Yang-Mills Theory Is Discussed with Some Detail

Yang-Mills theories and conﬁnement vs. Gravity and geometry∗

AntônioD. Pereira,1, † Rodrigo F. Sobreiro,1, ‡ and Anderson A. Tomaz1, § 1UFF − Universidade Federal Fluminense, Instituto de F´ısica, Campus da Praia Vermelha, Avenida General Milton Tavares de Souza s/n, 24210-346, Niterói, RJ, Brasil. (Dated: July 2015) These lectures contains a dense material about gauge theories and the first order formalism of gravity. A scenario where quantum gravity is realized as a Yang-Mills theory is discussed with some detail. In this scenario, gravity as a geometrodynamical effect is an emergent phenomenon.

DRAFT

∗ Lectures presented at the X Escola do Centro Brasileiro de Pesquisas F´ısicas. 13-24 July 2015. †Electronic address: [email protected]ff.br ‡Electronic address: [email protected]ff.br §Electronic address: [email protected]ff.br 2

Contents

INTRODUCTION 4

PART I: GAUGE THEORIES AND THE GRIBOV PROBLEM 5

I. Gauge theories and the gauge principle 5 A. Electromagnetism 5 B. Covariant formulation and U(1) symmetry 7 1. Definitions 7 2. Maxwell equations and gauge fixing 8 C. Non-Abelian gauge theories 9 1. Lie groups basics 9 2. Gauge field, field strength and Yang-Mills action 11

II. Quantization of gauge theories 12 A. Faddeev-Popov quantization 13 B. BRST quantization 15 C. Renormalization, physical states and unitarity 16 D. Asymptotic freedom and conﬁnement 17

III. Gribov ambiguities 18 A. Statement of the problem 18 B. Gribov’s solution 19 C. The Gribov-Zwanziger action and BRST soft breaking 22 D. The Reﬁned Gribov-Zwanziger action 23

PART II: GRAVITY AS A GAUGE THEORY 25

IV. Equivalence principle and geometrodynamics 25 A. General relativity in the metric formalism 25 B. First order formalism of gravity 27 1. Vierbein 27 2. Spin-connection 28 3. Curvature, torsion and hierarchy identities 29 4. Einstein-Hilbert action 30 C. Gravity as a gauge theory 31

V. Generalized theories 31 A. Lovelock theory 31 B. Mardones-Zanelli theory 32

VI. Quantization attempts 33

PART III: EFFECTIVE GRAVITY FROM YANG-MILLS THEORIES 34 VII. General ideas DRAFT 34 VIII. SO(5) Yang-Mills theory 36 A. Projection and rescaling 36 B. Contraction and symmetry breaking 37

IX. Induced gravity 38

X. Quantum aspects 38 A. Weinberg-Witten theorems, emergent gravity and spin-1 gravitons 38 B. Cosmological constant problem 39 C. Quantum predictions 39 1. Running parameters 40 2. Numerical estimates 41 3

XI. Classical aspects 42 A. Classical ﬁeld equations 42 B. Spherical symmetry static solutions 43 C. Cosmology 44 1. The Λ-CDM model 45 2. High curvature regime 45 D. Dark stuﬀ 48

A. Diﬀerential forms 48 1. Exterior product and p-forms 48 2. Exterior derivative 49 3. The Hodge dual operator 49 4. Integration of diﬀerential forms 50

B. Geometrical aspects of gauge theories 50 1. Principal bundles and connections 51 2. Universal bundles 52 3. Gravity geometrical structure 53

Acknowledgments 54

References 54

DRAFT 4

INTRODUCTION

Gauge theories[1–6] have been extremely successful in describing at least three of the four known fundamental interactions in Nature, at classical and quantum level. Except for gravity, the Standard Model is the state of the art of quantum field theory and particle physics where Yang-Mills theories [7, 8] is the main framework. Gravity, on the other hand, is not described by interacting fields but by a geometrodynamical theory for the spacetime itself [9–13]. In despite of the success of both type theories in describing Nature, some open issues remain to be understood. In particular, from the theoretical point of view, confinement and quantum gravity are, perhaps, the most important open problems of these theories. Confinement refers to a specific behavior of the strong sector of the Standard model, known as Quantum Chromo- dynamics (QCD) which describes, essentially, the dynamics of quarks and gluons. The SU(3) Yang Mills (YM) gauge theory perfectly describes this dynamics at high energies where quarks and gluons are almost free. It displays the so called asymptotic freedom characterizing the increasing of the coupling parameter as the energy decreases. Thus, as lower the energy is, as higher the force between quarks and gluons is. The low energy sector is highly non-perturbative and very difficult to be handled. Nevertheless, V. N. Gribov has found in in the late seventies [14, 15] that, at this region, the theory lacks of a consistent quantization at the non-perturbative level. This is a technical problem, known as Gribov ambiguities, which must be considered for quantum consistency. Remarkably, the improvement of the quantization leads naturally to confinement evidences in a highly non-trivial way. On the other hand, gravity, as a geometrodynamical phenomenon, lacks for a quantum version (even perturbatively). In many aspects, the principles of general relativity seem to be incompatible to the principles of quantum field theory, dooming their union to failure. Many alternative approaches to describe gravity at quantum levels have been proposed for almost one century. Widely known examples are string theory [16–19], loop quantum gravity [20, 21], asymptotic safety [22, 23], Hoˇrava-Lifshitz gravity [24–26], the vast scenarios of emergent gravities [27–40] and so on. In particular, emergent gravities possibly solve the problem by stating that gravity as a geometrodynamical phenomenon can not be quantized. Gravity should be an emergent feature of a more fundamental theory that have nothing to do with the spacetime dynamics at quantum level. In these lectures we discuss an interesting analogy between non-perturbative QCD and gravity [37–42]. In this proposal, gravity emerges from a Yang-Mills theory for the gauge symmetry group SO(5) in four dimensions. Thus, we will discuss the details of Yang-Mills theories and the Gribov problem as well as the first order formalism of gravity in order to reach the goal of understanding how gravity can emerge from a quantum gauge theory. Specifically, quantum gravity will be described by a Yang-Mills action in flat space while classical gravity will be obtained from it as a geometrodynamical theory for the spacetime and both sectors should not intersect. Before that, a dense discussion about Yang-Mills theories and the first order theory of gravity are discussed. The idea of these lectures is to provide a complete view of gauge theories and gravity in a conceptual way. Due to the vast amount of results known in these theories, many details are omitted in favor of conceptual discussions. A lot of technical passages are left as exercises of all kinds of difficulty levels. These exercise are recommended for students interested in explore the field. Previous notions in quantum field theory (QFT), differential geometry, group theory and general relativity is advisable, but not mandatory. Along the text many references are suggested and two appendices (about differential forms and fibre bundles) is included for self-consistency. At the end of the course, the student should acquire a lot of conceptual information about current research on gauge theories, confinement, gravity and quantum gravity. DRAFT 5

PART I: GAUGE THEORIES AND THE GRIBOV PROBLEM

In this part we focus on the construction of gauge theories. Starting with Maxwell equations as an Abelian gauge theory, we define gauge field, gauge transformations and the gauge principle. A few concepts in Lie groups and Lie algebras are introduced in order to discuss non-Abelian gauge theories, which are the main theme of this part. Quantization is discussed with some detail within Faddeev-Popov’s method. Then, BRST symmetry and its consequences, including BRST quantization, are carried out. The concepts of physical states, renormalizability, unitarity, asymptotic freedom and confinement are established. All these aspects are mainly found in quantum field theory text books, see for instance [1–4, 6, 43]. The issue of Gribov ambiguities is then addressed: a review since Gribov‘s seminal work back in 1978 is compiled with the most important results and concepts. The main references about the Gribov problem are [14, 15, 44–50].

I. GAUGE THEORIES AND THE GAUGE PRINCIPLE

A. Electromagnetism

The most known example of a gauge theory is, perhaps, electromagnetism [1, 51, 52]. In its non-covariant form, electromagnetism is described by Maxwell equations in vacuum, ρ ∇ · E = , o ∇ · B = 0 , ∂B ∇ × E + = 0 , ∂t ∂E ∇ × B − µ = µ j , (1) o o ∂t o where E and B are the electric and magnetic fields, respectively; o is the electric permittivity of the vacuum and µo is the magnetic permeability of the vacuum; the charge density is ρ while j = ρv is the current density with velocity v. Maxwell equations can be simplified by the introduction of potential fields. In fact, the second equation (magnetic Gauss law - first homogeneous equation) states that a solution for B is

B = ∇ × A , (2) where A is the vector potential field. Using this solution at Faraday law’s (second homogeneous equation) it is possible to show that the solution for the electric field is1 ∂A E = −∇φ − , (3) ∂t where φ is the electric scalar potential. Remarkably, the vector and scalar potentials can always be redefined in such a way that the electric and magnetic fields are left invariant (and so are the Maxwell equations, obviously). These redefinitions are called gauge transformations: ∂ξ DRAFTφ 7−→ φ − , ∂t A 7−→ A + ∇ξ , (4) with ξ = ξ(r, t) being the gauge transformation parameter. As opposed to coordinate transformations (for example, Lorentz transformations), gauge transformations are transformations that leave spacetime coordinates intact, it is a type of transformation that affects only fields. The fields φ and A are recognized as the gauge fields. The electric and

1 The fact that the homogeneous equations are used to define the potentials are not accidental. These equations have a deep geometrical meaning, which we will briefly discuss in this section and along this part. 6 magnetic fields are said to be gauge invariants. In terms of the gauge fields, the Maxwell equations (1) read (Exercise 1.1) ∂ ρ −∇2φ − ∇ · A = , ∂t o ∂2A ∂φ −∇2A + µ + ∇ ∇ · A + µ = µ j . (5) o o ∂t2 o o ∂t o We notice that the homogeneous equations are automatically satisfied through (2) and (3). Thus, (5) are obtained from the substitution of these solutions into the inhomogeneous Maxwell equations. Although equations (5) appear complicated with respect to the original equations (1), it is important to stress out that E and B carry six components of freedom while φ and A carry only four. The fact that the gauge fields can be redefined up to a scalar function establishes that they are not observables because there is an infinite number of equivalent fields leading to the same pair of electric and magnetic fields, recognized as the actual physical fields. Thus, no matter how one chooses ξ to solve a problem, the corresponding electric and magnetic fields will always be the same. Moreover, to solve a specific problem, it is possible to choose a unique class of gauge fields in order to simplify the equations. This procedure is called gauge fixing and is performed by the introduction of a constraint for the gauge potentials. For instance, the Coulomb gauge is characterized by transverse vector potentials, ∇ · A = 0, leading to ρ −∇2φ = , o ∂2A ∂ −∇2A + µ = µ j − µ ∇φ , (6) o o ∂t2 o o o ∂t Then, the scalar potential is the solution of the Poisson equation and is an instantaneous field, a non-physical feature, while the vector potential is the solution of a inhomogeneous wave equation. Another example (Exercise 1.2) is the 2 ∂φ Landau gauge , ∇ · A + oµo ∂t = 0, which leads to inhomogeneous wave equations for both, φ and A. The difference between physical and non-physical quantities can be summarized by the so called gauge principle

Gauge Principle: Physical observables are described by gauge invariant quantities.

Since E and B are gauge invariants, anything constructed with them are also gauge invariant, for instance, 2 2 the electromagnetic energy density oE /2+B /2µo. In this way, although physical, neither E or B are fundamental. The fundamental fields are φ and A, in the sense that they are “indivisible”. Hence, a gauge theory is constructed from the gauge fields (as fundamental objects) of the theory. However, due to the gauge symmetry, they can not be observables. Instead, any observable must be constructed from gauge fields and has to be gauge invariant. A very beautiful example (Exercise 1.3) of the gauge principle is the Aharonov-Bohm effect [53, 54]. Let us consider ˆ a long thin solenoid with constant magnetic field given by B = Bok inside the solenoid and 0 otherwise. A vector potential whose curl produces such magnetic field is A = Φ/(2πr)ϕ ˆ whereϕ ˆ is the polar angle in cylindrical coordinates and Z Φ = B · dS . (7) S is the magnetic flux through a surface whose boundary is a curve around the solenoid. Then, an electron beam is split in two, with each sub-beam passing on different sides of the solenoid (This can be done by the the double-slit apparatus). It can be shown thatDRAFT the interference pattern between both sub-beams on a screen will be shifted by a factor e I δ = exp A · dl . (8) ~ C with respect to the case with no solenoid. In (8), e is the electron charge and ~ is the Planck constant. Since this is an observable phenomenon, it should be consistent with the gauge principle. Indeed, (8) is gauge invariant: e I e I e I A · dl 7−→ A · dl + ∇ξ · dl . (9) ~ C ~ C ~ C

2 Also known as Lorentz gauge. 7

Since the last term vanishes identically, the shift (8) is gauge invariant. Moreover, due to Stokes theorem Z Z ln δ = ∇ × A · dS = B · dS = Φ , (10) S S where S is any surface with boundary C. Thus, the shift of the interference pattern, although given in terms of A, is a gauge invariant quantity. This illustrates the fact that, even though A is not physical, it can generate observable phenomena through gauge invariant objects constructed from it and starting from postulates which are based on gauge concepts.

B. Covariant formulation and U(1) symmetry

We will now be a bit more formal in constructing the classical electromagnetism as a gauge theory by writing it down in a manifestly covariant form, since it is a natural relativistic theory.

1. Deﬁnitions

We define Greek indexes as collective indexes for space and time, α, β, . . . ∈ {0, 1, 2, 3}, where the 0th coordinate µ stands for time. The derivative operator will be written in a simplified form as ∂µ = ∂/∂x . Moreover, from now on, 3 4 we adopt natural units and the 4-dimensional Euclidean metric will be assumed, i.e., δµν ≡ diag(1, 1, 1, 1). We consider a four dimensional gauge field Aµ and its gauge transformation 1 A 7−→ A + ∂ ξ . (11) µ µ e µ

This definition coincides with the gauge transformations (4) for A0 = φ (Exercise 1.4) and the factor 1/e is introduced for convenience. The gauge transformation (11) is a gauge transformation of a gauge connection under the action of an element of the U(1) group, i.e., the group of complex numbers with unit modulus. In fact, (11) can be rewritten as 1 A 7−→ A + e−ξ∂ eξ . (12) µ µ e µ where5 eξ ∈ U(1). Formally, the gauge field is a connection over an abstract topological space6 which we will call gauge space. Thus, a covariant derivative can be defined as

Dµ = ∂µ − ieAµ . (13) where e is the coupling parameter which is typically associated with the electron charge in QED. In general, covariant derivatives do not commute and deﬁne the curvature of the gauge space (commonly known as the ﬁeld strength) (Exercise 1.5):

[Dµ,Dν ] = −ieFµν = −ie (∂µAν − ∂ν Aµ) . (14) where the derivatives are supposed to act over a generic charged field, for instance, a spinor field describing the electron. It is simple to show that the field strength is gauge invariant (Exercise 1.6). The tensor Fµν is obviously anti-symmetric. Thus, it can be described by an anti-symmetric 4 × 4 matrix whose components coincide with the electric and magnetic fields (ExerciseDRAFT 1.7),  0 E1 E2 E3  −E1 0 B3 −B2 Fµν ≡   (15) −E2 −B3 0 B1  −E3 B2 −B1 0

3 From now on, unless the contrary is said, we will employ natural units c = ~ = 1. 4 Minkowskian metric would be preferable for perturbative treatments. However, we will deal with non-perturbative phenomena whether Wick’s rotation is not established. Thus, it is often chosen to start from Euclidean metric, where we can actually perform explicit computations in quantum ﬁeld theory. 5 Any complex number with unit norm can be written in this form. 6 See Ap. B. 8

The ﬁeld strength is clearly gauge invariant, a property which makes it an observable of the theory, as it should be due to its gauge invariant components.

2. Maxwell equations and gauge ﬁxing

In the covariant formulation, Maxwell equations (1) read

∂ν Fµν = jµ ,

µναβ∂αFµν = 0 , (16) where jµ is the four-current density, jµ ≡ (ρ, ρv). The first set of the Maxwell equations above coincide with the inhomogeneous equations of (1) (Exercise 1.8) while the second set coincide with the homogeneous equations of (1) (Exercise 1.9). In fact, analogously to (2) and (3), the solution of the second equation of (16) is (14). Moreover, this is a topological equation because the solution is the definition of the gauge potential, i.e. the definition of the gauge potential automatically satisfies this equation. These equations are recognized as the electromagnetic Bianchi identities. Maxwell equations (16) can be rewritten in terms of the gauge field (Exercise 1.10): the first of (16) leads to

∂µ∂ν Aν − Aµ = jµ , (17) while the second is automatically satisfied, as discussed. A useful way to simplify the equations (17) is to fix a gauge, which means that we can impose a constraint over Aµ in order to select a class of gauge potentials among all possible gauge potentials that could lead to a consistent field strength solution. Typicality, the linear covariant gauges are imposed: ∂µAµ = f, where f is a spacetime function. The special case f = 0 is recognized as the Landau gauge while the non-covariant gauge ∂iAi = 0 is the Coulomb gauge. In the case of the Landau gauge, for instance, Maxwell equations (17) simplify to

Aµ = −jµ , (18) It is very useful to deﬁne an action for electromagnetism. A typical choice is

Z 1 Z α S = d4x F F − j A + d4x b2 + b∂ A , (19) M 4 µν µν µ µ 2 µ µ where b is the Lautrup-Nakanishi field [43] which is a Lagrange multiplier enforcing the linear covariant gauge fixing. It is a straightforward exercise (Exercise 1.11) to show that the minimization of this action with respect to Aµ leads to a gauge fixed Maxwell equations and the minimization with respect to b leads to the gauge fixing condition7. In summary, electrodynamics, as described by the action (19), can be obtained by postulating: • Gauge principle with respect to the U(1) group - This postulate naturally leads to the existence of a gauge connection and a gauge curvature. • (Euclidean) Lorentz invariance of the action - This postulate ensures the principle of relativity. Moreover, Lorentz transformations form a group whose Euclidean version is the group of four-dimensional rotations in spacetime, the SO(4) group. • Further requirements as localityDRAFT8 and absence of higher order derivatives can be employed in a consistent way in order to avoid unwanted terms in (19).

• Any ﬁeld coupled to Aµ must respect all above postulates.

7 In field theory, the fields equations can be obtained through the generalized Euler-Lagrange equations: ∂L ∂L − ∂ = 0 . A µ A ∂Φ ∂ (∂µΦ ) where ΦA is a generic field with A being a collective index. 8 Roughly speaking, a local quantity in QFT is an object that only depends on a single spacetime point. 9

C. Non-Abelian gauge theories

From now on, due to the intricacies of non-Abelian theories (which will become evident as we proceed), we will neglect the coupling with external sources jµ or other ﬁelds.

1. Lie groups basics

Our aim in this section is to provide a brief overview of what we need about Lie groups and group theory. It is not our intent to provide detailed discussion on this matter. For detailed texts about Lie Groups, see for instance [55–57] and also [1] for an extended, yet summarized, introduction. To start with, let us enunciate the axioms of a group. Consider a set G of elements u and an operation ·, then G is said to be a group if its elements obey:

• Closure: If u1 and u2 ∈ G then u3 = u1 · u2 ∈ G.

• Associativity: If u1, u2 and u3 ∈ G, then (u1 · u2) · u3 = u1 · (u2 · u3).

• Identity: ∃ uo = 1 ∈ G u · 1 = 1 · u = u ∀u ∈ G.

−1 −1 −1 • Inverse: ∃ u ∈ G u · u = u · u = 1 ∀u ∈ G. Moreover, • If G is inﬁnite and continuous, the group is said to be a Lie Group.

• If [u1, u2] = 0 ∀ u1 and u2 ∈ G, the group is said to be an Abelian group, otherwise it is said to be a non-Abelian group. The notion of subgroup will also be important along this lectures: • If G is a group and H ∈ G is a smaller set than G, then H is a subgroup of G if H is also a group, i.e. if the elements of H obeys the four group postulates. Lie groups are typically, but not exclusively, N × N matrix groups (the case N=1 is a group formed by scalars, like U(1)). A useful property of Lie groups is that their elements can always be written as exponentials u = eξ, where ξ is the gauge parameter. When the gauge parameter is a function of the spacetime coordinates, the group is said to be a local group, otherwise the group is said to be a global group. The algebra of the group is the tangent space of the group at the identity[1]. For any element near the identity, we can write u ≈ 1 + ξ. We can expand ξ through a matrix basis T A as ξ = ξAT A where the capital Latin indexes A, B, . . . , H ∈ {1, 2,..., dim G}. The exponent ξ is said to belong to the algebra of G. It should be clear that if · is multiplication, its action with respect to ξ and T A is addition. It is important to have in mind that, in general, the dimension of the group does not coincide with the dimension of its elements, dim G 6= N. The basis elements T A are called generators. The characterization of the algebra is given by the commutation rules of the generators, [T A,T B] = f ABC T C , (20) where the skew-symmetric objectDRAFTf ABC is composed by the structure constants of the group. If f ABC = 0, then the group is obviously Abelian. Moreover, the structure constants obey the Jacobi relations (which can be derived from (20)) f ABC f CDE + f ADC f CEB + f AEC f CBD = 0 . (21) As a supplementary condition, the generators are normalized as 1 Tr(T AT B) = − δAB . (22) 2 A representation of a group is the realization of the elements u ∈ G as a linear operator U(u) acting on a linear space. Thus, given the elements of V , denoted by ψi, a representation is characterized by

ψ0i = U ijψj . (23) 10

It is demanded that

U(u1u2) = U(u1)U(u2) , U(u−1) = [U(u)]−1 , U(1) = 1 . (24)

Clearly, this is a mapping u 7−→ U which will be reﬂected on the generators of G. Hence, (24) implies that

U(T A + T B) = U(T A) + U(T B) , U(aT A) = aU(T A) , U([T A,T B]) = [U(T A),U(T A)] , (25) where a ∈ R. Thus, we also have U(u ≈ 1 + ξ) ≈ 1 + U(ξ) = 1 + ξAU(T A) . (26)

In gauge theories, the most relevant representations are the fundamental representation and the adjoint representation. For the fundamental representation, let us consider G as a group of N × N matrices. Then, an N-dimensional vector space V with elements ψI , transform under G as ψ0I = U IJ ψJ , where I, J, K, . . . ∈ 1, 2, 3,...N. In this case, the elements U are simply N × N matrices. The adjoint representation is a direct realization of the algebra of G. The action of an element u in the adjoint representation on an element of the algebra ξ is given by Ad(u)ξ = u−1ξu. Then, u−1ξu is required to belong to the adjoint representation as well. For u ≈ 1 + η it is then clear that (Exercise 1.12) Ad(η)ξ = [ξ, η]. Thus, if ξ and η are simply the generators, we have that Ad(T B)T A = f ABC T C . On the other hand, the deﬁnition of a representation (23) yields

[T A]BC = f ABC , (27) which means that in the adjoint representation the dimension of the elements coincide with the group dimension. It is a straightforward exercise (Exercise 1.13) to check that both, fundamental and adjoint, representations respect (24) and (25). Let us discuss a few important groups. We already discussed the unitary Abelian group U(1) which is the group of all complex numbers with unitary modulus, i.e. zz∗ = 1. Taking z = eiξ, where ξ ∈ R, it is almost trivial to check all four group axioms (Exercise 1.14). The generalization of the unitary group to matrix groups is realized by the group of unitary N ×N complex matrices, i.e. uu† = 1. This group is denoted by U(N). From its definition we have that det u = ±1 and dim U(N) = N 2 − 1. The restriction for positive determinants excludes discrete transformations such as reflections. This is the so called special unitary group, denoted by SU(N). It is easy to verify that unitary groups respect the group axioms (Exercise 1.15). Geometrically, the elements u ∈ SU(N) are rotations of complex vectors. The group of N × N orthogonal real matrices is denoted by O(N) and obeys uuT = 1 where uT is the transpose of u. Obviously, det u = ±1. It is clear that u ∈ O(N) represent rotations and reflections of real vectors in an N-dimensional Euclidean space and hence dim O(N) = N(N − 1)/2. The restriction to pure rotations leads to the special orthogonal groups SO(N). Again, the verification of the group axioms is very easy (Exercise 1.16). The algebra of SO(N) is related to the Euclidean metric. In fact, in the adjoint representation, the generators are matrices anti-symmetric in their indexes, T ab = −T ba, where a, b, c, . . . , h ∈ {1, 2, 3,...,N(N − 1)/2}, obeying ab DRAFTbc abcdef ef [T ,T ] = f T , 1 f abcdef = − δadδbe − δaeδbd δcf + δacδbe − δaeδbc δdf . (28) 2 All examples mentioned above assume · to be matrix multiplication. The group of translations, commonly denoted by RN , is an example of a group whose operation is addition. Hence, given an N-dimensional vector space V , the elements u ∈ RN are also N-dimensional vectors in the fundamental representation. The action of the group elements 0 i on the elements of V is given by ψ = ψ + u. In this case, the identity is the null vector ψo = 0 and the inverse is −u. Closure and associativity can be directly verified (Exercise 1.17). Because translations commute, the algebra of translations is characterized by

[P A,P B] = 0 . (29) 11

It is possible to define a more general group by considering rotations and translations in a single larger group, the so called Euclidean Poincarégroups ISO(N) = SO(N) n RN , where n is a semi-direct product9. The action of the Euclidean Poincarégroup on a vector ψ ∈ V is

ψ0 = Λψ + u , (30)

where ψ is written as a N-dimensional column vector, Λ = ΛabT ab ∈ SO(N) and u = uaP a ∈ RN . To show that ISO(N) is a group is also an easy exercise, yet bigger than the previous ones (Exercise 1.18). In the same way, the properties of the Lorentz group SO(1, 3) (which is the group of four-dimensional orthogonal matrices preserving the Minkowskian metric) and the general linear group GL(N, R) (which is the group of all N × N real matrices) can be easily checked (Exercise 1.19). In what follows, we will stick mainly to Lie groups which are semi-simple and compact10.

2. Gauge ﬁeld, ﬁeld strength and Yang-Mills action

The generalization from QED to non-Abelian gauge theories relies on its extension to a non-Abelian group, G. Following the Abelian case, we start by deﬁning the gauge ﬁeld as an algebra valued connection of the group space A A A † A in the adjoint representation, Aµ = Aµ T . The generators are chosen to be anti-hermitian, (T ) = −T , so [T A,T B] = f ABC T C . The respective covariant derivative which acts on objects with adjoint group indexes is

AB AB ABC C Dµ = δ ∂µ − κf Aµ , (31) where κ is the coupling parameter (analogous to e in QED). Equation (31), in a component independent notation, can be written as

Dµ = ∂µ + κ[Aµ, ] . (32) It is easy to show that (32) reduces to (31) (Exercise 1.20). The gauge transformations are now given by

1 A 7−→ u−1 A + ∂ u , (33) µ µ κ µ

where u ∈ G. It is convenient to consider inﬁnitesimal transformations through u ≈ 1 + ξ, which gives (Exercise 1.21)

1 AA 7−→ AA + DABξB . (34) µ µ κ µ

From (33) and (34) we can see that gauge transformations are no longer linear as in the Abelian case. This non-linear character is the main reason why non-Abelian theories are much more complicated than Abelian ones. Again, by commuting covariant derivatives, one ﬁnds the ﬁeld strength (curvature of the gauge space):

[Dµ,Dν ] = κFµν . (35) where DRAFTFµν = ∂µAν − ∂ν Aµ + κ[Aµ,Aν ] . (36) Two very important observations are in order now: First, the field strength has a non-linear piece, not present at the Abelian case. This feature means that the gauge field interacts with itself, for instance, gluons interact among each other (a property not observed in photons). Second, the field strength is not a gauge invariant object (Exercise 1.22),

−1 Fµν 7−→ u Fµν u . (37)

9 N The semi-direct product is a direct product for which one sector is a normal subgroup, which is the case of R . See also the next footnote 10 Roughly speaking, a compact group is a bounded and closed group. Semi-simple group is a group that can be decomposed in simple groups. Simple groups are groups with no normal subgroups, except for the identity and the group itself. Finally, a normal subgroup N of G is characterized by g−1Ng ⊆ N. For the actual formal deﬁnitions we refer to [55–57] 12

In fact, this is a typical transformation rule of matter ﬁelds in the adjoint representation. So, the electromagnetic analogues can not be associated with observables. The ﬁeld strength can be written in components of the algebra, A A Fµν = Fµν T , as (Exercise 1.23)

A A A ABC B C Fµν = ∂µAν − ∂ν Aµ + κf Aµ Aν . (38)

To construct a consistent action, we evoke: The gauge principle for the group G; Euclidean Lorentz invariance; Locality; Absence of higher order derivatives. It turns out that the simple generalization of (19) is indeed gauge invariant (Exercise 1.24):

Z 1 S = d4x F A F A , (39) YM 4 µν µν

where we have omitted the gauge fixed part (see next section) and a possible interacting term (as previously an- nounced). This action is known as Yang-Mills action [6, 7]. The non-linear character of the field strength leads to non-linear field equations which means that this kind of theories are physically richer as well as much complicated when compared to Abelian cases. In fact, variation of the Yang-Mills action with respect to the gauge field leads to (Exercise 1.25)

A DµFµν = 0 . (40) On the other hand, because of (38), a topological identity exists,

A µναβDν Fαβ = 0 , (41) which is recognized as the Bianchi identity and generalizes the homogenous electromagnetic ﬁeld equations in (16).

II. QUANTIZATION OF GAUGE THEORIES

Til now, we have deﬁned only the classical quantities of gauge theories. To study the quantum aspects of Yang- Mills theories we have to promote the ﬁelds to quantum operators or, equivalently, write down a consistent functional integral of the exponentiated classical action. We will follow the second option here [6]. If we naively try to consider11 Z Z = DAe−SYM , (42)

we will face inconsistencies right from the beginning. The main problem relies on the gauge symmetry: The functional integral sums over all possible field configurations in order to provide a probability of an event or an expectation value. However, the gauge symmetry establishes that there is an infinity number of equivalent fields related through gauge transformations. Therefore, each field is actually been infinitely overcounted. Hence, the probabilities are not being conserved. In fact, if one tries to compute the propagators of the theory (take the linearized version eq. (40) and try to find the associated Green function), a divergent Green function is found (Exercise 1.26). Thus, even the free theory is inconsistent (this is also true for QED). The solution is to consider only one configuration among all equivalent fields (those connected through gauge transformations) - for each infiniteDRAFT set of gauge fields related through gauge transformations, we must choose only one representative. This procedure is called gauge fixing and is implemented by imposing a constraint over the gauge field. The gauge fixing for the Abelian case was already discussed here, at classical level - It is easily generalized to the non-Abelian case, also at classical level. At quantum level however, it is not a trivial task. The way to do this in the path integral is through the Faddeev-Popov trick [58], which we discuss in the next section.

11 Since we are not going to perform any practical computation at this level, we omit the classical Schwinger sources [6]. 13

A. Faddeev-Popov quantization

The question here is: How to introduce a gauge ﬁxing in a consistent way in (42)? The idea is to integrate only over ﬁelds obeying a certain constraint. Let us consider the linear covariant gauges ∂µAµ = f. The trick is to write the unit in a non-trivial way Z u 1 = ∆(A) Duδ(f − ∂µAµ) , (43)

u where Aµ is the gauge orbit, which is deﬁned through 1 Au = u−1 A + ∂ u , (44) µ µ κ µ

while Aµ is a ﬁxed, yet arbitrary, gauge conﬁguration. It should be clear that the integral in (43) is gauge invariant because of the closure property of G. This means that ∆(A) = ∆(Au) . Therefore, Z Z −SYM (A) u Z = DAe ∆(A) Duδ(f − ∂µAµ) , (45)

From the gauge invariance of SYM , ∆(A) and DA, we can write

Z u Z u −SYM (A ) u u Z = DA e ∆(A ) Duδ(f − ∂µAµ) , (46)

u u 1 −1 Thus, performing the change of variables Aµ → u Aµ + κ ∂µ u = Aµ, we achieve Z Z −SYM (A) Z = DAe ∆(A) Duδ(f − ∂µAµ) . (47)

u It remains to determine ∆(A). This is easily done by changing the integration variable of (43) from u to f = ∂µAµ −f: Z Z −1 δu δu ∆ (A) = Duδ(f) = Dfδ(f) det = det , (48) δf δf f=0 which can be evaluated for infinitesimal gauge transformations u ≈ 1 + ξ (Exercise 1.27), δf 1 ∆(A) = det = det(∂µDµ) . (49) δξ f=0 κ Thus Z 1 Z Z = DAe−SYM det(∂ D )δ(f − ∂ A ) Du . (50) µ µ µ µ κ which can be normalized to finally obtain Z −SYM DRAFTZ = DAe det(∂µDµ)δ(f − ∂µAµ) . (51) This expression is supposedly free of gauge ambiguities. However, the determinant and the functional delta is not in a form which we can recognize a gauge fixing, as in (19). In fact, these terms can be very complicated to handle because they may ruin a simple algorithm to perform explicit perturbative computations such as the Feynman rules. Fortunately, both terms can be easily exponetiated with the help of additional fields. Let us start with the functional delta: Because gauge invariant quantities should not depend on f, we can integrate over f with a Gaussian weight12, Z 1 Z 1 Z Df exp Tr d4xf 2 δ(f − ∂ A ) = exp d4xTr(∂ A )2 , (52) α µ µ α µ µ

12 The sign of the Gaussian seems to be wrong. However, (22) still have to be employed, correcting the sign at the ﬁnal expression. 14

where α is a non-negative number13. It will be useful for the next sections to rewrite (52) as a normalized functional integral of an auxiliary bosonic algebra-valued field b (Exercise 1.28), 1 Z Z exp d4xTr(∂ A )2 ≡ Db exp [Tr (2b∂ A + αbb)] . (53) α µ µ µ µ The field b is recognized as the Lautrup-Nakanishi field [43]. Just like the Abelian case, it plays the role of a Lagrange multiplier enforcing the gauge fixing. The determinant is also simple lo exponentiate, it is a determinant with positive unit power, thus, it can be substituted by a Gaussian functional integral over Grassmann variables [6]. Recalling that ∂µDµ is an dim G × dim G matrix operator, we need a set of dim G2 independent variables, thus Z Z det(∂µDµ) = DcDc exp − c∂µDµc , (54)

where c and c are independent Grassmann variables. The local version of (51) is then Z Z = DADbDcDc e−SYMFP . (55)

where the action which must be considered for quantum theory is

SYMFP = SYM + Sgf , (56) with Z h α i S = d4x bA ∂ AA + bA + cA∂ DABcB . (57) gf µ µ 2 µ µ A few observations are in order: • The classical field equation of b results in the gauge fixing condition for f a = −αba (Exercise 1.29). Therefore, with the help of ba, it is more evident that the gauge fixed action actually enforces the proposed constraint; • The cases α = 0 and α = 1 are recognized as the Landau and Feynman gauges, respectively; • The algebra-valued fields c = cAT A and c = cAT A are the Faddeev-Popov ghosts and anti-ghost fields. They are fermionic fields with no Lorentz indexes (scalars), i.e., they are spin-0 bosons obeying Fermi statistics (because of the Faddeev-Popov determinant). In principle, they should violate the spin-statistics theorem [59] and, consequently, a causality violation is expected. However, what happens is that these fields are actually non-physical because they do not appear as asymptotic states [6], i.e., they are not observables of the theory. Nevertheless, together with the gauge fixing, they play an important role in canceling non-physical degrees of freedom of the gauge fields, ensuring the unitarity of the theory. In fact, there is a class of gauge fixings where µ the Faddeev-Popov fields decouple, the so called axial gauge nµAµ = 0, where n is a fixed vector [4]; • An important consequence of (56) is that it displays the Faddeev-Popov discrete symmetry, which means that SYMFP is invariant by c 7−→ c and c 7−→ −c. This symmetry characterizes the ghost number, see Table I. Since it is a symmetry of the action, all observables should also be invariant under the Faddeev-Popov symmetry, which means that this symmetry prevents ghost states to appear in external legs of any diagram - so, causality is safe indeed; • In the case where G is Abelian,DRAFT such as in QED, the Faddeev-Popov determinant can be eliminated by simple normalization of Z (Exercise 1.30). Thus, QED is naturally ghost-free. • The Yang-Mills action (56) has no mass parameters, i.e. the theory is massless and conformal at classical level. • It is important at this point to expend a few words about the mass dimensions of the fields in four-dimensional spacetime. Because ~ = 1, the action should be dimensionless in natural units. Thus, since [x] = −1 in mass dimension, [R d4x] = −4 and [∂] = 1. As a consequence, [L] = 4. A simple algebraic analysis leads to the second column of Table I (Exercise 1.31);

13 In a way, the arbitrariness of f is replaced by α. 15

Fields A b c c¯ κ Dimension 1 2 0 2 0 Ghost number 0 0 1 −1 0

TABLE I: Quantum numbers of the Yang-Mills ﬁelds in four-dimensional spacetime.

B. BRST quantization

A remarkable feature of the gauge ﬁxed action (56) is that, in despite of the broken gauge symmetry, a new symmetry shows up, the so called BRST symmetry [6, 43, 60, 61], due to C. Becchi, A. Rouet, R. Stora and, independently, I. V. Tyutin. BRST symmetry is characterized by a fermionic operator s, acting on the ﬁelds as

A AB B sAµ = −Dµ c , (58a) κ scA = f ABC cBcC , (58b) 2 scA = bA , (58c) sbA = 0 . (58d) It is a straightforward exercise to show that (Exercise 1.32)

sSY MGF = 0 . (59) From (58a - 58d) it is clear that BRST symmetry is actually a supersymmetry because it transforms bosons in fermions and vice versa. Moreover, the BRST operator is nilpotent, s2 = 0 (Exercise14 1.33), a very welcome feature. From (58a - 58d) and Table I we also have that [s] = 0 and that s has ghost number 1. Hence, when acting over an object, the BRST operator increases its ghost number by 1. It turns out that this very beautiful symmetry plays a fundamental role in non-Abelian gauge theories. As briefly discussed in the Appendix B, s has a deep geometrical meaning, it is the exterior derivative in the functional space of all gauge configurations along a gauge orbit. Moreover, the Faddeev-Popov ghost field is a Maurer-Cartan form in this very same space while the gauge field is a connection over a more primitive fiber bundle structure [2, 62, 63], which we have called gauge space. In fact, the Faddeev-Popov determinant corresponds to a Riemannian metric in the gauge functional space. Thus, independently of the Physics (independently of the chosen action), A, c and s exist as fundamental mathematical abstract structures. This fundamental appeal is the origin of the so called BRST quantization method where the BRST symmetry is taken as the fundamental principle to construct the invariant action. The main ingredients are the gauge field A, the ghost field c and the BRST operator s. The anti-ghost c and the Lautrup-Nakanishi b fields are introduced in order to account for the gauge fixing and also to ensure the Faddeev-Popov discrete symmetry. Before we proceed, let us enunciate two important results about the BRST operator: • The fact that s is a nilpotent operators allows one to define its cohomology, the BRST cohomology. The BRST cohomology problem is the problem of finding the solution of15 g sΘn = 0 , (60) g where Θn is typically and integrated object with specific ghost number g and dimension n. The solution of (60) is [43]

DRAFTg g g−1 g g Θ = Υ + sΘ sΥ = 0 and Υ 6= s(something) . (61) n n n n n g g−1 16 In (61), Υn is the non-trivial part of the cohomology and sΘn is the trivial sector of the cohomology .

14 Tip: Use the Jacobi identities (21). 15 This is actually the fundamental cohomology problem for any nilpotent operator. 16 It is commonly used the terminology BRST exact object for trivial objects and BRST closed object for non-trivial objects s(exact) = 0 ⇒ exact = s(something) , s(closed) = 0 ⇒ closed 6= s(something) . (62) 16

• BRST doublets consist on pair of fields transforming through sX = Y and sY = 0. Clearly, c and b are BRST doublets. A general result is that BRST doublets belong exclusively to the trivial sector of the cohomology [43]. In practice, any BRST doublet can be introduced because they are harmless to the physical content of the theory. