arXiv subjects

Simone Noja

Publications and source records attributed to Simone Noja.

At least 19 recordsLinked to original sources

arXiv2026

Genus 4 Supermoduli Space Is Not Projected

The supermoduli stack ${\mathfrak M}_g$, parametrizing smooth unpunctured super Riemann surfaces of genus $g$, is known to be non-projected for $g \ge 5$. We extend this result to the even-spin component of ${\mathfrak M}_4$. This is done by exhibiting an appropriate compact curve in the spin moduli space ${\mathcal S\mathcal M}_4^{+}$, and computing the restriction to it of the class obstructing the projectedness of ${\mathfrak M}_4^{+}$. This restriction turns out to be non-zero, proving that ${\mathfrak M}_4^{+}$ is non-projected.

math.AG↗

arXiv2026

Shifted Poisson unfoldings and quantum anomalies

Families of classical field theories often depend on geometric parameters. A basic question is whether the corresponding classical observables can be compared by flat parallel transport, and whether this comparison survives quantization. We study this problem for families of shifted Poisson structures. To such a family we attach a Poisson transverse controller $\mathbb U_\pi$, a homotopy stabilizer encoding transverse symmetries which preserve the Poisson Maurer--Cartan element up to coherent homotopy. Its flat splittings are precisely transversal shifted Poisson unfoldings, and they act on vertical symmetries, Poisson cohomology and local deformation theory. We then formulate the $\hslash$-adic lifting problem for quantized objects; its degree-two obstruction classes are the transport anomalies. The construction is realized for star-products, BV observables, factorization algebras and AKSZ theories. For the Poisson sigma model, anomaly-free quantization of the flat transport makes the Cattaneo--Felder/Kontsevich boundary product horizontal over the parameter space.

math.AG↗

arXiv2026

Compactified supermoduli space is almost never projected

We settle the projectedness problem for the compactified unpunctured supermoduli stack in every genus at least two. In genus two, the odd component is split, whereas the even component is non-projected. In every genus $g\geq 3$, both compactified parity components are non-projected.

math.AG↗

arXiv2026

Formal moduli and the splitting theory of supermanifolds

We develop a formal moduli theory for the splitting problem of complex supermanifolds. Starting from Green's obstruction tower, we construct a finite-step filtered dg Lie algebra which controls splittings by filtered Maurer-Cartan theory. We prove that the classical successive obstruction classes are recovered as the leading terms of adapted Maurer-Cartan representatives, and we transfer the theory to a minimal filtered $L_\infty$-model whose higher brackets give the intrinsic Kuranishi relations among the obstruction coordinates. We also prove that, in a precise filtered sense, the affine Atiyah class contains the entire Green obstruction tower: the Donagi-Witten component gives the primary obstruction, while the higher obstructions arise as successive projected defects and residual classes of the same Atiyah cocycle. We then pass to families, by constructing the fixed-retract formal moduli problem with prescribed split model and by identifying the formal neighbourhood of the split section with the fibrewise deformation theory of the split model; under standard finiteness, base-change, and descent hypotheses this yields relative tangent-obstruction and Kuranishi presentations. Finally, we work out explicit examples of non-split supergeometries, including cases with residual higher obstructions and a first nonlinear Kuranishi relation. These examples illustrate the interaction between Green obstructions, Atiyah classes, and higher $L_\infty$-operations.

math.AG↗

arXiv2026

A categorical and algebro-geometric theory of localization

We develop a categorical and algebro-geometric theory of localisation for cohomological theories with open--closed recollement. A class whose restriction to the open complement vanishes need not determine a preferred class on the closed stratum; the localisation triangle associates with it instead a torsor of supported refinements, whose secondary indeterminacy is governed by the connecting morphism from the open complement. We prove compatibility with excision, base change, proper pushforward, external products and local indices, and show that compatible supported constructions factor through this torsor. Under explicit Gysin hypotheses, injectivity of Euler multiplication gives a pre-Euler canonicity criterion, making the supported refinement unique before any coefficient localisation. Purity, concentration and Euler rigidification recover the usual Euler-denominator formulae. We also relate the secondary boundary group to link transgression, treat equivariant algebraic K-theory as a multiplicative analogue, and introduce Milnor localisation torsors for characteristic-class defects of singularities.

