Enriched Morita theory of monoids in a closed symmetric monoidal category
We develop Morita theory of monoids in a closed symmetric monoidal category, in the context of enriched category theory.
math.CT↗
arXiv subjects
Jae-Suk Park
Publications and source records attributed to Jae-Suk Park.
Affine group dg-schemes and linear representations I - Basic theory and Tannakian reconstructions
We develop a basic theory of affine group dg-schemes, their Lie algebraic counterparts and linear representations. We prove Tannaka type reconstruction theorems that an affine group dg-scheme can be recovered from the dg-tensor category of its linear representations as well as from the rigid dg-tensor category of its finite dimensional linear representations along with the forgetful functors to the underlying dg-tensor category of cochain complexes.
math.AG↗
Representable presheaves of groups on the homotopy category of cocommutative dg-coalgebras and Tannakian reconstruction
Motivated by rational homotopy theory, we study a representable presheaf of groups $\mathbf{\mathfrak{P}}$ on the homotopy category of cocommutative differential graded coalgebras, its Lie algebraic counterpart and its linear representations. We prove a Tannaka type reconstruction theorem that $\mathbf{\mathfrak{P}}$ can be recovered from the dg-category of its linear representations along with the forgetful dg-functor to the underlying dg-category of chain complexes.
math.AG↗
Homotopical Computations in Quantum Fields Theory
This paper is a mathematical study of quantum correlation functions in quantum field theory within a homotopy algebraic framework motivated from the BV quantization scheme. We characterize quantum correlation functions by algebraic homotopy theoretical methods which circumvent gauge fixing and perturbative Feynman diagrams. We show that there is a universal algebraic structure, closely related with that of the WDVV equation, governing quantum correlation functions of every quantum field theory in our framework up to a certain ambiguity. The algebraic structure is independent of the details of the quantum expectation, other than its existence with the prescribed symmetry, and comes with a concrete algorithm for explicit computations. We will also make proposals for the precise natures of quantum expectation and physical equivalence of quantum field theories.
math.QA↗
Enhanced homotopy theory for period integrals of smooth projective hypersurfaces
The goal of this paper is to reveal hidden structures on the singular cohomology and the Griffiths period integral of a smooth projective hypersurface in terms of BV(Batalin-Vilkovisky) algebras and homotopy Lie theory (so called, $L_\infty$-homotopy theory). Let $X_G$ be a smooth projective hypersurface in the complex projective space $\mathbf{P}^n$ defined by a homogeneous polynomial $G(\underline x)$ of degree $d \geq 1$. Let $\mathbb{H}=H^{n-1}_{\operatorname{prim}}(X_G, \mathbb{C})$ be the middle dimensional primitive cohomology of $X_G$. We explicitly construct a BV algebra $\mathbf{B\!\!V}_{\! \!X}=(\mathcal{A}_X,Q_X, K_X)$ such that its $0$-th cohomology $H^0_{K_X}(\mathcal{A}_X)$ is canonically isomorphic to $\mathbb{H}$. We also equip $\mathbf{B\!\!V}_{\! \!X}$ with a decreasing filtration and a bilinear pairing which realize the Hodge filtration and the cup product polarization on $\mathbb{H}$ under the canonical isomorphism. Moreover, we lift $C_{[γ]}:\mathbb{H} \to \mathbb{C}$ to a cochain map $\mathcal{C}_γ:(\mathcal{A}_X, K_X) \to (\mathbb{C},0)$, where $C_{[γ]}$ is the Griffiths period integral given by $ω\mapsto \int_γω$ for $[γ]\in H_{n-1}(X_G,\mathbb{Z})$. We use this enhanced homotopy structure on $\mathbb{H}$ to study an extended formal deformation of $X_G$ and the correlation of its period integrals. If $X_G$ is in a formal family of Calabi-Yau hypersurfaces $X_{G_{\underline T}}$, we provide an explicit formula and algorithm (based on a Gröbner basis) to compute the period matrix of $X_{G_{\underline T}}$ in terms of the period matrix of $X_G$ and an $L_\infty$-morphism $\underline κ$ which enhances $C_{[γ]}$ and governs deformations of period matrices.
