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E-raamat: Quantum Field Theory: Batalin-Vilkovisky Formalism and Its Applications

  • Formaat: 192 pages
  • Sari: University Lecture Series
  • Ilmumisaeg: 30-Aug-2019
  • Kirjastus: American Mathematical Society
  • ISBN-13: 9781470453688
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Quantum Field Theory: Batalin-Vilkovisky Formalism and Its ApplicationsSuurem pilt
  • Formaat: 192 pages
  • Sari: University Lecture Series
  • Ilmumisaeg: 30-Aug-2019
  • Kirjastus: American Mathematical Society
  • ISBN-13: 9781470453688
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This book originated from lecture notes for the course given by the author at the University of Notre Dame in the fall of 2016. The aim of the book is to give an introduction to the perturbative path integral for gauge theories (in particular, topological field theories) in Batalin-Vilkovisky formalism and to some of its applications. The book is oriented toward a graduate mathematical audience and does not require any prior physics background. To elucidate the picture, the exposition is mostly focused on finite-dimensional models for gauge systems and path integrals, while giving comments on what has to be amended in the infinite-dimensional case relevant to local field theory. Motivating examples discussed in the book include Alexandrov-Kontsevich-Schwarz-Zaboronsky sigma models, the perturbative expansion for Chern-Simons invariants of 3-manifolds given in terms of integrals over configurations of points on the manifold, the BF (background field) theory on cellular decompositions of manifolds, and Kontsevich's deformation quantization formula.
Preface ix
Chapter 1 Introduction
1(20)
1.1 Prologue
1(4)
1.2 Atiyah-Segal picture of quantum field theory
5(9)
1.2.1 Atiyah's axioms of topological quantum field theory
6(4)
1.2.2 Segal's QFT
10(2)
1.2.3 Example of a TQFT: Dijkgraaf-Witten model
12(2)
1.3 The idea of path integral construction of quantum field theory
14(4)
1.3.1 Classical field theory data
14(1)
1.3.2 Idea of path integral quantization
14(1)
1.3.3 Heuristic argument for gluing
15(1)
1.3.4 How to define path integrals?
15(2)
1.3.5 Towards Batalin-Vilkovisky (BV) formalism
17(1)
1.4 Plan of the exposition
18(3)
Chapter 2 Classical Chern-Simons theory
21(16)
2.1 Chern-Simons theory on a closed 3-manifold
21(4)
2.1.1 Fields
21(1)
2.1.2 Action
21(1)
2.1.3 Euler-Lagrange equation
21(1)
2.1.4 Gauge symmetry
22(1)
2.1.5 Chern-Simons invariant on the moduli space of flat connections
23(1)
2.1.6 Remark: more general G
24(1)
2.1.7 Relation to the second Chern class
24(1)
2.2 Chern-Simons theory on manifolds with boundary
25(12)
2.2.1 Phase space
25(1)
2.2.2 δScs, Euler-Lagrange equations
26(1)
2.2.3 Noether 1-form, symplectic structure on the phase space
26(1)
2.2.4 "Cauchy subspace"
27(1)
2.2.5 LM, Σ
28(1)
2.2.6 Reduction of the boundary structure by gauge transformations
29(1)
2.2.7 Lagrangian property of LM, Σ
30(1)
2.2.8 Behavior of SCS under gauge transformations, Wess-Zumino cocycle
31(1)
2.2.9 Prequantum line bundle on the moduli space of flat connections on the surface
32(1)
2.2.10 Two exciting formulae
33(1)
2.2.11 Classical field theory as a functor to the symplectic category
34(3)
Chapter 3 Feynman diagrams
37(38)
3.1 Gauss and Fresnel integrals
37(1)
3.2 Stationary phase formula
38(3)
3.3 Gaussian expectation values. Wick's lemma
41(3)
3.4 A reminder on graphs and graph automorphisms
44(3)
3.5 Back to integrals: Gaussian expectation value of a product of homogeneous polynomials
47(1)
3.6 Perturbed Gaussian integral
48(10)
3.6.1 Aside: Borel summation
53(1)
3.6.2 Connected graphs
54(1)
3.6.3 Introducing the "Planck constant" and bookkeeping by Euler characteristic of Feynman graphs
55(1)
3.6.4 Expectation values with respect to perturbed Gaussian measure
56(1)
3.6.5 Fresnel (oscillatory) version of perturbative integral
57(1)
3.6.6 Perturbation expansion via the exponential of a second order differential operator
57(1)
3.7 Stationary phase formula with corrections
58(3)
3.7.1 Laplace method
59(2)
3.8 Berezin integral
61(2)
3.8.1 Odd vector spaces
61(1)
3.8.2 Integration on the odd line
61(1)
3.8.3 Integration on the odd vector space
61(2)
3.9 Gaussian integral over an odd vector space
63(1)
3.10 Perturbative integral over a vector superspace
64(4)
3.10.1 "Odd Wick's lemma"
64(1)
3.10.2 Perturbative integral over an odd vector space
64(2)
3.10.3 Perturbative integral over a superspace
66(2)
3.11 Digression: the logic of perturbative path integral
68(7)
3.11.1 Example: scalar theory with φ3 interaction
69(1)