math.AG↗

arXiv2025

Cubic Dirac Operators and Dirac Cohomology for Basic Classical Lie Superalgebras

We study the Dirac cohomology of supermodules over basic classical Lie superalgebras, formulated in terms of cubic Dirac operators associated with parabolic subalgebras. Specifically, we establish a super-analog of the Casselman-Osborne theorem for supermodules with an infinitesimal character and use it to show that the Dirac cohomology of highest-weight supermodules is always non-trivial. In particular, we explicitly compute the Dirac cohomology of finite-dimensional simple supermodules for basic Lie superalgebras of type 1 with a typical highest weight, as well as of simple supermodules in the parabolic BGG category. We further investigate the relationship between Dirac cohomology and Kostant (co)homology, proving that, under suitable conditions, Dirac cohomology embeds into Kostant (co)homology. Moreover, we show that this embedding lifts to an isomorphism when the supermodule is unitarizable.

math.RT↗

arXiv2024

Adinkras and Pure Spinors

The nilpotence variety for extended supersymmetric quantum mechanics is a cone over a quadric in projective space. The pure spinor correspondence, which relates the description of off-shell supermultiplets to the classification of modules over the corresponding hypersurface ring, reduces to a classical problem of linear algebra. Spinor bundles, which correspond to maximal Cohen-Macaulay modules, serve as basic building blocks. Koszul duality appears as a deformed version of the Bernstein-Gel'fand-Gel'fand correspondence that we make fully concrete. We illustrate in numerous examples the close relationship between these connections and the powerful graphical technology of Adinkras, which appear as a decategorification of special complexes on quadrics. We emphasize the role of R-symmetry for recovering higher-dimensional gauge and gravity multiplets.

math-ph↗

arXiv2023

Poincar\'e Duality and Supergravity

We study relative differential and integral forms on families of supermanifolds and their cohomology. We prove a relative Poincar\'e--Verdier duality and show that it relates the cohomology of differential and integral forms, admitting a concrete geometric realization via Berezin fiber integration. We further introduce the Poincar\'e--dual integral form associated to an embedded even family and prove that it satisfies the correct localization property. We then apply these results to supergravity, focusing on the $3d$ case. In this setting, we show that relative Poincar\'e duality provides the natural framework for encoding the data needed to relate a superspace formulation to the physical spacetime, thereby yielding a rigorous definition of picture changing operators used in the physics literature. Building on this, after a careful analysis of the space of fields and the relevant constraints, we prove that the component, superspace, and geometric formulation of the theory are all equivalent. Finally, under suitable hypotheses, we argue that our construction illustrates a general principle governing the mathematical formulation of classical field theories on supermanifolds.

math-ph↗

arXiv2022

Six-dimensional supermultiplets from bundles on projective spaces

The projective variety of square-zero elements in the six-dimensional minimal supersymmetry algebra is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^3$. We use this fact, together with the pure spinor superfield formalism, to study supermultiplets in six dimensions, starting from vector bundles on projective spaces. We classify all multiplets whose derived invariants for the supertranslation algebra form a line bundle over the nilpotence variety; one can think of such multiplets as being those whose holomorphic twists have rank one over Dolbeault forms on spacetime. In addition, we explicitly construct multiplets associated to natural higher-rank equivariant vector bundles, including the tangent and normal bundles as well as their duals. Among the multiplets constructed are the vector multiplet and hypermultiplet, the family of $\mathcal{O}(n)$-multiplets, and the supergravity and gravitino multiplets. Along the way, we tackle various theoretical problems within the pure spinor superfield formalism. In particular, we give some general discussion about the relation of the projective nilpotence variety to multiplets and prove general results on short exact sequences and dualities of sheaves in the context of the pure spinor superfield formalism.

math-ph↗

arXiv2022

On BV Supermanifolds and the Super Atiyah Class

We study global and local geometry of forms on odd symplectic BV supermanifolds, constructed from the total space of the bundle of 1-forms on a base supermanifold. We show that globally 1-forms are an extension of vector bundles defined on the base supermanifold. In the holomorphic category, we prove that this extension is split if and only if the super Atiyah class of the base supermanifold vanishes. This is equivalent to the existence of a holomorphic superconnection: we show how this condition is related to the characteristic non-split geometry of complex supermanifolds. From a local point of view, we prove that the deformed de Rham double complex naturally arises as a de-quantization of the de Rham/Spencer double complex of the base supermanifold. Following \v{S}evera, we show that the associated spectral sequence yields semidensities on the BV supermanifold, together with their differential in the form of a super BV Laplacian.

