Quantum Field Theory: Batalin-Vilkovisky Formalism and Its Applications

Cover
Title page
Preface
Chapter 1. Introduction
	1.1. Prologue
	1.2. Atiyah-Segal picture of quantum field theory
		1.2.1. Atiyah’s axioms of topological quantum field theory
		1.2.2. Segal’s QFT
		1.2.3. Example of a TQFT: Dijkgraaf-Witten model
	1.3. The idea of path integral construction of quantum field theory
		1.3.1. Classical field theory data
		1.3.2. Idea of path integral quantization
		1.3.3. Heuristic argument for gluing
		1.3.4. How to define path integrals?
		1.3.5. Towards Batalin-Vilkovisky (BV) formalism
	1.4. Plan of the exposition
Chapter 2. Classical Chern-Simons theory
	2.1. Chern-Simons theory on a closed 3-manifold
		2.1.1. Fields
		2.1.2. Action
		2.1.3. Euler-Lagrange equation
		2.1.4. Gauge symmetry
		2.1.5. Chern-Simons invariant on the moduli space of flat connections
		2.1.6. Remark: more general ?
		2.1.7. Relation to the second Chern class
	2.2. Chern-Simons theory on manifolds with boundary
		2.2.1. Phase space
		2.2.2. ??_{??}, Euler-Lagrange equations
		2.2.3. Noether 1-form, symplectic structure on the phase space
		2.2.4. “Cauchy subspace”
		2.2.5. ?_{?,Σ}
		2.2.6. Reduction of the boundary structure by gauge transformations
		2.2.7. Lagrangian property of ?_{?,Σ}
		2.2.8. Behavior of ?_{??} under gauge transformations, Wess-Zumino cocycle
		2.2.9. Prequantum line bundle on the moduli space of flat connections on the surface
		2.2.10. Two exciting formulae
		2.2.11. Classical field theory as a functor to the symplectic category
Chapter 3. Feynman diagrams
	3.1. Gauss and Fresnel integrals
	3.2. Stationary phase formula
	3.3. Gaussian expectation values. Wick’s lemma
	3.4. A reminder on graphs and graph automorphisms
	3.5. Back to integrals: Gaussian expectation value of a product of homogeneous polynomials
	3.6. Perturbed Gaussian integral
		3.6.1. Aside: Borel summation
		3.6.2. Connected graphs
		3.6.3. Introducing the “Planck constant” and bookkeeping by Euler characteristic of Feynman graphs
		3.6.4. Expectation values with respect to perturbed Gaussian measure
		3.6.5. Fresnel (oscillatory) version of perturbative integral
		3.6.6. Perturbation expansion via the exponential of a second order differential operator
	3.7. Stationary phase formula with corrections
		3.7.1. Laplace method
	3.8. Berezin integral
		3.8.1. Odd vector spaces
		3.8.2. Integration on the odd line
		3.8.3. Integration on the odd vector space
	3.9. Gaussian integral over an odd vector space
	3.10. Perturbative integral over a vector superspace
		3.10.1. “Odd Wick’s lemma”
		3.10.2. Perturbative integral over an odd vector space
		3.10.3. Perturbative integral over a superspace
	3.11. Digression: the logic of perturbative path integral
		3.11.1. Example: scalar theory with ?³ interaction
		3.11.2. Divergencies!
		3.11.3. Regularization and renormalization
		3.11.4. Wilson’s picture of renormalization (“Wilson’s RG flow”)
Chapter 4. Batalin-Vilkovisky formalism
	4.1. Faddeev-Popov construction
		4.1.1. Hessian of ?_{??} in an adapted chart
		4.1.2. Stationary phase evaluation of Faddeev-Popov integral
		4.1.3. Motivating example: Yang-Mills theory
	4.2. Elements of supergeometry
		4.2.1. Supermanifolds
		4.2.2. \ZZ-graded (super)manifolds
		4.2.3. Differential graded manifolds (a.k.a. ?-manifolds)
		4.2.4. Integration on supermanifolds
		4.2.5. Change of variables formula for integration over supermanifolds
		4.2.6. Divergence of a vector field
	4.3. BRST formalism
		4.3.1. Classical BRST formalism
		4.3.2. Quantum BRST formalism
		4.3.3. Faddeev-Popov via BRST
		4.3.4. Remark: reducible symmetries and higher ghosts
	4.4. Odd-symplectic manifolds
		4.4.1. Differential forms on super (graded) manifolds
		4.4.2. Odd-symplectic supermanifolds
		4.4.3. Odd-symplectic manifolds with a compatible Berezinian. BV Laplacian.
		4.4.4. BV integrals. Stokes’ theorem for BV integrals.
	4.5. Algebraic picture: BV algebras. Master equation and canonical transformations of its solutions
		4.5.1. BV algebras
		4.5.2. Classical and quantum master equation
		4.5.3. Canonical transformations
	4.6. Half-densities on odd-symplectic manifolds. Canonical BV Laplacian. Integral forms
		4.6.1. Half-densities on odd-symplectic manifolds
		4.6.2. Canonical BV Laplacian on half-densities
		4.6.3. Integral forms
	4.7. Fiber BV integrals
	4.8. Batalin-Vilkovisky formalism
		4.8.1. Classical BV formalism
		4.8.2. Quantum BV formalism
		4.8.3. Faddeev-Popov via BV
		4.8.4. BV for gauge symmetry given by a non-integrable distribution
		4.8.5. Felder-Kazhdan existence-uniqueness result for solutions of the classical master equation
	4.9. AKSZ sigma models
		4.9.1. AKSZ construction
		4.9.2. Example: Chern-Simons theory
		4.9.3. Example: Poisson sigma model
		4.9.4. Example: ?? theory
Chapter 5. Applications
	5.1. Cellular ?? theory
		5.1.1. Abstract ?? theory associated to a dgLa
		5.1.2. Effective action induced on a subcomplex
		5.1.3. Geometric situation
		5.1.4. Remarks
	5.2. Perturbative Chern-Simons theory
		5.2.1. Perturbative contribution of an acyclic flat connection: one-loop part
		5.2.2. Higher loop corrections, after Axelrod-Singer
	5.3. Kontsevich’s deformation quantization via Poisson sigma model
		5.3.1. Associativity: a heuristic path integral argument
		5.3.2. Associativity from Stokes’ theorem on configuration spaces
		5.3.3. Kontsevich’s ?_{∞} morphism
Bibliography
Index
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