math.AG↗
Homotopy Theory of Probability Spaces I: Classical independence and homotopy Lie algebras
This is the first installment of a series of papers whose aim is to lay a foundation for homotopy probability theory by establishing its basic principles and practices. The notion of a homotopy probability space is an enrichment of the notion of an algebraic probability space with ideas from algebraic homotopy theory. This enrichment uses a characterization of the laws of random variables in a probability space in terms of symmetries of the expectation. The laws of random variables are reinterpreted as invariants of the homotopy types of infinity morphisms between certain homotopy algebras. The relevant category of homotopy algebras is determined by the appropriate notion of independence for the underlying probability theory. This theory will be both a natural generalization and an effective computational tool for the study of classical algebraic probability spaces, while keeping the same central limit. This article is focused on the commutative case, where the laws of random variables are also described in terms of certain affinely flat structures on the formal moduli space of a naturally defined family attached to the given algebraic probability space. Non-commutative probability theories will be the main subject of the sequels. (This work is a spin-off from the author's program to characterize path integrals of quantum field theory in terms of the symmetries of the quantum expectation which should satisfy a certain coherence with a particular weight filtration generated by the Planck constant $\hbar$. A similar idea is adopted here in a simplified form, without the $\hbar$-conditions, and, in return, many results in this paper will be used as background materials in the forthcoming work on homotopy theory of quantum fields).
math.PR↗
Homotopy Probability Theory II
This is the second of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. This paper outlines how the framework can assist in the development of homotopy probability theory, where a vector space of random variables is replaced by a chain complex of random variables. This allows the principles of derived mathematics to participate in classical and noncommutative probability theory. A simple example is presented.
math.PR↗
Homotopy Probability Theory I
This is the first of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. In this paper, cumulants are proved to coincide with morphisms of homotopy algebras. The sequel paper outlines how the framework presented here can assist in the development of homotopy probability theory, allowing the principles of derived mathematics to participate in classical and noncommutative probability theory.
math.PR↗
Algebraic Principles of Quantum Field Theory II: Quantum Coordinates and WDVV Equation
This paper is about algebro-geometrical structures on a moduli space $\CM$ of anomaly-free BV QFTs with finite number of inequivalent observables or in a finite superselection sector. We show that $\CM$ has the structure of F-manifold -- a linear pencil of torsion-free flat connection with unity on the tangent space, in quantum coordinates. We study the notion of quantum coordinates for the family of QFTs, which determines the connection 1-form as well as every quantum correlation function of the family in terms of the 1-point functions of the initial theory. We then define free energy for an unital BV QFT and show that it is another avatar of morphism of QFT algebra. These results are consequences of the solvability of refined quantum master equation of the theory. We also introduce the notion of a QFT integral and study some properties of BV QFT equipped with a QFT integral. We show that BV QFT with a non-degenerate QFT integral leads to the WDVV equation---the formal Frobenius manifold structure on $\CM$---if it admits a semi-classical solution of quantum master equation.
math-ph↗
Algebraic Principles of Quantum Field Theory I: Foundation and an exact solution of BV QFT
This is the first in a series of papers on an attempt to understand quantum field theory mathematically. In this paper we shall introduce and study BV QFT algebra and BV QFT as the proto-algebraic model of quantum field theory by exploiting Batalin-Vilkovisky quantization scheme. We shall develop a complete theory of obstruction (anomaly) to quantization of classical observables and propose that expectation value of quantized observable is certain quantum homotopy invariant. We shall, then, suggest a new method, bypassing Feynman's path integrals, of computing quantum correlation functions when there is no anomaly. An exact solution for all quantum correlation functions shall be presented provided that the number of equivalence classes of observables is finite for each ghost numbers. Such a theory shall have its natural family parametrized by a smooth-formal moduli space in quantum coordinates, which notion generalize that of flat or special coordinates in topological string theories and shall be interpreted as an example of quasi-isomorphism of general QFT algebra.
math-ph↗
Quantum backgrounds and QFT
We introduce the concept of a quantum background and a functor QFT. In the case that the QFT moduli space is smooth formal, we construct a flat quantum superconnection on a bundle over QFT which defines algebraic structures relevant to correlation functions in quantum field theory. We go further and identify chain level generalizations of correlation functions which should be present in all quantum field theories.
math.QA↗
Topological Sigma B Model in 4-Dimensions
We propose a 4-dimensional version of topological sigma B-model, governing maps from a smooth compact 4-manifold M to a Calabi-Yau target manifold X. The theory depends on on complex structure of X, while is independent of Kaehler metric of X. The theory is also a 4-dimensiona topological field theory in the sense that the theory is independent of variation of Riemannian metric of the source 4-manifold M, potentially leading to new smooth invariant of 4-manifolds. We argue that the theory also comes with a topological family parametrized by the extended moduli space of complex structures.
