DISSERTAÇÃO DE MESTRADO IFT–D.003/22 Worldsheet complex structure in the bosonic string sigma model Vinícius Bernardes da Silva Advisor Andrey Yuryevich Mikhaylov January 2022 Silva, Vinícius Bernardes da S586w Worldsheet complex structure in the bosonic string sigma model / Vinícius Bernardes da Silva. – São Paulo, 2022 84 f. Dissertação (mestrado) – Universidade Estadual Paulista (Unesp), Instituto de Física Teórica (IFT), São Paulo Orientador: Andrei Yuryevich Mikhaylov 1. Modelos de corda. 2. Física matemática. 3. Teoria das supercordas. 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). Dedico esta tese ao meu pai, Paulo (in memoriam). i Agradecimentos Quero agradecer meu orientador Andrei Mikhailov, que ao longo do curso de mestrado me ensinou muito sobre pesquisa. Seu jeito de pensar me influenciou em várias áreas da vida. Também agradeço meu colega durante esses dois anos, Eggon Viana, que me estimulou a querer melhorar sempre. Nossas conversas foram muito importantes, sobre física e sobre a vida. Agradeço a todos os colegas e professores que me acolheram na chegada ao IFT. Foi um desafio fazer a maior parte do mestrado de forma remota, então também os agradeço por proporcionaram boas conversas e convivência, mesmo à distância. As discussões com Lucas, Dennis, Lefundes, Vidal e João foram muito frutíferas. Muito obrigado à CAPES pelo financiamento confiado ao meu desenvolvimento ao longo do curso, que se conclui com esta dissertação. Finalmente, sou muito grato à minha família que me deu suporte constante. Especialmente, obrigado a Nana, Wali, Laura, Maira e Dani. ii Resumo O sigma-model da corda bosônica é uma teoria de calibre – é invariante sob reparametrizações da folha de mundo da corda –, portanto deve-se fixar o calibre para quantizá-lo. Uma das formas de se fazer isso é usando o formalismo BV, que permite a quantização de teorias com uma simetria fermiônica, como a simetria BRST. Define-se o espaço de fase BV, contendo os campos e seus anticampos. Para fixar um calibre, escolhemos uma subvariedade do espaço de fase BV. O calibre de Polyakov (ou calibre conforme) é definido ao fixar a estrutura complexa (ou métrica) na folha de mundo. No formalismo BV, esse procedimento define um número infinito de operadores bracket, que têm um papel nas deformações da teoria. Além disso, o operador BRST nesse calibre não é nilpotente, quando os campos estão off-shell, i.e. antes de usar equações de movimento. O anticampo da estrutura complexa surge neste calibre como o fantasma b, que é necessário para computar amplitudes, através de uma integração pelo espaço de moduli da folha de mundo. A transformação BRST do fantasma b é a expressão off-shell do tensor de energia-momento da teoria. Após ressomar os termos que dão origem ao operadores bracket, vemos que as deformações podem ser linearizadas, o que implica na fatorização holomórfica das amplitudes. O formalismo da supercorda de espinores puros é a generalização da corda bosôncia no calibre de Polyakov. Portanto, não há necessidade de fixar-se um calibre nesse formalismo. Diferentement da corda bosônica, o operador BRST é nilpotent mesmo off-shell. O fantasma b é definido de forma que a sua transformação BRST gera a expressão on-shell do tensor de energia-momento. Após a introdução no Capítulo 1, introduzimos fundamentos matemáticos no Capítulo 2. O formalismo BV é apresentado no Capítulo 3, e no Capítulo 4 o utilizamos para fixar fixar o calibre e quantizar o sigma-model da corda bosônica. Fazemos também comentários sobre os operadores bracket. No Capítulo 5 intro- duzimos a supercorda de espinores puros, e comparamos o formalismo com a corda bosônica. Palavras Chaves: Formalismo BV; Formalismo BRST; Corda bosôncia; Estrutura complexa; Supercorda de espinores puros. Áreas do conhecimento: Física; Teoria de cordas; Física Matemática. iii Abstract The bosonic string sigma model is a gauge theory – it is invariant under worldsheet diffeomorphisms, so it must be gauge fixed to be quantized. One way to do it is using BV formalism, which allows one to quantize theories with a fermionic symmetry, such as the BRST symmetry. One defines a BV phase space, with fields and their antifields. The gauge fixing procedure consists in a choice of submanifold in the BV phase space, and it is well defined off-shell. The Polyakov gauge (or conformal gauge) is defined by fixing the worldsheet complex structure (or metric) of the bosonic string. In BV formalism, this procedure defines an infinite number of derived brackets that play a role in deformations of the theory. Moreover, the BRST operator in this gauge in not nilpotent off-shell. This condition is substituted by constraints among the brackets. The antifield of the complex structure arises as the b ghost, which is needed to compute amplitudes, by integrating over the moduli space of the worldsheet. The BRST transformation of the b ghost is the energy-momentum of the theory defined off-shell. After resumming the terms that give rise to the higher brackets, we see that the deformations are linearized, and give rise to the holomorphic factorization. The pure spinor superstring is a generalization of the bosonic string in Polyakov gauge, so there is no need to gauge fix. Different from the bosonic string, the BRST operator is nilpotent off-shell. The b ghost is defined in such a way that its BRST transformation yields the energy-momentum tensor only on-shell. After the introduction in Chapter 1, we introduce the mathematical foundations in Chapter 2. The BV formalism is presented in Chapter 3, and in Chapter 4 we use it to gauge fix and quantize the bosonic string sigma-model. We make comments about the derived brackets. In Chapter 5 we introduce the pure spinor superstrings, and compare it with the bosonic string. Keywords: BV formalism; BRST formalism; Bosonic string; Complex structure; Pure spinor superstring. Fields of knowledge: Physics; String theory; Mathematical physics. iv Contents 1 Introduction 1 2 Mathematical foundations 3 2.1 Supermanifolds, Lie groups and Lie algebras . . . . . . . . . . . . . . 3 2.1.1 Exterior algebra and gradings . . . . . . . . . . . . . . . . . . 3 2.1.2 Supermanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.3 Lie groups and Lie algebras . . . . . . . . . . . . . . . . . . . 5 2.1.4 Lie superalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Spinors and supersymmetry algebra . . . . . . . . . . . . . . . . . . . 9 2.2.1 Poincaré group and algebra . . . . . . . . . . . . . . . . . . . 9 2.2.2 Clifford algebra and spinors . . . . . . . . . . . . . . . . . . . 10 2.2.3 Supersymmetry algebra . . . . . . . . . . . . . . . . . . . . . 12 2.3 Metric and complex structure . . . . . . . . . . . . . . . . . . . . . . 13 2.3.1 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.2 Complex structure . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.3 Complex structure in two dimensions . . . . . . . . . . . . . . 15 3 Gauge theories and BV formalism 17 3.1 Field theory and gauge symmetries . . . . . . . . . . . . . . . . . . . 17 3.1.1 Path integral and OPEs . . . . . . . . . . . . . . . . . . . . . 17 3.1.2 OPEs and Ward identities . . . . . . . . . . . . . . . . . . . . 18 3.1.3 Gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.4 BRST formalism . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Elements of BV formalism . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.1 BV phase space . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.2 Integration on Lagrangian submanifolds . . . . . . . . . . . . 24 3.2.3 Classical and quantum master equations . . . . . . . . . . . . 25 3.2.4 BRST-BV construction . . . . . . . . . . . . . . . . . . . . . . 26 3.2.5 Gauge fixing and conormal bundle . . . . . . . . . . . . . . . 27 3.3 Non-linearity and derived brackets . . . . . . . . . . . . . . . . . . . . 28 3.3.1 Derived brackets . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.2 Deformations of the action . . . . . . . . . . . . . . . . . . . . 31 v 4 Bosonic string sigma model 33 4.1 Formulation with metric . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.1 Action and gauge symmetries . . . . . . . . . . . . . . . . . . 33 4.1.2 BV construction . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.1.3 Gauge fixing: light-cone gauge . . . . . . . . . . . . . . . . . . 35 4.2 Complex structure as a field . . . . . . . . . . . . . . . . . . . . . . . 38 4.2.1 Expansion around Polyakov gauge . . . . . . . . . . . . . . . . 41 4.2.2 BRST transformation in Polyakov gauge . . . . . . . . . . . . 42 4.3 Energy-momentum tensor and OPEs . . . . . . . . . . . . . . . . . . 43 4.3.1 OPE with ghost current . . . . . . . . . . . . . . . . . . . . . 45 4.4 Integration over moduli space . . . . . . . . . . . . . . . . . . . . . . 46 4.4.1 Dolbeault operator and the µµ̄ fields . . . . . . . . . . . . . . 47 4.4.2 Action in terms of µµ̄ fields . . . . . . . . . . . . . . . . . . . 49 4.4.3 Holomorphic factorization . . . . . . . . . . . . . . . . . . . . 51 5 Pure spinor superstring 53 5.1 The target space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.2 The action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2.1 Symmetries and charges . . . . . . . . . . . . . . . . . . . . . 55 5.3 BRST operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.4 Energy-momentum tensor and the b ghost . . . . . . . . . . . . . . . 61 5.4.1 Definition of the b ghost . . . . . . . . . . . . . . . . . . . . . 63 5.4.2 Derivation of the b ghost and Y formalism . . . . . . . . . . . 64 6 Conclusions 67 A Differential geometry 69 A.1 Tangent and cotangent space . . . . . . . . . . . . . . . . . . . . . . . 70 A.2 Bundles, tensors and fields . . . . . . . . . . . . . . . . . . . . . . . . 73 A.3 Forms and densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A.4 Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 B Antifield of the complex structure 79 B.1 Antifields of mm̄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 B.2 Gauge symmetry of I⋆ . . . . . . . . . . . . . . . . . . . . . . . . . . 80 B.3 Antifields of µµ̄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Bibliography 84 vi Chapter 1 Introduction To quantize a gauge theory, we need to appropriately put a constraint on the fields. The BRST formalism is a correct approach that is well-known. But it is more insightful to use a formalism where deformations of the theory have geometrical meaning, such as the Batalin-Vilkovisky formalism. The BV formalism can be applied to any theory described by an action and a fermionic symmetry, such as the BRST symmetry. The space of fields is extended to the BV phase space, containing the fields and its antifields, and the action and BRST operator are encoded in the BV action. The gauge fixing procedure consists in a choice of submanifold in the BV phase space [1]. Once the choice is made, the action and the fermionic symmetry can be transformed into a gauge fixed action, plus a possibly infinite number of bracket operations in the space of fields, one of these being a BRST operator. The additional bracket operations are non-linear terms of the BV action, and their presence is not clear from the BRST point of view, though they are important when one considers deformations of the theory. The bosonic string sigma model is a gauge theory – invariant under diffeomor- phisms – that describes the dynamics of a freely propagating string. A BV action can be constructed with the action and the BRST operator that generates diffeomor- phisms [2]. To describe interaction of strings, one must introduce vertex operators that correspond to physical states. Their introduction is equivalent to small defor- mations of the free theory, and the BV formalism constrains them to preserve the BRST structure of the theory. The introduction of vertex operators is described as a deformation of the BV action, and their introduction changes not only the action, but possibly the BRST operator as well. In order to compute scattering amplitudes of strings, one must sum probabilities over the moduli space of the theory, which is the space of all topologies