A Note on Derived Geometric Interpretation of Classical Field Theories

In this note, we would like to provide a conceptional introduction to the interaction between derived geometry and physics based on the formalism that has been heavily studied by Kevin Costello. Main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, which can be, roughly speaking, thought of as a higher categorical refinement of an ordinary algebraic geometry, (ii) to understand how certain derived objects naturally appear in a theory describing a particular physical phenomenon and give rise to a formal mathematical treatment, such as redefining a perturbative classical field theory (or its quantum counterpart) by using the language of derived algebraic geometry, and (iii) how the notion of factorization algebra together with certain higher categorical structures come into play to encode the structure of so-called observables in those theories by employing certain cohomological/homotopical methods. Adopting such a heavy and relatively enriched language allows us to formalize the notion of quantization and observables in quantum field theory as well.Comment: 14 pages. This note serves as an introductory survey on certain mathematical structures encoding the essence of Costello's approach to derived-geometric formulation of field theories and the structure of observables in an expository manne

Stacky Formulations of Einstein Gravity

This is an investigation of "stacky" structures for Einstein gravity together with an alternative reformulation in the language of formal moduli problems. In the first part of the paper, we first revisit the aspects of (vacuum) Einstein gravity on a Lorentzian 3-manifold $M$ with cosmological constant $\Lambda=0$. Next, we shall provide a realization of the moduli space of Einstein's field equations as a certain stack. We indeed construct the stack of (vacuum) Einstein gravity in $n$-dimensional set-up with vanishing cosmological constant by using the homotopy theoretical formulation of stacks. With this new formulation, we also upgrade the equivalence of certain 2+1 quantum gravities with gauge theory to the isomorphism between the corresponding moduli stacks. The second part of the paper, on the other hand, is designed as a detailed survey on formal moduli problems. It is in particular devoted to formalize Einstein gravity in the language of formal moduli problems and to study the algebraic structure of observables in terms of factorization algebras.Comment: 54 page

Shifted Contact Structures on Derived Schemes and Their Local Theory

In this paper, we formally define the concept of a $``k$-shifted contact structure$"$ on a derived $\mathbb{K}$-scheme and study its local properties in the context of derived algebraic geometry. In this regard, we develop Darboux-type local models for $k$-shifted contact structures and present a Darboux-like theorem. More precisely, we prove that every $k$-shifted contact derived $\mathbb{K}$-scheme $({\bf X}, \alpha)$ with $k<0$ is locally equivalent to $(Spec A, \alpha_0)$ for $Spec A$ an affine derived $\mathbb{K}$-scheme and $A$ a commutative differential graded $\mathbb{K}$-algebra such that the pair $(A, \alpha_0)$ is in a Darboux-type form. Furthermore, we formulate the so-called symplectization $\mathcal{S}_{{\bf{X}}}(k)$ of a $k$-shifted contact derived $\mathbb{K}$-scheme $({\bf X}, \alpha)$ and give a canonical construction for the space $\mathcal{S}_{{\bf{X}}}(k)$.Comment: 19 page

Einstein gravitasyon kuramının staksal formülasyonları

This is a thesis on higher structures in geometry and physics. Indeed, the current work involves an extensive and relatively self-contained investigation of higher categorical and stacky structures in (vacuum) Einstein gravity with vanishing cosmological constant. In the first three chapters of the thesis, we shall provide a realization of the moduli space of Einstein’s field equations as a certain higher space (a stack). In this part of the thesis, the first aim is to present the construction of the moduli stack of vacuum Einstein gravity with vanishing cosmological constant in an n-dimensional setup. In particular, we shall be interested in the moduli space of 3D Einstein gravity on specific Lorentzian spacetimes. With this spirit, the second goal of this part is to show that once it exists, the equivalence of 3D quantum gravity with gauge theory in a particular setup, in fact, induces an isomorphism between the corresponding moduli stacks where the setup involves Lorentzian spacetimes of the form Mx R with M being a closed Riemann surface of genus g > 1. For our purposes, we shall employ a particular treatment that is essentially based on a formulation of stacks in the language of homotopy theory. The remainder of the thesis, on the other hand, is designed as a detailed survey on formal moduli problems, and it is particularly devoted to formalizing specific Einstein gravities in the language of formal moduli problems and L_infinity-algebras. Such an approach allows us to encode further higher structures in the theory if needed. To be more precise, this leads to the realization of the space of fields as a certain higher/derived stack (a formal moduli problem) endowed with more sensitive higher structure (encoding the possible higher symmetries/equivalences in the theory) once we ask the theory to possess higher symmetries. As a particular example, we use this approach to formulate specific 3D Einstein-Cartan-Palatini gravity. In addition, using local models for such higher structures and the algebra of functions on these higher spaces, we intend to study the algebraic structure of observables of 3D Einstein-Cartan-Palatini gravity as well.Bu tez, geometri ve fizikte ortaya çıkan çeşitli yüksek geometrik ve cebirsel yapıları ele almaktadır. Özel olarak, bu mevcut araştırma, vakum Einstein gravitasyon kuramında (kozmolojik sabit sıfır alınmaktadır) ortaya çıkan yüksek kategori teorisel ve "stacky" yapıları araştırarak, bu yapılar yardımıyla ortaya çıkan alternatif formülasyonları inceleyen bir çalışmalar bütünüdür. Bu tezin ilk üç bölümünde temel olarak, Einstein denklemlerinin moduli uzayının, özel bir yüksek uzay (stak) olarak nasıl yorumlanabileceği tartışılmaktadır. Stak dilini kullanan bir takım çalışmalardan yola çıkarak, bu tezde Einstein denklemlerinin moduli uzayı için benzer sonuçlar gösterilmiştir. Bu bağlamda, ilk olarak, bahsi geçen Einstein gravitasyonu için n boyutlu durumdaki ilgili moduli stakın inşası verilecektir. Bununla birlikte, n = 3 durumu özel olarak tartışılacaktır. Bu yeni formülasyonla birlikte, özellikle üç boyutlu kuantum gravitasyonu ve ayar kuramı arasındaki özel bir durumdaki denkliğin, aslında ilgili kuramlar için inşa edilen çözüm uzaylarının stak olarak izomorfik olmalarına yol açtığı gösterilmektedir. Buradaki özel durum, MxR tipinde Lorentz uzayzamanlarını kapsamaktadır (M, genus g > 1 kapalı Riemann yüzeyidir). Öte yandan tezin geri kalan bölümleri, formal moduli problemleri ve L_infinity-cebirleri üzerine detaylı bir literatür taraması içerecek şeklinde tasarlanmış olup, buradaki bölümlerde bu kavramların Einstein kuramı ile ilişkisi incelenmektedir. Bu yaklaşım sayesinde kuramların çözüm uzayları, kuramlarda ortaya çıkabilecek muhtemel yüksek simetrileri/denklikleri tespit etme konusunda daha hassas yüksek yapılara sahip olan, bir takım özel "derived" uzaylar (formal moduli problemleri) şeklinde görülebilmektedir. Tezin son bölümlerinde bu yaklaşımın özel bir durumu ve örneği olarak, 3D Einstein-Cartan-Palatini gravitasyon kuramının bu nesneler aracılığıyla formülasyonu üzerinde durulmuştur. Ayrıca bu bölümde yüksek uzayların yerel modelleri ve bu nesneler üzerindeki fonksiyonlar cebiri incelenerek, 3D Einstein-Cartan-Palatini gravitasyon kuramındaki gözlenebilirlerin cebirsel yapısı da çalışılmaktadır.Ph.D. - Doctoral Progra