math-ph↗

arXiv2021

On the Geometry of Forms on Supermanifolds

This paper provides a rigorous account on the geometry of forms on supermanifolds, with a focus on its algebraic-geometric aspects. First, we introduce the de Rham complex of differential forms and we compute its cohomology. We then discuss three intrinsic definitions of the Berezinian sheaf of a supermanifold - as a quotient sheaf, via cohomology of the super Koszul complex or via cohomology of the total de Rham complex. Further, we study the properties of the Berezinian sheaf, showing in particular that it defines a right $\mathcal{D}$-module. Then we introduce integral forms and their complex and we compute their cohomology, by providing a suitable Poincar\'e lemma. We show that the complex of differential forms and integral forms are quasi-isomorphic and their cohomology computes the de Rham cohomology of the reduced space of the supermanifold. The notion of Berezin integral is then introduced and put to the good use to prove the superanalog of Stokes' theorem and Poincar\'e duality, which relates differential and integral forms on supermanifolds. Finally, a different point of view is discussed by introducing the total tangent supermanifold and (integrable) pseudoforms in a new way. In this context, it is shown that a particular class of integrable pseudoforms having a distributional dependence supported at a point on the fibers are isomorphic to integral forms. Within the general overview several new proofs of results are scattered.

math.AG↗

arXiv2020

A Note on Super Koszul Complex and the Berezinian

We construct the super Koszul complex of a free supercommutative $A$-module $V$ of rank $p|q$ and prove that its homology is concentrated in a single degree and it yields an exact resolution of $A$. We then study the dual of the super Koszul complex and show that its homology is concentrated in a single degree as well and isomorphic to $\Pi^{p+q} A$, with $\Pi$ the parity changing functor. Finally, we show that, given an automorphism of $V$, the induced transformation on the only non-trivial homology class of the dual of the super Koszul complex is given by the multiplication by the Berezinian of the automorphism, thus relating this homology group with the Berezinian module of $V$.

math.AG↗

arXiv2020

The Universal de Rham/Spencer Double Complex on a Supermanifold

The universal Spencer and de Rham complexes of sheaves over a smooth or analytical manifold are well known to play a basic role in the theory of $\mathcal{D}$-modules. In this article we consider a double complex of sheaves generalizing both complexes for an arbitrary supermanifold, and we use it to unify the notions of differential and integral forms on real, complex and algebraic supermanifolds. The associated spectral sequences give the de Rham complex of differential forms and the complex of integral forms at page one. For real and complex supermanifolds both spectral sequences converge at page two to the locally constant sheaf. We use this fact to show that the cohomology of differential forms is isomorphic to the cohomology of integral forms, and they both compute the de Rham cohomology of the reduced manifold. Furthermore, we show that, in contrast with the case of ordinary complex manifolds, the Hodge-to-de Rham (or Fr\"olicher) spectral sequence of supermanifolds with K\"ahler reduced manifold does not converge in general at page one.

math.AG↗

arXiv2019

$A_\infty$-Algebra from Supermanifolds

Inspired by the analogy between different types of differential forms on supermanifolds and string fields in superstring theory, we construct new multilinear non-associative products of forms which yield an $A_\infty$-algebra.

hep-th↗

arXiv2018

Non-Projected Supermanifolds and Embeddings in Super Grassmannians

In this paper we give a brief account of the relations between non-projected supermanifolds and projectivity in supergeometry. Following the general results of arXiv:1706.01354, we study an explicit example of non-projected and non-projective supermanifold over the projective plane and we show how to embed it into a super Grassmannian. The geometry of super Grassmannians is also reviewed in details.

