nLab
A-model

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

Context

∞-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

Definition

Examples

Quantum field theory

functorial quantum field theory

Contents

String theory

Ingredients

Critical string models

Extended objects

Topological strings

Backgrounds

Phenomenology

Physics

physics, mathematical physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

Contents

Idea

What is called the A-model topological string is the 2-dimensional topological conformal field theory corresponding to the Calabi–Yau category called the Fukaya category of a symplectic manifold X. This is is effectively the Gromov–Witten theory of X.

The A-model arose in formal physics from considerations of superstring-propagation on Calabi-Yau spaces: it may be motivated by considering the vertex operator algebra of the 2dSCFT given by the supersymmetric sigma-model with target space X and then deforming it such that one of the super-Virasoro generators squares to 0. The resulting “topologically twisted” algebra may then be read as being the BRST complex of a TCFT.

One can also define an A-model for Landau–Ginzburg models. The category of D-branes for the corresponding open string theory is given by the Fukaya–Seidel category.

By homological mirror symmetry, the A-model is dual to the B-model.

Properties

Lagrangian

The action functional of the A-model is that associated by AKSZ theory to a Lagrangian submanifold in a target symplectic Lie n-algebroid which is the Poisson Lie algebroid of a symplectic manifold.

See the references on Lagrangian formulation.

Boundary theory / holography

On coisotropic branes in symplectic target manifolds that arise by complexification of phase spaces, the boundary path integral of the A-model computes the quantization of that phase space. For details see

and

Second quantization / effective background field theory

The second quantization effective background field theory defined by the perturbation series of the A-model string has been argued to be Chern-Simons theory. (Witten 92, Costello 06)

References

General

The A-model was first conceived in

An early review is in

The motivation from the point of view of string theory is reviewed for instance in

A summary of these two reviews is in

  • H. Lee, Review of topological field theory and homological mirror symmetry (pdf)

Action functional

Discussion of how the A-model Lagrangian arises in AKSZ theory:

around page 19 in

  • M. Alexandrov, M. Kontsevich, A. Schwarz, O. Zaboronsky, The geometry of the master equation and topological quantum field theory, Int. J. Modern Phys. A 12(7):1405–1429, 1997

section 5.3 of

Also

  • Noriaki Ikeda, Deformation of graded (Batalin-Volkvisky) Structures, in Dito, Lu, Maeda, Alan Weinstein (eds.) Poisson geometry in mathematics and physics Contemp. Math. 450, AMS (2008)

Discussion of how the second quantization effective field theory given by the A-model perturbation series is Chern-Simons theory is in

