DISSERTAÇÃO DE MESTRADO IFT–D.013/20 Cohomological Field Theory in the BV Formalism Iván Mauricio Burbano Aldana Orientador Nathan Jacob Berkovits December 16, 2020 Burbano Aldana, Iván Mauricio. B946c Cohomological field theory in the BV formalism / Iván Mauricio Burbano Aldana. – São Paulo, 2021 116 f. : il. Dissertação (mestrado) – Universidade Estadual Paulista (Unesp), Instituto de Física Teórica (IFT), São Paulo Orientador: Nathan Jacob Berkovits 1. Física matemática. 2. Campos de calibre (Física). 3. Homologia (Matemática). I. Título Sistema de geração automática de fichas catalográficas da Unesp. Biblioteca do Instituto de Física Teórica (IFT), São Paulo. Dados fornecidos pelo autor(a). Acknowledgments I would like to thank Nathan Berkovits, Pedro Vieira, and the SAIFR-Perimeter fellow- ship for giving me the opportunity of being part of the joint master’s program between the Instituto de Física Teórica and the Perimeter Institute for Theoretical Physics. I am spe- cially indebted to Nathan for his course in supersymmetry, which awakened my interest in field theory, and accepting to be my supervisor at the IFT. I would also like to thank Kevin Costello and Kasia Rejzner for their supervision and mentorship at PI. Without their guidance I would not have been able to navigate my way through this topic. They have set an example of the kind of mathematical physicist that I will strive to become in my future career. I am deeply indebted as well to Dalila Maria Pîrvu and Bruno de Souza Leão Torres for their thorough reading of parts of the manuscript and their insightful suggestions. I am also grateful to Luis Gabriel Caro Mendoza, Alicia Lima, and Jonas Neuser, for our fruitful discussions in mathematical physics. The writing of this essay would have been impossible without my fellow PSIons. I would like to specially thank Dalila Maria Pîrvu, my greatest companion during this last year. She always believed in me and offered her unconditional support through the challenges that I had to overcome. I will keep her teachings with me for the rest of my life. I am also grateful to Tales Rick Perche and Bruno de Souza Leão Torres, for their friendship during our time together in Brazil and Canada. I was incredibly lucky to get paired with them during the past year and a half. I would also like to thank Ghislaine Coulter-de Wit, Rémi Faure, Ian Holst, Julia Maristany, and Gloria Odak, for all of the amazing adventures we had together. Similarly, I would like to thank Luis Gabriel Mendoza Caro, Cassiano Daniel, Dean Valois, and the rest of the people at the IFT whose friendship made my time in Brazil unique. Special thanks to my parents for their support, advice, and patience during our long Skype conversations. Many thanks to Debbie Guenther and the rest of the PI/PSI team, as well as Luzinete Aparecida Martins, Jandira Ferreira de Oliveira, and the rest of the IFT/ICTP-SAIFR team, for their hospitality and disposition during my time in Canada and Brazil. The past year and a half have been academically very fruitful thanks to them. I am also grateful to the QFT and Mathematical Physics research group at Universidad de los Andes, led by Andrés Fernando Reyes-Lega, for preparing me for this challenge. Finally, I would like to thank Aiyalam Balachandran, Matilde Marcolli, Carlos Andrés Flórez Bustos, Sergio Miguel Adarve Delgado, Bruto Max Pimentel, Andrés Alejandro Plazas Malagón, and Roger Smith, for their mentoring and guidance at different stages of my scientific career. Finally, I would like to thank ICTP-SAIFR, ICTP-Trieste, FAPESP grant 2016/03143-7, and CAPES finance code 001, for partial financial support. i Resumo O formalismo de Batalin e Vilkovisky (BV) é um dos principais ingredientes de muitas das abordagens que temos para formulações matematicamente precisas de teoria quântica de campos (QFT) perturbativa. Vamos expor este formalismo através de uma reinterpretação cohomologica da equação de Schwinger-Dyson para uma teoria sem simetria de calibre. Isto nos levará a um ponto de vista alternativo das integrais de caminho. Seguindo em analogia com teorias de calibre, vamos codificar de maneira cohomológica as equações de movimento e a estrutura de calibre de teorias clássicas de campos. Os complexos de cocadeias resultantes vão ter como diferenciais os campos vetoriais Hamiltonianos de uma extensão BV da ação. Fixar o calibre nesta nos levará aos análogos quânticos dos complexos clássicos. Os elementos algébricos relevantes vão ser discutidos no contexto de “toy theories” de dimensão finita. Com estas vamos poder proceder de maneira análoga ao formalismo BV de uma partícula relativística, teoria de Yang-Mills, teoria de Chern-Simons, e teoria BF. Palavras Chaves: Formalismo BV; Simetrias de calibre; Espaços simpléticos ím- pares; Cohomologia. Áreas do conhecimento: Física; Física-Matemática; Teorias de Calibre. ii Abstract The Batalin-Vilkovisky (BV) formalism is a major ingredient in several ap- proaches towards mathematically sound formulations of perturbative quantum field theory (QFT). We will develop this framework by interpreting the Schwinger- Dyson equation for a theory without gauge symmetries in a cohomological fashion. This will lead us to an alternative point of view towards path integration. Mirror- ing this construction for gauge theories we will be able to encode the equations of motion and the gauge structure of classical field theories cohomologically. The differentials of the resulting cochain complexes will be Hamiltonian vector fields for an extended BV action. We will describe a method to gauge fix this action and obtain a quantum theory. The lifting of gauge invariance to the quantum theory will lead us to the quantum analogues of the classical cochain complexes we constructed. The relevant algebraic elements of this formalism will be devel- oped in the simple case of finite dimensional QFT analogues. These will allow us to proceed by analogy to the BV treatment of the relativistic particle, Yang-Mills theory, Chern-Simons theory, and BF theory. Keywords: BV Formalism; Gauge symmetries; Odd symplectic spaces; Cohomol- ogy. Areas of knowledge: Physics; Mathematical Physics; Gauge Theories. iii Contents 1 Introduction 1 2 Graded Linear Algebra 6 2.1 Graded Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Graded Symplectic Vector Spaces . . . . . . . . . . . . . . . . . . . . 14 2.3 Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Graded Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Graded Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6 Graded Poisson Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 27 2.7 BV Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 BV Formalism in Finite Dimensions 39 3.1 Gaussian Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Homological Picture of Integration: Antifields . . . . . . . . . . . . 43 3.3 Koszul Complex: Going On-Shell . . . . . . . . . . . . . . . . . . . . 45 3.4 Lie Algebras: Ghosts . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5 DG Lie algebras: Ghosts for ghosts . . . . . . . . . . . . . . . . . . . 52 3.6 Classical BV Formalism: Gauge Invariance and Observables . . . . 55 3.7 Gauge Fixing Fermion . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.8 Quantum BV Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.9 Deformations and Anomalies . . . . . . . . . . . . . . . . . . . . . . 63 4 Applications of the BV Formalism 68 4.1 The Hat Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 Free Real Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Extension to Infinite Dimensions . . . . . . . . . . . . . . . . . . . . 73 4.4 Diffeomorphism Symmetries . . . . . . . . . . . . . . . . . . . . . . 76 4.5 Equations of Motion for a Relativistic Extended Object . . . . . . . 80 4.6 Polyakov Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.7 Relativistic Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.8 Connection Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.9 Yang-Mills Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 iv 4.10 Chern-Simons Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.11 BF Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5 Conclusions 105 Bibliography 107 v Chapter 1 Introduction The standard model of particle physics (SM) has provided some of the most accurate predictions to date and some claim it is the most well-tested theory of physics (see for example 11.1 of [1], which makes the claim for quantum electrody- namics, a theory included in the SM). It is a particular instance of what has come to be known as quantum field theory (QFT). Applications of QFT can be found in other fields, such as condensed matter physics and cosmology [2, 3]. This makes QFT one of the pinnacles of modern physics. Much like other theories, practition- ers develop familiarity with the theory by examining different examples which have varying degrees of similarity among themselves. However, unlike theories such as quantum mechanics (QM), there is no universally accepted definition of what QFT actually is. A manifestation of this is that in QM we are able to describe sets of axioms [4, 5, 6] which accommodate several known examples and allow us to treat them within the same framework. These axioms are not only of structural interest, for they have allowed us to make general statements about the quantum behaviour of the universe. A simple example of this is Heisenberg’s Uncertainty Principle, which relates the noncommutative structure of quantum observables with the incompatibility of their measurements (see Theorem 12.4 of [4]). More- over, a happy by-product of this exercise has been the development of areas within mathematics. In our running example, functional analysis benefited greatly from its applications to QM. It is thus of fundamental importance to develop rigorous mathematical foundations for QFT that are flexible enough to accommodate the well known applications we currently have, but stringent enough to allow us to probe the theory further. There have been several attempts in this direction. Let us remark some of them and mention an incomplete account of the important consequences that have followed from each. The Wightman axioms [7] point out general properties that a QFT should have. These accommodate several free field theories while still allowing the study of general properties of QFTs, such as the CPT theorem, the connection between spin and statistics, and the reconstruction of full QFTs 1 Chapter 1. Introduction 2 from the knowledge of vacuum expectation values, to which a large amount of the applications in particle physics reduces to. The Osterwalder-Schrader axioms [8] yield a Euclidean counterpart to these, allowing the possibility of studying Wick rotations and the relationship between QFTs in Minkowski and Euclidean signatures. Finally, let us also remark the Haag-Kastler axioms [9, 10], which provide a generalization based on nets of algebras of observables now known as algebraic quantum field theory (AQFT). This formalism provides a suitable framework for the study of observables in the presence of gauge symmetries [11, 12], superselection sectors and particle statistics [13, 14], and the relationship between thermal states and time [15]. However, there is currently no proof that some of the important interacting field theories, like the SM, can be described through any of these frameworks. Although no universally accepted mathematically rigorous definition of QFT is currently available, the perturbative aspects of the theory are much better devel- oped. Alternative approaches to axiomatic QFT have focused on devising rigorous foundations for these. Such attempts have proven more successful at accommo- dating the existing examples. Among these we can mention the framework of factorization algebras, developed by Costello and Gwilliam [16, 17, 18], and the framework of perturbative algebraic quantum field theory (pAQFT), developed by Brunetti, Dütsch, Fredenhagen and Rejzner [19, 20, 21, 22, 23]. The former combines the assignment of factorization algebras to local regions in spacetime with the renormalization theory developed by Costello [16]. The latter extends the nets of observables