and metrics modulo rescaling and diffeomorphism on the worldsheet. In bosonic string theory, the metric is a field that is gauge fixed, so to perform the moduli space integration it is necessary to consider deformations that change the metric. These deformations are trivial, in the sense that the theory doesn’t change under them – they are equivalent 1 Chapter 1. Introduction 2 to a change in the gauge fixing condition. In BV formalism, the trivial deformations can be parameterized, and the fields that generate these deformations in bosonic strings are the b ghosts. The presence of non-linear terms with higher brackets affects the theory only up to equations of motion, so the BV formalism allows one to understand the off-shell aspects of the BRST structure of the bosonic string. Moreover, there are OPEs which are contact terms – which are delta functions on the worldsheet – that can be removed by resumming the deformations of the BV action. This procedure reduces the BRST structure to an action and a BRST operator only, and implies in the holomorphic factorization of the amplitudes [3]. In the pure spinor formalism of superstring theory, the action describes the worldsheet embedding in the supersymmetric spacetime, and a BRST-like operator constrains the physical states [4]. Though the model has a BRST-like structure, it doesn’t come from gauge fixing. Because there is no need to gauge fix, there are no higher brackets in the BV action of the theory, and the BRST-like operator is nilpotent off-shell by construction. On the other hand, the ghost field that generates deformations along the moduli space is a composite field that generates the energy-momentum only on-shell [5, 6]. Chapter 2 Mathematical foundations In this section we’ll present the mathematical definitions that will come to use in the subsequent chapters. Some of the content of this chapter is built upon basic differential geometry, which is reviewed in Appendix A. 2.1 Supermanifolds, Lie groups and Lie algebras 2.1.1 Exterior algebra and gradings The space of functions on a real manifold M is given by FunM = T (0,0)M . The elements satisfy an algebra defined by the algebra of multiplication of real numbers. For two functions F1, F2 ∈ FunM , there is another function F3 ∈ FunM given by F3 = F1F2. The product of real numbers is commutative, hence the algebra on FunM is associative as well. The wedge product defines an algebra of 1-forms, called the exterior algebra, which takes forms u1, u2 ∈ Ω1(M) to another form u1∧u2 ∈ Ω2(M). By the definition of wedge product, the algebra is anticommutative. Let’s define the space of all forms as Ω◦(M) = Ω0(M)⊕ Ω1(M)⊕ Ω2(M)⊕ · · · . (2.1.1) Then we can say that Ω◦(M) = Fun(T ∗M). The elements of Ω0(M) are functions M , and the elements of Ω1(M) are functions on M with values on the cotangent spaces. With the exterior algebra defined by the wedge product, elements of Ω◦(M) are considered functions on the cotangent bundle. An exterior algebra can also be defined without the notion of a wedge product. Let’s generalize the real linear space Rm of dimension m, whose elements satisfy the real commutative algebra. First we write the real space as Rm = Rm|0, and we say that it is a space with m dimensions of even parity. Then we define the real space R0|n of n dimensions of odd parity, which exactly like Rn, but the components 3 Chapter 2. Mathematical foundations 4 of its elements satisfy an anticommutative algebra, the exterior algebra. Thus for θ1, θ2 ∈ R0|n we have θ1θ2 = −θ2θ1. Finally we can define the real linear space Rm|n of m even dimensions and n odd dimensions, given by Rm|n = Rm|0 ⊕ R0|n. (2.1.2) Elements of Rm|n are written as (x1, · · · , xm, θ1, · · · , θn) ∈ Rm|n. (2.1.3) The exterior algebra can then be written as the algebra between each of the compo- nents, as [xi, xj]− = 0, [θa, θb]+ = 0, [xi, θa]− = 0, (2.1.4) where [, ]− is the commutator and [, ]+ the anticommutator. More generally, we can consider a linear space V which is decomposed in k others as V = V0 ⊕ · · · ⊕ Vk−1, (2.1.5) Along with a grading, which is a map gr : V → Z, that associates to each element v ∈ Vk to the integer label gr(v) = k. The grading of an element can also be called its degree. We define a product operation which sums the gradings: gr(uv) = gr(u) + gr(v). (2.1.6) An exterior algebra is a linear space with two gradings V = V0 + V1, with V0 = Rm|0 and V1 = R0|n. If we have two gradings, we may call it the parity of an element. 2.1.2 Supermanifolds A supermanifold is a manifold that is locally equal to Rm|n. It is equipped with charts that take it to linear spaces with odd dimensions, ϕ : U ⊂M → Rm|n, such that the transition functions τ = ϕ−1 ◦ ϕ̃ are smooth functions on an open subset of Rm|n. Given a chart, a supermanifold has coordinates (xi, θα), with gradings x̄ = 0, θ̄ = 1. (2.1.7) Chapter 2. Mathematical foundations 5 We say that a supermanifold is (m,n)-dimensional if it is locally equal to Rm|n. One example of a supermanifold are the odd tangent bundle and the odd cotangent bundle. Consider the tangent bundle TM of an even manifold M , i.e. (m, 0)- dimensional. The odd tangent bundle ΠTM is defined by flipping the parity of the tangent fiber. In coordinates, we write the elements as (x, dx), where x ∈ M have grading grx = 0, and the tangent vectors dx are odd: grdx = 1. The odd cotangent bundle ΠT ∗M of a manifold M is the cotangent bundle with flipped parity in the cotangent fiber. It can be seen as an incorporation of the wedge product algebra to the manifold. In fact, if we write the elements as (x, dx), and consider a function F ∈ FunΠT ∗M , it can be expanded in powers of the cotangent vectors as F (x, dx) = F0(x) + FA 1 (x)dxA + · · ·+ FA1···An n dxA1 · · · dxAn , (2.1.8) so it is a formal sum of forms Fi ∈ Ωi(M) on the base of ΠT ∗M . The de Rham operator on M is defined as a vector field on ΠT ∗M , given by d = dxA ∂ ∂xA . (2.1.9) For a vector field v ∈ TM , we define the iota operator ιv as a vector field on ΠT ∗M , by ιv = vA ∂ ∂dxA . (2.1.10) With these operators we are able to define the Cartan magic formula Lv = [ιv, d], (2.1.11) which holds for action of Lie derivative on forms on M . The commutator takes into account the odd parity of d and ιv (if we take v to be an odd vector field, then ιv has even parity). 2.1.3 Lie groups and Lie algebras A group G is a set with a map G×G→ G, (g1, g2) 7→ g1 · g2, called the product, that satisfies associativity, i.e. (g1 · g2) · g3 = g1 · (g2 · g3), (2.1.12) Chapter 2. Mathematical foundations 6 an element 1 ∈ G called the identity, that satisfies 1 · g = g · 1 = g, (2.1.13) and a map G→ G, g 7→ g−1 called the inverse, which has the property g · g−1 = g−1 · g = 1. (2.1.14) A Lie group is a manifold equipped with a product, an identity and an inverse that satisfy the above mentioned properties, and such that the product and the inverse are smooth maps. The general linear group GL(n) is the set of linear transformations on Rn. There is also the general Lie group over the complex numbers, GL(n,C), which is the set of linear transformations on Cn. The coordinates of Rn induce the parameterization of GL(n) as n× n matrices. This allows the straighforward definitions of the determinant, trace and transpose of an element of the group, which are independent of the choice of coordinates. The orthogonal group O(n) ⊂ GL(n) is defined by imposing the condition det g = ±1 for every g ∈ O(n). The special orthogonal group SO(n) ⊂ O(n) is defined by containing only elements with positive determinant. The Lie algebra of a Lie group G is the tangent space around the identity: g = T1G. (2.1.15) The elements v ∈ g can be written, for some (g, dg) ∈ ΠTG, as v = g−1dg. (2.1.16) The elements of the Lie algebra can also be seen as small deformations of the group identity. Take a chart of G in which g = gAtA, where tA form the basis of the tangent space, and are called generators. Let there be a curve t 7→ gv(t) on G such that g(0) = 1. It defines the Lie algebra vector v as v = ∂ ∂t ∣∣∣∣ t=0 gv(t). (2.1.17) Chapter 2. Mathematical foundations 7 The Lie algebra vectors act as linear maps on G, as vg = ∂ ∂t ∣∣∣∣ t=0 (gv(t) · g). (2.1.18) The commutator is a map [ · , · ] : g× g→ g defined from the group product by the action on G, as [u, v]g. In terms curves g1(t) and g2(t), the definition is [v1, v2] = ∂ ∂t1 ∂ ∂t2 ( g1(t1) · g2(t2)− g2(t2) · g1(t1) ) t1=t2=0 = v1v2 − v2v1. (2.1.19) It is proportional to the term that goes with t1t2 in the expansion of group elements around the identity in the group product, because the linear terms don’t couple the tangent vectors, and thus don’t define operations in the algebra. In coordinates, the commutator can be written in terms of structure constants fABC , as in [u, v] = fABCu BvCtA. (2.1.20) The commutator satisfies antisymmetry [u, v] = − [v, u] (2.1.21) and the Jacobi identity [u, [v, w]] + [v, [w, u]] + [w, [u, v]] = 0. (2.1.22) An example of Lie algebra is gl(n), the algebra of GL(n). The elements v ∈ gl(n) are represented as n-dimensional matrices, and the commutator is the commutator of the matrices. 2.1.4 Lie superalgebras A Lie superalgebra is a Lie algebra g which is the sum of subalgebras gi with different gradings [7] g = g0 ⊕ · · · ⊕ gn, (2.1.23) Chapter 2. Mathematical foundations 8 where an element v ∈ gk has grading k. The commutator is then a map [ · , · ] : g× g→ g that sums the gradings of the elements of g, as gr([u, v]) = gr(u) + gr(v). (2.1.24) The symmetry relation is generalized to [u, v] = (−1)ūv̄[v, u] (2.1.25) where ū and v̄ are the signs of the gradings of the elements of the algebra, and the Jacobi identity is [u, [v, w]] + (−1)ū(v̄+w̄)[v, [w, u]] + (−1)w̄(ū+v̄)[w, [u, v]] = 0. (2.1.26) One example of Lie superalgebra is gl(m|n), which is a generalization of gl(m) to linear transformations on Rm|n. We define this algebra as the sum of two subalgebras with different gradings: gl(m|n) = gl0(m|n)⊕ gl1(m|n). (2.1.27) The even elements u = gl0(m|n) = gl(m)× gl(n) are block diagonal matrices u = ( A 0 0 B, ) (2.1.28) where A ∈ gl(m) and B ∈ gl(n). The odd elements v ∈ gl1(m|n) are off diagonal matrices v = ( 0 C D 0 ) . (2.1.29) The commutator [u, v] is the commutator of the matrices, and the result is [u, v] = ( 0 (A−B)C (B − A)D 0 ) , (2.1.30) which is an element of grading 1. Chapter 2. Mathematical foundations 9 2.2 Spinors and supersymmetry algebra 2.2.1 Poincaré group and algebra The group GT of translations on Rm is a m-dimensional group, parameterized by translation vectors a ∈ Rm as g(a) ∈ GT, with the group product given by g(a1) · g(a2) = g(a1 + a2). (2.2.1) The generators Pm of translation are called momenta. For a small parameter a, we define g(a) = 1− iamPm. (2.2.2) From the group product, we can derived the translation algebra [Pm, Pn] = 0. (2.2.3) The generalization of SO(n) for a space with a metric of signature (m,n) (instead of the Euclidean signature (n, 0)) is the Lorentz group SO(m,n). Its fundamental representation consists of (m+ n)-dimensional matrices that satisfy ΛTΛ = g, (2.2.4) where g is the metric. Parameterizing group elements by these matrices, as g(Λ), the group product is written as g(Λ1) · g(Λ2) = g(Λ1Λ2). (2.2.5) To obtain the Lorentz algebra, we parameterize small transformations around the identity as Λmn = δmn + ωmn, where ωmn is antisymmetric due to the orthogonality condition. The Lorentz generators Mmn are then given by g(1 + ω) = 1− i 2 ωmnM mn. (2.2.6) The Lorentz algebra is then given by [Mmn,Mpq] = i(gnpMmq − gmpMnq − gnqMmp + gmqMnp). (2.2.7) Chapter 2. Mathematical foundations 10 The Poincaré group in n dimensions the product of translation group and Lorentz group: Gn P = Gn T × SO(p, n− p). Its elements are written as g(a,Λ), and the group product is g(a1,Λ1) · g(a2,Λ2) = g(a1 + Λ1a2,Λ1Λ2). (2.2.8) We have translation group and Lorentz group as subgroups. The Poincaré algebra is parameterized by the Pm and Mmn, and the translation and Lorentz algebras are subalgebras. The commutator left to determine the Poincaré algebra is given by [Pm,Mpq] = i(gmpP q − gmqP p). (2.2.9) 2.2.2 Clifford algebra and spinors The Clifford algebra