Topological Quantum Mechanics Research Articles

On protected defect correlators in 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document}≥ 4 theories

We study and compute supersymmetric observables for line defects in 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} ≥ 4 theories. Our setup is a novel supersymmetric configuration involving line operators and local operators living on a linked circle. The algebra of the local operators is described by a topological quantum mechanics. For operators belonging to conserved current multiplets, we propose an exact formula for their correlation functions based on a Ward identity for integrated correlators. Our formula gives a general recipe to compute the bremsstrahlung function for any 13\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{3} $$\\end{document}-BPS lines in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 6 SCFTs. We apply our relation to the 12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{2} $$\\end{document}-BPS Wilson loop in the ABJM model, showing the validity of previous computations. Furthermore, our construction allows us to explore higher points correlators. As an example, we compute the two-point function of the stress tensor multiplet correlators in ABJM theory in the presence of the Wilson line. We also present some perturbative checks of our formulae.

Homotopy transfer for QFT on non-compact manifolds with boundary: A case study

In this work we report a homological perturbation calculation to construct effective theories of topological quantum mechanics on R⩾0. Such calculation can be regarded as a generalization of Feynman graph computations. The resulting effective theories fit into a derived BV algebra structure, which generalizes BV quantization. Besides, our construction may serve as the simplest example of a process called “boundary transfer”, which may help study bulk-boundary correspondence.

Covariant action for conformal higher spin gravity

Conformal higher spin (HS) gravity is a HS extension of Weyl gravity and is a family of local HS theories, which was put forward by Segal and Tseytlin. We propose a manifestly covariant and coordinate-independent action for these theories. The result is based on an interplay between HS symmetries and deformation quantization: a locally equivalent but manifestly background-independent reformulation, known as the parent system, of the off-shell multiplet of conformal HS fields (Fradkin–Tseytlin fields) can be interpreted in terms of Fedosov deformation quantization of the underlying cotangent bundle. This brings into the game the invariant quantum trace, induced by the Feigin–Felder–Shoikhet cocycle of Weyl algebra, which extends Segal’s action into a gauge invariant and globally well-defined action functional on the space of configurations of the parent system. The same action can be understood within the worldline approach as a correlation function in the topological quantum mechanics on the circle.

Mathematical structures of cohomological field theories

A mathematical framework of cohomological field theories (CohFTs) is formulated in the language of bigraded manifolds. Algebraic properties of operators in CohFTs are studied. Methods of constructing CohFTs, with or without gauge symmetries, are discussed. In particular, a generalization of the Mathai-Quillen formalism is given. Examples such as topological quantum mechanics, topological sigma model, topological M-theory, and topological Yang-Mills theory can be obtained uniformly using this new formalism.

Correction: Two field-theoretic viewpoints on the Fukaya-Morse A∞ category

We study an enhanced version of the Morse degeneration of Fukaya $A_\infty$ category with higher compositions given by counts of gradient flow trees. The enhancement consists in allowing morphisms from an object to itself to be chains on the manifold. Higher compositions correspond to counting Morse trees passing through a given set of chains. We provide two viewpoints on the construction and on the proof of the $A_\infty$ relations for the composition maps. One viewpoint is via an effective action for the $BF$ theory computed in a special gauge. The other is via higher topological quantum mechanics.

Two field-theoretic viewpoints on the Fukaya-Morse $$A_\infty $$ category

We study an enhanced version of the Morse degeneration of Fukaya \(A_\infty \) category with higher compositions given by numbers of gradient flow trees. The enhancement consists in allowing morphisms from an object to itself to be chains on the manifold. Higher compositions correspond to counting Morse trees passing through a given set of chains. We provide two viewpoints on the construction and on the proof of the \(A_\infty \) relations for the composition maps. One viewpoint is via an effective action for the BF theory computed in a special gauge. The other is via higher topological quantum mechanics.