considered in AQFT and treats them using Epstein-Glaser renormalization [24]. These approaches share several common ingredients [25]. Among them is the treatment of gauge symmetries through the Batalin-Vilkovisky (BV) formalism [26, 27, 28, 29]. The BV formalism is one of the most general methods we have (if not the most, as claimed in 9.3 of Chapter 1 of [16]) to study quantum theories with gauge symmetries. In what follows we will attempt to give an account of this formalism as a tool for the treatment of such theories in a unified fashion. Roughly speaking, the formalism tackles two problems: the lack of rigorous foundations for path integrals and the evaluation of said integrals on theories with gauge redundancies. At the classical level, it reduces to a Lagrangian reformulation of the BRST procedure. In it, the usual phase spaces are replaced by the spaces of field con- figurations, with constraint surfaces of the former being instantaneous snapshots at some spatial hypersurface of the on-shell configurations in the latter. Since no Chapter 1. Introduction 3 such surface is required in the Lagrangian picture, the BV formalism provides an approach to quantization that deals better with covariance. The space of field configurations is extended to include ghosts and their corresponding antifields, which gives it the structure of a shifted Poisson manifold. As a consequence, the action will induce a shifted Hamiltonian vector field on the extended manifold of field configurations, which plays the distinguished role of encoding the equations of motion, symmetry structures, gauge redundancies, and their representations on the theory. This is done by requiring that the action satisfies the classical master equation. Solving this equation will lead to a modified action from which the gauge-fixed actions obtained by more heuristic methods can be recovered. At the quantum level, the formalism does an order h̄ perturbation on this Hamiltonian vector field. Requiring that the classical gauge symmetry lifts to the quantum theory, i.e. that the path integral measure be gauge invariant, leads to the quantum master equation. Finding solutions of this will in general require quantum corrections to the solution of the classical master equation. Obstructions to finding solutions then correspond to gauge anomalies in the theory. Moreover, naive interpretations of the quantum master equation will in general lead to ill- defined quantities. Instead, making sense of the quantum master equation will require a renormalization procedure. Much of the work in the frameworks of factorization algebras and pAQFT has been focused on the proper mathematical formulation of the coupling between the BV formalism and renormalization. Both at the classical and quantum levels, the corresponding master equations are equivalent to the requirement that the Hamiltonian vector field induced by the action, or its quantum perturbation, are differentials in the cohomological sense. Consequently, the BV formalism naturally brings the techniques of cohomological algebra into field theory. In particular, the resulting cohomology groups will encode the structure of the classical and quantum gauge invariant observables, as well as the information needed to study perturbative expansions. The computa- tional power of homological algebra has played a major role in new developments in field theory. An example of this is a proof of the perturbative renormalizability of Yang-Mills (YM) theory on flat four-manifolds with coefficients in semi-simple Lie algebras (see [30] or Theorem 1.0.1 of [16]). In the BV formalism we assign two kind of graduations to our fields. One of them determines whether the field will behave as a commuting or anticommuting variable. The other determines the physical origin of the field and will play an important role in extracting cohomological information from our theories. The Chapter 1. Introduction 4 associated mathematical structures that describe these type of objects are those of graded algebra. We will start our discussion in Chapter 2 by giving a thorough review of this language. In here most of the structures that will appear in the BV formalism are presented and the essential theorems regarding the mathematical interpretation of the classical and quantum master equations are proven. In Chapter 3, we begin by introducing the point of view of the BV formalism towards path integrals. This will be done by trying to find a method to compute finite dimensional gaussian integrals without actually needing to integrate. We will stick to the finite dimensional case throughout this chapter. Not all of the important physical and mathematical aspects of the BV formalism can be described in this case. However, we can justify our approach in several ways. On the one hand, there are several aspects that such models still exhibit and are most easily understood without the difficult subtleties that accompany more realistic field theories. On the other, the formal “generalized Einstein summation convention” of DeWitt [31] makes several of the formulas that will follow identical to the ones presented in usual expositions of the formalism in the physics literature (cf. [32, 33]). Finally, our considerations will be enough to proceed by analogy with several examples of proper field theories. Our approach will be based on a cohomological interpretation of the Schwinger-Dyson equation [see 32, (15.25)]. The classical limit of this construction will yield a cohomology theory which encodes both the equations of motion of the theory and its infinitesimal symmetry structure. This is known as the Koszul complex. Mirroring this we will try to build a cohomological theory of the Lie algebras of our gauge symmetries. This will lead us to the Chevalley-Eilenberg complex. The classical theory will be complete once we unify this two cohomology theories and construct the BV action. Finally, we discuss a procedure for gauge fixing such an action through a gauge fixing fermion. Independence from the gauge fixing fermion of the path integral measure will lead us to the quantum BV formalism. Finally, we will discuss the applications of this formalism in Chapter 4. After a quick finite dimensional example, we will discuss the extension of the formalism to infinite dimensional theories. We will do so in a rigorous fashion for the scalar field. Afterwards, we will do a brief heuristic discussion of this extension for more general field theories. We will conclude by exhibiting examples of the BV formalism as an alternative of the BRST formalism for diffeomorphism invariant theories and theories of connections on principal bundles. For the first, we will discuss the theory of extended quantum objects. We will particularly focus on Chapter 1. Introduction 5 the relativistic particle to exemplify the BV formalism. For the second we will study Yang-Mills theory and Chern-Simons theory. We will also discuss abelian BF theory to see how the BV formalism deals with reducible gauge symmetries. Before we start, let me comment on some important topics that will not be discussed in this work. First of all, the cohomological interpretation of path integrals in the BV formalism will only be discussed for theories without gauge symmetries. Its purpose will be to make natural the construction of the Koszul complex from a physics viewpoint. In particular, once we have constructed the classical BV complex, we will introduce its quantum counterpart and its physical significance using heuristically the usual point of view towards the path integral. Secondly, although motivated by the use of the BV formalism in factorization algebras and pAQFT, a discussion of these is outside the scope of this thesis. Finally, we will not attempt to find solutions of the quantum master equation or study the problem of renormalization in this context. Chapter 2 Graded Linear Algebra 2.1 Graded Vector Spaces The formalism we will explore is based on an extension of linear algebra based on a grading. After giving a general definition we will follow with some foresight to the reader on its importance. Definition 2.1.1 ([Special case of 34, Definition 3]). Let G be an Abelian group. A (G-)graduation on a vector space V is a collection of subspaces { Vi ∣∣ i ∈ G } such that V = ⊕ i∈G Vi. A vector space equipped with such a graduation is said to be a (G-)graded vector space. The elements in Vi are called homogeneous of degree i. This yields a map pV : ⋃ i∈G Vi \ {0} → G, defined by v ∈ VpV(v) for all homogeneous v. A basis α of V consisting solely of homogeneous elements is said to be adapted. If there is a subset S ⊆ G such that Vi = {0} for all i ∈ G \ S, we say that V is concentrated in S. Remark 2.1.1. To avoid cluttering our notation, we will use p ≡ pV whenever the graded vector space in question is clear. Example 2.1.1. Z�2Z-graded vector spaces are called super vector spaces. We will prefer the snotation | · | ≡ p for these spaces and we will use the term parity instead of degree. We will also denote the graduation using lower indices {V0, V1}. The elements of V0 are said to even bosonic while the elements of V1 are said to be fermionic. Having this structure will be useful when we construct algebras from V. We will use the bosonic elements as commuting and the fermionic elements as anticommuting. Remark 2.1.2. We should note that during this work our use of bosonic and fermionic is not tied in principle to any notion of spin. These two are usually related by the spin-statistics theorem [see 35, Theorem 4-10]. However, we will encounter in our procedure auxiliary fields that do not comply with this theorem. 6 Chapter 2. Graded Linear Algebra 7 Example 2.1.2. Z-graded vector spaces are at the core of the theory of cohomology. We will see this in Definition 2.1.3. In these we usually use the notation deg ≡ p. We will use them to distinguish fields with different physical meanings. The elements in degree 0 will correspond to the fields we start with. In terms of these we will have our initial description of the physics of the system. Associated to each of these we will build new fields, called antifields, which will sit in degree 1. This will be of importance in order to give us a Poisson structure to work with. These antifields will play a role with respect to the fields similar to the one played by the momenta with respect to the positions in classical mechanics. Their grading will be chosen so that the cohomology will capture certain physically important data. In order to couple symmetries to our theory, we will interpret the parameters of such symmetries as new fields, called ghosts. These will sit in degree −1, guaranteeing that they fit along with the other infinitesimal transformations of the theory (see (3.82)). In order to recover a Poisson structure, we will introduce antifields for these too, which will in turn sit in degree 2. Redundancies in these symmetries, their redundancies, and so on, will populate the degrees less than −1, while their antifields will sit in degrees bigger than 2. Example 2.1.3. The reader may by now suspect we will need Z�2Z×Z-graduations. This is indeed the case. However, since Z and Z�2Z graduations are interesting on their own, we will simply develop the theory for general Abelian groups, which adds little difficulty. Having however both graduations is in fact crucial since they will have to interact together in our theory. For example, in order to obtain a nice interpretation to our cohomologies, we will assign to the antifields a different parity than their corresponding fields. On the other hand, bosonic symmetries of some theory (see Definition 2.1.2 for the degree/parity of a transformation) will be described by fermionic ghosts. Z�2Z×Z-graded vector spaces will be called bigraded vector spaces. The degree map has two projections. We will use the notation | · | for the Z�2Z projec- tion and deg for the Z one. These are then defined by p(v) = (|v|, deg v) for all homogeneous v ∈ V. Every super vector space can be understood as a bigraded vector space by setting deg = 0. From now on, we will always do this implicitly whenever no other bigraduation is given. Similarly, every Z-graded vector space