in the space Rn, with metric gmn, is a superalgebra gcl(n). An element of grading k is an antisymmetric tensor Γm1···mk . We represent elements of degree 1 as p-dimensional matrices Γm, where p = 2n/2−1 or p = 2(n+1)/2−1 depending whether the dimension n is even or odd. In the matrix representation, the elements of degree k are defined by the antisymmetrized product of degree 1 matrices. In particular, elements of degree 2 is given by the matrix commutator Γmn = 1 2 [Γm,Γn]. The algebra commutator { · , · } is defined so that it subtracts the grading by 2: gr({u, v}) = gr(u) + gr(v)− 2. In the matrix representation, the algebra commutator of elements of odd gradings is represented by the matrix anticommutator, whereas for elements of opposite grading it is the matrix commutator. The Clifford algebra is defined by the relation on degree 1 elements {Γm,Γn} = 2gmn. (2.2.10) The algebra commutator on degree 2 elements then yields the Lorentz algebra [Γmn,Γpq] = gnpΓmq − gnqΓmp − gmpΓnq + gmqΓnp, (2.2.11) where the i factor can be made present by properly rescaling the matrices. The p-dimensional space where the Γm matrices act is called spinor space Spin(Rn), and its elements θα (α = 1, · · · , p) are called spinors. Because Γmn satisfy the Lorentz algebra, they are generators of the Lorentz group in the spinor representation. On a Chapter 2. Mathematical foundations 11 spinor θ, a rotation Λ = 1 + ω acts as δθ = 1 2 ωmnΓ mnθ. (2.2.12) We take translation to act on the spinor space as δθ = 0. We can take the Γmαβ to be symmetric. For two spinors ξ and η, we have ξΓmη = (−1)ξ̄η̄ηΓmξ. (2.2.13) In 10 dimensions (m = 1, · · · , 10) the matrices satisfy the Fierz identity, (Γm)α(β(Γm)γδ) = 0, (2.2.14) where () indicates symmetrization of the indices. The identity can also be written, for spinors ξ, η, ψ, as (ξΓmη)ψΓm + (−1)ξ̄η̄+ξ̄ψ̄(ηΓmψ)ξΓm + (−1)ψ̄ξ̄+ψ̄η̄(ψΓmξ)ηΓm = 0. (2.2.15) Another useful identity is the decomposition of the product ΓmΓn = 1 2 {Γm,Γn}+ 1 2 [Γm,Γn] = gmn + Γmn. (2.2.16) The spinor bundle of the manifold M is a manifold Spin(M) with elements (m, θ), where m ∈ M and θ ∈ Spin(Rn), and θ transforms as a spinor under change of coordinates. One can also define a spinor bundle with k sectors: Spink(M), with k spinor linear spaces at each point m ∈M . One example of spinor bundle with two sectors is the one over Rn, with the sector of spinors Spin(Rn) and the space of antispinors Spin(Rn), or spinors with opposite chirality. We can also define the two sectors as SpinL(Rn) and SpinR(Rn), for left and right spinors. We write θα for left spinors, and θα̇ for right spinors. We define the space of Dirac spinors as the sum of left and right spinor spaces: SpinD(Rn) = SpinL(Rn) + SpinR(Rn). (2.2.17) The Dirac matrices are defined as matrices that act on SpinD(Rn) through Pauli Chapter 2. Mathematical foundations 12 matrices ΓmL on SpinL(Rn) and ΓmR on SpinR(Rn) Γm = ( ΓmL 0 0 ΓmR ) . (2.2.18) 2.2.3 Supersymmetry algebra The supersymmetry algebra on an n-dimensional space is a superalgebra which is the direct sum of two linear spaces gSUSY = gP + gQ, (2.2.19) where gP is the Poincaré algebra on an n-dimensional space, and gQ is a p-dimensional space with odd grading, where p = 2n/2−1 or p = 2(n+1)/2−1 depending whether n is even or odd. The odd parity spinor generator Qα is an element of gQ. The algebra between Qα and Poincaré generators is given by [Q,Pm] = 0, [Q,Mmn] = ΓmnQ. (2.2.20) For a spinor bundle on a 2-dimensional manifold, we define [Q,Q] = 2ΓmPm. (2.2.21) If we have left and right sectors we also impose independence between the sectors as [QL, QR] = 0. (2.2.22) The representation of supersymmetry transformation on a spinor bundle parame- terized by (x, θ) is given by δxm = 1 2 ξΓmθ, δθ = ξ. (2.2.23) We can also have two supersymmetry generators Q+ and Q− over the same Chapter 2. Mathematical foundations 13 bundle sector, with [Q+, Q−] = 0, [Q+, Q+] = 2ΓmPm, [Q−, Q−] = − 2ΓmPm. (2.2.24) We call Q− a supersymmetry transformation with flipped parity. The representation of this algebra on a spinor bundle parameterized by (x, θ) is δ+x m = 1 2 ξΓmθ, δ+θ = ξ, δ−x m = − 1 2 ξΓmθ, δ−θ = ξ. (2.2.25) 2.3 Metric and complex structure 2.3.1 Metric A metric is a symmetric tensor field of rank (0, 2). More precisely, the space of metrics on a manifold M is given by S2(M) = T (0,2)M/ ∼, (2.3.1) with the equivalence relation u1 ⊗ u2 ∼ 1 2 (u1 ⊗ u2 + u2 ⊗ u1) , (2.3.2) which makes the antisymmetric products equivalent to zero. A metric g ∈ S2(M) defines an inner product between vector fields U, V ∈ Γ(TM). The metric is written in coordinates as g = gijdxi ⊗ dxj, (2.3.3) and the vector fields are written in coordinates as U = Ui ∂ ∂xi , V = Vi ∂ ∂xi . (2.3.4) Chapter 2. Mathematical foundations 14 The inner product can be defined in terms of the action on the basis as g(U, V ) = gijU kV l(dxi ⊗ dxj) ( ∂ ∂xk , ∂ ∂xl ) , (2.3.5) where we used linearity of inner products to move Uk and V l out, and also wrote g as a linear combination of the S2(M) basis. Then we take the action of the basis of metrics on the basis of vector fields to be (dxi ⊗ dxj) ( ∂ ∂xk , ∂ ∂xl ) = 〈 dxi, ∂ ∂xk 〉〈 dxj, ∂ ∂xl 〉 = δikδ j l . (2.3.6) The inner products thus gets written in components as g(U, V ) = gijU iV j. (2.3.7) A metric g ∈ S2(M) also defines a linear map g : TM → T ∗M , which takes a tangent vector field to a cotangent vector field. It is given in components by g(V ) = gijV kdxi ⊗ dxj ( ∂ ∂xk ) = gijV idxj. (2.3.8) More generally, the metric can act on a tensor of rank (p, q) and take it to a tensor of rank (p− 1, q + 1). It does so by acting with a component of its basis dxi on a component of the vector field basis ∂j, yielding the inner product ⟨i, ∂j⟩ = δij. The inverse of the metric g ∈ S2(M) is a tensor field g−1 ∈ T (2,0)M such that gg−1 = 1, where 1 ∈ T (1,1)M is the identity operator given by 1 = n∑ i=1 dxi ⊗ ∂ ∂xi . (2.3.9) The inverse is written in components as g−1 = gij ∂ ∂xi ⊗ ∂ ∂xj , (2.3.10) so its definition in components is given by gijgjk = δki . Chapter 2. Mathematical foundations 15 2.3.2 Complex structure A complex structure is a (1, 1) tensor field written as I = I ij ∂ ∂xi ⊗ dxj, (2.3.11) that satisfies the constraint I2 = −1. (2.3.12) It has no real eigenvalues, because its eigenvalues must satisfy λ2 = −1. If it is defined on a complex manifold, or if the real manifold where it’s defined on is complexified, than we can say its eigenvalues are ±i. We can see that it can be understood as a rotation by π/2 at each point, since applying it twice is equivalent to a reflection of the coordinates. 2.3.3 Complex structure in two dimensions On a 2-dimensional manifold (which we will also call a “worldsheet”), a complex structure can be used to define the Hodge dual operator ⋆ : Ω1(M)→ Ω1(M), which maps 1-forms to 1-forms by the expression ⋆u = I ·u, (2.3.13) or, in coordinates, ⋆(uadσa) = Iabuadσ b. (2.3.14) The Hodge dual must be connected with a definition of inner product by the property u ∧ ⋆v = ⟨u, v⟩vol, (2.3.15) where vol is the volume form. Thus complex structure must be related to a metric. We have u ∧ I ·u = Ibcuaubdσ adσc = Ibcϵ acuaubdσ1dσ2, (2.3.16) Chapter 2. Mathematical foundations 16 so the connection with the metric is Iαβϵ βγ = √ det ggαγ. (2.3.17) Chapter 3 Gauge theories and BV formalism 3.1 Field theory and gauge symmetries 3.1.1 Path integral and OPEs We consider a field theory as a manifold Φ and a measure µ defined on it. The manifold Φ is the space of fields, and it contains fields ϕ ∈ Φ (notice that we write it lowercase for the fields, and uppercase for the space of fields). The fields ϕ : Σ→ N are maps from the worldsheet Σ, which is a two dimensional manifold, to a target space N . The path integral is defined by a measure of integration µ on Φ, e.g. a density on Φ, which we write as µ = [dϕ] exp(−S(ϕ)/ℏ), (3.1.1) where S : Φ → R is the action. The path integral is defined as integration over µ and it yields a number defined as P = ∫ [dϕ] exp(−S(ϕ)/ℏ). (3.1.2) We call such a theory a sigma model. Correlation functions are computed by inserting operators in the path integral, and we take operators to be functions of the fields and its derivatives. The correlation function between operators O1, · · · ,On is defined as ⟨O1 · · · On⟩ = ∫ [dϕ]O1 · · · On exp(−S(ϕ)/ℏ). (3.1.3) Another way to define correlation functions is to think of operators as deformations 17 Chapter 3. Gauge theories and BV formalism 18 of the measure. Consider the following trasformation of the action S 7→ S + ϵO1 · · · On, (3.1.4) for some parameter ϵ. For small values of ϵ, the measures is transformed by µ 7→ µ(ϵ) = µ (1− ϵO1 · · · On) , (3.1.5) so that the correlation function can be written as function of the deformation, like ⟨O1 · · · On⟩ = lim ϵ→0 1 ϵ ∫ (µ− µ(ϵ)). (3.1.6) 3.1.2 OPEs and Ward identities A free theory is a theory with a quadratic action on the fields. Let’s write the worldsheet coordinates as (z, z̄), and write fields as ϕa(z) (we write only z for short, but it can depend on both z and z̄). The action of a free theory depends on a local linear operator Aij(z) as S = ∫ d2z 1 2 ϕiAijϕ j. (3.1.7) In a free theory, the correlation function between two fields is the inverse of the quadratic factor, as ⟨ϕi(z)ϕj(z′)⟩ = (A−1)ij(z)δ2(z − z′). (3.1.8) Some of what follows is presented in [8]. We can consider a correlation function between two local operators (local functions of fields) very close to each other, and expand the result in powers of their distance. This is called the operator product expansion between two fields, or OPE. Consider the operator P (ϕ)(z, z̄) located at (z, z̄) in the worldsheet, and an operator Q(ϕ)(0) located at (z, z̄) = 0. Their operator product expansion is given by P (z, z̄)Q(0) = V0(z, z̄) + V(1,0) z + V(0,1) z̄ + V(1,1) zz̄ + V(2,0) z2 + · · · , (3.1.9) where the operator V0(z, z̄) is a polynomial in z and z̄, and V(m,n) are independent of (z, z̄). To determine the operators on the right hand side, one has to compute the correlation function on the left hand side, using Wick theorem to express it in terms Chapter 3. Gauge theories and BV formalism 19 of correlation functions of the fields in coordinates ϕi. Consider an action S which is symmetric under the vector field v ∈ VectΦ in the space of fields. For every operator U(z) that we insert in the path integral, we derive the following identity 0 = ∫ v(Ue−S) = ∫ ((vU)− (vS)U) e−S, (3.1.10) which is the statement that the expected value of the symmetry transformation on U has expected value zero: ⟨vU⟩ = 0, because of the symmetry vS = 0. Now let the action be S = ∫ L, for a 2-form L on the worldsheet, and let v(ξ) be the generator of a global symmetry, with parameter ξ. The gauged generator ṽξ is defined by promoting the parameter ξ to a function ξ(z) on the worldsheet. The variation of the action is then vS = ∫ ξdj, (3.1.11) for some 1-form j, which is the current of symmetry generated by v. When the fields are on-shell, we have vS = 0, and thus the current is conserved: dj = 0. An important expression is the relation between OPE of an operator U with the current j, which yields the action of the symmetry transformation on U , as j(z)U(0) = δvU z + other orders. (3.1.12) A countour integral of the OPE selects only the factor of the simple pole, so we can write ∮ jU = δU. (3.1.13) We see that we can compute how fields transform under a symmetry transformation by computing their OPEs with the current. 