On Globalized Traces for the Poisson Sigma Model

A globalized version of a trace formula for the Poisson Sigma Model on the disk is presented by using its formal global picture in the setting of the Batalin–Vilkovisky formalism. This global construction includes the concept of zero modes. Moreover, for the symplectic case of the Poisson Sigma Model with cotangent target, the globalized trace reduces to a symplectic construction which was presented by Grady, Li and Li for 1-dimensional Chern–Simons theory (topological quantum mechanics). In addition, the connection between this formula and the Nest–Tsygan theorem and the Tamarkin–Tsygan theorem is explained.

Superconformal Line Defects in 3D

We review the recent progress in the study of line defects in three-dimensional Chern–Simons-matter superconformal field theories, notably the ABJM theory. The first part is focused on kinematical defects, supporting a topological sector of the theory. After reviewing the construction of this sector, we concentrate on the evaluation of topological correlators from the partition function of the mass-deformed ABJM theory and provide evidence on the existence of topological quantum mechanics living on the line. In the second part, we consider the dynamical defects realized as latitude BPS Wilson loops for which an exact evaluation is available in terms of a latitude Matrix Model. We discuss the fundamental relation between these operators, the defect superconformal field theory and bulk physical quantities, such as the Bremsstrahlung function. This relation assigns a privileged role to BPS Wilson operators, which become the meeting point for three exact approaches: localization, integrability and conformal bootstrap.

Geometry of Localized Effective Theories, Exact Semi-classical Approximation and the Algebraic Index

In this paper we propose a general framework to study the quantum geometry of \(\sigma \)-models when they are effectively localized to small quantum fluctuations around constant maps. Such effective theories have surprising exact descriptions at all loops in terms of target geometry and can be rigorously formulated. We illustrate how to turn the physics idea of exact semi-classical approximation into a geometric set-up in this framework, using Gauss–Manin connection. As an application, we carry out this program in details by the example of topological quantum mechanics, and explain how to implement the idea of exact semi-classical approximation into a proof of the algebraic index theorem. The proof resembles much of the physics derivation of Atiyah–Singer index theorem and clarifies the geometric face of many other mathematical constructions.

On a gravity dual to flavored topological quantum mechanics

In this paper, we propose a generalization of the AdS2/CFT1 correspondence constructed by Mezei, Pufu and Wang in [1], which is the duality between 2d Yang-Mills theory with higher derivatives in the AdS2 background, and 1d topological quantum mechanics of two adjoint and two fundamental U(N ) fields, governing certain protected sector of operators in 3d ABJM theory at the Chern-Simons level k = 1. We construct a holographic dual to a flavored generalization of the 1d quantum mechanics considered in [1], which arises as the effective field theory living on the intersection of stacks of N D2-branes and k D6-branes in the Ω-background in Type IIA string theory, and describes the dynamics of the protected sector of operators in mathcal{N} = 4 theory with k fundamental hypermultiplets, having a holographic description as M-theory in the AdS4× S7/ℤk background. We compute the structure constants of the bulk theory gauge group, and construct a map between the observables of the boundary theory and the fields of the bulk theory.

Twisted characters and holomorphic symmetries

We consider holomorphic twists of arbitrary supersymmetric theories in four dimensions. Working in the BV formalism, we rederive classical results characterizing the holomorphic twist of chiral and vector supermultiplets, computing the twist explicitly as a family over the space of nilpotent supercharges in minimal supersymmetry. The BV formalism allows one to work with or without auxiliary fields, according to preference; for chiral superfields, we show that the result of the twist is an identical BV theory, the holomorphic beta gamma system with superpotential, independent of whether or not auxiliary fields are included. We compute the character of local operators in this holomorphic theory, demonstrating agreement of the free local operators with the usual index of free fields. The local operators with superpotential are computed via a spectral sequence and are shown to agree with functions on a formal mapping space into the derived critical locus of the superpotential. We consider the holomorphic theory on various geometries, including Hopf manifolds and products of arbitrary pairs of Riemann surfaces, and offer some general remarks on dimensional reductions of holomorphic theories along the (n-1)-sphere to topological quantum mechanics. We also study an infinite-dimensional enhancement of the flavor symmetry in this example, to a recently studied central extension of the derived holomorphic functions with values in the original Lie algebra, that generalizes the familiar Kac–Moody enhancement in two-dimensional chiral theories.