hep-th↗
Semi-Classical Quantum Fields Theories and Frobenius Manifolds
We show that a semi-classical quantum field theory comes with a versal family with the property that the corresponding partition function generates all path integrals and satisfies a system of 2nd order differential equations determined by algebras of classical observables. This versal family gives rise to a notion of special coordinates that is analogous to that in string theories. We also show that for a large class of semi-classical theories, their moduli space has the structure of a Frobenius super-manifold.
hep-th↗
Deformations of coisotropic submanifolds and strong homotopy Lie algebroids
In this paper, we study deformations of coisotropic submanifolds in a symplectic manifold. First we derive the equation that governs $C^\infty$ deformations of coisotropic submanifolds and define the corresponding $C^\infty$-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies. This is a non-commutative and non-linear generalization of the well-known description of the local deformation space of Lagrangian submanifolds as the set of graphs of {\it closed} one forms in the Darboux-Weinstein chart of a given Lagrangian submanifold. We then introduce the notion of {\it strong homotopy Lie algebroid} (or {\it $L_\infty$-algebroid}) and associate a canonical isomorphism class of strong homotopy Lie algebroids to each pre-symplectic manifold $(Y,ω)$ and identify the formal deformation space of coisotropic embeddings into a symplectic manifold in terms of this strong homotopy Lie algebroid. The formal moduli space then is provided by the gauge equivalence classes of solutions of a version of the {\it Maurer-Cartan equation} (or the {\it master equation}) of the strong homotopy Lie algebroid, and plays the role of the classical part of the moduli space of quantum deformation space of coisotropic $A$-branes. We provide a criterion for the unobstructedness of the deformation problem and analyze a family of examples that illustrates that this deformation problem is obstructed in general and heavily depends on the geometry and dynamics of the null foliation.
math.SG↗
BV Quantization of Topological Open Membranes
We study bulk-boundary correlators in topological open membranes. The basic example is the open membrane with a WZ coupling to a 3-form. We view the bulk interaction as a deformation of the boundary string theory. This boundary string has the structure of a homotopy Lie algebra, which can be viewed as a closed string field theory. We calculate the leading order perturbative expansion of this structure. For the 3-form field we find that the C-field induces a trilinear bracket, deforming the Lie algebra structure. This paper is the first step towards a formal universal quantization of general quasi-Lie bialgebroids.
hep-th↗
Flat family of QFTs and quantization of d-algebras
Exploiting the path integral approach al la Batalin and Vilkovisky, we show that any anomaly-free Quantum Field Theory (QFT) comes with a family parametrized by certain moduli space M, which tangent space at the point corresponding to the initial QFT is given by the space of all observables. Furthermore the tangent bundle over M is equipped with flat quantum connection, which can be used to determine all correlation functions of the family of QFTs. We also argue that considering family of QFTs is an inevitable step, due to the fact that the products of quantum observables are not quantum observables in general, which leads to a new "global" perspective on quantum world. We also uncover structure of $d$-algebra in the large class of d-dimensional QFT. This leads to an universal quantization machine for d-algebras decorated by algebro-differential-topology of (d+1)-manifolds as well as a new perspective on differential-topology of low dimensions. This paper is a summary of a forthcoming paper of this author.
hep-th↗
Topological Open Membranes
We study topological open membranes of BF type in a manifest BV formalism. Our main interest is the effect of the bulk deformations on the algebra of boundary operators. This forms a homotopy Lie algebra, which can be understood in terms of a closed string field theory. The simplest models are associated to quasi-Lie bialgebras and are of Chern-Simons type. More generally, the induced structure is a Courant algebroid, or ``quasi-Lie bialgebroid'', with boundary conditions related to Dirac bundles. A canonical example is the topological open membrane coupling to a closed 3-form, modeling the deformation of strings by a C-field. The Courant algebroid for this model describes a modification of deformation quantization. We propose our models as a tool to find a formal solution to the quantization problem of Courant algebroids.
hep-th↗
Higgs Bundles and Four Manifolds
It is known that the Seiberg-Witten invariants, derived from supersymmetric Yang-Mill theories in four-dimensions, do not distinguish smooth structure of certain non-simply-connected four manifolds. We propose generalizations of Donaldson-Witten and Vafa-Witten theories on a Kähler manifold based on Higgs Bundles. We showed, in particular, that the partition function of our generalized Vafa-Witten theory can be written as the sum of contributions our generalized Donaldson-Witten invariants and generalized Seiberg-Witten invariants. The resulting generalized Seiberg-Witten invariants might have, conjecturally, information on smooth structure beyond the original Seiberg-Witten invariants for non-simply-connected case.
hep-th↗