formalizing at least aspects of the observations in

Revised on June 13, 2013 21:45:33 by Urs Schreiber (131.174.43.123)
Edit | Back in time (13 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: HomePage, physics, BV-BRST formalism, 2-spectral triple, brane, conformal field theory, AQFT, quantum field theory, FQFT, smooth Lorentzian space, (infinity,n)-category of cobordisms, extended cobordism, physicscontents, topological T-duality, T-duality, AKSZ sigma-model, sigma-model, GUT, string theory, higher category theory and physics, string field theory, path integral, factorization algebra, On the Classification of Topological Field Theories, Dijkgraaf-Witten theory, orientifold, Green-Schwarz mechanism, FFRS-formalism, topological quantum field theory, quantum anomaly, cobordism hypothesis, QFT with defects, gauge theory, electromagnetism, electromagnetic field, Kalb-Ramond field, supergravity C-field, electric charge, magnetic charge, RR field, field strength, Yang-Mills theory, Yang-Mills field, HQFT, integrable system, quantization, geometric quantization, BPS state, cobordism category, extended topological quantum field theory, classical physics, classical mechanics, force, wave, Gromov-Witten invariants, functorial quantum field theory - contents, Axiomatic field theories and their motivation from topology, (1,1)-dimensional Euclidean field theories and K-theory, geometric model for elliptic cohomology, (2,1)-dimensional Euclidean field theories and tmf, (2,1)-dimensional Euclidean field theory, D'Auria-Fre formulation of supergravity, supergravity, classical field theory, Supergravity and Superstrings - A Geometric Perspective, Poisson sigma-model, quantum mechanics, action functional, supersymmetry, Lagrangian, Euler-Lagrange equation, Jordan algebra, Chern-Simons theory, quantum harmonic oscillator, perturbation theory, renormalization, Hamiltonian, gravity, gravity as a BF theory, first-order formulation of gravity, Plebanski formulation of gravity, landscape of string theory vacua, Hamiltonian mechanics, Reshetikhin-Turaev construction, spectral triple, Berkovits superstring, multisymplectic geometry, classical limit, thermodynamics, dilaton, Bell's theorem, Dyson formula, gauge fixing, target space, Fukaya category, S-duality, net of C-star-systems, conformal net, infinity-Chern-Simons theory - contents, Seiberg-Witten theory, Haag-Kastler axioms, Topological Quantum Field Theories from Compact Lie Groups, entanglement, Wightman axioms, quantum operation, modular theory, supersymmetric quantum mechanics, Bisognano-Wichmann theorem, topological Yang-Mills theory, density matrix, quantum information, hermitian matrix, Maslov index, Osterwalder-Schrader theorem, superselection theory, finite quantum mechanics in terms of dagger-compact categories, Geometric and topological structures related to M-branes, semiclassical approximation, mirror symmetry, Stokes phenomenon, deformation quantization, graphical quantum channel, quantum state, Kerr spacetime, standard model of particle physics, observable, phase space, mathematical physics, The Dirac Electron, variational calculus, DHR category, Gleason's theorem, books and reviews in mathematical physics, special relativity, Kochen-Specker theorem, PCT theorem, spin-statistics theorem, Reeh-Schlieder theorem, general relativity, 2d TQFT, TCFT, A-model, B-model, state, Fell's theorem, Witten genus, no-go theorem, black hole, fermion, boson, gauge group, Dirac charge quantization and generalized differential cohomology, conserved current, AdS-CFT, books about string theory, Courant sigma-model, Bekenstein-Hawking entropy, generalized second law of thermodynamics, phenomenology, Wess-Zumino-Witten model, NS5-brane, beta-gamma system, Schwarzschild spacetime, 11-dimensional supergravity, kilogram, dual heterotic string theory, variational bicomplex, Calabi-Yau object, parallel transport, differential string structure, quantum gravity, Einstein-Hilbert action, Avogadro constant, holographic principle of higher category theory, Kaluza-Klein mechanism, Chern-Simons element, gauge transformation, gravitino, graviton, BF-theory, cosmological constant, SCFT, electric-magnetic duality, supergravity Lie 6-algebra, quantization via the A-model, holographic principle, Mathematical Theory of Quantum Fields, quantum electrodynamics, differential fivebrane structure, relativistic particle, Green-Schwarz action functional, Euclidean quantum field theory, Stone-von Neumann theorem, mechanical system, Yetter model, superparticle, Landau-Ginzburg model, quantum sheaf cohomology, second quantization, heterotic string theory, kinematics and dynamics, Chan-Paton bundle, S-matrix, charge sector, Chern-Simons gravity, Haag's theorem, Khovanov homology, causal locality, string, sewing constraint, photon, time slice axiom, charge of a conserved current, picture of mechanics, path integral as a pull-push transform, homological quantum field theory, Noether's theorem, reductions deformations resolutions in physics, spinning particle, Peierls bracket, Mathematical Topics Between Classical and Quantum Mechanics, Renormalization and Effective Field Theory, configuration space, effective quantum field theory, Quantization of Gauge Systems, Noether identities, gaugino, Bohr topos, M2-brane, Gruppenpest, gluon, gauge boson, Three Roles of Quantum Field Theory, fermionic path integral, worldvolume, background field, particle, timelike curve, Lorentz force, Nambu-Goto action, differential