arXiv research

Constructs moduli stacks of quiver bundles and applies to Higgs bundles.

Constructs moduli stacks for quiver connections and extends non-Abelian Hodge theory.

The paper studies higher geometric structures and connections on manifolds, constructing moduli stacks and proving equivalence criteria.

Researchers compute differential K-theory for moduli stacks.

New homotopy theory reveals the structure of stable curves.

Study moduli spaces of elliptic PDEs using derived $C^{\infty}$-geometry.

Geometric compactification for complex structures on Lie groups.

Constructs cohomology decompositions for symmetric stacks.

The higher gauge field in 11-dimensional supergravity -- the C-field -- is constrained by quantum effects to be a cocycle in some twisted version of differential cohomology. We argue that it should indeed be a cocycle in a certain twisted nonabelian differential cohomology. We give a simple and natural characterization…

The BPS decomposition theorem splits cohomology of symmetric stacks into invariant parts.

We prove that the second Hochschild cohomology group of the moduli stack of stable $n$-pointed genus $g$ curves vanishes for all but finitely many $(g,n)$.

In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, as well as the "underlying" diffeological space of any stack. More precisely, we consider diffeological spaces as so-called concrete sheaves and show that the Grothendieck construction sending these sheaves to stacks has a left…

Computes infinitesimal automorphisms for $L$-valued Higgs bundles, leading to DM stacks.

Study flat connections with logarithmic singularities on complex plane curves.

Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective threefold Y that admits a no…

Constructs diffeological moduli stacks for Higgs and flat bundles on Kähler manifolds

Normal forms and moduli stacks for flat connections on complex manifolds.

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 $\infty$-topos equipped with any stable factorization systems. In the intended applications of this theory, the factorization systems are …

Fix a smooth projective curve over a field of characteristic zero and a finite set of punctures. Let G be a connected linear algebraic group. We prove that the moduli of G-bundles with logarithmic connections having fixed residue classes at the punctures is an algebraic stack of finite type.

Constructs the moduli stack of elliptic curves as an orbifold.

Study Kähler groups from orbifold compactifications of curve moduli.

Develops moduli theory for Calabi-Yau pairs, constructing a projective space.

New potentials found for sheaves on Calabi-Yau 4-folds.

Quillen connection links Riemann surfaces to projective structures.

The paper establishes a Lagrangian correspondence linking different geometric structures on complex varieties.

New Poisson structures found on Higgs bundle moduli spaces.

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 …

Short note shows unbounded dimensions in Fano K-moduli spaces.

The Liouville symplectic form connects various moduli spaces in algebraic geometry.

Study of quantum decorated character stacks and their quantizations.

New theory proves representability of PDE solutions without complex machinery.

Using stable log maps, we introduce log twisted differentials extending the notion of abelian differentials to the Deligne-Mumford boundary of stable curves. The moduli stack of log twisted differentials provides a compactification of the strata of abelian differentials. The open strata can have up to three connected c…

Proves properness of K-moduli spaces for Fano varieties.

In this article, we derive many properties of étale stacks in various contexts, and prove that étale stacks may be characterized categorically as those stacks that arise as prolongations of stacks on a site of spaces and local homeomorphisms. Moreover, we show that the bicategory of étale differentiable stacks and loca…

We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with [Jia17,BL18], the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension a…

Analyzes quantization of flux observables in gauge theories.

Let $X$ be a compact connected Riemann surface of genus at least two, and let ${G}$ be a connected semisimple affine algebraic group defined over $\mathbb C$. For any $δ\in π_1({G})$, we prove that the moduli space of semistable principal ${G}$--bundles over $X$ of topological type $δ$ is simply connected. In contrast,…

We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $Θ$-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove…

In this short note, we compute the Betti numbers of the moduli stack of flat SU(3)-bundles over a Klein bottle. We also handle the general compact group case over RP^2. In all cases the cohomology is found to be equivariantly formal, supporting a conjecture from the author's doctoral thesis. Our results also verify con…

For G a complex reductive group and X a smooth projective or convex quasi-projective polarized G-variety we construct a formal map in quantum K-theory from the equivariant quantum K-theory $QK^G(X)$ to the quantum K-theory of the git quotient $QK(X//G)$ assuming the quotient $X//G$ is a smooth Deligne-Mumford stack wit…

The paper proves a new version of dimensional reduction in cohomological Donaldson-Thomas theory.

These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at Zhejiang University in July, 2008. Their goal is to introduce and motivate basic concepts and constructions (such as orbifolds and stacks) important in the study of moduli spaces of curves and abelian varieties thro…

New theory captures framing anomaly in gauge theory.

The first part of this text is a gentle exposition of some basic constructions and results in the extended prequantum theory of Chern-Simons-type gauge field theories. We explain in some detail how the action functional of ordinary 3d Chern-Simons theory is naturally localized ("extended", "multi-tiered") to a map on t…

We construct a compactification of the moduli spaces of abelian differentials on Riemann surfaces with prescribed zeroes and poles. This compactification, called the moduli space of multi-scale differentials, is a complex orbifold with normal crossing boundary. Locally, our compactification can be described as the norm…

Abstract: New geometric incarnation of isomonodromy functors.

We use Morse theory of the Yang-Mills functional to compute the Betti numbers of the moduli stack of flat U(3)-bundles over a compact nonorientable surface. Our result establishes the antiperfection conjecture of Ho-Liu, and provides evidence for the equivariant formality conjecture of the author.

Study of K-moduli of prime Fano threefolds of genus twelve, proving boundary purely divisorial.