Taming defects in mathcal{N} = 4 super-Yang-Mills

We study correlation functions involving extended defect operators in the four- dimensional mathcal{N} = 4 super-Yang-Mills (SYM). The main tool is supersymmetric localization with respect to the supercharge \U0001d4ac introduced in [1] which computes observables in the \U0001d4ac- cohomology. We classify general defects of different codimensions in the mathcal{N} = 4 SYM that belong to the \U0001d4ac-cohomology, which form 1 -BPS defect networks. By performing the \U0001d4ac- localization of the mathcal{N} = 4 SYM on the four-dimensional hemisphere, we discover a novel defect-Yang-Mills (dYM) theory on a submanifold given by the two-dimensional hemisphere and described by (constrained) two-dimensional Yang-Mills coupled to topological quantum mechanics on the boundary circle. This also generalizes to interface defects in mathcal{N} = 4 SYM by the folding trick. We provide explicit dictionary between defect observables in the SYM and those in the dYM, which enables extraction of general frac{1}{16} -BPS defect network observables of the SYM from two-dimensional gauge theory and matrix model techniques. Applied to the D5 brane interface in the SU(N ) SYM, we explicitly determine a set of defect correlation functions in the large N limit and obtain precise matching with strong coupling results from IIB supergravity on AdS5× S5.

Non-Abelian mirror symmetry beyond the chiral ring

Mirror symmetry is a type of infrared duality in 3D quantum field theory that relates the low-energy dynamics of two distinct ultraviolet descriptions. Though first discovered in the supersymmetric context, it has far-reaching implications for understanding nonperturbative physics in general 3D quantum field theories. We study mirror symmetry in 3D $\mathcal{N}=4$ supersymmetric field theories whose Higgs or Coulomb branches realize $D$- and $E$-type Kleinian singularities in the $ADE$ classification, generalizing previous work on the $A$-type case. Such theories include the $SU(2)$ gauge theory coupled to fundamental matter in the $D$-type case and non-Lagrangian generalizations thereof in the $E$-type case. In these cases, the mirror description is given by a quiver gauge theory of affine $D$- or $E$-type. We investigate the mirror map at the level of the recently identified 1D protected subsector described by topological quantum mechanics, which implements a deformation quantization of the corresponding $ADE$ singularity. We give an explicit dictionary between the monopole operators and their dual mesonic operators in the $D$-type case. Along the way, we extract various operator product expansion (OPE) coefficients for the quantized Higgs and Coulomb branches. We conclude by offering some perspectives on how the topological subsectors of the $E$-type quivers might shed light on their non-Lagrangian duals.

Homotopy algebras in higher spin theory

Motivated by string field theory, we explore various algebraic aspects of higher spin theory and Vasiliev equation in terms of homotopy algebras. We present a systematic study of unfolded formulation developed for the higher spin equation in terms of the Maurer-Cartan equation associated to differential forms valued in L-infinity algebras. The elimination of auxiliary variables of Vasiliev equation is analyzed through homological perturbation theory. This leads to a closed combinatorial graph formula for all the vertices of higher spin equations in the unfolded formulation. We also discover a topological quantum mechanics model whose correlation functions give deformed higher spin vertices at first order.

An angular dependent supersymmetric quantum mechanics with a Z2-invariant potential

We generalize the conformally invariant topological quantum mechanics of a particle propagating on a punctured plane by introducing a potential that breaks both the rotational and the conformal invariance down to a Z2 angular-dependent discrete symmetry. We derive a topological quantum mechanics whose localization gauge functions give interesting self-dual equations. The model contains an order parameter and exhibits a critical phenomenon with the existence of two ground states above a critical scale. Unlike the ordinary O(2)-invariant Higgs potential, an angular-dependence is found and saddle points, instead of local maxima, appear, posing subtle questions about the existence of instantons. The supersymmetric quantum mechanical model is constructed in both the path integral and the operatorial frameworks.