can be understood as a bigraded vector space by setting | · | = 0. However, for the signs below to coincide with those usually used in the derived geometry literature, it is better to Chapter 2. Graded Linear Algebra 8 set | · | ≡ deg mod 2 in these type of vector spaces. Let us go back to consider a general Abelian group G. We will always use additive notation for these. Example 2.1.4. Any vector space V can be regarded as a G-graded vector space by setting V0 = V and Vi = {0} for all i ∈ G \ {0}. We will do this whenever a vector space is not already equipped with a grading. Whenever we introduce new mathematical objects we would like to identify the structure preserving maps. Definition 2.1.2. A linear map T : V → W between two G-graded vector spaces is said to be homogeneous of degree p(T) = k ∈ G if TVi ⊆W i+k. The set of such maps is denoted by Homk(V, W). If the degree of a homogeneous map is omitted, we imply it has degree 0. Theorem 2.1.1. Given two G-graded vector spaces V and W, Hom(V, W) = ⊕ k∈G Homk(V, W) (2.1) makes Hom(V, W) a graded vector space. Proof. Let v ∈ Vi. The grading of W guarantees that for every j ∈ G there exist unique wj ∈ W j such that { wj ∣∣ j ∈ G and wj 6= 0 } is finite and Tv = ∑j∈G wj. For every k ∈ G let Tkv := wi+k. We then have ∑k∈G Tkv = ∑k∈G wi+k = ∑j∈G wj = Tv. Repeating this for every v ∈ V we obtain a degree k linear Tk : V → W for every k ∈ Z such that T = ∑k∈G Tk. This shows that the space of linear maps is indeed the sum of the grading we are trying to impose. This sum is clearly direct since it is impossible for a non-zero homogeneous linear map to have two degrees. Corollary 2.1.2. Let V be a G-graded vector space. Then (V∗)i ∼= (V−i)∗ canoni- cally. Proof. First, recall that V∗ := Hom(V, F), with F the field where V is defined. Since F is a G-graded vector space concentrated at degree 0, the previous theorem establishes the grading on V∗. Now, every f ∈ (V−i)∗ can be canonically extended to V by setting it equal to 0 at all other degrees, i.e. declaring that f (v) = 0 if p(v) 6= −i. The resulting map is in (V∗)i since f (V−i) ⊆ F = F0 = F−i+i, while f (V j) = {0} = Fj+i for all j ∈ G \ {i}. Chapter 2. Graded Linear Algebra 9 On the other hand, every map f ∈ (V∗)i satisfies the above inclusions. It can thus be restricted to a map in (V−i)∗. Remark 2.1.3. Let V, W, Z be graded vector spaces and T ∈ Hom(V, W)i while S ∈ Hom(W, Z)j. Then it is clear that S ◦ T ∈ Hom(V, Z)i+j. At the core of cohomology theory is the notion of a cochain complex. In order to continue our discussion in general terms we will assume from now on that our graded vector spaces are equipped with a homomorphism deg : G → Z. Precomposing this with the degree map, we obtain a map deg : ⋃ i∈G Vi \ {0} → Z, which we will denote by the same symbol. We will call this the cohomological degree. This of course induces a Z-graduation on V. The resulting Z-graded vector space will be denoted Vdeg and we have Vi deg = span deg−1(i). (2.2) In agreement with Example 2.1.3, in the case of G = Z�2Z×Z we will always assume this deg map is the projection of the degree map onto the second compo- nent. Definition 2.1.3. Let V be a graded vector space. A differential is a degree k map d : V → V such that deg k = 1 annd d2 = 0. The pair (V, d) is called a cochain complex. We will use diagrams of the form · · ·Vi−1 deg Vi deg Vi+1 deg · · · ,di−1 di di+1 (2.3) to describe them. Elements in im d are called exact, while elements in ker d are called closed. Note that Bk := im d∩Vk deg ⊆ ker d∩Vk deg =: Zk for all k ∈ Z. We thus define the cohomology Hk(V, d) := Zk �Bk in degree k. We are also interested in creating new such objects out of previous ones. We have already seen how to do this by taking spaces of linear maps. We will now consider direct sums, tensor products, and shifts. Theorem 2.1.3. Given two graded vector spaces V and W, the following gives the direct sum the structure of a graded vector space (V ⊕W)i = Vi ⊕W i (2.4) Proof. This is clear from the associativity and commutativity of the direct sum. Chapter 2. Graded Linear Algebra 10 Theorem 2.1.4. Given two graded vector spaces V and W, the following decom- position defines a grading of their tensor product (V ⊗W)i := ⊕ j∈G (V j ⊗W i−j). (2.5) Proof. This is clear from the distributivity of the tensor product over the direct sum [see 36, Corollary 2.2 of Chapter XVI]. The tensor product will be our most useful tool to produce algebras out of graded vector spaces. In order to do this we need to be able to multiply several elements together. This is achieved through the isomorphism (U ⊗V)⊗W → U ⊗ (V ⊗W) (u⊗ v)⊗ w 7→ u⊗ (v⊗ w). (2.6) Moreover, we want our algebras to reflect the fermionic and bosonic nature of their elements. For this we will need to modify the usual commutativity isomorphism to a graded commutativity isomorphism. In order to do this, we need a super vector space structure. Following the same procedure we used to introduce cochain complexes, will from now on assume our graded vector spaces are equipped with a homomorphism | · | : G → Z�2Z. Precomposing this with the degree map, we obtain a map | · | : ⋃ i∈G Vi → Z�2Z, which we will denote by the same symbol. We will call this the parity. Then the graded commutativity isomorphism takes the form V ⊗W →W ⊗V v⊗ w 7→ (−1)|w||v|w⊗ v (2.7) Thus far we have given the category of super vector spaces with even linear maps the structure of a symmetric monoidal category [see 37, Chapter XI]. Although we will refrain from the use of this language, this is the mathematical structure behind the intricate sign manipulations that will follow. As a simple example of how this signs appear, consider the identification of V∗∗ ∼= V. In order for us to do this, we have to stablish the action of V on V∗ via a pairing V ⊗V∗ → F. (2.8) Chapter 2. Graded Linear Algebra 11 However, we already have an action of V∗ on V, given by the pairing V∗ ⊗V → F f ⊗ v 7→ f (v). (2.9) Thus, we obtain the map we are seeking by application of (2.7) V ⊗V∗ → V∗ ⊗V → F, (2.10) under which v⊗ f 7→ (−1)| f ||v| f ⊗ v 7→ (−1)| f ||v| f (v) =: v( f ). (2.11) A similar example is in the identification of V∗ ⊗W∗ with (V ⊗W)∗. In this case, the pairing is obtained by application of assocativity and graded commuta- tivity (V∗ ⊗W∗)⊗ (V ⊗W)→ (V∗ ⊗V)⊗ (W∗ ⊗W)→ F, (2.12) under which ( f ⊗ g)⊗ (v⊗ w) 7→ (−1)|g||v|( f ⊗ v)⊗ (g⊗ w) 7→ (−1)|g||v| f (v)g(w) =: ( f ⊗ g)(v⊗ w). (2.13) As a last example for this remark, note the fact that Hom(V, W) is isomorphic to both W ⊗V∗ and V∗ ⊗W whenever the dimensions of V and W are finite. We will make the identification Hom(V, W) ∼= W ⊗V∗, so that our linear maps will act from the left. Indeed, if (v1, . . . , vn) is a basis of V and (v1, . . . , vn) is its dual, this identification takes the form f (v) = vi(v) f (vi) =: ( f (vi)⊗ vi)(v). (2.14) On the other hand, it introduces a sign when consider the dual of a linear map. Indeed, Hom(V, W)→W ⊗V∗ → V∗ ⊗W → V∗ ⊗W∗∗ → Hom(W∗, V∗) (2.15) defines the dual f ∗ ∈ Hom(W∗, V∗) of f ∈ Hom(V, W). In the second and third arrow of this chain we have to be careful to introduce the correct signs. In Chapter 2. Graded Linear Algebra 12 particular, we obtain from the first two arrows f 7→ f (vi)⊗ vi 7→ (−1)|vi|(| f |+|vi|)vi ⊗ f (vi). (2.16) In here we used the fact that |vi| = |vi|, which is clear form Corollary 2.1.2 and the fact that | · | : G → Z�2Z is a homomorphism. Through the third and fourth arrows, this defines a map f ∗ whose action on a g ∈W∗ is f ∗(g) = (−1)|vi|(| f |+|vi|) f (vi)(g)vi = (−1)(|vi|+|g|)(| f |+|vi|)g( f (vi))vi = (−1)| f ||g|g( f (vi))vi = (−1)| f ||g|g ◦ f . (2.17) In the last maniputation of the signs we used that g( f (vi)) 6= 0 implies |g|+ | f |+ |vi| = 0. Remark 2.1.4. As consistency of notation suggests, in the G = Z�2Z×Z case we will choose the homomorphism G → Z�2Z so that | · | coincides with the projection of p onto the Z�2Z factor. Definition 2.1.4. Let V be a graded vector space and k ∈ G. We define the shifted graded vector space V[k] as the vector space V with the grading V[k]i := Vi+k, (2.18) for all i ∈ G. Remark 2.1.5. Given a graded vector space V and k ∈ Z, we have VpV(v) 3 v ∈ V[k]pV[k](v) = VpV[k](v)+k, (2.19) that is, pV[k] = pV − k. In super vector spaces this is trivial since addition and subtraction coincide. In this case the only possible shift we have is ΠV ≡ V[1]. However, the sign in this formula is an important (although arbitrary) choice we have made for Z-graded vector spaces. Example 2.1.5. Let V be a vector space with no grading. Then V[−k] is the graded vector space which has V in degree k and {0} in all other degrees. Let V be a graded vector space and k ∈ G. In the following several signs that will appear stem from noticing that V[k] is isomorphic to both F[k] ⊗ V and V ⊗ F[k]. However, the isomorphism between the latter is fixed by the Chapter 2. Graded Linear Algebra 13 isomorphism (2.7). We will adopt by convention to make the identification V[k] ∼= F[k] ⊗ V. We can keep track of the signs coming from this choice by the use of a symbol sk which generates F[k] and thus has p(sk) = −k. This symbol is fermionic or bosonic depending on the parity |k|. It implements the identification by denoting the counterpart in V[k] of v ∈ V by sk ⊗ v ∈ F[k]⊗V. The fact that V[k][l] = V[k + l] can be obtained by identifying sk ⊗ sl ∼= sk+l. As a first consequence of this choice, note that the identification V[k]∗ ∼= V∗[−k] is obtained through the pairing V∗[k]⊗V[−k]→ F[k]⊗V∗ ⊗F[−k]⊗V → F[k]⊗F[−k]⊗V∗ ⊗V → V∗ ⊗V → F, (2.20) which yields f ⊗ v→ sk ⊗ f ⊗ s−k ⊗ v→ (−1)|k|| f |V∗ sk ⊗ s−k ⊗ f ⊗ v → (−1)|k|| f |V∗ f ⊗ v→ (−1)|k|| f |V∗ f (v). (2.21) In other words, we have the identification V∗[k]→ V[k]∗ f 7→ (−1)|k|| f | f . (2.22) As a second consequence, note the isomorphism of finite dimensional vector spaces Hom(V[k], W[l])→W[l]⊗ (V[k])∗ →W[l]⊗V∗[−k] → F[l]⊗W ⊗F[−k]⊗V∗ → F[l − k]⊗W ⊗V∗ → Hom(V, W)[l − k]. (2.23) Using a basis (v1, . . . , vn) for V and taking (v1, . . . , vn) to be its dual, we have f 7→ f (vi)⊗ vi 7→ (−1)|k||vi|V f (vi)⊗ vi 7→ (−1)|k||vi|V sl ⊗ f (vi)⊗ s−k ⊗ vi 7→ (−1)�� ��|k||vi|V+|k|(| f |Hom(V,W)+�� �|vi|V)sl−k ⊗ f (vi)⊗ vi 7→ (−1)|k|| f |Hom(V,W) f . (2.24) Remark 2.1.6. For bigraded vector spaces V a shift by k = (k1, k2) ∈ Z�2Z×Z Chapter 2. Graded Linear Algebra 14 will be denoted by V[k] ≡ V[(k1, k2)] =: V[k1, k2]. Most of the shifts that will appear in the BV formalism will in fact satisfy k1 ≡ k2 mod 2. Therefore, we will from now on use the notation (n) = (n, n) for all n ∈ Z. Similarly, we will use V[n] := V[n, n]. 2.2 Graded Symplectic Vector Spaces We will begin by introducing the symplectic structure from which we will develop the BV formalism. Definition 2.2.1 ([slightly generalized version of 38, Definition 2.4.1]). A symplec- tic structure of degree k ∈ G on a graded vector space V over a field F is a bilinear map ω : V ×V → F such that: 1. ω(Vi, V j) ⊆ Fi+j+k := F, i + j + k = 0 {0}, i + j + k 6= 0, (2.25) 2. for all homogeneous v, u ∈ V we have ω(v, u) = −(−1)|v||u|ω(u, v), (2.26) and 3. for all homogeneous v ∈ V \ {0} the map ω(v, · ) : V−p(v)−k → F in (V∗)p(v)+k is not zero. Remark 2.2.1 ([extension of 38, Remark 2.4.2]). Note that in Definition 2.2.1 we do not require that the degree k map · [ : V → V∗ v 7→ v[ := ω(v, · ), (2.27) is invertible. In the finite dimensional case, the fact that the map is not null already assures that it is an isomorphism via the dimension theorem [see 39, Theorem 2.3]. In particular, we get the constraint dim Vi = dim(V∗[k])i = dim(V∗)i+k = dim(V−i−k)∗ = dim V−i−k, (2.28) Chapter 2. Graded Linear Algebra 15 and the degree −k isomorphism · ] := ( · [)−1 : V∗ → V. (2.29) Remark 2.2.2. By the definition of a tensor product, establishing a bilinear map ω : V ×V → F is equivalent to establishing a linear map usually denoted by the same symbol ω : V ⊗V → F. With the grading we have given to tensor products, condition 1. corresponds to this linear map being of degree k. Example 2.2.1. Let V be a bigraded vector space. Then E = V ⊕ V∗[−1] has a degree (−1) symplectic form σ(Φ + Φ∗, Ψ + Ψ∗) := Ψ∗(Φ)− (−1)|Φ ∗|V∗ [−1]|Ψ|VΦ∗(Ψ) (2.30) for all homogeneous Φ, Ψ ∈ V and Φ∗, Ψ∗ ∈ V∗. To study its degree, let us further assume that the entries are homogeneous pV (Φ) = pV∗[−1](Φ ∗) = k, pV (Ψ) = pV∗[−1](Ψ ∗) = l. (2.31) notice that if the symplectic form doesn’t vanish we must have pV∗[−1](Ψ ∗) + (−1) = pV∗(Ψ∗) = −pV (Φ), (2.32) or pV∗[−1](Φ ∗) + (−1) = pV∗(Φ∗) = −pV (Ψ). (2.33) Of course both equations are equivalent due to the restrictions in the degrees above. This means that the symplectic form is non-trivial only for elements satisfying k + l + (−1) = 0. We thus conclude that p(σ) = (−1). On the other hand, the sign between both terms is chosen so that the correct skew symmetry is attained. Let (Φ∗1, . . . , Φ∗n) be an adapted basis for V and (Φ1, . . . , Φn) be its dual. The notation in here is chosen since the first basis yields linear coordinates on V∗ while the second yields linear coordinates on V. Then σ(Φ∗A, ΦB) = δB A = −(−1)|Φ ∗ A|V |ΦB|V∗ [−1]σ(ΦB, ΦA) = −(−1)|Φ ∗ A|V (|ΦB|V∗+1)σ(ΦB, Φ∗A) = −σ(ΦB, Φ∗A), (2.34) while σ(Φ∗A, Φ∗B) = σ(ΦA, ΦB) = 0. To express the musical isomorphism induced Chapter 2. Graded Linear Algebra 16 by this symplectic structure, let (Φ1, . . . , Φn, Φ∗1 , . . . , Φ∗n) be the basis of E∗ dual to (Φ∗1 , . . . , Φ∗n, Φ1, . . . , Φn). Then, this symplectic form induces the isomorphism of degree (−1). [ : E → E∗ Φ∗A 7→ Φ∗A ΦA 7→ −ΦA (2.35) E is called the shifted cotangent bundle of V . Let us anticipate that V will be the structure containing the fields we will start with and the ghosts in various degrees. On the other hand V∗[−1] will contain all of the antifields. 