arXiv subjects

Kadri İlker Berktav

Publications and source records attributed to Kadri İlker Berktav.

13 recordsLinked to original sources

arXiv2026

Shifted Symplectic Fibrations and Derived Thurston Theorem

In classical symplectic geometry, under mild conditions, Thurston proved that one can construct a compatible symplectic form on the total space of a symplectic fibration with a connected symplectic base. Here we prove a derived symplectic analogue of this result, called the derived Thurston theorem. More precisely, we show that if a morphism $π: X \rightarrow S$ of derived stacks has a shifted symplectic fibration structure and the target stack $S$ admits a shifted symplectic structure, then, under certain conditions, one can construct a shifted symplectic structure on a dense Zariski-open derived substack of the source stack $X$, compatible with $π$ in a sense similar to the classical case. In this derived setting, we also present numerous examples of shifted symplectic fibrations, develop an affine model construction for such fibrations, and provide applications of the derived Thurston theorem.

math.SG↗

arXiv2026

Equivariant Quotients of Derived Symplectic Spaces and Legendrian Intersection Theorem

The classical transversality lemma of contact geometry constructs a contact structure on a hypersurface transverse to a Liouville vector field using point-set topology and local flows. This paper translates the classical transversality lemma into the context of derived algebraic geometry and provides the derived Legendrian intersection theorem, along with various applications to moduli theory. In brief, we first prove that taking the quotient of a derived symplectic space descends the symplectic data to a contact structure, avoiding the transverse hypersurface requirement, such that the fundamental vector field of a weight 1 $\mathbb{G}_m$-action in the derived setting replaces the classical Liouville vector field. Second, we establish the derived Legendrian intersection theorem using base change, an $\infty$-categorical descent cube, and $\mathbb{G}_m$-equivariant lifts along the symplectification projection. As applications of the main results, we first examine the derived geometry of the discriminant loci of 1-jet bundles and show that these loci carry a $(-1)$-shifted contact structure. In addition, we show that our results apply to certain moduli problems, including projective Higgs bundles, $\ell$-adic local systems, and Lie 2-groups, and produce further examples of contact derived moduli stacks.

math.AG↗

arXiv2026

A Derived Legendrian Category for Shifted Contact Stacks

For every integer $s$ and every $n \geq 1$, we construct a symmetric monoidal $(\infty,n)$-category $\mathrm{Leg}_n^s$ whose objects are $s$-shifted contact derived Artin stacks and whose $i$-morphisms are $i$-fold Legendrian correspondences. The construction is based on the fact that an $s$-shifted contact structure on a derived Artin stack $X$ is an $s$-shifted symplectic structure on $X$ relative to $B\mathbb{G}_m$, twisted by the weight-one line bundle, and that Legendrian structures are the corresponding relative Lagrangian structures. This lets us obtain the $(\infty,n)$-categories $\mathrm{Leg}_n^s$ as twisted forms over $B\mathbb{G}_m$ of the $(\infty,n)$-categories of iterated Lagrangian correspondences of Calaque, Haugseng and Scheimbauer, with the fiber product over $B\mathbb{G}_m$ as tensor product. For an $s$-shifted contact stack $X$, we then obtain the derived Legendrian category $\mathcal{F}_{c}(X)$. Its objects are Legendrian morphisms into $X$, and its mapping categories are categories of Legendrians in derived Legendrian intersections, which carry $(s-1)$-shifted contact structures. The AKSZ construction gives oriented extended topological field theories with values in $\mathrm{Leg}_n^s$, as well as shifted contact structures on the quotient mapping stacks $[\mathrm{Map}(Y,\widetilde X)/\mathbb{G}_m]$. Our definitions of contact and Legendrian structures agree with those of Berktav where both are defined, for every shift.

math.AG↗

arXiv2026

AKSZ Construction for Shifted Contact Structures

This paper establishes the AKSZ theorem for shifted contact structures and its applications. In brief, to resolve certain obstructions, we first define the quotient mapping stack as the quotient of the symplectified mapping space by the constant multiplicative group action. We then prove that if $X$ is an $n$-shifted contact derived Artin stack and $Y$ is an $\mathcal{O}$-compact, $d$-oriented derived stack, the quotient mapping stack $$[\mathrm{Map}(Y, \widetilde{X})/\mathbb{G}_m]$$ admits an $(n-d)$-shifted contact structure. In addition, by formalizing the derived analogue of the graded contact AKSZ formalism, we also introduce the notion of weak shifted contact structures in derived algebraic geometry and prove that under global trivialization of the contact line bundle, the unmodified mapping stack inherits a weak contact structure. Extending this setup to spaces with boundary, we demonstrate that derived fillings naturally induce Legendrian morphisms between quotient mapping stacks, and that topological gluing of cobordisms evaluates to derived Legendrian intersections. Furthermore, we trace the transgression of the canonical shifted $1$-form to prove that our quotient mapping stacks satisfy the derived Classical Master Equation (CME). As applications of our quotient mapping stack formalism, we define the derived analogues of specific topological field theories, including the Jacobi, Courant-Jacobi, and Loop Space Sigma Models. Next, we establish the contact analogue of the PTVV theorem for moduli spaces of perfect complexes. Finally, by composing this geometric construction with the perverse linearization in a companion paper, we elevate these moduli spaces to generate Cohomological Contact Extended Topological Field Theories.

math.AG↗

arXiv2026

Shifted Contact Structures on Exact Symplectic Fibrations

Within the framework of classical contact geometry, the first author introduced the notion of contact symplectic structure --the data of an exact symplectic fibration with a contact base-- and proved, under mild conditions, that the total space of such a fibration inherits a contact structure compatible with the fibration map, thereby establishing the contact Thurston theorem. This paper provides a derived contact version of that result by incorporating our prior work on the derived symplectic Thurston theorem. In this paper, we prove, under certain conditions, that if a morphism $π: X \rightarrow S$ of derived stacks has a shifted exact symplectic fibration structure and the target stack $S$ admits a shifted contact structure, then one can construct a shifted contact structure on the source stack $X$, compatible with $π$ in a sense similar to the smooth case. Our framework relies on the theory of relative shifted structures; hence our result, called the derived contact Thurston theorem, in fact establishes a relative-to-absolute type construction. As an application, we present examples of our relative-to-absolute construction formalism in the derived contact setting, including conormal stacks, quotient mapping stacks, and affine exact symplectic fibrations.

math.SG↗

arXiv2025

A Mathematical Introduction to Geometric Quantization

These notes are based on a series of lectures by Kadri İlker Berktav from May 2024 to November 2024, providing a detailed exposition of geometric quantization formalism and its essential components. They are organized into three parts: background in symplectic geometry, basics of geometric quantization formalism, and an application related to Edward Witten's work in knot theory and topology.

math-ph↗

arXiv2025

Legendrian Structures in Derived Geometry

This is the third installment in a series of papers on the subject of derived contact structures. In this paper, we formally introduce the notion of a Legendrian structure in the derived context and provide natural constructions. We then present affine models and prove a Legendrian-Darboux theorem for the Legendrians in contact derived schemes.

math.SG↗

arXiv2025

On Shifted Contact Derived Artin Stacks

This is a sequel of our previous work, arXiv:2209.09686, on the development of derived contact geometry, in which we formally introduced shifted contact structures on derived stacks and proved some results for $k$-shifted contact derived schemes, with $k<0$. In this paper, we extend these results from derived schemes to derived Artin stacks. In brief, we first show that for $k<0$, every $k$-shifted contact derived Artin stack admits a contact Darboux atlas. Secondly, we canonically describe the symplectification of a derived Artin stack equipped with a $k$-shifted contact structure, where $k<0$. Lastly, we give several constructions of contact derived stacks using certain cotangent stacks and shifted prequantization structures.

math.AG↗

arXiv2025

Shifted Contact Structures and Their Local Theory

In this paper, we formally define the concept of shifted contact structures on derived (Artin) stacks and study their local properties in the context of derived algebraic geometry. In this regard, for negatively shifted contact derived $\mathbb{K}$-schemes, we develop a Darboux-like theorem and formulate the notion of symplectification.

math.AG↗

arXiv2023

Notes on Derived Geometric Formulations in Physics

This is an overview of higher structural constructions in physics. The main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, (ii) to understand how derived objects naturally appear in physics and give rise to formal mathematical treatment, and (iii) to investigate how factorization algebras together with certain higher categorical structures come into play to encode the structure of observables in physics. Adopting such a heavy and relatively enriched language allows us to formalize the notions of quantization and observables in quantum field theory as well. This document is organized to examine each task listed above in an expository manner.

math.AG↗

arXiv2023

Stacks in Einstein Gravity and a Stacky Equivalence of 3D Quantum Gravity with Gauge Theory

In this paper, we examine stacky structures in Einstein's theory of gravity. In brief, we first give a construction of the moduli stack of solutions to (vacuum) Einstein field equations on $n$-dimensional spacetimes, with vanishing cosmological constant. Using a similar approach, we also study Einstein's gravity on families of manifolds and define another stack encoding this situation as well. Secondly, we focus on the gauge theoretical interpretation of 3D gravity and the concept of equivalence of 3D quantum gravity with gauge theory. By equivalence, we essentially mean the existence of an isomorphism between the phase spaces of 3D gravity and the associated gauge theory. In this regard, we show that once it exists, the equivalence induces an isomorphism between the corresponding moduli stacks.

math.AG↗

arXiv2022

Notes on Moduli theory, Stacks and 2-Yoneda's Lemma

This note is a survey on the basic aspects of moduli theory along with some examples. In that respect, one of the purposes of this current document is to understand how the introduction of stacks circumvents the non-representability problem of the corresponding moduli functor $\mathcal{F}$ by using the 2-category of stacks. To this end, we shall briefly revisit the basics of 2-category theory and present a 2-categorical version of Yoneda's lemma.

math.AG↗

arXiv2019

An Introduction to Geometric Quantization and Witten's Quantum Invariant

This note, in a rather expository manner, serves as a conceptional introduction to the certain underlying mathematical structures encoding the geometric quantization formalism and the construction of Witten's quantum invariants, which is in fact organized in the language topological quantum field theory.

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