3.1.3 Gauge theories A gauge group is a group that is parameterized by functions on the world- sheet, so its dimension is infinite. The elements of such a group are called gauge transformations. If a quantum field theory has an action which is invariant under a gauge group, Chapter 3. Gauge theories and BV formalism 20 it is said to have a gauge symmetry, and it is called a gauge theory. It is often the case that the correlation functions of such theories are ill-defined, because the path integral sums over redundant field configurations, and is thus infinite, in a sense. To deal with gauge theories we must do a process called gauge fixing. Let H be the gauge group. Each element h ∈ H is a gauge transformation that defines an automorphism on the space of fields Φ: ϕ ∈ Φ 7→ h(ϕ) ∈ Φ. (3.1.14) The statement that the action is invariant under H is written as S(h(ϕ)) = S(ϕ), for all h ∈ H,ϕ ∈ Φ. (3.1.15) The field configurations ϕ and h(ϕ) are said to be physically equivalent. The aim of gauge fixing is to switch the integral over the whole space of fields to an integral which sums only fields which are not physically equivalent among each other. We will call by gauge slice such a space of fields that are not physically equivalent. More rigorously, we define a gauge slice Γ ∈ Φ as a manifold such that, for all ϕ ∈ Γ, we have h(ϕ) /∈ Γ, for any transformation h ∈ H. This means that any gauge transformation takes a configuration out of the gauge slice. One way to define a gauge slice is with a constraint. The constraint is defined by a function K : Φ→ V , for some linear space V . Then the gauge slice Γ ⊂ Φ is defined by K(ϕ) = 0 iff ϕ ∈ Γ. (3.1.16) The constraint K really defines a gauge slice if it is perpendicular to any gauge transformation, i.e. va ∂ ∂ϕa K = 0, (3.1.17) for element v of the Lie algebra of H, which acts as a small gauge transformation. The correct way to write the path integral as an integration only over a gauge slice is to change coordinates of the space of fields, so the integration along one gauge slice can be factored out. To change the coordinates correctly we need to know the Jacobian of the transformation, which is non trivial for spaces with infinite dimensionality. To do this properly we need to use BRST formalism to introduce Chapter 3. Gauge theories and BV formalism 21 ghost fields, and then we can generalize the procedure using BV formalism. 3.1.4 BRST formalism To change coordinates in the space of fields Φ, we consider an extension of this space to the BRST field space ΦBRST . The BRST field space is defined by ΦBRST = Φ× ΠTH H . (3.1.18) Here ΠTH is a tangent space with flipped parity, we denote its elements as (h, dh). Its action on H is naturally defined.1 The equivalence relation on eq. (3.1.18) is given by (ϕ, h, dh) ∼ (h0 ·ϕ, h0 ·h, h0 · dh), (3.1.19) for any h0 ∈ H. If we choose h0 = h−1 to parameterize the space, we identify the isomorphism ΦBRST = Φ× Πh, (3.1.20) with ΦBRST ∋ (ϕ, h−1dh) = (ϕ, c). The c coordinates are generators of the gauge transformation (i.e. elements of the gauge algebra), but with flipped parity, and are called ghosts. There are vector fields on Φ×ΠTH which are equivalent to zero in ΦBRST. They are the generators of small perturbations around h = 1, and are parameterized by ghosts c0 as h0 = 1 + c0 = 1 + (h0) −1dh0. These null vector fields are given by vc0 = c0 ·ϕ ∂ ∂ϕ + c0 ·h ∂ ∂h + c0 · c ∂ ∂c . (3.1.21) The de Rham operator on H is a vector field on ΠTH given by d = dhα ∂ ∂hα , (3.1.22) and it can be trivially defined on Φ×ΠTH with the same expression, transversal to 1The action on elements of ΠTH can be defined as either by the right or by the left. We take action by the left as the equivalence relation. Then the action by the right remains as an operation over ΦBRST. Chapter 3. Gauge theories and BV formalism 22 the direction along Φ. To descend its definition to ΦBRST, we add it to a suitable null vector field vc0 that makes the sum transversal to the direction along H. The resulting vector is proportional to what we define as the BRST operator QBRST = c ·ϕ ∂ ∂ϕ + 1 2 [c, c] ∂ ∂c , (3.1.23) where we used c · c = 1 2 [c, c]. The BRST quantization of a gauge theory with action S ∈ FunΦ and gauge group H consists in choosing a gauge fixing constraint K ∈ FunΦ, along with the definition of a BRST action SBRST ∈ FunΦBRST|K=0 which correctly contains the factor coming from the Jacobian in the path integral. Schematically, one needs to find the action such that ∫ [δϕ]e−S = ( ∫ [δϕ′]) ∫ [δϕ0]e −SBRST , where ϕ0 parameterizes the gauge fixed fields, and ϕ′ parameterizes the degrees of freedom that decouple from the action. The introduction of ghosts to gauge fix makes the path integral well defined. However, to recover the correct degrees of freedom, the physical states must be elements of the cohomology of the QBRST operator. A clear way to understand the BRST construction is using the BV formalism, which will be discussed in the next section. 3.2 Elements of BV formalism Now we will introduce the definitions and results of the Batalin-Vilkovisky formalism. 3.2.1 BV phase space To do the quantization using BV formalism, we start with a space of fields Φ and an action S ∈ FunΦ. If we are using the BRST formalism we consider the action S with the gauge symmetry, and the BRST space of fields ΦBRST, with ghosts. The BV phase space is a cotangent space with flipped parity ΦBV = ΠT ∗Φ. Its elements are (ϕ, ϕ⋆), where ϕ are the fields (including ghosts) and ϕ⋆ are covectors in the space of fields, with flipped parity, and are called antifields. In ΦBV there is a naturally defined symplectic form ω with odd parity, which can Chapter 3. Gauge theories and BV formalism 23 be written, using Darboux coordinates, as ω = δϕ⋆a ∧ δϕa, (3.2.1) where δ is the differential in the space of fields. The inversion of the odd symplectic form defines the odd Poisson bracket, or BV bracket, as {F ,G} = F ( ←− δ δϕa −→ δ δϕ⋆a − ←− δ δϕ⋆a −→ δ δϕa ) G, (3.2.2) for any functions F ,G ∈ ΦBV. The relation between left and right derivatives is −→ δ δϕ F = (−1)(F̄+1)ϕ̄F ←− δ δϕ . (3.2.3) The BV bracket has the (anti)symmetry property {F ,G} = (−1)F̄Ḡ+F̄+Ḡ{G,F}, (3.2.4) which means that it is symmetric, but has odd parity. It satisfies the Jacobi identity as {F , {G,H}}+ (−1)(F̄+1)(Ḡ+H̄){G, {H,F}}+ (−1)(H̄+1)(F̄+Ḡ){H, {F ,G}} = 0, (3.2.5) which is the usual Jacobi identity, but taking anticommutations into account. Another meaningful operator on an odd cotangent space is the odd Laplacian operator, or BV Laplacian, defined as ∆ = (−1)Ā ∂ ∂ϕA ∂ ∂ϕ⋆A , (3.2.6) where Ā is the parity of the coordinate ϕA in the BV phase space. It is related to the BV bracket by the following property ∆(FG) = (∆F)G + (−1)F̄F(∆G) + (−1)F̄{F ,G}. (3.2.7) Given a function H ∈ FunΦBV in BV phase space, one defines the Hamiltonian Chapter 3. Gauge theories and BV formalism 24 vector field {H,−} as {H,−} = (−1)(H̄+1)Ā δH δϕA δ δϕ⋆A + (−1)(H̄+1)(Ā+1) δH δϕ⋆A δ δϕA . (3.2.8) An important property is that, for any H ∈ FunΦBV, we have L{H,−}ω = 0, (3.2.9) where ω = δϕAδϕ⋆A is the symplectic form. 3.2.2 Integration on Lagrangian submanifolds A Lagrangian submanifold ΦLAG ⊂ ΦBV is a submanifold with half the dimension- ality of ΦBV, and where the odd symplectic form vanishes. The trivial Lagrangian submanifold is the one obtained putting all antifields to zero: ϕ⋆ = 0. A half-density ρ1/2 is a density with weight 1/2. In particular, the volume form of the space of fields Φ is a half-density on the BV phase space ΦBV = ΠT ∗Φ. The restriction of any half-density of the BV phase space to a Lagrangian submanifold becomes a volume form. We can thus consider, for a half-density ρ1/2 and a Lagrangian submanifold ΦLAG, the integral∫ ΦLAG ρ1/2. (3.2.10) Consider another Lagrangian submanifold Φ′ LAG that is an infinitesimal deformation of the original one. Let {Ψ,−} be the Hamiltonian vector field on ΦBV that generates the variation, for an odd function Ψ ∈ FunΦBV. This deformation preserves the condition ω = 0, and we call such a function a gauge fermion. By Stokes theorem, the integral changes by ∫ L{Ψ,−}ΦLAG ρ1/2 = ∫ ΦLAG L{Ψ,−}ρ1/2. (3.2.11) The action of the Lie derivative on the half density is L{Ψ,−}ρ1/2 = 1 2 (div{Ψ,−})ρ1/2 + {Ψ, ρ1/2} = − (∆Ψ)ρ1/2 + {Ψ, ρ1/2}. (3.2.12) Chapter 3. Gauge theories and BV formalism 25 By using the relation between the BV bracket and BV Laplacian, we get the identity L{Ψ,−}ρ1/2 = −∆(Ψρ1/2) + Ψ∆ρ1/2. (3.2.13) The first term is a total derivative, so the variation of the integral is L{Ψ,−} ∫ ΦBV ρ1/2 = ∫ ΦBV Ψ∆ρ1/2. (3.2.14) 3.2.3 Classical and quantum master equations The trivial Lagrangian submanifold, i.e. putting antifields to zero ϕ⋆ = 0, is the original space of fields. The path integral of a quantum field theory with action S0 ∈ FunΦ is given by the integration of the measure µ = [δϕ] exp(−S0/ℏ). (3.2.15) Consider a half-density ρ1/2 such that its restriction to ϕ⋆ = 0 is the measure µ. Then the path integral is given by eq. (3.2.10). Under a deformation of the trivial Lagrangian submanifold, the integral remains constant if the measure satisfies ∆ρ1/2 = 0. (3.2.16) This is the quantum master equation. Let’s define the half-density in terms of a BV action S ∈ FunΦBV, which equal to the original action when one restricts ϕ⋆ = 0, as ρ1/2 = [δϕδϕ⋆]1/2 exp(S/ℏ). (3.2.17) In terms of the action, the quantum master equation is written as 1 2 {S, S}+ ℏ∆S = 0. (3.2.18) It is a condition on the BV action, so that we can perform the path integral in any Lagrangian submanifold, and obtain the same correlation functions. A change in Lagrangian submanifold can be seen as an exchange of degrees of freedom from the fields to the antifields, so it can be used to do gauge fixing of the fields. The classical master equation is the one obtained by putting ℏ → 0 in the Chapter 3. Gauge theories and BV formalism 26 quantum master equation, and is given by {S, S} = 0. (3.2.19) 3.2.4 BRST-BV construction A theory in BRST formalism consists of a BRST space of fields Φ, containing fields and ghosts, a classical action Scl ∈ Φ (the nomenclature helps to remind that this action usually doesn’t depend on ghosts), and an odd vector field Q ∈ VectΦ, which is nilpotent: Q2 = 1 2 [Q,Q] =0 (3.2.20) (the brackets are for anticommutator, because Q is odd). The BRST operator Q is a symmetry of the action. Writing the fields in Φ in coordinates as ϕa(σ), where σ are worldspace M coordinates, and a are indices in the target space N , we write the symmetry condition as QS = ∫ M Qa(ϕ) δ δϕa S(ϕ) = 0. (3.2.21) To apply BV formalism in such a theory, we consider a BV action given by S(ϕ, ϕ⋆) = Scl(ϕ) + ⟨Q(ϕ), ϕ⋆⟩. (3.2.22) The inner product ⟨⟩ contracts tangent fields with cotangent fields (antifields). We will often write it as S = Scl + Q̂, where Q̂ = ⟨Q, ϕ⋆⟩ = ∫ M Qaϕ⋆a (3.2.23) is the generating function in BV phase space of the BRST vector field. This BV action is linear in antifields, so it is useful to split the classical master equation in orders of antifields and see which conditions on Scl and Q it implies. The classical master equation is {S, S} = {Scl + Q̂, Scl + Q̂} = 2{Scl, Q̂}+ {Q̂, Q̂} = 0, (3.2.24) where we already used {Scl, Scl} = 0, because Scl doesn’t depend on antifields. At Chapter 3. Gauge theories and BV formalism 27 zeroth order in antifields we have {Scl, Q̂} = −QScl, (3.2.25) so the symmetry of the classical action under the BRST transformation is implied. At first order in antifields we have {Q̂, Q̂} = − ⟨[Q,Q], ϕ⋆⟩, (3.2.26) so the classical master equation also constrains Q to be nilpotent. We are thus allowed to apply BV formalism with this action for a BRST system. 3.2.5 Gauge fixing and conormal bundle The last step in applying BV formalism to a BRST theory is gauge fixing. If the classical action has gauge symmetries, it makes the path integral ill-defined, so we have to constrain some degrees of freedom of the fields. BV formalism allows us to do that by deforming the Lagrangian submanifold to another where the fields are constrained. Consider a space of fields Φ. A gauge fixing condition is a function K : Φ→ V , where V is some linear space, that defines a gauge fixed space of fields Φgf in the following way: is K(ϕ) = 0, ϕ ∈ Φgf. From the gauge condition K = 0 we get a constraint on tangent vectors of Φgf: δϕµ ∂KA ∂ϕµ = 0, (3.2.27) where A are components of coordinates in V . We need to find a Lagrangian submanifold ΦLAG in BV phase space ΦBV = ΠT ∗Φ such that Φgf is in ΦLAG. We now construct such a space: the conormal bundle of Φgf is a space ΠN∗Φgf ⊂ ΠT ∗Φ defined by ⟨δϕ, ϕ⋆⟩ = 0, (3.2.28) where δϕ ∈ ΠTΦgf and ϕ∗ ∈ ΠN∗Φ. It is a submanifold of BV phase space such that its base coincides with Φgf, and the cotangent fiber is constrained to be transversal to variations along Φgf. If we define the symplectic potential as α = ⟨δϕ, ϕ∗⟩, we can obtain the symplectic form by ω = δα. In the conormal bundle ΠN∗Φgf, we have α = 0, and thus also Chapter 3. Gauge theories and BV formalism 28 ω = 0. In order to be sure that the conormal bundle N∗Φgf is be a Lagrangian submanifold, we still have to check that it has the same dimensionality as Φ. We write the dimension of the constrained space as dimΦgf = dimΦ− dimK(Φ), (3.2.29) so the dimension of the fiber of the conormal bundle has to be the dimension of the gauge fixing constraint K. We thus parameterize the antifields by Ψ ∈ ΠV , as ϕ⋆µ = ΨA ∂KA ∂ϕµ . (3.2.30) This parameterization is consistent with eq. (3.2.28), because of the constraint in eq. (3.2.27). 