2.3 Graded Algebras Definition 2.3.1. A graded algebra A is an algebra with a collection of subspaces{ Ai ∣∣ i ∈ G } that makes it a graded vector space and that satisfies AiAj ⊆ Ai+j. A trivial example of these algebras is the ring of polynomial functions on a vector space. Example 2.3.1. Let V be a vector space. Then S•V := ⊕k∈NSnV is a commutative bigraded algebra with the symmetric tensor product and the grading obtained by declaring it concentrated in {0} ×Z and setting (S•V)(0,i) = SiV i ≥ 0 {0} i < 0. (2.36) Once a basis (v1, . . . , vn) has been chosen for V, one can identify this graded algebra with the algebra S•V ∼= F[v1, . . . , vn]�〈vivj − vjvi 〉 (2.37) of polynomials with coefficients in F, the field over which V is defined, on the symbols v1, . . . , vn which satisfy vivj = vjvi for all i, j ∈ {1, . . . , n}. The ring of polynomial function on V is defined to be S•(V∗). Given elements f1, . . . , fk ∈ V∗, the element f1 · · · fk ∈ S•(V∗) is usually interpreted as a map F : V → F v 7→ f1(v) · · · fn(v). (2.38) Chapter 2. Graded Linear Algebra 17 This is however not adequate for an extension to anticommuting objects. A more convenient interpretation is obtained as G : SkV → F v1 · · · vk 7→ 1 k! ∑ σ∈Sn f1(vσ(1)) · · · fk(vσ(k)). (2.39) Both actions carry the same information. Indeed, F(v) = G(v · · · v). On the other hand, G(v1 · · · vk) = 1 k! Dv1 · · ·Dvk F. We can proof this by induction on n. Taking k = 1, we have that F = f1 is linear and thus Dv1 F = f1(v1) = G(v1). Now, assume that we have proven it for k− 1. Then Dv1 · · ·Dvk F = D1 · · ·Dvk−1 k ∑ i=1 f1 · · · fi−1 fi(vk) fi+1 · · · fk = k ∑ i=1 ∑ σ∈Sk−1 f1(vσ(1)) · · · fi−1(vσ(i−1)) fi(vk) fi+1(vσ(i)) · · · fn(vσ(k−1)). (2.40) Now, for every i ∈ {1, . . . , k} and σ ∈ Sk−1 we can define µ ∈ {1, . . . , k} by µ(j) :=  σ(j) j ∈ {1, . . . , i− 1} k j = i σ(j− 1) j ∈ {i + 1, . . . , k}, (2.41) which gives us a bijection {1, . . . , k} × Sk−1 → Sk. We thus conclude Dv1 · · ·Dvn F = ∑ µ∈Sn f1(vµ(1)) · · · fn(vµ(k)) = k!G(v1 · · · vn). (2.42) Another standard example of such algebras comes from the exterior algebra. This example shows how the grading helps us keep track of which elements commute and which anticommute. Example 2.3.2. Let V be a vector space. Then ∧• V := ⊕ k∈N ∧k V is a commuta- tive graded algebra with the wedge product and the graduation obtained by (∧• V )(i,j) =  ∧j V, j ≥ 0 and i ≡ j mod 2 {0} i < 0 or i 6≡ j mod 2. (2.43) Once a basis (v1, . . . , vn) has been chosen for V, one can identify this graded Chapter 2. Graded Linear Algebra 18 algebra with the algebra ∧• V ∼= F[v1, . . . , vn]�〈vivj + vjvi 〉 (2.44) of polynomials with coefficients in F, the field over which V is defined, on the symbols v1, . . . , vn which satisfy vivj = −vjvi for all i, j ∈ {1, . . . , n}. In particular,∧• V∗ mimics the ring of polynomial functions over a vector space V whose symbols anticommute. Heuristically speaking, in the commuting case this ring of polynomial functions usually only approximates the algebra of smooth functions on the vector space. However, in the anticommuting case functions always have to be of this form. This is because their Taylor series expansion in any of the variables terminates at order 1 f (v) = a + bv, (2.45) since v2 = 0. Example 2.3.3. The previous examples generalize to graded vector spaces V. We define their symmetric algebra to be the commutative graded algebra S•V := V⊗•� 〈{ v⊗ w− (−1)|v||w|w⊗ v ∣∣∣∣∣ v, w ∈ ⋃ i∈G Vi }〉 . (2.46) We will systematically avoid any tensor product signs in such a space from now on. Given homogeneous v1, . . . , vn ∈ V, we declare p(v1 · · · vn) = ∑n i=1 p(vi). This gives S•V its grading. Moreover, we define the subspaces SkV = span { v1 · · · vk | v1, . . . , vk ∈ V are homogeneous } . (2.47) Given the intuition gained from the previous example, we define the ring of polynomial functions on V to be O(V) := S•(V∗). We will use this algebra extensively as an approximation to the algebra of smooth functions on V. It is when speaking of this space that algebraic considerations will be most transparent. Similarly, we can define the antisymmetric algebra of graded vector spaces ∧• V := V⊗•� 〈{ v⊗ w + (−1)|v||w|w⊗ v ∣∣∣∣∣ v, w ∈ ⋃ i∈Z Vi }〉 . (2.48) Remark 2.3.1 ([see 40, Section 2.1]). Let V be a graded vector space with an adapted Chapter 2. Graded Linear Algebra 19 basis (v1, . . . , vn). As is standard in discussions of the BV formalism [see 23, for a counterexample], if we use the generic symbol φ ∈ V for an element of V, then we denote the elements of the basis dual to (v1, . . . , vn) by (φ1, . . . , φn). One should be careful with this convention, specially in field theory. In this case the discrete indices transform to continuous indices, so that the coordinates on V will be written as φ(x). It is then common to omit the x symbol from this coordinates to simplify formulas. This however leads us to using the same symbol φ for generic elements of V and the coordinates on V. The interpretation should be clear from context. Remark 2.3.2. Note that although S•V = ⊕ k∈N SkV, this is not the grading we have given to this space above. Remark 2.3.3. Using the antisymmetric algebra we can recast the definition of a graded symplectic form in a simpler fashion. Namely, it is a linear map ω : V ∧V → V of degree k such that for all v ∈ V \ {0} the map ω(v, · ) 6= 0. Remark 2.3.4. Let V be a bigraded vector space. Notice that using the graded commutativity (2.7), we have the isomorphisms (V[1])⊗n → F[1]⊗V ⊗ · · · ⊗F[1]⊗V︸ ︷︷ ︸ n times → · · · → F[1]⊗V ⊗ · · · ⊗F[1]⊗V︸ ︷︷ ︸ n− 2 times ⊗F[2, 2]⊗V ⊗V → F[n]⊗V⊗n → V⊗n[n], (2.49) under which v1 ⊗ · · · ⊗ vn → s(1) ⊗ v1 ⊗ · · · ⊗ s(1) ⊗ vn 7→ (−1)|vn−1|s(1) ⊗ v1 ⊗ · · · ⊗ s(1) ⊗ vn−2 ⊗ s(2) ⊗ vn−1 ⊗ vn 7→ (−1)|vn−1|s(1) ⊗ v1 ⊗ · · · ⊗ s(1) ⊗ vn−3 ⊗ s(3) ⊗ vn−2 ⊗ vn−1 ⊗ vn 7→ · · · → (−1)|vn−1|+|vn−3|+···s(n) ⊗ v1 ⊗ · · · ⊗ vn 7→ (−1)∑n k=1(n−k)|vk|v1 ⊗ · · · ⊗ vn. (2.50) This is known as the décalage (shift in French) isomorphism [see 41, Section 1]. Notice that symmetrization using the degrees in V[1] corresponds to antisym- metrization using the degree in V. This descends into an isomorphism Sn(V[1])→ (∧n V ) [n]. (2.51) Chapter 2. Graded Linear Algebra 20 Making use of this identification we can obtain a degree 0 isomorphism dec : Hom (∧n V, W ) → Hom(Sn(V[1]), W[1])[n− 1]. (2.52) Indeed, we already found a map (2.23) which assigns to f ∈ Hom ( ∧n V, W) the function (−1)n| f | f ∈ Hom (( ∧n V)[n], W[1])[n− 1]. This can then be precom- posed with the décalage isomorphism to obtain a map in Hom(Sn(V[1]), W[1])[n− 1] given by dec( f )(v1 · · · vn) = (−1)n| f |+∑n k=1(n−k)|vk| f (v1 · · · vn). (2.53) This map will be instrumental in order to understand gauge symmetries with- ing the BV formalism. For example, in the case of a bosonic gauge symmetry described by some Lie algebra, this map allows us to understand the Lie bracket at the level of the fermionic ghosts that enter the theory. Much like on functions, we can pullback the regular functions on a graded vector space via homogeneous linear transformations. Definition 2.3.2. Let V and W be graded vector spaces and ϕ : V → W be a homogeneous linear transformation. Then, let f1 · · · fn ∈ W∗ be homogeneous. We define the pullback of f1 · · · fn ∈ O(V) via ϕ to be ( f1 ◦ ϕ) · · · ( fn ◦ ϕ) ∈ O(V). This extends to a homogeneous map of graded algebras ϕ∗ : O(W)→ O(V) Definition 2.3.3. A left derivation X of degree k ∈ G of a graded algebra A is a homogeneous linear map X : A → A of degree k such that for all homogeneous a, b ∈ A X(ab) = (Xa)b + (−1)|k||a|a(Xb) (2.54) A right derivation satisfies X(ab) = a(Xb) + (−1)|k||b|(Xa)b (2.55) instead. The graded vector space of left derivations is denoted by DerA. Theorem 2.3.1. Given a left derivation X of a graded algebra A, we define Xr : A → A by Xra = (−1)|X|(|a|+1)Xa (2.56) for all a ∈ A. Then this is a right derivation of degree |X|. Chapter 2. Graded Linear Algebra 21 Proof. Indeed, for all a, b ∈ A we have Xr(ab) = (−1)|X|(|a|+|b|+1)((Xa)b + (−1)|X||a|a(Xb)) = (−1)|X||b|(Xra)b + aXrb. (2.57) Remark 2.3.5. Heuristically speaking, we want to have ∂ ∂θ (ψθ) = (−1)|ψ||θ|ψ ∂θ ∂θ = (−1)|ψ||θ|ψ = (−1)(|ψθ|+|θ|)|θ|ψ = (−1)(|ψθ|+1)|θ|ψ, ∂r ∂θ (ψθ) = ψ ∂θ ∂θ = ψ. (2.58) In the last step of the first requirement we used the fact that for x ∈ Z2 we have x2 = x. This heuristics is the reason behind our definition of Xr. 2.4 Graded Lie Algebras Another example of graded algebraic structures we will encounter is that of a graded Lie algebra. Definition 2.4.1 ([42, see, Definition 3.1]). A graded Lie bracket of degree k ∈ G on a graded vector space g is a bilinear map [ · , · ] : g× g→ g such that 1. [gi, gj] ⊆ gi+j+k, 2. [X, Y] = −(−1)(|X|+|k|)(|Y|+|k|)[Y, X] for all homogeneous X, Y ∈ g, and 3. for all homogeneous X, Y, Z ∈ g we have the graded Jacobi identity 0 = (−1)(|X|+|k|)(|Z|+|k|)[X, [Y, Z]] + (−1)(|Y|+|k|)(|X|+|k|)[Y, [Z, X]] + (−1)(|Z|+|k|)(|Y|+|k|)[Z, [X, Y]]. (2.59) A graded vector space equipped with a graded Lie bracket of degree k is called a graded Lie algebra of degree k. If no degree is mentioned, we will assume that graded Lie algebras are of degree 0. Chapter 2. Graded Linear Algebra 22 Remark 2.4.1. Super Lie algebras corresponding to the choice of G = Z�2Z are extremely important to supersymmetric theories. Using these we can not only describe bosonic symmetries which preserve the parity of a field, but also fermionic symmetries which mix bosonic and fermionic components. On the other hand, Z-graded Lie algebras allow us to characterize symmetries that may be trivial in a way we will make precise in Section 3.3. Indeed, an element with a certain degree i will characterize the trivial symmetries corresponding to the degree i + 1. In order to do this however we need a map between these two degrees. This leads us immediately to the notion of a differential graded Lie algebra. Definition 2.4.2 ([see 43, Definition 3.1]). A dg (differential graded) Lie algebra g is a graded Lie algebra with a differential d : g→ g of degree k ∈ G which makes g a cochain complex, and d[X, Y] = [dX, Y] + (−1)|X|[X, dY], (2.60) for all X, Y ∈ g. Remark 2.4.2. Much like in Remark 2.3.3, we can define a graded Lie bracket of degree k more simply as a degree 0 map [ · , · ] : g[−k] ∧ g[−k]→ g[−k] satisfying the graded Jacobi identity 0 = (−1)|X|g[−k]|Z|g[−k] [X, [Y, Z]] + (−1)|Y|g[−k]|X|g[−k] [Y, [Z, X]] + (−1)|Z|g[−k]|Y|g[−k] [Z, [X, Y]]. (2.61) Remark 2.4.3. Notice that a graded Lie algebra of degree 0 is almost a graded alge- bra by viewing the Lie bracket as a product. Property 1. ensures that the degrees work as expected in a graded algebra. However, Lie algebras are distinguished by the algebras we are