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Yiannis Loizides

Publications and source records attributed to Yiannis Loizides.

At least 19 recordsLinked to original sources
arXiv2026

Weightings and Lie groupoids

This note originated with a mini-course taught at KU Leuven by the author in June 2026. We review the definition of weightings (or quasi-homogeneous structures), the weighted normal bundle, and the weighted deformation space. We emphasize the algebraic approach and functorial properties, using sheaves of ideals and the character spectrum for sheaves of algebras. We review the definition of a multiplicative weighting on a Lie groupoid $G\rightrightarrows M$, and give a self-contained proof of the three-way equivalence between (i) multiplicative weightings of $G$ along $M$, (ii) Lie filtrations of the Lie algebroid $A$ of $G$, (iii) dg weightings of the NQ manifold $A[1]$ along $M$. The corresponding weighted deformation space is a weighted variant of the tangent/adiabatic groupoid and is used in the study of hypoelliptic operators.

math.DG↗
arXiv2026

Slices for reductive group actions in algebraic and holomorphic symplectic geometry

Symplectic slice theorems elucidate the local structure of symplectic manifolds carrying Hamiltonian actions of compact Lie groups. We generalize these theorems in two natural settings. The first is based on the idea that complex reductive algebraic groups are the natural complex-geometric counterparts of compact Lie groups. Using new definitions of Poisson and symplectic slices, we prove analogues of the classical symplectic slice theorems for Hamiltonian actions of complex reductive algebraic groups. These analogues have versions in the complex-algebraic and holomorphic categories, and make extensive use of Slodowy slices and decomposition classes in complex reductive Lie algebras. The starting point for our second setting is the fact that Hamiltonian Lie group actions are special cases of Hamiltonian symplectic groupoid actions. We generalize the classical symplectic slice theorems to the latter case.

math.SG↗
arXiv2026

Integral points and volume of integral-integral affine manifolds

We give an elementary proof that, for a closed manifold with an integral-integral affine structure, its total volume and number of integral points coincide. The proof uses rational Ehrhart theory and elementary Fourier analysis to estimate the difference between the total volume and the number of integral points.

math.DG↗
arXiv2024

A fixed-point formula for Dirac operators on Lie groupoids

We study equivariant families of Dirac operators on the source fibers of a Lie groupoid with a closed space of units and equipped with an action of an auxiliary compact Lie group. We use the Getzler rescaling method to derive a fixed-point formula for the pairing of a trace with the K-theory class of such a family. For the pair groupoid of a closed manifold, our formula reduces to the standard fixed-point formula for the equivariant index of a Dirac operator. Further examples involve foliations and manifolds equipped with a normal crossing divisor.

math.DG↗
arXiv2023

Index formula for Hamiltonian loop group spaces

We study K-theory classes of Hamiltonian loop group spaces represented by admissible Fredholm complexes. We prove various equivariant index formulae in this context. In a sequel to this article we show that, when specialized to a family of non-local elliptic boundary value problems over the moduli space of framed flat connections on a surface, one obtains a gauge theory analogue of the Teleman-Woodward index formula.

math.SG↗
arXiv2023

Families of elliptic boundary problems and index theory of the Atiyah-Bott classes

We study a natural family of non-local elliptic boundary problems on a compact oriented surface $Σ$ parametrized by the moduli space $\mathcal{M}_Σ$ of flat $G$-connections with framing along $\partial Σ$. This family generalizes one introduced by Atiyah and Bott for closed surfaces. In earlier work we constructed an analytic index morphism out of a subring of the K-theory of $\mathcal{M}_Σ$. In this article we apply that morphism to the K-class of the Fredholm family and derive cohomological formulas. The main application is to calculate K-theory intersection pairings on symplectic quotients of $\mathcal{M}_Σ$; the latter are compact moduli spaces of flat connections on surfaces with boundary, where the boundary holonomies lie in prescribed conjugacy classes. The results provide a gauge theory analogue of the Teleman-Woodward index formula.

math.SG↗
arXiv2022

Riemannian foliations and geometric quantization

We introduce geometric quantization for constant rank presymplectic structures with Riemannian null foliation and compact leaf closure space. We prove a quantization-commutes-with-reduction theorem in this context. Examples related to symplectic toric quasi-folds, suspensions of isometric actions of discrete groups, and K-contact manifolds are discussed.

math.SG↗
arXiv2022

Log symplectic manifolds and $[Q,R]=0$

We show, under an orientation hypothesis, that a log symplectic manifold with simple normal crossing singularities has a stable almost complex structure, and hence is Spin$_c$. In the compact Hamiltonian case we prove that the index of the Spin$_c$ Dirac operator twisted by a prequantum line bundle satisfies a $[Q,R]=0$ theorem.

math.SG↗
arXiv2022

Semi-classical analysis of piecewise quasi-polynomial functions and applications to geometric quantization