3.3 Non-linearity and derived brackets Until now we have only discussed the construction of a BV action which is linear in antifields, and have a BRST-like structure. However, we saw that under a change of coordinates equivalent to a deformation of Lagrangian submanifold, the BV action might acquire non-linear terms on antifields. Let’s consider a general BV action, and expand it in powers of antifields, as S = S0 + S1 + S2 + · · · , (3.3.1) where Sk is of order k in antifields, as Sk = 1 k σA1···Ak k ϕ⋆A1 · · ·ϕ⋆Ak . (3.3.2) The classical master equation imposes constraints between the Sk terms. We have 0 = {S, S} = {S0 + S1 + · · · , S0 + S1 + · · · }. (3.3.3) When we split the equation order by order in power of antifields, we get the list of Chapter 3. Gauge theories and BV formalism 29 the conditions {S0, S1} = 0, {S0, S2}+ 1 2 {S1, S1} = 0, {S0, S3}+ {S1, S2} = 0, {S0, S4}+ {S1, S3}+ 1 2 {S2, S2} = 0, · · · . (3.3.4) The generalization can be made at order n, for even n, as n/2∑ i=0 {Si, Sn−i} = 0, (3.3.5) and for odd n as 1 2 {S(n+1)/2, S(n+1)/2}+ (n−1)/2∑ i=0 {Si, Sn−i} = 0. (3.3.6) At first order, the condition states that the BRST operator Q = {S1,−} is a symmetry of S0, as {S0, S1} = −Qa δS0 δφa = 0. (3.3.7) To look at the other equations, we can use the condition that the action S0 is on-shell. We can easily do it by taking {S0, Sk} = 0. (3.3.8) We are then putting to zero the equations of motion that are perpendicular to the directions of the σk indices. That can be seen by writing the condition in coordinates: σ AB1···Bk−1 k δS0 δϕA = 0. (3.3.9) 3.3.1 Derived brackets We can see that the Sk term defines a k-bracket on the field, i.e. a map from k functions F1, · · · , Fk ∈ Φ on the space of fields to numbers. The Sk-bracket is given Chapter 3. Gauge theories and BV formalism 30 by {F1, · · · , Fk}Sk := {{· · · {S1, F1}, · · · }, Fk}. (3.3.10) and is called a derived bracket. In particular, {F}S1 = {S1, F} is simply the action of the vector field generated by S1 on the function F (in BRST, the vector field is Q). Aĺso, the second order term defines a Poisson-like bracket {F,G}S2 = {{S2, F}, G}, (3.3.11) which may or may not satisfy Jacobi identity. Let’s check the symmetry of the S2-bracket. Using Jacobi identity of BV bracket, we compute {F,G}S2 = − (−1)F̄ Ḡ+Ḡ{{G,S2}F} − (−1)F̄+Ḡ{{F,G}, S2} = (−1)(F̄+1)(Ḡ+1){G,F}S2 (3.3.12) where we used that {F,G} = 0, because none of the functions depends on antifields. We can now check what are the conditions from the classical master equation on the derived brackets. The constraint at first order in antifields is {S0, S2}+ 1 2 {S1, S1} = 0, (3.3.13) and it implies Q2F = {{F}S1}S1 = {S1, {S1, F}} = − 1 2 {F, {S1, S1}} = {F, {S0, S2}}. (3.3.14) We see that this condition makes Q nilpotent only if the on-shell condition {S0, S2} is applied. It doesn’t mean necessarily putting all fields on-shell, but only the fields in the direction of σ2. Another important condition is the one at third order, that is {S0, S4}+ {S1, S3}+ 1 2 {S2, S2} = 0. (3.3.15) If S3 = 0, and if impose the {S0, S4} = 0 on-shell condition, then {S2, S2} = 0 Chapter 3. Gauge theories and BV formalism 31 implies that the S2 bracket satisfies Jacobi identity. 3.3.2 Deformations of the action If we want to compute correlation functions between operators, for a theory with an action S0 ∈ ΦBV, we have to consider a suitable deformation, like S0 7→ S0 + ϵV0, (3.3.16) where V0 ∈ ΦBV is an operator, and ϵ is a small real parameter. In BV formalism, a deformation of the theory consists in a deformation of the BV action, which contains more information than just the classical information. Let S ∈ ΦBV be the BV action, and V ∈ ΦBV the deformation operator, so that the deformation is S 7→ S + ϵV. (3.3.17) Every term of the BV action gets deformed. We split the action like S = ΣkSk, so we also split the deformation operator in orders of antifields as V = V0 + V1 + V2 + · · · . (3.3.18) Then each term gets deformed by Sk 7→ Sk + ϵVk. (3.3.19) This amounts to a deformation of the classical action S0 by V0, but also a deformation of the BRST operator S2 = Q̂ by V2, and so on. The classical master equation imposes a restriction on what deformation operators V are allowed. Let’s take ϵ to be a small parameter, so we can compute the linear deviation from the classical master equation. It is 1 2 {S + ϵV, S + ϵV } ≈ 1 2 {S, S}+ ϵ{S, V } = 0, (3.3.20) so that the deformation operator must satisfy {S, V } = 0. At each order, the Chapter 3. Gauge theories and BV formalism 32 condition becomes {S0, V1}+ {S1, V0} = 0, 1 2 {S0, V2}+ {S1, V1}+ 1 2 {S2, V1} = 0, {S0, V3}+ {S1, V2}+ {S2, V1}+ {S3, V0} = 0, · · · . (3.3.21) We can generalize it: for even n, {Sn/2, Vn/2}+ 1 2 n∑ i=0 {Si, Vn−i} = 0, (3.3.22) and, for odd n, n∑ i=0 {Si, Vn−i} = 0. (3.3.23) As an example, take the deformation of the BRST operator as V2 = KAϕ⋆A, for a vector field K ∈ VectΦBV. The following relation must hold: KA δS0 δϕA +QA δV0 δϕA = 0, (3.3.24) i.e. the BRST variation of the deformation of the classical action must cancel the K variation of the classical action. Another example: consider a linear BV action S = S0 + Q̂. Then, after putting fields on-shell, we must have {S1, V1} = − ⟨[Q,K], ϕ⋆⟩ = 0, (3.3.25) so the BRST operator and its deformation must commute. There is a special type of transformation. If the deformation operator is the generating function of a Hamiltonian vector field, i.e. V = ⟨{Ψ,−}, ϕ⋆⟩ = {Ψ, ϕ⋆}, (3.3.26) then the deformed action is BRST trivial, because it is equivalent to a deformation of Lagrangian submanifold. In other words, it only changes the fixed gauge. Chapter 4 Bosonic string sigma model 4.1 Formulation with metric 4.1.1 Action and gauge symmetries The bosonic string sigma model describes the propagation of quantum strings on spacetime. A configuration is described by a worldsheet Σ, which is a smooth 2-dimensional surface embedded in a D-dimensional spacetime. The worldsheet embedding can be described by a parameterization x : V ⊂ R2 → M , where M is spacetime, which takes worldsheet coordinates σa to spacetime coordinates xi(σa). The amplitude of a worldsheet configuration is exp(−SNG), where SNG is the Nambu-Goto action. This action is proportional to the area of the worldsheet surface. With a given parameterization, it is written as SNG(x) = T ∫ Σ d2σ √ det ( ∂x ∂σα · ∂x ∂σβ ) , (4.1.1) where the inner product is given by the spacetime metric. The proportionality constant T is the string tension, which is sometimes written as T = 1 2πα′ , (4.1.2) where α′ is then taking as the weighing factor of the action. There is an another action that describes the same dynamics. Consider a metric g on the worldsheet. The Polyakov action depends on the spacetime coordinates x and on the worldsheet metric γ, and is defined by SP(x, γ) = T 2 ∫ Σ d2σ √ det γγαβ∂αx · ∂βx. (4.1.3) The classical equations of motion for the worldsheet metric implies that it becomes the induced spacetime metric on-shell. Putting only the metric on-shell, the Polyakov 33 Chapter 4. Bosonic string sigma model 34 action reduces to the Nambu-Goto action. This means that the correlation functions involving only the x fields are equivalent for both actions. In other words, we recover the path integral with the Nambu-Goto action if we integrate the Polyakov path integral over worldsheet metric configurations. Because of the introduction of the metric as a field, the Polyakov action is invariant under diffeomorphisms, i.e. change of coordinates of the worldsheet. The generators of diffeomorphisms are vector fields v ∈ VectΣ on the worldsheet, and the action of this symmetry on the fields is Lvx = vα∂αx, Lvγαβ = vγ∂γγαβ + ∂αv γγγβ + ∂βv γγαγ. (4.1.4) There is an additional gauge symmetry on the Polyakov action. For any scalar field ϕ ∈ FunΣ, the Weyl transformation is defined as γαβ(σ) 7→ eϕ(σ)γαβ(σ), (4.1.5) i.e. it is a rescaling of the metric independently at each worldsheet point. It is a symmetry precisely because the worldsheet has two dimensions. The generators of the trasformation are also functions on the worldsheet — we can think of them as infinitesimal versions of the transformation. The action of the generator can be written as δϕγαβ = ϕγαβ, (4.1.6) and it defines the trivial Lie algebra of Weyl transformations: [δϕ1 , δϕ2 ] = 0. 4.1.2 BV construction To construct the BRST field space, we introduce the diffeomorphism ghosts c ∈ ΠVectΣ, which are vector fields on the worldsheet with odd parity, and the Weyl ghosts θ ∈ FunΣ, which are functions on the worldsheet. The BRST operator is then a vector field Q ∈ VectΦ in the BRST space of fields, given by Q = ∫ d2σ ( Lcxm δ δxm + (Lcγαβ + θγαβ) δ δγαβ + Lccα δ δcα + Lcθ δ δθ ) , (4.1.7) Chapter 4. Bosonic string sigma model 35 where we used that 1 2 [c, c]α = cβ∂βc α = 1 2 Lccα (4.1.8) and [c, θ]α = cα∂αθ = Lcθ. (4.1.9) The BV phase space of the theory of bosonic strings is given by D fields x ∈ FunΣ, the metric g on the worldsheet, the diffeomorphism ghost c ∈ ΠVectΣ, the Weyl ghost θ ∈ FunΣ, and their antifields. The antifield of x is a D-plet scalar density of odd parity x⋆. The antifield of the metric is a symmetric (2, 0)-tensor density (γ⋆)αβ on the worldsheet. The antifield of the ghost is a vector field density c⋆ on the worldsheet, with even parity (the parity flipping for being BRST ghost and BV antifield add up). Finally, the antifield of Weyl ghost is simply a density θ⋆ on the worldsheet, with flipped parity. The BV action for a theory in flat space is thus S = ∫ d2σ ( T 2 √ det γγab∂ax m∂bx m(x) + Lcxmx⋆m + (Lcγαβ + θγαβ)(γ ⋆)αβ + Lccαc⋆α + Lcθθ⋆ ) . (4.1.10) 4.1.3 Gauge fixing: light-cone gauge As an example of gauge fixing in BV formalism, we will consider the light-cone gauge used in Chapter 1 of [9]. Let’s fix the light-cone gauge for an infinite string. First we define the light-cone coordinates x± = 1√ 2 (x0 ± x1). (4.1.11) Then we fix diffeomorphism in σ0 direction by imposing x+ = σ0, (4.1.12) Chapter 4. Bosonic string sigma model 36 we fix Weyl symmetry putting det γ = −1. (4.1.13) To fix diffeomorphism in σ1 direction we define dl = γ11dσ1 √ − det γ = γ11dσ1 (4.1.14) which is invariant under σ1 diffeormorphism, so we can fix σ1 = β(σ0) ∫ σ1 0 dl = β(σ0) ∫ σ1 0 γ11dσ1, (4.1.15) where β is some function that fixes the proportionality for each σ0. Differentiating with respect to σ1 we get 1 = β(σ0)γ11(σ 0, σ1), so we get the constraint ∂1γ11 = 0. (4.1.16) The line defined by σ1 = 0 along the string is arbitrarily fixed when we choose coordinates for the worldsheet. Thus we still have a residual symmetry due to the transformation δσ1 = f(σ0), for a function f . To fix it, we impose γ01(σ 0, 0) = 0, (4.1.17) which fixes perpendicularity between σ0 and σ1 at the origin. Now we are able to write the gauge fixing constraint K = 0 as K1 K2 K3 K4  =  −detγ − 1 x+ − σ0 ∂1γ11 γ01δ(σ 1)  (4.1.18) The variations of the constraints are δK1 δγαβ = γαβδ(σ − σ′), δK2 δx+ = δ(σ − σ′), δK3 δγ11 = − ∂1δ(σ − σ′), δK4 δγ01 = δ(σ1)δ(σ − σ′). (4.1.19) Chapter 4. Bosonic string sigma model 37 Then the conormal bundle gets the antifields parameterized by ΨA, as x⋆+ = Ψ2, (γ⋆)11 = ∂1Ψ3 −Ψ1γ 11, (γ⋆)01 = Ψ4δ(σ 1)−Ψ1γ 01, (γ⋆)00 = Ψ1γ 00, (4.1.20) and c∗ = 0 and θ∗ = 0. From det γ = −1 we get the following form for the metric: γαβ = ( −(1− γ201)γ−1 11 γ01 γ01 γ11 ) , (4.1.21) and the inverse can be written as γαβ = ( −γ11 γ01 γ01 (1− γ201)γ−1 11 ) . (4.1.22) We recall that γ01(σ0, 0) = 0, and γ11 is a function of σ0 only. The light-cone gauge action is then the BV action restricted to the conormal bundle. Let’s write it as SLC = SP + Sgh. (4.1.23) The first term comes from the Polyakov action restriction on the submanifold, and it is SP = T 2 ∫ d2σ ( − 2γ11∂0x − + 2γ01∂1x − + γ11∂0x i∂0x i − 2γ01∂0x i∂1x i − (1− γ201)γ−1 11 ∂1x i∂1x i ) . (4.1.24) The second term is the one that comes from Q̂, and it becomes Sgh = ∫ d2σ ( − c0Ψ2 + θ(γ11∂1Ψ3 + 2Ψ1 + γ01Ψ4δ(σ 1)) − (cα∂αγ11 + 2∂1c αγ1α)∂1Ψ3 − (cγ∂γγαβγ αβ + 2∂αc α)Ψ1 − (cα∂αγ01 + 2∂0c αγα1)Ψ4δ(σ 1) ) , (4.1.25) Chapter 4. Bosonic string sigma model 38 where we used γαβγαβ = 2. We see that the action acquires a term involving ghosts and antifields, which took the degrees of freedom from the gauge fixed fields. In BV formalism they appear from the deformation of the Lagrangian submanifold. The higher order terms of the BV action also change when deforming the path integral, and it modifies the way that the BRST symmetry acts on the fields. We’ll explore it in the next section. 