studying in that associativity fails. Indeed, this failure is precisely measured through the graded Jacobi identity. To see this, let us put it in the form [X, [Y, Z]] = −(−1)|Y||X|+|X||Z|[Y, [Z, X]]− (−1)|Z||Y|+|X||Z|[Z, [X, Y]] = (−1)|Y||X|[Y, [X, Z]] + [[X, Y], Z]. (2.62) In this form we also see that the Jacobi identity guarantees that [X, · ] is a derivation of degree |X| of this “algebra”. All graded Lie algebras can be obtained through graded Lie algebras of degree 0 via the following theorem. Chapter 2. Graded Linear Algebra 23 Theorem 2.4.1. Let L be a graded Lie algebra of degree k. Then L[s] is a graded Lie algebra of degree k + s. Proof. This is a simple consequence of Remark 2.4.2. Graded algebras induce two important examples of a graded Lie algebras. Theorem 2.4.2 ([see 42, Proposition 4.1]). Let A be a graded algebra, and define its commutator [ · , · ] : A×A → A (a, b) 7→ ab− (−1)|a||b|ba. (2.63) Then [ · , · ] is a graded Lie bracket on A. Theorem 2.4.3. Let A be a graded algebra and define [ · , · ] : DerA×DerA → DerA (X, Y) 7→ XY− (−1)|X||Y|YX. (2.64) Then [ · , · ] is a graded Lie bracket on DerA. Proof. From Remark 2.1.3 it is clear that the degrees work. We just need to show that the Leibniz rule remains valid. Consider homogeneous a, b ∈ A and X, Y ∈ DerA. We then have (XY)(ab) =X((Ya)b + (−1)|Y||a|a(Yb)) =(X(Ya))b + (−1)|X|(|a|+|Y|)(Ya)(Xb) + (−1)|Y||a|(Xa)(Yb) + (−1)(|Y|+|X|)|a|a(X(Yb)). (2.65) To compute [X, Y](ab) we need to mirror the above computation exchanging the roles of X and Y, multiplying by (−1)|X||Y| and subtracting the result. The mirror of (X(Ya))b + (−1)(|X|+|Y|)|a|a(X(Yb)), (2.66) yields the Leibniz rule ([X, Y]a)b + (−1)(|X|+|Y|)|a|a([X, Y]b), (2.67) during the computation of [X, Y](ab). We thus only need to show that the terms coming from (−1)|X|(|a|+|Y|)(Ya)(Xb) + (−1)|Y||a|(Xa)(Yb) vanish. Chapter 2. Graded Linear Algebra 24 The mirror of (−1)|X|(|a|+|Y|)(Ya)(Xb) is of the form (−1)|Y|(|a|+|X|)+|X||Y|(Xa)(Yb) = (−1)|Y||a|(Xa)(Yb). (2.68) This will cancel the term (−1)|Y||a|(Xa)(Yb). Conversely, the mirror of this term is (−1)|X||a|+|X||Y|(Ya)(Xb), which cancels (−1)|X|(|a|+|Y|)(Ya)(Xb). 2.5 Graded Vector Fields Following the trend in noncommutative geometry, we proceed to define vector fields on graded vector spaces by extending their algebraic description on usual manifolds [see 44, Problem 19.12]. Definition 2.5.1 ([see 45, Definition 4.7.2]). Let V be a graded vector space. Graded derivations of O(V) are called graded vector fields on V. Theorem 2.5.1. Let V be a graded vector space and X : V∗ → O(V) a homoge- neous map of degree k ∈ G. Then there is a unique extension of X into a left vector field of degree k on V. Similarly, there is a unique extension into a right vector field Xr. Proof. We will proof this only for left derivations since the proof for right deriva- tions is identical. The extension is easily constructed by induction on m in the de- composition O(V) = S•V∗ = ⊕ m∈N SmV∗. Namely, it acts trivially on S0V∗ = F and we have already defined it on S1V∗ = V∗. Assuming that it has been defined for k = m− 1, recall that SmV∗ is spanned by elements of the form f F, with f ∈ V∗ and F ∈ Sm−1v∗. We then define X( f F) := (X f )F + (−1)|k|| f | f X(F). (2.69) The fact that the resulting linear map has degree k is immediate. We can now show the graded Leibniz rule also via induction. Indeed, if it has been shown for all pairs (F, G) ∈ SmV∗ × SlV∗ with m + l < N, then we have for f ∈ V∗ X( f FG) = (X f )FG + (−1)|X|| f | f X(FG) = (X f )FG + (−1)|X|| f | f X(F)G + (−1)|X|(| f |+|F|) f FXG = X( f F)G + (−1)|X||| f F| f FXG. (2.70) Chapter 2. Graded Linear Algebra 25 Example 2.5.1. Let (x1, . . . , xn) be an adapted basis of V∗ with V a bigraded vector space over a field F. Consider ∂ / ∂xi : V∗ → F defined by ∂xi ∂xj = δi j . (2.71) In other words, let (∂ / ∂x1 , . . . ∂ / ∂xn ) be the dual basis. Notice that deg ∂ / ∂xi = −deg xi while | ∂ / ∂xi | = |xi|. We will always extend this to a left vector field on V. The extension onto a right vector field will be denoted ∂r ∂xi := ( ∂ ∂xi ) r (2.72) In fact, the most generic vector fields can be built from this example. Theorem 2.5.2. Let X be a graded vector field on a graded vector space V with a graded basis (x1, . . . , xn) of its dual. Then X = X(xi) ∂ ∂xi . (2.73) Proof. In light of Theorem 2.5.1 this is clear from the trivial computation X(xi) ∂xj ∂xi = X(xj). (2.74) Remark 2.5.1. Given Theorem 2.4.3, we see that the graded vector fields on a graded manifold form a graded Lie algebra. Theorem 2.5.3. Let V be a graded vector space and (x1, . . . , xn) be a basis of its dual. We then have [∂ / ∂xi , ∂ / ∂xj ] = 0. Proof. Due to Theorem 2.5.1, we only need to prove this on V∗. We then have[ ∂ ∂xi , ∂ ∂xj ] xk = ∂ ∂xi δ j k − (−1)|x| i|x|j ∂ ∂xj δi k = 0. (2.75) Chapter 2. Graded Linear Algebra 26 Theorem 2.5.4. Let V be a graded vector space with (x1, . . . , xn) an adapted basis of its dual. Then a vector field F := Fi ∂ / ∂xi induces the right vector field FrG = (−1)Fi(|xi|+1) ∂G ∂xi Fi, (2.76) for all G ∈ O(V) Proof. ( Fi ∂ ∂xi ) r G = (−1)|F|(|G|+1)Fi ∂G ∂xi (2.77) Reordering to put the left hand side in the form ∂rG / ∂xi Fi yields an overall sign with exponent |F|(|G|+ 1) + |Fi|(|G|+ |xi|) + |xi|(|G|+ 1). (2.78) Using the fact that in Z�2Z we have x2 = x, we have |xi|(|G|+ 1) = |xi|(|G|+ |xi|). We can thus rewrite this exponent in the form |F|(|G|+ 1) + (|Fi|+ |xi|)(|G|+ |xi|). (2.79) Noticing that |Fi|+ |xi| = |F|, the exponent reduces to |F|(��|G|+ 1 +��|G|+ |x i|). (2.80) Definition 2.5.2 ([see 46, Definition 3.3]). A cohomological vector field is a graded vector field of degree (1) which commutes with itself. Remark 2.5.2 ([see 46, Remark 3.4]). Since for a cohomological vector field δ we have 0 = [δ, δ] = δδ− (−1)1×1δδ = 2δ2, we conclude that δ is a differential for the graded algebra O(V). Notice that the choice of degree (1) is fundamental for this result. Indeed, |(1)| = 1 guarantees that a cohomological vector field squares to 0, while deg(1) = 1 guarantees that it raises the cohomological degree by 1. Definition 2.5.3 ([see 46, Definition 3.6]). A graded vector space equipped with a cohomological vector field is called a differential graded space or dg space for short. Chapter 2. Graded Linear Algebra 27 Remark 2.5.3. A dg space (V, δ) induces a cochain complex (see Definition 2.1.3) · · · O(V)i−1 deg O(V)i deg O(V)i+1 deg · · · .δi−1 δi δi+1 (2.81) In other words, the space O(V) equipped with δ and the graduation given by deg instead of p, is a cochain complex. 2.6 Graded Poisson Vector Spaces Definition 2.6.1 ([see 47, Section 2.3]). A graded Poisson bracket { · , · } of degree k ∈ G on a graded algebra A is a graded Lie bracket of degree k such that for all homogeneous a ∈ A the induced map {a, · } : A → A is a derivation of degree p(a) + k. A graded algebra equipped with a Poisson bracket of degree k is called a graded Poisson algebra of degree k. The sign conventions between the Lie brackets and derivations are compatible in the Poisson bracket. Theorem 2.6.1. Let A be a graded Poisson algebra of degree k. Then for all homogeneous a ∈ A the map { · , a} : A → A is a right derivation of degree p(a) + k. Proof. This is clear from the fact that for all b ∈ A {b, a} = −(−1)(|a|+|k|)(|b|+|k|){a, b} = −(−1)(|a|+|k|)(|b|+1)+(|a|+|k|)(|k|−1){a, b} = −(−1)(|a|+k)(|k|−1){a, · }rb, (2.82) i.e. { · , a} = −(−1)(|a|+|k|)(|k|−1){a, · }r. Example 2.6.1. Let V be a graded symplectic vector space of degree k. We then have the musical isomorphism of degree −k, ] : V∗ → V of (2.29). We define the bilinear { · , · } : V∗ ×V∗ → F ( f , g) 7→ ω( f ], g]) ∈ F�� ��� �: | f |−�k+|g|−k+�k | f |]+|g|]+k ⊆ O(V)| f |+|g|−k. (2.83) We can extend this to a Poisson bracket of degree −k via the Leibniz rule. Namely, for all f ∈ V∗ we have the degree p( f )− k linear map { f , · } : V∗ → F ⊆ O(V). Chapter 2. Graded Linear Algebra 28 Via Theorem 2.5.1, this extends to a derivation { f , · } : O(V)→ O(V). Using the symmetry of the Poisson bracket we want to obtain, this defines a degree p(F)− k linear map {F, · } : V∗ → O(V) f 7→ {F, f } := −(−1)(|F|−k)(| f |−k){ f , F}. (2.84) for every F ∈ O(V). Via Theorem 2.5.1, this extends to a vector field {F, · } : O(V) → O(V). This defines the Poisson bracket { · , · } of O(V). Notice that it satisfies the correct skew symmetry since for all f , g ∈ V∗ { f , g} = ω( f ], g]) = −(−1)| f ]|V |g]|V ω(g], f ]) = −(−1)(| f |V∗−k)(|g∗|V−k){g, f } (2.85) Definition 2.6.2. A graded Poisson vector space of degree k ∈ G is a graded vector space V equipped with a graded Poisson bracket { · , · } of degree k on O(V). Definition 2.6.3. Let V be a graded Poisson vector space of degree k ∈ G and f ∈ O(V). The graded vector field X f := { f , · } of degree p( f ) + k is called the Hamiltonian vector field of f . Theorem 2.6.2 (Classical Master Equation). Let A be a bigraded Poisson algebra of degree (1). The Hamiltonian vector field to a smooth “function” S ∈ A0 is cohomological if and only if the master equation {S, S} = 0 is satisfied. Proof. By the graded Jacobi identity 0 = (−1)| f |+1{S, {S, f }}+ (−1)| f |+1{ f , {S, S}} − {S, { f , S}} = 2(−1)| f |+1{S, {S, f }}+ (−1)| f |+1{ f , {S, S}}, (2.86) i.e. {S, · }2 = −1 2{ · , {S, S}}. Thus, δ2 = 0 if and only if {S, S} = 0. Example 2.6.2. Continuing with Example 2.6.1, let us now consider the degree (-1) graded symplectic vector space E of Example 2.2.1. Given the musical isomor- phism, we characterize the graded Poisson bracket on E by {ΦA, Φ∗B} = −σ(ΦA, Φ∗B) = δA B , (2.87) with all other combinations of the basis elements equal to 0. Thus, {ΦA, · } = ∂ ∂Φ∗A , {Φ∗A, · } = − ∂ ∂ΦA . (2.88) Chapter 2. Graded Linear Algebra 29 We then conclude that for F ∈ O(E) we have {F, ΦA} = −(−1)(|F|+1)(|ΦA|+1) ∂F ∂Φ∗A = − ∂rF ∂Φ∗A {F, Φ∗A} = (−1)(|F|+1)(|Φ∗A|+1) ∂F ∂ΦA = ∂rF ∂ΦA , (2.89) so that {F, · } = ∂rF ∂ΦA ∂ ∂Φ∗A − ∂rF ∂ΦA ∂ ∂ΦA . (2.90) In other words, if F, G ∈ O(E) {F, G} = ∂rF ∂ΦA ∂G ∂Φ∗A − ∂rF ∂Φ∗A ∂G ∂ΦA . (2.91) In particular, for S ∈ O(E)0 we have ∂rS ∂Φ∗A ∂S ∂ΦA = (−1)|Φ ∗ A|+|ΦA| ∂S ∂Φ∗A ∂rS ∂ΦA = (−1)|Φ ∗ A|+|ΦA|+|ΦA||Φ∗A| ∂rS ∂ΦA ∂S ∂Φ∗A = (−1)−1+|ΦA|(|ΦA|−1) ∂rS ∂ΦA ∂S ∂Φ∗A = − ∂rS ∂ΦA ∂S ∂Φ∗A , (2.92) so that its master equation is 0 = ∂rS ∂ΦA ∂S ∂Φ∗A . (2.93) 2.7 BV Algebras Definition 2.7.1 ([see 48, p. 2.2.2]). A BV algebra is a graded algebra A equipped with a degree (1) map ∆ : A → A, the BV Laplacian, which satisfies ∆2 = 0 and is second order, i.e. for all F, G, H ∈ A we have ∆(FGH) = ∆(FG)H + (−1)|F|F∆(GH) + (−1)(|F|+1)|G|G∆(FH) − ∆FGH − (−1)|F|F∆GH − (−1)|F|+|G|FG∆H. (2.94) Every BV algebra has a graded Poisson bracket of degree 1. Chapter 2. Graded Linear Algebra 30 Theorem 2.7.1. Let A be a BV algebra. Define { · , · } : A×A → A by {F, G} = (−1)|F|∆(FG)− (−1)|F|∆FG− F∆G (2.95) for all F, G ∈ A. Then for all F, G ∈ A we have ∆{F, G} = {∆F, G}+ (−1)|F|+1{F, ∆G}. (2.96) Proof. For all F, G ∈ A we can express the BV Laplacian of FG in terms of the new bracket ∆(FG) = ∆FG + (−1)|F|F∆G + (−1)|F|{F, G}. (2.97) Applying this expansion twice we then obtain 0 = ∆2(FG) =�� ��* 0 (∆2F)G +(((( ((( (( (−1)|F|+1∆F∆G + (−1)|F|+1{∆F, G}+���� �� �� (−1)|F|∆F∆G + F�� ��* 0 (∆2G) + {F, ∆G}+ (−1)|F|∆{F, G}. (2.98) Theorem 2.7.2. Let A be a BV algebra. Then { · , · } is a graded Poisson bracket of degree (1) on A. Proof. The bilinearity is clear from the linearity of ∆. The degree is clear from the degree of ∆. The second order guarantees that for every F ∈ A the map {F, · } is a derivation. Indeed, applying the definition of the Poisson bracket we have {F, GH} = (−1)|F|∆(FGH)− (−1)|F|∆FGH − F∆(GH). (2.99) The second order condition on the first term then yields {F, GH} = (−1)|F|∆(FG)H +��� ��F∆(GH) + (−1)(|F|+1)|G|+|F|G∆(FH) − (−1)|F|∆FGH − F∆GH − (−1)|G|FG∆H − (−1)|F|∆FGH −���� �F∆(GH). (2.100) We notice that we have two terms (−1)|F|∆FGH. Let us rewrite one of them as (−1)|F|+|G|(|F|+1)G∆FH. The remaining terms proportional to H on the right yield Chapter 2. Graded Linear Algebra 31 precisely ( (−1)|F|∆(FG)− (−1)|F|∆FG− F∆G ) H = {F, G}H. (2.101) If we rewrite (−1)|G|FG∆H as (−1)|G|+|F||G|GF∆H, the other terms will be pro- portional to G on the left and equal to (−1)(|F|+1)|G|G ( (−1)|F|∆(FH)− (−1)|F|∆FH − F∆H ) = (−1)(|F|+1)|G|G{F, H}. (2.102) We conclude that {F, GH} = {F, G}H + (−1)(|F|+1)|G|G{F, H}. (2.103) The graded antisymmetry can be directly computed {F, G} = (−1)|F|∆(FG)− (−1)|F|∆FG− F∆G = (−1)|F|+|F||G|∆(GF)− (−1)|F|+(|F|+1)|G|G∆F − (−1)(|G|+1)|F|∆GF = −(−1)(|F|+1)(|G|+1)+|G|∆(GF) + (−1)(|F|+1)(|G|+1)G∆F + (−1)(|G|+1)(|F|+1)+|G|∆GF = −(−1)(|F|+1)(|G|+1){G, F}. (2.104) Finally, the Jacobi identity is a consequence of applying the BV Laplacian to the Leibniz rule (2.96). Indeed, we can use the definition of the bracket on the second term of ∆({F, GH}) = {∆F, GH}+ (−1)|F|+1{F, ∆(GH)}, (2.105) to obtain ∆({F, GH}) = {∆F, GH}+ (−1)|F|+1{F, (∆G)H} + (−1)|F|+1+|G|{F, G(∆H)}+ (−1)|F|+1+|G|{F, {G, H}} (2.106) We can now use (2.103) to expand the first, second and third brackets. In particular, in the expansion of the first and second bracket we will recover {∆F, G}H + (−1)|F|+1{F, ∆G}H = ∆{F, G}H. (2.107) Adding in the term (−1)|F|+|G|+1{F, G}∆H coming from the expansion of the Chapter 2. Graded Linear Algebra 32 third term leads us to ∆({F, G}H)− (−1)|F|+|G|+1{{F, G}, H}. (2.108) Similarly, using the terms left from the expansion of the first and third bracket we will also recover (−1)|F||G|G{∆F, H}+ (−1)|F|+��|G|+1+(|F|+�1)|G|G{F, ∆H} = (−1)|F||G|G∆{F, H}. (2.109) Then, adding the term (−1)|F|+1+(|F|+1)(|G|+1)∆G{F, H} left from the expansion of the second bracket we have (−1)|F||G|+|G|∆(G{F, H})− (−1)|F||G|{G, {F, H}. (2.110) Addition of (2.108) and (2.110) leads us to ∆{F, GH} − (−1)|F|+|G|+1{{F, G}, H} − (−1)|F||G|{G, {F, H}. (2.111) Putting everything together we have ∆{F, GH} = ∆({F, GH})− (−1)|F|+|G|+1{{F, G}, H} − (−1)|F||G|{G, {F, H}}+ (−1)|F|+|G|+1{F, {G, H}}. (2.112) From this we conclude that {F, {G, H}} = {{F, G}, H}+ (−1)(|F|+1)(|G|+1){G, {F, H}}, (2.113) which is another way to express the graded Jacobi identity (2.59) for |k| = 1 (see (3.2) of [42]). From now on we will always assume that every BV algebra is equipped with this bracket, called the BV bracket. Even though the BV Laplacian is not a deriva- tive for the ordinary product, it does satisfy a graded Leibniz rule for the BV bracket (2.96). Therefore, we will be able to use it to deform cohomological Hamil- tonian vector fields. Theorem 2.7.3. Let A be a BV algebra and S ∈ A0. Then Q = {S, · } − h̄∆ is a Chapter 2. Graded Linear Algebra 33 differential if and only if the quantum master equation 1 2 {S, S}+ h̄∆S = 0 (2.114) is satisfied. Proof. As we already computed in the proof of Theorem 2.6.2 {S, · }2 = −1 2 { · , {S, S}}. (2.115) Therefore Q2 =− 1 2 { · , {S, S}} − h̄{S, ∆ · } − h̄∆{S, · }+ h̄2 � �� 0 ∆2 =− 1 2 { · , {S, S}} −���� �h̄{S, ∆ · } − h̄{∆S, · }+���� �h̄{S, ∆ · }. (2.116) Finally, note that deg{S, S} = 1 and for all f ∈ A we have { f , {S, S}} = −(−1)(1+1)(| f |+1){{S, S}, f }. (2.117) We conclude that Q2 = { 1 2{S, S} − h̄∆S, · }. Definition 2.7.2 ([see 38, Construction 2.4.9]). Let V be a bigraded Poisson vector space of degree (1). We will define the BV Laplacian inductively, as the linear map ∆BV : O(V)→ O(V) satisfying the initial data 1. ∆BV f = ∆BV1 = 0, 2. ∆BV( f g) = (−1)| f |{ f , g}, for all f , g ∈ V∗, and the identity ∆BV(FG) = ∆BVFG + (−1)|F|F∆BVG + (−1)|F|{F, G}, (2.118) for all F, G ∈ O(V). Theorem 2.7.4. Let V be a Poisson vector space of degree (1). Then ∆BV makes O(V) a BV algebra. Proof. Trivially, ∆BV is of degree (1) on S0V∗ = F and S1V∗ = V∗. It inherits its degree from the Poisson bracket in S2V∗. Assuming that it acts with degree (1) on Chapter 2. Graded Linear Algebra 34 SkV∗, consider f ∈ V∗ and F ∈ SkV∗. Then, ∆BV( f F) =��� ��:0 (∆BV f )F + (−1)| f | f ∆BVF + (−1)| f |{ f , F}. (2.119) We conclude that it acts with degree (1) on Sk+1V∗ by noticing that due to the degree of the Poisson bracket p({ f , F}) = p( f ) + p(F) + (1) = p( f F) + (1), (2.120) and due to our induction hypothesis p( f ∆BVF) = p( f ) + p(∆BVF) = p( f ) + p(F) + (1) = p( f F) + (1). (2.121) We thus establish that p(∆BV) = 1. Next, we need to show that ∆2 BV = 0. We will do this using an induction like the one above. It is trivial to see that ∆2 BVF = 0 for all F ∈ SkV∗ with k ∈ {1, 2, 3}. Now, assume that ∆2 BVF = 0 for every F ∈ Sk(V). Take f ∈ V∗. We then have ∆2 BV( f F) = (−1)| f |∆BV( f ∆BVF + { f , F}) = f�� ��* 0 ∆2 BVF + { f , ∆BVF}+ (−1)| f |∆BV{ f , F}. (2.122) We thus see that ∆2 BV( f F) = 0 is equivalent to ∆BV{ f , F} = (−1)| f |+1{ f , ∆BVF}. (2.123) This should not come as a suprise, given the proof of Theorem 2.7.1. To proof the latter, we note that F will be a linear combination of elements of the form gG, with g ∈ V∗ and G ∈ Sk−1V∗. Due to linearity, we will proof (2.123) for F = gG. The left hand side is of the form ∆BV{ f , gG} = ∆BV({ f , g}G + (−1)(| f |+1)|g|g{ f , G}) = (−1)| f |+|g|+1{ f , g}∆BVG + (−1)| f |+|g|+1{{ f , g}, G} + (−1)(| f |+�1)|g|+��|g|g∆BV{ f , G} + (−1)(| f |+�1)|g|+��|g|{g, { f , G}}. (2.124) Chapter 2. Graded Linear Algebra 35 On the other hand, the right hand side is of the form (−1)| f |+1{ f , ∆BV(gG)} = (−1)| f |+|g|+1{ f , g∆BVG} + (−1)| f |+|g|+1{ f , {g, G}}. (2.125) We thus get 6 terms in total, with half depending on ∆BV. Let us first deal with the independent ones. Let us group the terms independent of ∆BV on one side. The term proportional to {{ f , g}, G} can be rewritten in terms of {G, { f , g}}. The resulting sign has exponent | f |+ |g|+ ��1 + (|G|+ 1)(| f |+ |g|) + ��1 =��� ��| f |+ |g|+ |G|| f |+ � �| f |+ |G||g|+ � �|g|. (2.126) Multiplying this term by a sign with exponent | f ||G|+ |g|+ |G|+ 1, yields an exponent |G||g|+ |g|+ |G|+ 1 = (|G|+ 1)(|g|+ 1). (2.127) Similarly, the term propotional to {g, { f , G}} can be rewritten in terms of {g, {G, f }}. This yields an overall sign with exponent | f ||g|+ (|G|+ 1)(| f |+ 1) + 1. Multiply- ing this term with the sign we used before then yields a total sign with exponent | f ||g|+ (|G|+ 1)(| f |+ 1) + ��1 + | f ||G|+ |g|+ |G|+ ��1 = | f ||g|+ | f |+ |g|+ 1 = (|g|+ 1)(| f |+ 1). (2.128) Finally, the term proportional to { f , {g, G}} has to be moved to the other side of the equation. Multiplying it by the sign we have used so far yields the sign with exponent | f |+ � �|g|+ | f ||G|+ � �|g|+ |G|+ 1 = (| f |+ 1)(|G|+ 1). (2.129) The grouping of these three terms together then yields (−1)(|G|+1)(|g|+1){G, { f , g}}+ (−1)(|g|+1)(| f |+1){g, {G, f }} + (−1)(| f |+1)(|G|+1){ f , {g, G}}, (2.130) which vanishes due to the Jacobi identity. We are thus left with three ∆BV-dependent terms. Multiplying by | f |+ |g|+ 1 Chapter 2. Graded Linear Algebra 36 and noting that | f |+ |g|+ 1+ | f ||g| = (| f |+ 1)(|g|+ 1), we see that what remains to be shown is { f , g}∆BVG + (−1)(| f |+1)(|g|+1)g∆BV{ f , G} = { f , g∆BVG}. (2.131) Using the Leibniz rule, the right hand side can be expanded to { f , g∆BVG} = { f , g}∆BVG + (−1)(| f |+1)|g|g{ f , ∆BVG}. (2.132) We are thus left with showing that (−1)| f |+1g∆BV{ f , G} = g{ f , ∆BVG}. (2.133) Using computation (2.122) with F 7→ G, we conclude this is equivalent to g∆2 BV( f G) = 0. This is true due to our induction hypothesis since f G ∈ SkV. Finally, we will show that it is a second order differential operator. This is an immediate consequence of the graded Leibniz rule of the graded Poisson bracket. Indeed, one can solve (2.118) to express the Poisson bracket in terms of ∆BV. Namely, for all F, G ∈ O(V) we have that {F, G} = (−1)|F|∆BV(FG)− (−1)|F|∆BVFG− F∆BVG (2.134) Then, by comparing the outcomes of {F, GH} = (−1)|F|∆BV(FGH)− (−1)|F|∆BVFGH − F∆BV(GH), (2.135) and the Leibniz rule {F, GH} = {F, G}H + (−1)(|F|+1)|G|G{F, H} = (−1)|F|∆BV(FG)H − (−1)|F|∆BVFGH − F∆BVGH + (−1)|F|+(|F|+1)|G|G∆BV(FH)− (−1)|F|+(|F|+1)|G|G∆BVFH − (−1)(|F|+1)|G|GF∆BVH, (2.136) we conclude that ∆BV is second order. Example 2.7.1. Continuing with Example 2.6.2, the Laplacian obtained in the Chapter 2. Graded Linear Algebra 37 construction above can be written in coordinates as ∆BV = (−1)|Φ A| ∂2 ∂ΦA∂Φ∗A . (2.137) Indeed, we have ∆BV(ΦBΦ∗C) = (−1)|Φ A| ∂2 ∂ΦA∂Φ∗A ( ΦBΦ∗C ) = (−1)|Φ A|+(|ΦA|+1)|ΦB|δB AδA C = (−1)|Φ B|δB C , (2.138) which coincides with (−1)|Φ B|{ΦB, Φ∗C}. Moreover, we can expand ∆BV(FG) = (−1)|Φ A| ∂ ∂ΦA ( ∂F ∂Φ∗A G + (−1)|Φ ∗ A||F|F ∂G ∂Φ∗A ) = (−1)|Φ A| ∂2F ∂ΦA∂Φ∗A G + (−1)|Φ A|+|ΦA|(|F|+|Φ∗A|) ∂F ∂Φ∗A ∂G ∂ΦA + (−1)|Φ ∗ A||F|+|ΦA| ∂F ∂ΦA ∂G ∂Φ∗A + (−1)|Φ ∗ A||F|+|ΦA||F|+|ΦA|F ∂2G ∂ΦA∂Φ∗A (2.139) We immediately recognize the actions of ∆BV in the first and last terms. With respect to the last term, we can also use |ΦA|+ |Φ∗A| = 1 to simplify the exponent in the sign. As for the second and third term we can modify the derivatives acting on F to right derivatives in the hope of recovering the BV bracket. Doing so yields ∆BV(FG) = (∆BVF)G + (−1)|F|F(∆BVG) + (−1)|Φ A|+|ΦA|(|F|+|Φ∗A|)+|Φ∗A|(|F|+1) ∂rF ∂Φ∗A ∂G ∂ΦA + (−1)|Φ ∗ A||F|+|ΦA|+|ΦA|(|F|+1) ∂rF ∂ΦA ∂G ∂Φ∗A (2.140) Studying the exponent of the signs in the last two factors we see that the |F| dependent term will be |F|(|ΦA|+ |Φ∗A|) = |F|. In the second term the remaining exponent is |ΦA|+ |ΦA||Φ∗A|+ |Φ∗A| = (|ΦA|+ 1)(|Φ∗A|+ 1) + 1 = 1, (2.141) while the exponent of the remaining exponent of the last term is |ΦA|+ |ΦA| = 0. Chapter 2. Graded Linear Algebra 38 We thus recover the BV bracket and we have ∆BV(FG) = (∆BVF)G + (−1)|F|F(∆BVG) + (−1)|F|{F, G}. (2.142) Given our realization that any degree (1) vector space is a BV algebra, we may now wonder whether any degree (1) Poisson algebra leads to a BV algebra. Answers to this question and the uniqueness of the associated BV Laplacian can be found in [49]. Chapter 3 BV Formalism in Finite Dimensions 3.1 Gaussian Integration In the path integral formulation of quantum field theory, expectation values are of the form 〈F〉 = 1 Z ∫ Dφ e− 1 h̄ SF, (3.1) with the normalization factor Z = ∫ Dφ e− 1 h̄ S. (3.2) This integral only has heuristic meaning as we shall now explain. Every field configuration φ has a probabilistic weight 1 Z e− 1 h̄ S, where S is the Euclidean action functional. Thus, if F is an observable, i.e. a suitable (in a sense that will be discussed latter) function of the fields, its expectation value is given by the sum of its value at each field configuration weighed by the probability of such a configuration. Of course, the space of all field configuration is continuous and the sum needs to be interpreted as an integral over this space against some measure Dφ. This is where the pitfall of the path integral formulation lies in. Such a measure in general does not exist. Before seeing how the BV formalism addresses this issue, let us reflect on why path integrals may seem reasonable at a first glance. For this purpose, we recall the following theorem: Theorem 3.1.1 (Riesz’s Representation Theorem). Let X be a locally compact Hausdorff space. If I is a positive linear functional on the compactly supported continuous functions Cc(X) on X. Then, there exists a measure space structure (X, Σ, µ) such that I( f ) = ∫ dµ f (3.3) for all f ∈ Cc(X). 39 Chapter 3. BV Formalism in Finite Dimensions 40 Proof. [see 50, Theorem 12.36] Thus, if 〈 · 〉 is to be a positive linear functional, it seems that the above theorem guarantees the existence of a measure that will aid us in its calculation. However, the devil lies in the details. A topological vector space, as we expect our space of field configurations to be, is locally compact if and only if it is finite dimensional [see 51, Theorems 1.21 and 1.22]. This is not the case unless our spacetime has a finite number of points. Thus, Theorem 3.1.1 does not guarantee the existence of such a measure in field theory. Thus, we are now faced against the problem of finding another way to calculate such a functional. Our approach is based on the fact that linear functionals are determined, up to a constant, by their kernel [see 52, Proposition 1.1.1]. In the case where the integral does make sense, the divergence gives us a method of calculating elements of such a kernel through Stokes’ theorem [see 53, equation (∗)]. Thus, such a divergence allows us to study these functionals. The treatment below is based on the work in [54]. Let us illustrate this idea by studying the case where Theorem 3.1.1 applies. We will follow [17, 54]. Namely, consider a scalar field theory on a finite spacetime M = { p1, . . . , pn }. The field configurations are given by real functions on M. This space is clearly identifiable with a real vector space V of dimension1 n. Since this space is finite dimensional, we can associate to our expectation value a probability measure µ. Let (φ1, . . . , φn) be a basis for the dual of V for the rest of this chapter. Let us further assume µ is absolutely continuous with respect to dλ := Dφ ≡ dnφ := dφ1 ∧ · · · ∧ dφn and that the Radon-Nykodim derivative dµ / dλ is always positive . Then, taking our action to be S := −h̄ log ( N dµ dλ ) , (3.4) for some N ∈ (0, ∞), we obtain2 〈F〉 = 1 Z ∫ Dφ e− 1 h̄ SF, Z := ∫ Dφ e− 1 h̄ S. (3.5) 1Of course, such a structure can arise in several different ways. For example, n-fields living in a spacetime consisting of only one point would be described by the same type of vector space. 