Motivated by applications to multiplicity formulas in index theory, we study a family of distributions $Θ(m;k)$ associated to a piecewise quasi-polynomial function $m$. The family is indexed by an integer $k \in \mathbb{Z}_{>0}$, and admits an asymptotic expansion as $k \rightarrow \infty$, which generalizes the expansion obtained in the Euler-Maclaurin formula. When $m$ is the multiplicity function arising from the quantization of a symplectic manifold, the leading term of the asymptotic expansion is the Duistermaat-Heckman measure. Our main result is that $m$ is uniquely determined by a collection of such asymptotic expansions. We also show that the construction is compatible with pushforwards. As an application, we describe a simpler proof that formal quantization is functorial with respect to restrictions to a subgroup.

math.CA↗
arXiv2022

Singular Lie filtrations and weightings

We study weightings (a.k.a. quasi-homogeneous structures) arising from manifolds with singular Lie filtrations. This generalizes constructions of Choi-Ponge, Van Erp-Yuncken, and Haj-Higson for (regular) Lie filtrations.

math.DG↗
arXiv2021

Differential Geometry of Weightings

We describe the notion of a \emph{weighting} along a submanifold $N\subset M$, and explore its differential-geometric implications. This includes a detailed discussion of weighted normal bundles, weighted deformation spaces, and weighted blow-ups. We give a description of weightings in terms of subbundles of higher tangent bundles, which leads us to notions of multiplicative weightings for Lie algebroids and Lie groupoids.

math.DG↗
arXiv2021

Connes-Moscovici Residue Cocycle For Some Dirac-Type Operators

The residue cocycle associated to a suitable spectral triple is the key component of the Connes-Moscovici local index theorem in noncommutative geometry. We review the relationship between the residue cocycle and heat kernel asymptotics. We use a modified version of the Getzler calculus to compute the cocycle for a class of Dirac-type operators introduced by Bismut, obtained by deforming a Dirac operator by a closed 3-form B. We also compute the cocycle in low-dimensions when the 3-form B is not closed.

math.OA↗
arXiv2021

Geometric quantization of b-symplectic manifolds

We introduce a method of geometric quantization for compact $b$-symplectic manifolds in terms of the index of an Atiyah-Patodi-Singer (APS) boundary value problem. We show further that b-symplectic manifolds have canonical Spin-c structures in the usual sense, and that the APS index above coincides with the index of the Spin-c Dirac operator. We show that if the manifold is endowed with a Hamiltonian action of a compact connected Lie group with non-zero modular weights, then this method satisfies the Guillemin-Sternberg ``quantization commutes with reduction'' property. In particular our quantization coincides with the formal quantization defined by Guillemin, Miranda and Weitsman, providing a positive answer to a question posed in their paper.

math.SG↗
arXiv2021

A KK-theoretic perspective on deformed Dirac operators

We study the index theory of a class of perturbed Dirac operators on non-compact manifolds of the form $\mathsf{D}+\mathrm{i}\mathsf{c}(X)$, where $\mathsf{c}(X)$ is a Clifford multiplication operator by an orbital vector field with respect to the action of a compact Lie group. Our main result is that the index class of such an operator factors as a KK-product of certain KK-theory classes defined by $\mathsf{D}$ and $X$. As a corollary we obtain the excision and cobordism-invariance properties first established by Braverman. An index theorem of Braverman relates the index of $\mathsf{D}+\mathrm{i}\mathsf{c}(X)$ to the index of a transversally elliptic operator. We explain how to deduce this theorem using a recent index theorem for transversally elliptic operators due to Kasparov.

math.KT↗
arXiv2019

Norm-square localization and the quantization of Hamiltonian loop group spaces

In an earlier article we introduced a new definition for the `quantization' of a Hamiltonian loop group space $\mathcal{M}$, involving the equivariant $L^2$-index of a Dirac-type operator $\mathscr{D}$ on a non-compact finite dimensional submanifold $\mathcal{Y}$ of $\mathcal{M}$. In this article we study a Witten-type deformation of this operator, similar to the work of Tian-Zhang and Ma-Zhang. We obtain a formula for the index with infinitely many non-trivial contributions, indexed by the components of the critical set of the norm-square of the moment map. This is the main part of a new proof of the $[Q,R]=0$ theorem for Hamiltonian loop group spaces.

math.SG↗
arXiv2019

Quasi-polynomials and the singular $[Q,R]=0$ theorem

In this short note we revisit the `shift-desingularization' version of the $[Q,R]=0$ theorem for possibly singular symplectic quotients. We take as starting point an elegant proof due to Szenes-Vergne of the quasi-polynomial behavior of the multiplicity as a function of the tensor power of the prequantum line bundle. We use the Berline-Vergne index formula and the stationary phase expansion to compute the quasi-polynomial, adapting an early approach of Meinrenken.

math.SG↗
arXiv2019

Geometric K-homology and the Freed-Hopkins-Teleman theorem

We describe a map from the equivariant twisted K-homology of a compact, connected, simply connected Lie group $G$ to the Verlinde ring. Our map is described at the level of `D-cycles' for the geometric twisted K-homology of $G$, and is inverse to the Freed-Hopkins-Teleman isomorphism. As an application, we show that two possible definitions of the `quantization' of a Hamiltonian loop group space are compatible with each other.

math.KT↗