4.2 Complex structure as a field There is another formulation of the bosonic string, which is formulated in a smaller space of fields: the worldsheet metric field is replaced by a complex structure on the worldsheet. The action does not depend on the scale of the metric at each point, so we can write it in terms of the complex structure. This removes Weyl symmetry, so we don’t have this gauge symmetry anymore. This construction follows [2]. We recall the relation between Hodge operator and complex structure by acting on the form dx, where d is the de Rham operator in the worldsheet. We have ⋆dx = Iαβ∂αx dσb. (4.2.1) We are thus able to write the Polyakov action as SP(x, I) = T 2 ∫ Σ dxm ⋆ dxnGmn, (4.2.2) where we have explicitly written the spacetime metric as Gmn. Now the BRST operator can be reduced to contain only action of diffeomorphisms, and become Q = ∫ d2σ ( Lcxm δ δxm + LcIαβ δ δIαβ + Lccα δ δcα ) . (4.2.3) The BV phase space has a very similar structure to the formulation with the metric: removing Weyl ghost and its antifield, and switching the metric with the complex structure. However, the antifield of the complex structure is the subtlest one, because the complex structure satisfies a constraint. Let’s parameterize the space of all complex structures on a two dimensional Chapter 4. Bosonic string sigma model 39 manifold (or worldsheet). We’ll work here with zz̄ coordinates, which are given by z = σ1 + iσ2, z̄ = σ1 − iσ2, (4.2.4) i.e. a change of coordinates to a complexified chart. We will also use the notation zα = (z1, z2) = (z, z̄). Consider the space of mixed (1, 1) tensors of the worldsheet, whose elements are I = Iαβ dz β ∂ ∂zα . (4.2.5) The constraint I2 + 1 = 0 defines a submanifold of which the elements are complex structures. To find a parametrization for I, we parameterize a general (1, 1) tensor with four independent parameters in zz̄ coordinates as I = ( iρ im −im̄ −iρ̄. ) . (4.2.6) The constraint is then given by( −ρ2 +mm̄ im(ρ− ρ̄) im̄(ρ− ρ̄)m̄ −ρ̄2 +mm̄ ) = ( −1 0 0 −1 ) , (4.2.7) which is solved by I = ( i √ 1 +mm̄ im −im̄ −i √ 1 +mm̄ ) . (4.2.8) The complex structure which corresponds to a flat metric is I(0) = ( i 0 0 −i ) , (4.2.9) from the relation Iαβϵβγ = √ det ggαγ. We get I = I(0) by taking mm̄ = 0. We can expand the non-polynomial component in power series on the fields m and m̄. Using the expansion √ 1 +mm̄ = 1 + 1 2 mm̄− 1 4 (mm̄)2 + · · · , (4.2.10) Chapter 4. Bosonic string sigma model 40 we can split I in different orders on the m and m̄ parameters, as I = I(0) + I(1) + I(2) + I(4) + · · · (4.2.11) with I(0) = ( i 0 0 −i ) , I(1) = ( 0 im −im̄ 0 ) , I(2) = ( 1 2 mm̄ 0 0 1 2 mm̄ ) , (4.2.12) and so on. Notice that there are no terms of odd order, except for the first order term. Because of the constraint of the complex structure, its antifield has a gauge symmetry. In Appendix B.1 we describe it and gauge fix it. We are then able to write the complex structure as I = i √ 1 +mm̄ ( ∂ ∂z dz − ∂ ∂z̄ dz̄ ) + im ∂ ∂z̄ dz − im̄ ∂ ∂z dz̄, (4.2.13) and the antifield of the complex structure as I⋆ = dzdz̄ 1 4 ( −b dz ∂ ∂z̄ + b̄ dz̄ ∂ ∂z ) . (4.2.14) The components of I⋆ are called b and b̄ to agree with the standard notation, as they turn out to be the ghosts that generate the energy-momentum tensor. The BV bracket between mm̄ and bb̄ show that they are Darboux coordinates: {m(z), b(z′)} = 4iδ2(z − z′), {m(z), b̄(z′)} = 0, {m̄(z), b(z′)} = 0, {m̄(z), b̄(z′)} = 4iδ2(z − z′). (4.2.15) Thus b is the antifield of m and b̄ is the antifield of m̄, apart from a factor of 4i. We are able to write the odd symplectic form in Darboux coordinates as ω = ∫ ( δxδx⋆ + δcδc⋆ + δc̄δc̄⋆ − i 4 δmδb− i 4 δm̄δb̄ ) . (4.2.16) Chapter 4. Bosonic string sigma model 41 4.2.1 Expansion around Polyakov gauge To write the Polyakov action in terms of mm̄ let’s use the relation dx = dz∂x+ dz̄∂̄x, where ∂ = ∂/∂z and ∂̄ = ∂/∂z̄, and the complex structure parameterized as in eq. (4.2.13). Then I · dx can be computed as ιIdx. We also note that the relation between the dzdz̄ and the measure on the worldsheet d2z = dτdσ is dzdz̄ = (dτ + idσ)(dτ − idσ) = −2id2z. (4.2.17) Then the classical action is written as Scl = ∫ d2z (√ 1 +mm̄∂x∂̄x+ 1 2 m(∂x)2 + 1 2 m̄(∂̄x)2 ) . (4.2.18) To write the BRST operator in terms of mm̄ and bb̄, we must write the Lie derivative of the complex structure I in terms of the mm̄ parameters, and substitute the gauge fixed I⋆ in terms of bb̄. We get Q̂ = ∫ d2z ( Lcxx⋆ + 1 2 Lccc⋆ + √ 1 +mm̄(b∂̄c+ b̄∂c̄) − 1 2 ( c(∂m)b+ c̄(∂̄m)b− (∂c)mb+ (∂̄c̄)mb ) − 1 2 ( c(∂m̄)b̄+ c̄(∂̄m̄)b̄+ (∂c)m̄b̄− (∂̄c̄)m̄b̄ )) . (4.2.19) We can rewrite the BRST operator with the derivatives acting on bb̄ instead of on mm̄. It is Q̂ = ∫ d2z ( Lcxx⋆ + 1 2 Lccc⋆ + √ 1 +mm̄(b∂̄c+ b̄∂c̄) + 1 2 ( c(∂b)m+ c̄(∂̄b)m+ 2(∂c)bm ) + 1 2 ( c(∂b̄)m̄+ c̄(∂̄b̄)m̄+ 2(∂̄c̄)b̄m̄ )) . (4.2.20) We can now change coordinates of the BV phase space, and let bb̄ be fields and mm̄ be the antifields. This change of coordinates parameterizes the BV phase space around a different conormal bundle. By putting the antifields to zero, we obtain the Polyakov gauge, or conformal gauge. When m = m̄ = 0, the complex structure is the one related to a flat metric. Chapter 4. Bosonic string sigma model 42 We rewrite the BV action as a power series in the antifields S = S0 + S1 + S2 + · · · , (4.2.21) where the term Sk is of kth power on x⋆, c⋆ and mm̄. We see right away that x⋆ and c⋆ appear only linearly, so the non-linear terms come from the expansion of Scl and Q̂ in terms of mm̄. The term S0 is S0 = ∫ d2z ( ∂x∂̄x+ b∂̄c+ b̄∂c̄ ) (4.2.22) and it is the action we get upon fixing Polyakov gauge. The first order term is given by S1 = ∫ d2z ( Lcxx⋆ + 1 2 Lccc⋆ + 1 2 m(∂x)2 + 1 2 m̄(∂̄x)2 + 1 2 ( c(∂b)m+ c̄(∂̄b)m+ 2(∂c)bm ) + 1 2 ( c(∂b̄)m̄+ c̄(∂̄b̄)m̄+ 2(∂̄c̄)b̄m̄ )) . (4.2.23) It defines the BRST operator in Polyakov gauge, which generates BRST symmetry on S0. The last term that we want to write is S2, given by S2 = ∫ d2z 1 2 ( ∂x∂̄x+ b∂̄c+ b̄∂c̄ ) mm̄. (4.2.24) This coefficient of mm̄ is a bivector, which defines Poisson brackets in the space of fields. 4.2.2 BRST transformation in Polyakov gauge The Polyakov gauge was defined as the choice of Lagrangian submanifold given by fixing the complex structure. The fields in this gauge are x, c, c̄, b and b̄, and the action is S0, given in eq. (4.2.22). The equations of motion that come from this Chapter 4. Bosonic string sigma model 43 action are ∂∂̄x = 0, ∂c̄ = ∂̄c = 0, ∂b̄ = ∂̄b = 0. (4.2.25) The term S1 of the BV action, given by eq. (4.2.23), defines the BRST transfor- mation of the fields. The BRST vector field Q on the space of fields is defined as S1 = Qa(ϕ)ϕ⋆a. Then the x, c and c̄ fields transform as Qx = c∂x+ c̄∂̄x, Qc = c∂c+ c̄∂̄c, Qc̄ = c∂c̄+ c̄∂̄c̄. (4.2.26) To find the transformations of b, b̄ we first carefully write the relevant term in S1 as (QI)I⋆ = − 2i ∫ d2z ((QI)zz̄(I ⋆) z̄ zz̄z + (QI)z̄z(I ⋆) z zz̄z̄ ) = − 1 2 ∫ d2z ( (Qm)b+ (Qm̄)b̄ ) = 1 2 ∫ d2z ( (Qb)m+ (Qb̄)m̄ ) . (4.2.27) We used the normalization (4.2.15) of the Darboux coordinates of bb̄ in the BV phase space, and also the relation dzdz̄ = −2id2z. Then, paying attention to the 1/2 term in front of the transformation, we get Qb = c∂b+ c̄∂̄b+ 2(∂c)b+ (∂x)2, Qb̄ = c∂b̄+ c̄∂̄b̄+ 2(∂̄c̄)b̄+ (∂̄x)2. (4.2.28) The BRST operator defined by these transformations is only nilpotent on-shell, as consequence from the relations between S0, S1 and S2 that come from the classical master equation. 4.3 Energy-momentum tensor and OPEs The energy-momentum tensor of the theory is defined to be the result of the variation of S0 with respect to the worldsheet metric. To compute the variation we Chapter 4. Bosonic string sigma model 44 must consider the minimal coupling of a metric, which recovers the Polyakov action. Then we compute the variation with respect to the metric – or, in our case, to the complex structure, since the action does not depend on the scale of the metric. And then we fix again the complex structure to the one we fixed in Polyakov gauge. So we are able to define the two independent components of the energy-momentum tensor as T = ( δS δm ) ϕ⋆=0 , T̄ = ( δS δm̄ ) ϕ⋆=0 , (4.3.1) where S = S0 +S1 + · · · is the BV action. This shows us that the energy-momentum tensor is defined in terms of S1, and is related with the BRST operator acting on the b fields, as T = Qb, T̄ = Qb̄. (4.3.2) However the definition of energy-momentum that is often considered is the one defined on-shell. So the standard definition is obtained by using eqs. (4.2.25) in the expressions of eq. (4.2.28), so we get T = (∂x)2 + c∂b+ 2(∂c)b, T̄ = (∂̄x)2 + c̄∂̄b̄+ 2(∂̄c̄)b̄. (4.3.3) Now we can compute OPEs involving the (normal ordered) energy momentum tensor. As we’re working in BV formalism, we are able to compute the OPEs off-shell, and put the result on-shell afterwards. We’ll see that this procedure yields different results: we can get contact terms in the OPEs. From the action S0 in eq. (4.2.22) we compute the basic OPEs xm(z)xn(0) = − 1 2 δmn ln |z|2 + regular, b(z)c(0) = 1 z + regular, b̄(z)c̄(0) = 1 z̄ + regular, (4.3.4) from which we can compute OPEs of composite fields. Chapter 4. Bosonic string sigma model 45 4.3.1 OPE with ghost current The ghost number current is defined as j = : bc :, j̄ =: b̄c̄ :, (4.3.5) and comes from the U(1) symmetry b→ eiab, c→ e−iac, and analogously for b̄ and c̄. Then the on-shell OPE of T with j is given by T (z)j(0) = 3 z3 + j z2 + ∂j z + regular (4.3.6) We see the conformal weight of j, i.e. the factor multiplying j on the second order pole, is 1. Also, the action of translation on j is ∂j, which we see from the first order pole. This kind of transformation determines that j is a primary operator, in the language of conformal theories [8]. Using the off-shell expression for the energy-momentum tensor, we have the following contributions to the OPE: T (z)j(0) = c(z)∂b(z)b(0)c(0) + c̄(z)∂̄b(z)b(0)c(0) + 2∂c(z)b(z)b(0)c(0). (4.3.7) The first term is obtained by performing contractions, and expanding around z = 0. As we are off-shell, the fields are not holomorphic, and we must expand in both z and z̄ coordinates. We get c(z)∂b(z)b(0)c(0) = 1 z3 − : b(0)c(z) : z2 − : ∂b(z)c(0) : z = 1 z3 − : bc : z2 − ∂ : bc : z − z̄ z2 : b∂̄c : . (4.3.8) The third term of eq. (4.3.7) is 2∂c(z)b(z)b(0)c(0) = 2 z3 + 2 : b(z)c(0) : z2 + 2 : b(0)∂c(z) : z = 2 z3 + 2 : bc : z2 + 2∂ : bc : z + 2 z̄ z2 : ∂̄bc : . (4.3.9) Finally, the second term, which comes strictly from the off-shell contribution, is given by c̄(z)∂̄b(z)b(0)c(0) = : bc̄ : δ2(z). (4.3.10) Chapter 4. Bosonic string sigma model 46 Summing all of the terms, the off-shell Tj OPE yields T (z)j(0) = 3 z3 + j z2 + ∂j z + z̄ z2 (2 : ∂̄bc : − : b∂̄c :) + δ2(z) : bc̄ : . (4.3.11) When we put the off-shell contraction on-shell, we almost recover the on-shell contraction: T (z)j(0) = 3 z3 + j z2 + ∂j z + δ2(z) : bc̄ : . (4.3.12) The additional term, which is proportional to a delta function, is called a contact term. 4.4 Integration over moduli space The BV quantization of the Polyakov