2For the reader that has not yet been introduced to measure theory, this can be taken as the starting point of our discussion. The material above served the purpose of illustrating how in this case Theorem 3.1.1 yields to a generic integral of the form (3.1). All of the required material is however found in [50]. Chapter 3. BV Formalism in Finite Dimensions 41 The constant N is added because we want to identify S with the action of our system, which usually doesn’t lead to automatically normalized measures like µ. As usual in field theory, let us now specialize to the case of S = Q + b, (3.6) for some positive-definite quadratic form Q and some polynomial b of order higher3 than 3. We will also assume that S is increasing at infinity and we will denote by A the positive-definite symmetric matrix for which Q = 1 2 φi Aijφ j. We want to define a divergence divµ such that e− 1 h̄ S divµ F = div ( e− 1 h̄ SF ) (3.7) for suitable vector fields F on V. In particular, if F = Fi ∂ / ∂φi is a vector field with polynomial components, the rapid decrease of the exponential guarantees 〈 divµ F 〉 = 1 Z ∫ Dφ e− 1 h̄ S divµ F = 1 Z ∫ Dφ div ( e− 1 h̄ SF ) = 0. (3.8) Equation (3.7) is immediately solved for these vector fields divµ F = e 1 h̄ S div ( e− 1 h̄ SF ) = −1 h̄ ∂S ∂φi Fi + ∂Fi ∂φi . (3.9) Thus, if we let Vect(V) be the set of all polynomial vector fields, this defines a map divµ : Vect(V)→ O(V). (3.10) Applying (3.8), we conclude that〈 ∂S ∂φi Fi 〉 = h̄ 〈 ∂Fi ∂φi 〉 . (3.11) This is known as the Schwinger-Dyson equation ([see 32, (15.25)] or, for its appear- ance in pAQFT, [see 22, Remark 7.7]). As an example of the use of this formula, let us take b = 0, which corresponds to a free theory. Now consider a set of indices i1, . . . , in, j ∈ {1, . . . , n}. We can 3At this level considering polynomials of order one is futile since we can always complete the square. Chapter 3. BV Formalism in Finite Dimensions 42 compute −h̄ divµ ( φi1 · · · φin ∂ ∂φj ) = φi Aijφ i1 · · · φin − h̄ n ∑ r=1 φi1 · · · φ̂ir · · · φin δir j . (3.12) Thus, if we let Aij be the inverse matrix to Aij, i.e. Aij Ajk = δi k , we conclude φiφi1 · · · φin = −h̄ divµ ( φi1 · · · φin Aij ∂ ∂φj ) + h̄ n ∑ r=1 Aiir φi1 · · · φ̂ir · · · φin . (3.13) By taking the expectation value of both sides and doing a simple relabeling, we go back to a special case of the Schwinger-Dyson equation 〈 φi1 · · · φin 〉 = h̄ n ∑ r=2 Ai1ir 〈 φi2 · · · φ̂ir · · · φin 〉 . (3.14) This recurrence relation instructs us to pair the element φi1 which any element φir in the list (φi2 , . . . , φin), remove these two and replace them by Ai1ir , and take the expectation value of the rest. By the symmetry of our µ, it is clear that if n is odd, the expectation value is null. On the other hand, if n is even, given that 〈1〉 = 1 we can continue our recursion relation to〈 φi1 · · · φin 〉 = h̄n/2 ∑ P∈Pair(n) ∏ {r,s}∈P Airis , (3.15) where the set of pairings Pair(n) is defined to be the set of partitions of {1, . . . , n} into sets of 2 elements. This is of course just Wick’s theorem! We have found a way of computing expectation values without actually needing to integrate. Although there is a measure µ in this case, knowledge of divµ was the only thing we needed to proof this theorem. Chapter 3. BV Formalism in Finite Dimensions 43 3.2 Homological Picture of Integration: Antifields Let us now build a homological picture of this construction. To do this note that we can understand the map 〈 · 〉 : O(V)→ R through the induced diagram O(V) O(V)�ker 〈 · 〉 R, 〈 · 〉 [1] 7→1 (3.16) In here the first arrow is the canonical projection into the quotient, while the second is the isomorphism induced by 〈 · 〉 using the first isomorphism theorem [see 52, Proposition 1.1.1]. The latter is fixed by the normalization 〈1〉 = 1. Thus, we see that the computation of 〈 · 〉 can be achieved by computing O(V)�ker 〈 · 〉. In the previous section we attempted to do this using the operator divµ, for which im divµ ⊆ ker 〈 · 〉. The computation of the quotient O(V)�im divµ then corresponded to the Schwinger-Dyson equation (3.11). Comparing with the Defi- nition 2.1.3, we see this would correspond to the cohomology in the cohomological degree of O(V) of a cochain complex with · · ·Vect(V)→ O(V)→ 0 · · · . (3.17) To obtain such a complex, note that we can identify Vect(V) ∼= O(V)⊗V (3.18) by identifying vectors with the directional derivatives along them. Thus, if F ∈ O(V) and φ ∈ V, we identify the vector field F∂φ with F ⊗ φ. Moreover O(V) and Vect(V) ∼= O(V)⊗ V are the smooth functions in cohomological degree 0 and −1 of the degree (−1) symplectic vector space E = V ⊕V∗[−1]. In light of Example 2.7.1, we have that O(E) is a BV algebra and in particular has a degree (1) Poisson structure. In this identification we can use coordinates φ∗i := ∂ / ∂φi for V∗[−1] such that our vector fields on V can be written as φ∗i Fi. We will keep this convention for the rest of the chapter. These are called antifields to the fields φi. Under these identifications, our divergence (3.9) is a degree (1) map − h̄ divµ = ∂S ∂φi ∂ ∂φ∗i − h̄ ∂2 ∂φi∂φ∗i = {S, · } − h̄∆BV : O(E)−1 → O(E)0 (3.19) Chapter 3. BV Formalism in Finite Dimensions 44 Via this formula, it extends to a degree (1) algebra homomorphism Q on O(E). The corresponding cochain complex is then of the form · · · O(V)⊗V ∧V O(V)⊗V O(V) 0 · · · , Q Q Q (3.20) which we will call the quantum Koszul complex. In fact, Q can be decomposed into the Hamiltonian graded vector field δ := ∂S / ∂φi ∂ / ∂φ∗i = {S, · } and the BV Laplacian ∆BV = ∂2/∂φi∂φ∗i . Via Theorem 2.6.2, δ is cohomological. In this case it is clear that the classical master equation is satisfied since S only depends on the φ coordinates. By this same fact, the quantum master equation is also satisfied. We conclude by Theorem 2.7.3 that Q is a differential. In the free case we showed in (3.13) that H0(O(E), Q) = ker Q0 �im Q−1 = O(V)�im Q−1 ∼= R. (3.21) Then we conclude our expectation value map is precisely the projection 〈 · 〉 : O(E)→ H0(O(E), Q), (3.22) once we fix 〈1〉 = 1. Let us remark that in our finite dimensional case, the cochain complex O(E) is nothing but a disguise for the de Rham complex. Namely, the path integral measure yields a map O(V)⊗ m times︷ ︸︸ ︷ V ∧ · · · ∧V → ΩN−m(V), (3.23) given by contracting a given multivector field with the top form Dφ e− 1 h̄ S. For example, under this map the vector field φ∗i Fi is mapped to N ∑ i=1 (−1)i−1e− 1 h̄ SFi dφ1 ∧ · · · ∧ d̂φi ∧ · · · ∧ dφN . (3.24) Applying −h̄d to this form we obtain N ∑ i=1 (−1)i−1e− 1 h̄ S ( ∂S ∂φj Fi − h̄ ∂Fi ∂φj ) dφj ∧ dφ1 ∧ · · · ∧ d̂φi ∧ · · · ∧ dφN , (3.25) Chapter 3. BV Formalism in Finite Dimensions 45 which is simply Dφ e− 1 h̄ SQ(φ∗i Fi). Since both Q and −h̄d are extended to O(E) and Ω(V) using the Leibniz rule, we see that we have a commutative diagram · · · O(V)⊗V ∧V O(V)⊗V O(V) ΩN−2(V) ΩN−1(V) ΩN(V) Q Q −h̄d −h̄d (3.26) However, in infinite dimensions we do not have a notion of top form. This is the reason why this change of perspective plays an important role in QFT. At this point, we note that the quantum Koszul complex makes sense even in the case where V is a bigraded vector space concentrated in Z�2Z× {0} (i.e. a super vector space in cohomological degree 0) as long as we introduce the proper sign conventions to the BV bracket (2.91) {S, · } = ∂rS ∂φi ∂ ∂φ∗i , (3.27) and Laplacian (2.137) ∆BV = (−1)|φ i| ∂2 ∂φi∂φ∗i . (3.28) As a consequence, the vector fields φ∗i Fi act naturally on G ∈ O(V) from the right via {G, φ∗i Fi} = ∂rG ∂φi Fi. (3.29) We will also define εi := |φi| and ε∗i := |φ∗i | = εi + 1. We will extend to this case from now on to include fermionic theories into our discussion. 3.3 Koszul Complex: Going On-Shell Let us first begin with a theorem from differential geometry which will proof useful for us. Theorem 3.3.1 ([see 32, Theorem 1.1]). Let N be a regular zero set of a smooth map x : M → Rn on a manifold M of dimension n + m. Then every F ∈ C∞(M) vanishing on N is of the form F = xiGi for some G1, . . . , Gn ∈ C∞(M). Proof. First, note that by the Regular Level Set Theorem [see 44, Theorem 9.9], N is a regular submanifold. Moreover, every p ∈ N is covered by a chart (U, φ = Chapter 3. BV Formalism in Finite Dimensions 46 (x, y)) adapted to N [see 44, Lemma 9.10]. Consider the coordinate representation Fφ := F ◦ φ−1 of F. Then for all a ∈ x(U) ⊆ Rn and b ∈ y(U) ⊆ Rm we have Fφ(a, b) = ∫ 1 0 dt d dt Fφ(ta, b) = ∫ 1 0 dt ∂F ∂xi ( φ−1(ta, b) ) ai = (xi ◦ φ−1)(a, b) ∫ 1 0 dt ∂F ∂xi ( φ−1(ta, b) ) . (3.30) Thus, if one defines Gi(p) := ∫ 1 0 dt ∂F ∂xi ( φ−1(tx(p), y(p)) ) , (3.31) We conclude that F = xiGi on U. On the other hand, if p ∈ M is not in N, there must be a component xi which does not vanish at this point. In particular, there must a neighborhood of p where the component xi does not vanish (take for example (xi)−1(R \ {0})). We thus have that F = xiG, where G = F/xi, on this neighborhood. We have thus shown that every p ∈ M has a neighborhood U ⊆ M where F = xiG(U) i for some G1, . . . , Gn ∈ C∞(U). Recall that every manifold is paracompact [see 55, Theorem 4.77]. We thus have a locally finite open cover {Uα | α ∈ I } of M for which in each α ∈ I there are Gα 1 , . . . , Gα n ∈ C∞(Uα) such that F = xiGα i on Uα. Moreover, we have a partition of unity { ρα | α ∈ I } subbordinate to this cover [see 44, Theorem 13.7]. Using this we have that for each α ∈ I the support of ραGα is in Uα, and thus the function can be extended to all of M [see 44, Proposition 13.2]. We can then define the global functions Gi := ∑α∈A ραGα i ∈ C∞(M). Finally, we have xiGi = ∑ α∈A xiGα i ρα = ∑ α∈A Fρα = F. (3.32) Now, consider the limit h̄ → 0 of our previous discussion. In this limit our differential Q reduces to the cohomological Hamiltonian vector field δ := {S, · } = ∂rS / ∂φi ∂ / ∂φ∗i . Let us study the cohomology in degree 0. First of all, note that every element O(E)0 deg = O(V) is automatically closed since it is independent of the antifields. On the other hand, for a generic vector field φ∗i Fi ∈ O(V)⊗V, the element δ(φ∗i Fi) = ∂rS / ∂φi Fi corresponds simply to the infinitesimal change due to this vector field on the action function from the right. In particular, by definition Chapter 3. BV Formalism in Finite Dimensions 47 of the equations of motion4 ∂S / ∂φi = 0, the image of δ in degree 0 consists of the functions on O(V) which vanish on-shell5. Moreover, by Theorem 3.3.1, under certain regularity assumptions, we can assume that all functions which vanish on-shell are of the form ∂rS / ∂φi Fi. We thus conclude that H0(O(E), δ) corresponds to identifying the functions of our fields in V if they coincide on-shell. As a consequence, we will call these the on-shell functions. This construction is known as the Koszul complex of the ideal generated by the equations of motion ∂S / ∂φi in O(V). Let us now turn to the study of H−1(O(E), δ). As we already noticed, the action of δ on the vector fields O(E)−1 deg = O(V)⊗ V is simply to let the vector field act on the action functional S. Therefore, the closed elements Z−1, which is the set of vector fields φ∗i Fi for which ∂rS / ∂φi Fi = 0, are precisely symmetries of the action. On the other hand, a generic bivector field 1 2 φ∗i φ∗j Fij, where we can choose Fij = (−1)ε∗i ε∗j Fji without loss of generality, is sent to the symmetry δ(1 2 φ∗i φ∗j Fij) = ∂rS / ∂φi φ∗j Fij = (−1)ε∗j εi φ∗j ∂rS / ∂φi Fij. The fact that this is indeed a symmetry is a consequence of δ2 = 0, i.e. the classical master equation. We can however explicitly check this by noting that δ2 ( 1 2 φ∗i φ∗j Fij ) = δ ( ∂rS ∂φi φ∗j Fij ) = (−1)ε∗j εi ∂rS ∂φj ∂rS ∂φi Fij. (3.33) Then the coefficient µij := (−1)ε∗j εi Fij is graded antisymmetric µij = (−1)ε∗j εi Fij = (−1)εjεi+εi Fij = (−1)εjεi+εi+ε∗j ε∗i Fji = (−1)εjεi+��� �: 1 (εi+ε∗i )+εjε∗i Fji = −(−1)εjεi µji, (3.34) while the term ∂rS / ∂φj ∂rS / ∂φi is graded symmetric. Therefore, the contraction vanishes. These symmetries are trivial because they vanish on-shell and act trivially on the solutions of the equations of motion. The degree −1 cohomology H−1(O(E), δ) then yields all of the non-trivial symmetries. Let us now remark on the graded Lie algebra structure of this complex. By the Definition 2.6.1, the BV bracket gives O(E) the structure of a graded Lie algebra of degree (1). In particular, the set of vector fieldsO(E)−1 deg is a super Lie algebra. The BV bracket structure is compatible with the usual structure of super Lie algebra on 4Remember, the equations of motion are obtained by demanding that the action be stationary under an arbitrary local variation (vector field). 5We will reserve the use of “on-shell” for the solutions of the equations of motion Chapter 3. BV Formalism in Finite Dimensions 48 vector fields O(V)⊗V obtained by the commutator. Indeed, consider F = φ∗i Fi and G = φ∗j Fj {φ∗i Fi, φ∗j Gj} = φ∗i ∂rFi ∂φj Gj − (−1)ε∗i |F i|+εiε∗j Fiφ∗j ∂Gj ∂φi . (3.35) If we attempt to put the last term into the form φ∗j ∂rGj/∂φi Fi, the overall exponent in the sign becomes (recall the definition in Theorem 2.3.1) ε∗i |Fi|+ εiε∗j + |Fi|(ε∗j + |Gj|+ εi) + εi(|Gj|+ 1). (3.36) If we now assume that our vector fields are homogeneous, so that |F| = |Fi|+ εi and |G| = |Gi|+ εi, we have ε∗i (|F|+ εi) + � ��εiε∗j + (|F|+ εi)(|G|+ εi + 1) + εi(|G|+�� �� εj + 1) = ε∗i (� �@ @ |F|+��εi) + |F|(|G|+�� ��HHHHεi + 1) + εi(ZZ|G|+�� �� εi + 1) +HHHεi|G| = |F||G|. (3.37) We thus conclude that {φ∗i Fi, φ∗j Gj} = φ∗i ∂rFi ∂φj Gj − (−1)|F||G|φ∗j ∂rGj ∂φi Fi = { · , [G, F]}, (3.38) where [G, F] is the commutator of the right vector fields G = { · , φ∗i Gi} and F = { · , φ∗i Fi}. On top of this, O(E)[1] is in fact a dg Lie algebra with differential {S, · }. 3.4 Lie Algebras: Ghosts In physics, Lie groups usually represent symmetries of physical systems. These are usually referred to as continuous symmetries. The set of infinitesimal devia- tions from the identity of such groups has the structure of a Lie algebra. In the case that fermionic symmetries are present, one in fact requires a super Lie algebra. One can reconstruct some of the Lie group elements by a process of infinite iter- ation of these infinitesimal deviations. This process is known as exponentiation. Depending on the group, this process yields more or less information. However, it is always true that this process reconstructs a neighborhood of the identity. To continue our excursion into the BV formalism, we will recast the definition of a super Lie algebra in the language of graded manifolds. In particular, we Chapter 3. BV Formalism in Finite Dimensions 49 will see that the super Lie brackets on a super vector space g are in one to one correspondance with the cohomological vector fields on g[1]. This section and the next are extensions of the material found in [56]. For the rest of this chapter, let us fix g to be a super Lie algebra. As we saw, the Lie algebra of vector fields in the Koszul complex was in cohomological degree −1. Consequently, we would like to reinterpret the Lie bracket on g as a structure in g[1] by making use of (2.53). This yields the degree (1) map dec([ · , · ]) : S2(g[1])→ g[1] (3.39) We can now consider the dual of this map using (2.17) δ[ · , · ] := dec([ · , · ])∗ : g∗[−1]→ S2(g[1])∗. (3.40) Replacing the codomain by S2(g[1]∗) ⊆ O(g[1]) using (2.13) and applying Theo- rem 2.5.1, this extends to a degree (1) vector field on g[1]. To be explicit, let us calculate this map on a basis ( X1, . . . , Xm:=dim g ) of g. We will also denote the dual basis by ( c1, . . . , cm) and define εa = |ca|g[1]∗ = |Xa|g[1] = |Xa|g + 1. We will refer to the elements ca ∈ O(g[1]) as ghosts. Let f a bc be the associated structure constants, i.e. f a bc = ca([Xb, Xc]). (3.41) We will keep these conventions throughout the rest of this chapter. Given Defini- tion 2.4.1, the structure constants are graded antisymmetric in the bottom indices f a bc = ca([Xb, Xc]) = −(−1)(ε b+1)(εc+1)ca([Xc, Xb]) = −(−1)(ε b+1)(εc+1) f a cb = −(−1)(ε b+1)(εc+1) f a cb (3.42) and satisfy the Jacobi identity 0 = ca((−1)(ε b+1)(εd+1)[Xb, [Xc, Xd]] + (−1)(ε d+1)(εc+1)[Xd, [Xb, Xc]] + (−1)(ε c+1)(εb+1)[Xc, [Xd, Xb]]) = (−1)(ε b+1)(εd+1) f a be f e cd + (−1)(ε d+1)(εe+1) f a de f e bc + (−1)(ε c+1)(εb+1) f a ce f e db . (3.43) Chapter 3. BV Formalism in Finite Dimensions 50 According to (2.53), the resulting shifted Lie bracket is then dec([ · , · ])(XbXc) = (−1)εa+1[Xb, Xc] = (−1)εb+1 f a bc Xa. (3.44) Following (2.17), its dual is then (dec([ · , · ])∗ca)(XbXc) = (−1)εa+εb+1 f a bc = f a bc (−1)εc . (3.45) In here we used the fact that [ · , · ] has degree (0) and thus f a bc = 0 unless 0 = εa + 1 + εb + 1 + εc + 1 = εa + εb + εc + 1. (3.46) One can express dec([ · , · ])∗ca ∈ S2(g[1])∗ as an element of S2(g[1]∗) following (2.13). Let us start by computing 1 2 f a de (−1)εe cecd(XbXc) = 1 2 f a de (−1)εe ((−1)εbεd δe bδd c + (−1)� �εeεd+��εeεb δd bδe c). (3.47) The cancellation of the signs on the second term of the right hand side is due to the constraints imposed by the Kronecker deltas. Now, using the graded antisymmetry of the structure constants, we can see that the coefficient f a de (−1)εe is in fact graded symmetric. This should be the case to ensure that the term on the left hand side makes sense in O(g[1]). This is easily seen by multiplying (3.42) by (−1)εc and noticing that the overall sign on the right hand side will end up having exponent (εb + 1)(εc + 1) + 1 + εc = εbεc + εb. (3.48) As a result, both of the terms in the right hand side of (3.47) yield the same outcome. In particular, focusing on the form obtained from the second term, we conclude that 1 2 f a de (−1)εe cecd(XbXc) = f a cb (−1)εc = dec([ · , · ])∗ca)(XbXc). (3.49) The resulting vector field is δ[ · , · ] = 1 2 f a bc (−1)εc cccb ∂ ∂ca . (3.50) It turns out that δg extends to a graded Hamiltonian vector field. To obtain a Chapter 3. BV Formalism in Finite Dimensions 51 Poisson bracket, we first extend to the space E = g[1]⊕ g∗[−2]. Much like in sec- tion 3.2, this is a symplectic vector space of degree (−1). In particular, we have the coordinates (c1, . . . , cm, c∗1, · · · , c∗m) on E dual to the basis (X1, . . . , Xm, c1, . . . , cm), in which we can define S[ · , · ] = 1 2 c∗a f a bc (−1)εb+1cccb. (3.51) The new elements c∗a ∈ O(E) will be referred to as the antifields of our ghosts ca. Let us also set ε∗a := |c∗a |g∗[−2]∗ = εa + 1. This function allows us to us to define the graded Hamiltonian vector field δg = {Sg, · } on E . Notice that the extended δg agrees with the previous one on O(g[1]), which can be considered to be a graded subalgebra of O(E) via pullback with the projection g[1]⊕ g∗[−2]→ g[1]. Indeed, we have {S[ · , · ], ca} = −1 2 f a bc (−1)εb+1+ε∗a (ε b+εc)cccb. (3.52) Using the conditions (3.46) we can reduce the exponent of the overall sign to εb + 1 + ε∗a(ε a + 1) + 1 = εb + (εa + 1)2 = εb + εa + 1 = εc, (3.53) which yields {S[ · , · ], ca} = 1 2 f a bc (−1)εc cccb = δ[ · , · ]ca. (3.54) This new graded vector field is cohomological due to Theorem 2.6.2 since S[ · , · ] satisfies the classical master equation. The latter is easily computed in the form (2.93). To begin with, note that the coefficient f a bc (−1)εb+1 = f a bc (−1)εc+εa inherits the graded symmetry of f a bc (−1)εc . Consequently, the two terms in the derivative ∂rS[ · , · ] ∂cd = 1 2 c∗a( f a dc (−1)εd+1cc + f a bd (−1)εb+1+εdεb cb), (3.55) coincide. This yields ∂rS[ · , · ] ∂cd = c∗a f a dc (−1)εd+1cc, (3.56) and {S[ · , · ], S[ · , · ]} = c∗a f a bc (−1)εb+1cc f b de (−1)εd+1cecd = c∗a(−1)εa+εc+εd+1 f a bc f b de cccecd (3.57) In here we have used (3.46) to make the overall sign independent of the index b. To take the graded symmetrization in (c, e, d) we need to sum over the permutations Chapter 3. BV Formalism in Finite Dimensions 52 of these indices. In fact, since the structure constants f b de (−1)εd+1 are already graded symmetric in its two lower indices, we only need to sum over cyclic permutations. We conclude that the classical master equation is satisfied if and only if (−1)εa+εc+εd+1 f a bc f b de + (−1)εd(εc+εe)+εa+εd+εe+1 f a bd f b ec + (−1)εc(εe+εd)+εa+εe+εc+1 f a be f b cd (3.58) vanishes. Multiplying the whole expression by (−1)εa+εdεc , we obtain the graded Jacobi identity (3.42). The resulting cochain complex (O(g[1]), δ[ · , · ]) is called the Chevalley-Eilenberg complex for cohomology of g. It is useful to also remark that in the case where all symmetries are bosonic the expression for the action simplifies to S[ · , · ] = − 1 2 c∗a f a bc cbcc. (3.59) 3.5 DG Lie algebras: Ghosts for ghosts Recall that the Koszul complex discussed in Section 3.3 described the struc- ture of the Lie algebra of all infinitesimal symmetries as its closed elements in cohomological degree -1. Correspondingly, in the previous section we explored Lie algebras that correspond to infinitesimal gauge symmetries. In preparation to see how these fit into the Koszul complex, we studied their Chevalley-Eilenberg complex, which shifted them to be concentrated in cohomological degree -1 and encoded their structure through an action. However, in Section 3.3 we also found that the Koszul complex forms a shifted degree dg Lie algebra. The cohomological structure in this case encoded the possibility of having trivial symmetries. Now, our gauge theories may in general also contain symmetries that act trivially in this sense. Therefore, we have to extend our description of gauge structures to allow for dg Lie algebras. Let g be a dg Lie algebra concentrated in Z�2Z×Z≤0 with Lie bracket [ · , · ] and a degree (1) differential d. We can encode the Lie bracket using the same action (3.51). To encode d into an action, we can follow the same procedure: use (2.53) to obtain a degree (1) map dec(d) : g[1]→ g[1], (3.60) Chapter 3. BV Formalism in Finite Dimensions 53 use (2.17) to construct its dual, δd := dec(d)∗ : (g[1])∗ → (g[1])∗ ⊆ O(g[1]), (3.61) and, finally, apply Theorem 2.5.1 to obtain a degree (1) vector field on g[1]. Let us compute this using the same coordinates as above. However, now we have the possibility deg ca 6= 1. The coordinates with deg ca = i are said to be the ghosts for the ghosts of cohomological degree i + 1. Let us define the matrix representation Da b by dXb = Da bXa. Since d has degree (1), we know that Da b vanishes unless 0 = εa + 1 + εb + 1 + 1 = εa + εb + 1. (3.62) Application of (2.53) yields [see 41, Section 2] dec(d)(Xa) = −dXa = −Db aXb. (3.63) Its dual is then given according to (2.17) (dec(d)∗ca)(Xb) = (−1)εa ca(dec(d)(Xb)) = (−1)εa+1Da b = (−1)εa+1Da b = (−1)εa+1Da ccc(Xb). (3.64) Finally, Theorem 2.5.1 yields the degree (1) vector field δd := (−1)εa+1Da bcb ∂ ∂ca . (3.65) We can extend this to a cohomological vector field on E := g[1]⊕ g∗[−2] by considering the action Sd := −c∗a Da bcb. (3.66) Indeed, with this action we have {Sd, ca} = (−1)εbε∗a Da bcb. (3.67) Using (3.62), we can set εb = εa + 1 = ε∗a , so that εbε∗a = (εa + 1)2 = εa + 1. This shows that {Sd, ca} = δdca. Moreover, this vector field is cohomological since {Sd, Sd} = 2c∗a Da bDb ccc = 0, (3.68) given that d2 = 0. Chapter 3. BV Formalism in Finite Dimensions 54 Finally, let us combine the actions obtained so far into Sg := Sd + S[ · , · ]. (3.69) To see that this action satisfies the classical master equation we only need to show that {Sd, S[ · , · ]} = 0. The left hand side can be expanded to − c∗a Da b f b cd (−1)εc+1cdcc + 2(−1)εbε∗a+εaε∗c Da bcbc∗c f c da (−1)εd+1cd. (3.70) We have used the graded symmetry of f c de (−1)εd+1 to see that the two terms obtained in ∂S[ · , · ] / ∂ca are