subspace. The notion of holonomy is natura…
Formula for Euler characteristic of moduli spaces of Abelian differentials.
problem Computing the Euler characteristic of moduli spaces of Abelian differentials.
method Intersection theory on the smooth compactification by multi-scale differentials, Euler sequence for cotangent bundle, and tools in the Chow ring.
result Formula for the full Chern polynomial of the cotangent bundle.
Estimates the rational homological dimension of Riemann surfaces with boundary and marked points.
problem Estimating the homological dimensions of Riemann surfaces with boundary and marked points.
method Developed an estimate for the rational homological dimension of Riemann surfaces with possible boundary and marked points.
result Provided an estimate for the rational homological dimension of Riemann surfaces with boundary and marked points.
Finite presentations for skein algebras linked to gauge field theory.
problem Understanding finite presentations for skein algebras and their relationship to gauge field theory.
method Provided finite presentations and deduced properties of stated skein algebras.
result Stated skein algebras are Koszul and isomorphic to quantum moduli algebras in gauge field theory.
Study presents a twistor correspondence for specific geometric structures.
problem Twistor theory for almost-Grassmannian manifolds.
method Utilizes moduli of curves-with-boundary for global correspondence.
result Foundational results in complex setting, global correspondence for real Grassmannian.
We compute the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we study their…
Study connects G2-structures to flat connections on compact 3-manifolds.
problem Understanding moduli spaces of G2-structures and flat connections. method Equivalence between moduli spaces of G2-structures and flat connections on compact 3-manifolds. result Equivalence between moduli spaces of G2-structures and flat connections on compact 3-manifolds. We prove diffeomorphisms of polygonal linkage moduli spaces to Euclidean spaces.
problem Moduli spaces of self-avoiding polygonal linkages and configurations.
method Construct Lyapunov-Reeb functions to show diffeomorphisms.
result Moduli spaces are diffeomorphic to Euclidean spaces.
We describe the induced geometry on several classes of Kodaira moduli spaces of rational curves in twistor spaces. By constructing connections and frames on the moduli spaces we build and review twistor theories pertaining to relativistic and non-relativistic geometries. Focussing on the cases of three- and five-dimens…
Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on …
Surveying stability and deformation of Einstein metrics.
problem Stability and deformation of Einstein metrics.
method Study of the spectrum and eigentensors of the Lichnerowicz Laplacian.
result Recent results on stability and deformation theory of Einstein metrics.
Study new ECS structures in 5D Minkowski compactifications of M-theory.
problem Characterize geometries of supersymmetric compactifications to 5D Minkowski space.
method Define and classify ECS structures, relate to hypermultiplet moduli.
result Classify ECSs and find their moduli, relating to hypermultiplet moduli.
This article constructs the moduli stack of torsionfree G-jet-structures in homotopy type theory with one monadic modality. This yields a construction of this moduli stack for any ∞-topos equipped with any stable factorization systems. In the intended applications of this theory, the factorization systems are …
We calculate certain homotopy groups of the moduli spaces for representations of a compact oriented surface in the Lie groups GL(n,C) and U(p,q). Our approach relies on the interpretation of these representations in terms of Higgs bundles and uses Bott--Morse theory on the corresponding moduli spaces.
The article develops deformation theory for ACyl associative submanifolds in ACyl G2-manifolds.
problem Deformation theory of ACyl associative submanifolds in ACyl G2-manifolds.
method Study of moduli spaces with fixed and varying asymptotic data, computing virtual dimensions.
result The moduli space of ACyl associative submanifolds embeds as a Lagrangian submanifold in the moduli space of holomorphic curves.
Holomorphic supergravity theory simplifies anomaly cancellation in heterotic moduli.
problem Anomaly cancellation in heterotic moduli space.
method Formulated a ten-dimensional version of Kodaira-Spencer gravity, quantized fluctuations, and showed partition function simplification.
result Holomorphic supergravity theory simplifies anomaly cancellation and relates to type I Kodaira-Spencer theory.
Gluing instantons on 4-manifolds with group action.
problem Gluing invariant ASD connections on 4-manifolds with group action.
method Equivariant gluing theory for instanton moduli spaces.
result Construction of Γ-invariant ASD connections on connected sums. The proper action functional of (4k+3)-dimensional U(1)-Chern-Simons theory including the instanton sectors has a well known description: it is given on the moduli space of fields by the fiber integration of the cup product square of classes in degree-(2k+2) differential cohomology. We first refine this statement from …
The paper introduces elliptic quasi-modular forms via moduli spaces.
problem Developing a theory of elliptic quasi-modular forms.
method Using moduli spaces and the Gauss-Manin connection.
result Presented a succinct theory of elliptic quasi-modular forms.
We consider euclidean D-branes wrapping around manifolds of exceptional holonomy in dimensions seven and eight. The resulting theory on the D-brane---that is, the dimensional reduction of 10-dimensional supersymmetric Yang-Mills theory---is a cohomological field theory which describes the topology of the moduli space o…
Finiteness predicts dualities in quantum gravity.
problem Finiteness of quantum gravity amplitudes in fully compactified theories.
method Relating moduli space compactifiability to duality group representations.
result Finiteness requires compact moduli spaces and semisimple duality group representations.
Analyzes quantization of flux observables in gauge theories.
problem Lifting Poisson-brackets of flux observables to higher moduli stacks.
method Systematic analysis of canonical quantization and flux quantization laws.
result Topological quantum observables form homology Pontrjagin algebra of loop space.
The moduli space metric and its Kahler potential for well-separated non-Abelian vortices are obtained in U(N) gauge theories with N Higgs fields in the fundamental representation.