T-duality, Einstein equation, fundamental particle, superstring, Fermi theory of beta decay, T-fold, Quantum Fields and Strings, second law of thermodynamics, Cauchy surface, type II string theory, self-dual higher gauge theory, what to contribute, mechanics, QCD, Dirac interaction picture, classical state, Jordan-Lie-Banach algebra, Local Quantum Physics -- Fields, Particles, Algebras, spinning string, supersymmetry and Calabi-Yau manifolds, spontaneously broken symmetry, sigma-model -- exposition of classical sigma-models, sigma-model -- exposition of higher gauge theories as sigma-models, sigma-model -- exposition of quantum sigma-models, sigma-model -- exposition of a general abstract formulation, sigma-model -- exposition of second quantization of sigma-models, criticism of string theory, 4-dimensional supergravity, black brane, NS-brane, WZW-type superstring field theory, higher dimensional Chern-Simons theory, c-theorem, contents of contents, hole argument, D7-brane, D-brane, Kapustin-Witten TQFT, instanton, 3-dimensional supergravity, Yang-Mills instanton, moment of inertia, 6d Chern-Simons theory, string theory - contents, Characteristic forms and geometric invariants, Yang-Mills instanton Floer homology, vacuum, 6d (2,0)-supersymmetric QFT, universality class, Geometric Algebra, rigid body dynamics, angular momentum, angular velocity, Euler-Arnold equation, Foundations of Mechanics, Hamiltonian dynamics on Lie groups, symplectic field theory, M-brane, 3d quantum gravity, symplectic infinity-groupoid, geometric quantization of symplectic groupoids, Wick's lemma, duality in string theory, ABJM theory, modular functor, event horizon, Seiberg duality, counterterm, BLG model, Steady State, effective action functional, collective field theory, M5-brane, false vacuum, flux, higher spin gauge theory, super Yang-Mills theory, D4-brane, little string theory, D3-brane, D1-brane, D5-brane, D0-brane, BFSS matrix model, 7-dimensional supergravity, N=4 D=4 super Yang-Mills theory, D2-brane, B6-field, C6-field, gauged supergravity, higher gauge field, physics resources, Feynman diagram, higher dimensional WZW model, 1-dimensional Chern-Simons theory, 4d WZW model, quantum lattice system, minimal coupling, Kirchhoff's laws, topological M-theory, Born-Oppenheimer approximation, on-shell recursion, Freund-Rubin compactification, axion, p-adic physics, quantum observable, coset WZW model, 1d WZW model, quantum hadrodynamics, braid group statistics, higher electric background charge coupling, F-theory, M-theory on G2-manifolds, type II supergravity, infinite-dimensional Chern-Simons theory, axio-dilaton, observable universe, N=2 D=4 super Yang-Mills theory, N=4 D=3 super Yang-Mills theory, 3d Chern-Simons theory, teleparallel gravity, topological string, M-theory, Freed-Witten anomaly, thermodynamic limit, Planck's constant, twisted smooth cohomology in string theory, topological M5-brane, M9-brane, flux compactification, general covariance, quantum cosmology, technicolor, string phenomenology, Higgs field, The Geometry of Physics - An Introduction, Maxwell's equations, Harmonic oscillator, KLT relations, AGT correspondence, Einstein-Yang-Mills-Dirac-Higgs theory, C-star algebraic deformation quantization, Dirac charge quantization, kinetic action, instanton in QCD, G2-MSSM, magnetic field, MSSM, holomorphic Chern-Simons theory, cosmology, moduli stabilization, scattering amplitude, Turaev-Viro model, black holes in string theory, standard model of cosmology, open/closed string duality, model (in theoretical physics), particle physics, string theory results applied elsewhere, twistor string theory, quark-gluon plasma, cosmic inflation, Einstein-Maxwell theory, cosmic microwave background radiation, geometric quantization by push-forward, dark matter, higher gauge transformation, Einstein-Yang-Mills theory, geometry of physics, factorization algebra of observables, theory (physics), matter, confinement, virtual particle, electric field, Liouville integrable system, field (physics), Levin-Wen model, prequantum field theory, quark, QFT on non-commutative spacetime, MOND, BPTS-instanton, nuclear force, superposition, Aharonov-Bohm effect, ghost field, duality in physics, 7d Chern-Simons theory, electromagnetic potential, continuum mechanics, 5d Chern-Simons theory, self-dual Yang-Mills theory, electromagnetic field strength, higgsino, 2d Chern-Simons theory, electron, topological membrane, geometric quantization of non-integral forms, prequantum 0-bundle, Bn-geometry, Einstein-Yang-Mills-Dirac theory, neutrino, electroweak field, interaction, 4d Chern-Simons theory, Guillemin-Sternberg geometric quantization conjecture, free field theory, extended Lagrangian, double dimensional reduction, renormalization scheme, string scattering amplitude, Factorization algebras in perturbative quantum field theory, Moyal quantization, level (Chern-Simons theory), Chern-Simons invariant, field bundle, effect algebra, local quantum field theory, principle of least action, light, scalar field, Big Bang, W-boson, extended geometric quantization of 2d Chern-Simons theory, quantum decoherence, Hilbert's sixth problem, 4d TQFT, type II supergravity Lie 2-algebra, string theory FAQ, gravitational constant, semiclassical state, phase and phase space in physics, 1d Dijkgraaf-Witten theory, 3-category of fermionic conformal nets, gauged WZW model, cosmic Galois group, black hole firewall problem, Diaconescu-Moore-Witten anomaly