Chains of topological oscillators with instantons and calculable topological observables in topological quantum mechanics

We extend to a possibly infinite chain the conformally invariant mechanical system that was introduced earlier as a toy model for understanding the topological Yang–Mills theory. It gives a topological quantum model that has interesting and computable zero modes and topological invariants. It confirms the recent conjecture by several authors that supersymmetric quantum mechanics may provide useful tools for understanding robotic mechanical systems (Vitelli et al.) and condensed matter properties (Kane et al.), where trajectories are allowed or not by the conservation of topological indices. The absences of ground state and mass gaps are special features of such systems.

One-dimensional Chern–Simons theory and theÂgenus

We construct a Chern-Simons gauge theory for dg Lie and L-infinity algebras on any one-dimensional manifold and quantize this theory using the Batalin-Vilkovisky formalism and Costello's renormalization techniques. Koszul duality and derived geometry allow us to encode topological quantum mechanics, a nonlinear sigma model of maps from a 1-manifold into a cotangent bundle T*X, as such a Chern-Simons theory. Our main result is that the partition function of this theory is naturally identified with the A-genus of X. From the perspective of derived geometry, our quantization construct a volume form on the derived loop space which can be identified with the A-class.

Instantons, quivers and noncommutative Donaldson–Thomas theory

We construct noncommutative Donaldson–Thomas invariants associated with abelian orbifold singularities by analyzing the instanton contributions to a six-dimensional topological gauge theory. The noncommutative deformation of this gauge theory localizes on noncommutative instantons which can be classified in terms of three-dimensional Young diagrams with a colouring of boxes according to the orbifold group. We construct a moduli space for these gauge field configurations which allows us to compute its virtual numbers via the counting of representations of a quiver with relations. The quiver encodes the instanton dynamics of the noncommutative gauge theory, and is associated to the geometry of the singularity via the generalized McKay correspondence. The index of BPS states which compute the noncommutative Donaldson–Thomas invariants is realized via topological quantum mechanics based on the quiver data. We illustrate these constructions with several explicit examples, involving also higher rank Coulomb branch invariants and geometries with compact divisors, and connect our approach with other ones in the literature.

New observables in topological instantonic field theories

Instantonic theories are quantum field theories where all correlators are determined by integrals over the finite-dimensional space (space of generalized instantons). We consider novel geometrical observables in instantonic topological quantum mechanics that are strikingly different from standard evaluation observables. These observables allow jumps of special type for the trajectory (at the point of insertion of such observables). They do not (anti)commute with evaluation observables and raise the dimension of the space of allowed configurations, while the evaluation observables lower this dimension. We study these observables in geometric and operator formalisms. Simple examples are explicitly computed; they depend on the linking of points. The new “arbitrary jump” observables may be used to construct correlation functions computing, e.g., the linking numbers of cycles, as we illustrate on Hopf fibration. We expect that such observables could be generalized in an interesting way to instantonic topological theories in all dimensions.

QSPR/QSAR Study of Mercaptans by Quantum Topological Method

A new molecular quantum topological index QT was constructed by molecular topological methods and quantum mechanics (QM), which together with Gibbs free energy(G), Constant volume mole hot melting(CV) that were calculated by density functional theory (DFT) at the B3LYP/6-31G(d) level of theory for mercaptans. Index QT can not only efficiently distinguish molecular structures of mercaptans, but also possess good applications of QSPR/QSAR (quantitative structure-property/activity relationships). And most of the correlation coefficients of the models were over 0.99. The LOO CV (leave-one-out cross-validation) method was used to testify the stability and predictive ability of the models. The validation results verified the good stability and predictive ability of the models employing the cross-validation parameters: RCV, SCVand FCV, which demonstrated the wide potential of the index QT for applications to QSPR/ QSAR.