math.AG↗

arXiv2017

Projective Superspaces in Practice

We study the supergeometry of complex projective superspaces $\mathbb{P}^{n|m}$. First, we provide formulas for the cohomology of invertible sheaves of the form $\mathcal{O}_{\mathbb{P}^{n|m}} (\ell)$, that are pull-back of ordinary invertible sheaves on the reduced variety $\mathbb{P}^n$. Next, by studying the even Picard group $\mbox{Pic}_0 (\mathbb{P}^{n|m})$, classifying invertible sheaves of rank $1|0$, we show that the sheaves $\mathcal{O}_{\mathbb {P}^{n|m}} (\ell)$ are not the only invertible sheaves on $\mathbb{P}^{n|m}$, but there are also new genuinely supersymmetric invertible sheaves that are unipotent elements in the even Picard group. We study the $\Pi$-Picard group $\mbox{Pic}_\Pi (\mathbb{P}^{n|m})$, classifying $\Pi$-invertible sheaves of rank $1|1$, proving that there are also non-split $\Pi$-invertible sheaves on supercurves $\mathbb{P}^{1|m}$. Further, we investigate infinitesimal automorphisms and first order deformations of $\mathbb{P}^{n|m}$, by studying the cohomology of the tangent sheaf using a supersymmetric generalisation of the Euler exact sequence. A special special attention is paid to the meaningful case of supercurves $\mathbb{P}^{1|m}$ and of Calabi-Yau's $\mathbb{P}^{n|n+1}$. Last, with an eye to applications to physics, we show in full detail how to endow $\mathbb{P}^{1|2}$ with the structure of $\mathcal{N}=2$ super Riemann surface and we obtain its SUSY-preserving infinitesimal automorphisms from first principles, that prove to be the Lie superalgebra $\mathfrak{osp} (2|2)$. A particular effort has been devoted to keep the exposition as concrete and explicit as possible.

math.AG↗

arXiv2017

Non Projected Calabi-Yau Supermanifolds over $\mathbb{P}^2$

We start a systematic study of non-projected supermanifolds, concentrating on supermanifolds with fermionic dimension 2 and with the reduced manifold a complex projective space. We show that all the non-projected supermanifolds of dimension $2|2$ over $\mathbb{P}^2$ are completely characterised by a non-zero 1-form $\omega$ and by a locally free sheaf $\mathcal{F}$ of rank $0|2$, satisfying $Sym^2 \mathcal{F} \cong K_{\mathbb{P}^2}$. Denoting such supermanifolds with $\mathbb{P}^{2}_\omega(\mathcal{F})$, we show that all of them are Calabi-Yau supermanifolds and, when $\omega \neq 0$, they are non-projective, that is they cannot be embedded into any projective superspace $\mathbb{P}^{n|m}$. Instead, we show that every non-projected supermanifolds over $\mathbb{P}^2$ admits an embedding into a super Grassmannian. By contrast, we give an example of a supermanifold $\mathbb P^{2}_\omega(\mathcal F)$ that cannot be embedded in any of the $\Pi$-projective superspaces $\mathbb P^{n}_{\Pi}$ introduced by Manin and Deligne. However, we also show that when $\mathcal F$ is the cotangent bundle over $\mathbb{P}^2$, then the non-projected $\mathbb{P}^2_\omega(\mathcal F)$ and the $\Pi$-projective plane $\mathbb P^{2}_{\Pi}$ do coincide.

math.AG↗

arXiv2017

Supergeometry of $\Pi$-Projective Spaces

In this paper we prove that $\Pi$-projective spaces $\mathbb{P}^n_\Pi$ arise naturally in supergeometry upon considering a non-projected thickening of $\mathbb{P}^n$ related to the cotangent sheaf $\Omega^1_{\mathbb{P}^n}$. In particular, we prove that for $n \geq 2$ the $\Pi$-projective space $\mathbb{P}^n_\Pi$ can be constructed as the non-projected supermanifold determined by three elements $(\mathbb{P}^n, \Omega^1_{\mathbb{P}^n}, \lambda)$, where $\mathbb{P}^n$ is the ordinary complex projective space, $\Omega^1_{\mathbb{P}^n}$ is its cotangent sheaf and $\lambda $ is a non-zero complex number, representative of the fundamental obstruction class $\omega \in H^1 (\mathcal{T}_{\mathbb{P}^n} \otimes \bigwedge^2 \Omega^1_{\mathbb{P}^n}) \cong \mathbb{C}.$ Likewise, in the case $n=1$ the $\Pi$-projective line $\mathbb{P}^1_\Pi$ is the split supermanifold determined by the pair $(\mathbb{P}^1, \Omega^1_{\mathbb{P}^1} \cong \mathcal{O}_{\mathbb{P}^1} (-2)).$ Moreover we show that in any dimension $\Pi$-projective spaces are Calabi-Yau supermanifolds. To conclude, we offer pieces of evidence that, more in general, also $\Pi$-Grassmannians can be constructed the same way using the cotangent sheaf of their underlying reduced Grassmannians, provided that also higher, possibly fermionic, obstruction classes are taken into account. This suggests that this unexpected connection with the cotangent sheaf is characteristic of $\Pi$-geometry.

math.AG↗