3.11.2 Divergencies!
70(1)
3.11.3 Regularization and renormalization
71(1)
3.11.4 Wilson's picture of renormalization ("Wilson's RG flow")
72(3)
Chapter 4 Batalin-Vilkovisky formalism
75(72)
4.1 Faddeev-Popov construction
75(9)
4.1.1 Hessian of Spp in an adapted chart
78(1)
4.1.2 Stationary phase evaluation of Faddeev-Popov integral
79(3)
4.1.3 Motivating example: Yang-Mills theory
82(2)
4.2 Elements of supergeometry
84(11)
4.2.1 Supermanifolds
84(2)
4.2.2 Z-graded (super)manifolds
86(1)
4.2.3 Differential graded manifolds (a.k.a. Q-manifolds)
87(5)
4.2.4 Integration on supermanifolds
92(1)
4.2.5 Change of variables formula for integration over supermanifolds
93(1)
4.2.6 Divergence of a vector field
94(1)
4.3 BRST formalism
95(5)
4.3.1 Classical BRST formalism
95(1)
4.3.2 Quantum BRST formalism
96(1)
4.3.3 Faddeev-Popov via BRST
97(3)
4.3.4 Remark: reducible symmetries and higher ghosts
100(1)
4.4 Odd-symplectic manifolds
100(6)
4.4.1 Differential forms on super (graded) manifolds
100(1)
4.4.2 Odd-symplectic supermanifolds
101(2)
4.4.3 Odd-symplectic manifolds with a compatible Berezinian. BV Laplacian
103(1)
4.4.4 BV integrals. Stokes' theorem for BV integrals
104(2)
4.5 Algebraic picture: BV algebras. Master equation and canonical transformations of its solutions
106(3)
4.5.1 BV algebras
106(2)
4.5.2 Classical and quantum master equation
108(1)
4.5.3 Canonical transformations
109(1)
4.6 Half-densities on odd-symplectic manifolds. Canonical BV Laplacian. Integral forms
109(4)
4.6.1 Half-densities on odd-symplectic manifolds
109(2)
4.6.2 Canonical BV Laplacian on half-densities
111(1)
4.6.3 Integral forms
112(1)
4.7 Fiber BV integrals
113(2)
4.8 Batalin-Vilkovisky formalism
115(14)
4.8.1 Classical BV formalism
115(1)
4.8.2 Quantum BV formalism
116(2)
4.8.3 Faddeev-Popov via BV
118(3)
4.8.4 BV for gauge symmetry given by a non-integrable distribution
121(5)
4.8.5 Felder-Kazhdan existence-uniqueness result for solutions of the classical master equation
126(3)
4.9 AKSZ sigma models
129(18)
4.9.1 AKSZ construction
129(5)
4.9.2 Example: Chern-Simons theory
134(3)
4.9.3 Example: Poisson sigma model
137(3)
4.9.4 Example: BF theory
140(7)
Chapter 5 Applications
147(32)
5.1 Cellular BF theory
147(15)
5.1.1 Abstract BF theory associated to a dgLa
147(2)
5.1.2 Effective action induced on a subcomplex
149(5)
5.1.3 Geometric situation
154(7)
5.1.4 Remarks
161(1)
5.2 Perturbative Chern-Simons theory
162(8)
5.2.1 Perturbative contribution of an acyclic flat connection: one-loop part
162(3)
5.2.2 Higher loop corrections, after Axelrod-Singer
165(5)
5.3 Kontsevich's deformation quantization via Poisson sigma model
170(9)
5.3.1 Associativity: a heuristic path integral argument
173(1)
5.3.2 Associativity from Stokes' theorem on configuration spaces
174(1)
5.3.3 Kontsevich's L∞ morphism
175(4)
Bibliography 179(6)
Index 185
Pavel Mnev, University of Notre Dame, IN. Steklov Institute of Mathematics, St. Petersburg, Russia.
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