action is built upon the gauge symmetry of diffeomorphisms (and Weyl transformations, if we work with the metric instead of complex structure). Thus when choosing a Lagrangian submanifold we fix the parameterization of the worldsheet, specifying the complex structure I(0). However we cannot map all complex structures by diffeomorphisms. The space of complex structures modulo diffeomorphisms is called moduli space. For smooth manifolds we can parameterize the moduli space by the length of the closed curves that cannot be dragged one into the other. For the sphere, every closed curve can be dragged into a point, thus the moduli space is trivial – it has only one element. As for a torus, there are two independent curves – the moduli space has two real dimensions. For higher genus, the real dimension of the moduli space is 6g − 6. So to compute correlation function of some operator O (e.g. the product of local operators) on the worldsheet, we must construct the correlation function on a Lagrangian submanifold, corresponding to a point in moduli space, and then integrate over all moduli. For that, we must define an integration measure We define a mapping Ω from operators to functions of µ and µ̄, given by Ω(O)µ,µ̄ = 〈 exp (∫ (µT + µ̄T̄ ) ) O 〉 , (4.4.1) where on the right hand side the ⟨⟩ brackets mean integration on the Lagrangian submanifold, and T = Qb, T̄ = Qb̄ are components of the energy-momentum tensor. The parameters µ and µ̄ are the ones that parameterize variations of the fixed complex structure, I(0). To get the integration measure of the moduli space, we Chapter 4. Bosonic string sigma model 47 expand the exponential in power series, and pick the term with the correct dimen- sionality, and the same number of b and b̄. For instance, for the torus the integration measure is 〈∫ µQb ∫ µ̄Qb̄ O 〉 , (4.4.2) whereas for g > 1 the measure is〈( 3g−3∏ i=1 ∫ µT ∫ µ̄T̄ ) O 〉 . (4.4.3) The expression defines a measure on the moduli space, when we take µ, µ̄ to be infinitesimal variations around the fixed Lagrangian submanifold. The deformations around the Lagrangian submanifold are generated by the gauge fermion ψ = − ∫ (µb+ µ̄b̄), (4.4.4) which a function on the BV phase that depends only on b and b̄. It defines the Hamiltonian vector field {ψ,−} = µ δ δm + µ̄ δ δm̄ , (4.4.5) which changes m 7→ m+ µ and m̄ 7→ m̄+ µ̄. The Polyakov gauge is slightly changed, and it induces a reparameterization of the BV phase space. We will later introduce a new parameterization of the space of complex structures, and thus a new parameterization of the BV phase space. The new fields will be also written as µ and µ̄, and they will be useful to understand the deformations along the moduli space. 4.4.1 Dolbeault operator and the µµ̄ fields We have parameterized the complex structure with the fields m and m̄, which are Darboux coordinates to the b and b̄ fields. To introduce the µµ̄ parameterization, we will introduce the Dolbeault operators, which are eigenvectors of the complex struc- ture. We will describe the space of complex structures as the space of deformations of these operators. Chapter 4. Bosonic string sigma model 48 In zz̄ coordinates we can write the complex structure locally as I = ( i 0 0 −i ) , (4.4.6) so it has eigenvalues ±i. We define the Dolbeault operators as ∂ := dz ∂ ∂z , ∂̄ := dz̄ ∂ ∂z̄ , (4.4.7) which satisfy d = ∂ + ∂̄, where d is the de Rham operator. Their action on the X fields are eigenfunctions of the complex structures, as I∂X = i∂X, I∂̄X = −i∂̄X. (4.4.8) Then we can write Polyakov action in terms of ∂ and ∂̄, which gives S = −i 2 ∫ ∂X ∧ ∂̄X, (4.4.9) which has the form of the Polyakov action with a flat metric. To parameterize locally the possible gauge fixing choices we can fix the complex structures to different values. It should be equivalent to fixing the complex structure to be flat in different coordinates. Let’s parameterize the family of complex structures by functions µ and µ̄, and call the complex structure I [µ]. For µ = µ̄ = 0 we get the flat one, I [0]. The Dolbeault operators ∂ and ∂̄ are eigenvectors of I [0]. We can then define deformed Dolbeault operators by I [µ]∂[µ] =i∂[µ], I [µ]∂̄[µ] = − i∂̄[µ]. (4.4.10) We fix their normalization by asking that d = ∂[µ] + ∂̄[µ] (4.4.11) continues to hold. The way we want to parameterize ∂[µ] and ∂̄[µ] is the following. Define the vector Chapter 4. Bosonic string sigma model 49 fields v[µ] = ∂ ∂z + µ̄ ∂ ∂z̄ , (4.4.12) v̄[µ] = ∂ ∂z̄ + µ ∂ ∂z . (4.4.13) The Dolbeault operators are mixed 2-tensors, so we define them as ∂[µ] = u[µ]v[µ], ∂̄[µ] = ū[µ]v̄[µ], (4.4.14) for a family of forms u[µ] and ū[µ]. Upon the definition of eq. (4.4.13), the forms are fixed by eq. (4.4.11). They are given by u[µ] = dz − µdz̄ 1− µµ̄ , ū[µ] = dz̄ − µ̄dz 1− µµ̄ . (4.4.15) So the Dolbeault operators are explicitly given by ∂[µ] = 1 1− µµ̄ (dz − µ̄dz̄) ( ∂ ∂z + µ̄ ∂ ∂z̄ ) , ∂̄[µ] = 1 1− µµ̄ (dz̄ − µdz) ( ∂ ∂z̄ + µ ∂ ∂z ) . (4.4.16) 4.4.2 Action in terms of µµ̄ fields The Polyakov action is defined in terms of the complex structure I [µ] as Scl = − 1 4 ∫ dx ∧ I [µ]dx, (4.4.17) where we make the dependence on µ and µ̄ explicit. Then it is written in terms of the Dolbeault operators as Scl = − 1 4 ∫ (∂[µ]x+ ∂̄[µ]x) ∧ I [µ](∂[µ]x+ ∂̄[µ]x) = i 2 ∫ ∂[µ]x ∧ ∂̄[µ]x. (4.4.18) Using eq. (4.4.16) to write the dependence on µ, µ̄ explicitly, we get Scl = ∫ d2z ( 1 + µµ̄ 1− µµ̄ ∂x ∂z ∂x ∂z̄ + µ 1− µµ̄ ( ∂x ∂z )2 + µ̄ 1− µµ̄ ( ∂x ∂z̄ )2 ) . (4.4.19) Chapter 4. Bosonic string sigma model 50 Then we compare it with the Polyakov action in terms of the m, m̄ coordinates (eq. (4.2.18)), which can be written in terms of the Dolbeault operators as Scl = ∫ d2z ( √ 1 +mm̄ ∂x ∂z ∂x ∂z̄ + m 2 ( ∂x ∂z )2 + m̄ 2 ( ∂x ∂z̄ )2 ) . (4.4.20) We can then relate the coordinates m, m̄ with the coordinates µ, µ̄. The relation is m = 2µ 1− µµ̄ , m̄ = 2µ̄ 1− µµ̄ . (4.4.21) This makes a clear connection between the Darboux coordinates m, m̄ and the Dolbeault parameters µ, µ̄. Other useful relations are √ 1 +mm̄ = 1 + µµ̄ 1− µµ̄ (4.4.22) and µ = m 1 + √ 1 +mm̄ , µ̄ = m̄ 1 + √ 1 +mm̄ . (4.4.23) The µµ̄ fields parameterize the complex structure as I = i 1− µµ̄ ( 1 + µµ̄ 2µ −2µ̄ −1− µµ̄ ) . (4.4.24) There must exist fields B and B̄ which are Darboux conjugates to µ and µ̄, as {µ(z), B(w)} = δ(z − w), {µ(z), B̄(w)} = 0, {µ̄(z), B(w)} = 0 {µ̄(z), B̄(w)} = δ(z − w). (4.4.25) In Appendix (B.3), we show that B and B̄ parameterize I⋆ by I⋆ = 1− µµ̄ 1 + µµ̄ ( 0 B̄ − µ2B −B + µ̄2B̄ 0 ) . (4.4.26) Now let’s write the BV action in terms of µµ̄ and BB̄ coordinates. The classical Chapter 4. Bosonic string sigma model 51 action is Scl = ∫ d2z ( 1 + µµ̄ 1− µµ̄ ∂x∂̄x+ µ 1− µµ̄ (∂x)2 + µ̄ 1− µµ̄ (∂̄x)2 ) , (4.4.27) and the BRST generator becomes Q̂ = ∫ d2z ( Lcxx⋆ + 1 2 Lccc⋆ + (B − µ̄2B̄)∂̄c+ (B̄ − µ2B)∂c̄ + µ(c · ∂ + 2(∂c))B + µ̄(c · ∂ + 2(∂̄c̄))B̄ ) . (4.4.28) The BV expansion around Polyakov gauge is S0 = ∫ d2z(∂x∂̄x+B∂̄c+ B̄∂c̄), S1 = ∫ d2z(Lcxx⋆ + 1 2 Lccc⋆ + µ ( (∂x)2 + (c · ∂ + 2(∂c))B ) + µ̄ ( (∂̄x)2 + (c · ∂ + 2(∂̄c̄))B̄ ) , S2 = ∫ d2z ( 2µµ̄∂x∂̄x− µ2B∂c̄− µ̄2B̄∂̄c ) , (4.4.29) which is generalized as Sk = ∫ d2z(−1)(k+1)/2(µµ̄)(k−1)/2 ( µ(∂x)2 + µ̄(∂̄x)2 ) , odd k, Sk = ∫ d2z(2µµ̄)k/2∂x∂̄x, even k. (4.4.30) We can see that the structure of S0 and S1 in µµ̄ coordinates is the same as for mm̄ coordinates. Thus the two parameterizations of the field space are equal to the view of on-shell BRST formalism, when the higher terms vanish. 4.4.3 Holomorphic factorization Now we’ll make some comments about the results of [3]. They shine a light in the importance of the µµ̄ coordinates. Consider the classical action, written in terms of a linear operator on the world- sheet, Scl = 1 2 ∫ d2z 1 1− µµ̄ ( ∂x ∂̄x )( 2µ 1 + µµ̄ 1 + µµ̄ 2µ̄ )( ∂x ∂̄x ) . (4.4.31) Chapter 4. Bosonic string sigma model 52 The µ and µ̄ fields parameterize the deformations of the action in Polyakov gauge along the moduli space. We write the deformation as the operator insertion exp(δS(µ, µ̄)), where δS = Scl(µ, µ̄)− Scl(0, 0). (4.4.32) The action in the Polyakov gauge gives rise to the OPE ∂xm(z)∂̄xn(w) = 1 2 gmnδ(z − w), (4.4.33) which is a contact term, i.e. it is a delta function on the wordlsheet. We can resum these terms by acting with an operator on the path integral deformation, using the identity exp ( 1 2 Gαβ ∂ ∂∂αx ∂ ∂∂βx ) exp ( −1 2 Aαβ∂αx∂βx ) = exp ( −1 2 ( A(1 +GA)−1 )αβ ∂αx∂βx ) , (4.4.34) where A is the quadratic operator in δS. We take A = − 2 1− µµ̄ ( µ µµ̄ µµ̄ µ̄ ) , G = ( 0 1 1 0 ) . (4.4.35) To write the resummed deformation, we define a normal ordering × × × × that means that all contact terms are removed. The new way to write the deformation is eδS = × × exp (∫ d2z(µ(∂x)2 + µ̄(∂̄x)2) ) × ×. (4.4.36) Under this normal ordering, the classical action then depends linearly on deformations of the complex structure. Moreover, in the expansion of the path integral in powers of µ and µ̄, which parameterize a deformation over the moduli space around the Polyakov gauge, there are only terms µk and µ̄k. Every time there should be µµ̄ in the expansion, there is a contact term, then the normal ordering removes it. This result is called holomorphic factorization: the amplitudes are sums of holomorphic functions – which depend on µ, plus antiholomorphic ones, depending on µ̄. Thus, even though the µ and µ̄ coordinates are not Darboux conjugates to the b and b̄ fields, they are the coordinates that decouple in the deformations. Chapter 5 Pure spinor superstring The pure spinor superstring is a formalism that describes the propagation of a string in a supersymmetric spacetime. The formalism consists of an action, which is invariant under supersymmetry, and a BRST-like fermionic operator that defines physical states. In this chapter we describe the formalism. 5.1 The target space In the flat space pure spinor superstring description, the target space locally the product of the supersymmetric spacetime M , which is a fermionic spinor bundle with two sectors (left and right) over an even 10 dimensional manifold (spacetime), with the space of the so called pure spinors. We now describe each of this objects with more detail. Consider the supermanifold M10|32 as the spacetime with 10 even dimensions and 32 odd dimensions. We write its elements as (x, θL, θR), where xm are the even coordinates and θαL and θα̇R are the odd coordinates, split in two parts: one called left moving, with 16 coordinates, and the other called right moving, with the remaining 16 degrees of freedom. Consider the group G = SO(10,C)× C× SO(10,C)× C. (5.1.1) The spacetime M10|32 can be written as a coset space M10|32 = E G , (5.1.2) where E is a principal bundle generated by the free action of G on M10|32. The pure spinor bundle over M10|32 is defined by the equivalence relation PS(M) = E × V G (5.1.3) 53 Chapter 5. Pure spinor superstring 54 where V is given by V = V0 ∼ × V0 ∼ , (5.1.4) where V0 = Spin(C) is a spinor bundle on 10d spacetime, with 16d complex spinors λα as elements, and the equivalence relation is the pure spinor constraint, given by λΓmλ = 0, (5.1.5) where Γmαβ are the Pauli matrices. The action of G rotates and rescales either the left and right moving spinors. The elements of PS(M) are (x, θL, θR, λL, λR). 5.2 The action The pure spinor string is a sigma model on a 2-dimensional bosonic worldsheet Σ, on the target space PS(M). The fields are functions on the worldsheet with values on PS(M). We call the space of fields Φ ∋ Σ→ PS(M). To define the action, we consider the extension of the space of fields to contain momenta of the fields. The Hamiltonian phase space of fields ΦH consists of the fields (x, θL, θR, λL, λR), and the momenta (P, pL, pR, wL, wR), which are 1-forms on the worldsheet. The inner product in the phase space involves an integral over a spatial slice, which we take to be S1 for closed strings (topologically a circle). The symplectic potential is given by α = PAδϕ A = ∫ S1 (Pmδx m + pLδθL + pRδθR + wLδλL + wRδλR), (5.2.1) and the symplectic form is ω = δα. The action is given by S0 = ∫ dτ+dτ− ( 1 2 ∂+x∂−x+ pL+∂−θL + pR−∂+θR − wL+∂−λL − wR−λR ) . (5.2.2) The action depends on the momenta pL,R and wL,R only through one component each, because of dτ+dτ+ = 0, etc. We write, for short, pL+ = p+, pR− = p−, wL+ = w+ and wR− = w−. Chapter 5. Pure spinor superstring 55 Varying the action with respect to ∂τθL, we get the momentum constraint pθL = − 2 ∂S0 ∂∂τθL = p+. (5.2.3) Similarly, the momenta of the other spinor fields are p−, w+ and w−. For the momentum of x, the constraint is Pm = − 2 ∂S0 ∂∂τxm = −1 2 ∂τxm. (5.2.4) The action is quadratic, so the equations of motion are simple: ∂+∂−x m = 0, ∂−θL = 0, ∂+θR = 0, ∂−p+ = 0, ∂+p− = 0, ∂−λL = 0, ∂+λR = 0, ∂−w+ = 0, ∂+w− = 0. (5.2.5) Under the on-shell condition, the left fields depend only on τ+ and the right fields depend only on τ−. Because of the pure spinor constraint λΓλ = 0, the momenta w± have a gauge symmetry under the transformation δw+ = Λm+ΓmλL, δw− = Λm−ΓmλR. (5.2.6) for worldsheet 1-form, spacetime vectors, Λm = Λm+dτ+ and Λm = Λm−dτ−. 5.2.1 Symmetries and charges For a vector field v ∈ VectΦ given by v = ∫ vA(ϕ) δ δϕA , (5.2.7) we define its generating function on the Hamiltonian phase space ΦH as v̂ = ∫ vA(ϕ)PA + u(ϕ) (5.2.8) Chapter 5. Pure spinor superstring 56 where u ∈ FunΦ is an arbitrary function of the fields, and doesn’t change the action of v on Φ, but changes the action in the momenta PA. The generating functions are also called charges of the symmetries. The supersymmetry transformation, in a negative momentum representation, on the left sector and right sector, with fermionic spinor parameters ξL and ξR respectively, is given by the vector fields SL(ξL) = ∫ ( −1 2 ξLΓ mθL δ δxm + ξL δ δθL ) , SR(ξR) = ∫ ( −1 2 ξRΓ mθR δ δxm + ξR δ δθR ) , (5.2.9) so their charges are of the form ŜL(ξL) = ∫ ( −1 2 ξLΓ mθLPm + ξLp+ ) + uL(ξL), ŜR(ξR) = ∫ ( −1 2 ξRΓ mθRPm + ξRp− ) + uR(ξR). (5.2.10) The action on the fields and momenta is computed by S = {Ŝ,−}, and the transfor- mations are, for the left supersymmetry only, SLx m = − 1 2 ξLΓ mθL, SLθL = ξL, SLp+ = − 1 2 ξLΓ mPm − δuL δθL , (5.2.11) and SLθR = SλL = SλR = 0. Also, we assume uL is a function of x and θL only, so that we have SLp− = SLw+ = SLw− = 0. We have used the antisymmetry of the Poisson brackets in the Hamiltonian field space: {x, P} = −{P, x} and {θ, p+} = {p+, θ} (the sign from anticommuting θ and p+ cancels the antisymmetry of the brackets). Another point is that ( −→ δ /δθL)uL = −uL( ←− δ /δθL). The action of left sector supersymmetry transformation on the action is SLS0 = ∫ (−SLxm∂+∂−xm + SLp+∂−θL + p+∂−SLθL) = ∫ ( 1 2 (ξLΓmθL)∂+∂−x m + SLp+∂−θL ) , (5.2.12) For the action to be invariant under supersymmetry, we choose the action on p+ to Chapter 5. Pure spinor superstring 57 be SLp+ = 1 2 ξLΓ m∂+xm, (5.2.13) which fixes δuL δθL = − 1 2 ξLΓ m(Pm + ∂+xm) = − 1 4 ξLΓ m∂σxm. (5.2.14) We integrate it to obtain uL = ∫ 1 4 (ξLΓ mθL)∂σxm, (5.2.15) So the left supersymmetry generator becomes ŜL = ∫ ( 1 2 (ξLΓ mθL)∂+xm + ξLp+ ) . (5.2.16) Analogously, the right supersymmetry charge becomes ŜR = ∫ ( 1 2 (ξRΓ mθR)∂−xm + ξRp− ) . (5.2.17) Now we’ll consider the a BRST-like operator Q that is the generator of super- symmetry in the positive momentum representation, and take the pure spinor fields as generators. It is the sum of QL, which acts on the left sector, and QR, which acts on the right sector. They are given by QL = ∫ ( 1 2 λLΓ mθL δ δxm + λL δ δθL ) , QR = ∫ ( 1 2 λRΓ mθR δ δxm + λR δ δθR ) . (5.2.18) The charge of the left supersymmetry on the Hamiltonian function is Q̂L = ∫ ( 1 2 (λLΓ mθL)Pm + λLp+ ) + vL, (5.2.19) Chapter 5. Pure spinor superstring 58 which defines the following transformations on the fields and momenta QLx m = 1 2 λLΓ mθL, QLPm = − δvL δxm , QLθL = λL, QLp+ = 1 2 λLΓ mPm + δvL δθL , QLλL = 0, QLw+ = −1 2 θLΓ mPm − p+ − δvL δλL . (5.2.20) The left BRST transformation on the action gives us QLS0 = ∫ ( 1 2 (λLΓ m∂−θL)(Pm + ∂+xm) + δvL δθL ∂−θL + 1 2 (∂−λLΓ mθL)(Pm + ∂+xm) + δvL δλL ∂LλL ) . (5.2.21) To make the action invariant, we take the conditions δvL δθL = − 1 2 λLΓ m(Pm + ∂+xm) = 1 4 λLΓ m∂σxm, δvL δλL = − 1 2 θLΓ m(Pm + ∂+xm) = 1 4 θLΓ m∂σxm, (5.2.22) which fixes vL = 1 4 (λLΓ mθL)∂σxm. But, more generally, we can have vL = 1 4 (λLΓ mθL)∂σxm + v0L, (5.2.23) with δv0L δθL ∂−θL + δv0L δλL ∂−λL = 0, (5.2.24) so we can take v0L to be any function of θL and λL. The charge of QL is then Q̂L = ∫ ( −1 2 (λLΓ mθL)∂+xm + λLp+ ) + v0L. (5.2.25) Similarly, the charge of QR is Q̂R = ∫ ( −1 2 (λRΓ mθR)∂−xm + λRp− ) + v0R. (5.2.26) We can see that left supersymmetry and left BRST transformations commute by Chapter 5. Pure spinor superstring 59 computing the Poisson brackets of their generating functions: {ŜL, Q̂L} = − 1 2 {ξLp+, (λLΓmθL)∂+xm}+ 1 2 {(ξLΓmθL)∂+xM , λLp+} = 0. (5.2.27) The nilpotency of the action of Q in the Hamiltonian phase space can then be checked by computing the Poisson brackets of the generating function with itself: {Q̂L, Q̂L} = (λLΓ mλL)∂+xm = 0, (5.2.28) which is zero because the λL ghost is a pure spinor. 5.3 BRST operator Now we are going to define the BRST operator Q, that is used to define the physical states of the theory. We start defining the supersymmetric momentum as a 1-form Πm = Πm +dτ+ +Πm −dτ− given by Π = dx+ u, where u is a function of θL,R fixed so Πm ± is invariant under the SL supersymmetry. We have SLΠ m + = − ∂+ ( 1 2 ξLΓ mθL ) + ξL δum+ δθL (5.3.1) and analogously for Πm − . To have δΠ = 0 we fix it to be, on-shell, [4] Πm + = ∂+x m + 1 2 θLΓ m∂+θL, Πm − = ∂−x m + 1 2 θRΓ m∂−θR. (5.3.2) We also define the supersymmetric covariant derivative constraint, or Green-Schwarz constraint, as d+ = p+ − 1 2 ∂+xmΓ mθL − 1 6 (θLΓm∂+θL)Γ mθL, d− = p− − 1 2 ∂−xmΓ mθR − 1 6 (θRΓm∂−θR)Γ mθR, (5.3.3) Chapter 5. Pure spinor superstring 60 which is related to the supersymmetric momentum by {d+, d+} = − ΓmΠ m + , {d−, d−} = − ΓmΠ m − , {d+, d−} = 0. (5.3.4) We can parameterize the fiber of the phase space with the fields (Π+, d±, w±) or (Π−, d±, w±). In terms of these fields, the p± momenta are expressed as p+ = d+ + 1 2 Πm+Γ mθL − 1 12 (θLΓ m∂+θL)ΓmθL, p− = d− + 1 2 Πm−Γ mθR − 1 12 (θRΓ m∂−θR)ΓmθR. (5.3.5) The BRST operator is then defined by the BRST charge, which is the generating function on the phase space, given by Q̂ = Q̂L + Q̂R, with Q̂L = ∫ λLd+, Q̂R = ∫ λRd−. (5.3.6) These charges are the same as the ones of eqs. (5.2.25) and (5.2.26), with v0L = −(1/6)ΓmθL(θLΓm∂+θL), etc. The action of QL on the fields is QLx m = 1 2 λLΓ mθL, QLθL = λL, QLλL = 0, (5.3.7) and QLθR = QLλR = 0. Now we write the transformation on the momenta. On the Green-Schwarz constraint instead, we use QLd+ = {λLd+, d+}, and get QLd+ = − λLΓmΠm+, (5.3.8) and QLd− = 0. On the momenta of the pure spinor ghost, the transformation is given by QLw+ = − d+, (5.3.9) Chapter 5. Pure spinor superstring 61 and QLw− = 0. On the supersymmetric momenta of x we get QLΠm+ = λLΓ m∂+θL (5.3.10) and QLΠm− = 0. To verify the nilpotency of QL, we act twice on each field. On x, we need to use the pure spinor constraint Q2 Lx m = 1 2 QL(λLΓ mθL) = 1 2 λLΓ mλL = 0. (5.3.11) On θL and λL, the QλL = 0 implies the nilpotency. For the pure spinor momentum, we have Q2 Lw+ = −QLd+ = λLΓ mΠm+, (5.3.12) which is a gauge transformation on w+, so it can be compensated to zero. On the Green-Schwarz constraint, we get Q2 Ld+ = −QL(λLΓ mΠm+) = −λLΓm(λLΓm∂+θL). (5.3.13) We use Fierz identity to rewrite it as Q2 Ld+ = 1 2 ∂+θLΓ m(λLΓ mλL) = 0, (5.3.14) which is zero due to the pure spinor constraint. Finally, on the supersymmetric momentum of x we get Q2 LΠm+ = QL(λLΓ m∂+θL) = λLΓ m∂+λL = 1 2 ∂+(λLΓ mλL) = 0, (5.3.15) where again the pure spinor constraint guarantees nilpotency. 5.4 Energy-momentum tensor and the b ghost The energy-momentum can be computed by acting with worldsheet diffeomor- phism on the action, as LvS0 = ∫ vα∂βT β α . The expression is given in terms of Lie derivatives acting on the fields. Under a diffeomorphism generated by a worldsheet Chapter 5. Pure spinor superstring 62 vector field v, the action changes by LvS0 = ∫ (−Lvxm∂+∂−xm + LvpL+∂−θL + pL+∂−LvθL + LvpR−∂+θR + pR−∂+LvθR − (w, λ)), (5.4.1) where (w, λ) represent the ghost terms, which have a similar structure to the fermionic spinor terms. We are now expressing left and right momenta explicitly. Applying the Lie derivatives, we see that the energy-momentum tensor depends on components that are not present in the action: T++ = − 1 2 ∂+x m∂+xm − pL+∂+θL − pR+∂+θR + wL+∂+λL + wR+∂+λR, T−− = − 1 2 ∂−x m∂−xm − pR−∂−θR − pL−∂−θL + wR−∂−λR + wL−∂−λL, (5.4.2) and T+− = T−+ = 0. The components pL−, pR+, wL− and wR+ vanish when we put all spinor fields on-shell. Hiding L,R superscripts of the momenta again, we get T++ = − 1 2 ∂+x m∂+xm − p+∂+θL + w+∂+λL, T−− = − 1 2 ∂−x m∂−xm − p−∂−θR + w−∂−λR, (5.4.3) In terms of Π± and d± we can write it as T++ = − 1 2 Πm +Πm+ − d+∂+θL + w+∂+λL, T−− = − 1 2 Πm −Πm− − d+∂+θL + w+∂+λL. (5.4.4) The momenta w± of the pure spinor ghosts have a gauge symmetry. We can express the gauge invariant degrees of freedom of these fields in term of some composite field expressions [5]. The Lorentz current for the ghosts is (NL)mn = 1 2 w+ΓmnλL, (NR)mn = 1 2 w−ΓmnλR, (5.4.5) and the ghost current is JL = w+λL, JR = w−λR. (5.4.6) Chapter 5. Pure spinor superstring 63 The energy-momentum tensor can be written in terms of these gauge invariant quantities as T++ = − 1 2 Πm +Π m + − d+∂+θL + 1 10 : Nmn L NLmn : −1 8 : JLJL : +∂+JL, T−− = − 1 2 Πm −Π m − − d−∂−θR + 1 10 : Nmn R NRmn : −1 8 : JRJR : +∂−JR, (5.4.7) which is the Sugawara form of the energy-momentum tensor. 5.4.1 Definition of the b ghost In the bosonic string sigma model, the b ghost arised from the antifield of the complex structure, after gauge fixing to Polyakov gauge. In the case of the pure spinor superstring, we did not fix a gauge – the model is a generalization of the already gauge fixed bosonic string. But, to compute amplitudes, we need to integrate over the moduli space as well, and thus we need to construct the proper Beltrami differential. To do that we must have an expression for the b ghost. We will define it using the relations Qb++ = T++, Qb−− = T−−. (5.4.8) The energy-momentum is a worldsheet 2-form with ghost number 0. The BRST operator carries ghost number 1, and is fermionic, so that the b field must be a fermionic worldsheet 2-form with ghost number −1. The only fields with negative ghost number are w±, with ghost number −1, but they are not gauge invariant. Let’s define fermionic spinor fields Gα ++ and Gα̇ −−, 2-forms on the worldsheet, by the expressions QG++ = λLT++, QG−− = λRT−−. (5.4.9) Then we can define the b field as a fractions b++ = G++ λL , b−− = G−− λR . (5.4.10) To make the fraction well-defined, we pick an arbitrary bosonic spinor Cα, and define Chapter 5. Pure spinor superstring 64 the b ghost as b++ = CαG α ++ Cβλ β L , b−− = CαG α −− Cβλ β R (5.4.11) From now on we’ll hide the L subscripts of the spinor fields. To define the b ghost that generates the energy-momentum tensor in Sugawara form, in eq. (5.4.7), the expression for G++ is given by [5] Gα ++ = 1 2 Πm+(Γ md+) α − 1 4 (NmnΓ mn∂+θ) α − 1 4 J∂+θ α − 1 4 ∂2+θ α. (5.4.12) The BRST transformation on this expression yields the energy-momentum tensor in terms of Nmn and J , instead of w. Then the b ghost is expressed as b = Cα(2Π m + (Γ md+) α −Nmn(Γ mn∂+θ) α − J∂+θα − ∂2+θα 4Cβλβ . (5.4.13) 5.4.2 Derivation of the b ghost and Y formalism Instead of defining the b ghost that generates the energy-momentum tensor in Sugawara form, we can define the expression that generates the energy-momentum tensor in eq. (5.4.4). This expression for T is given in terms of the momentum w+ of the pure spinor ghost, which is not gauge-invariant because of the pure spinor constraint. We start with the ansatz that G++ is a polynomial on the fields. As it is a 2-form, we consider a sum of products of the 1-forms w+, d+, Π+ and ∂+θ. To keep the ghost number zero, for each w+ we also multiply the term by λ. We also impose the condition that each product of fields must be fermionic. From the possible expressions, let’s choose G++ = 1 2 Πm +Γmd+ + λ(w+∂+θ)− 1 2 w+Γ m(λΓm∂+θ). (5.4.14) Now we act with the BRST transformation on G++, which gives QLG++ = 1 2 (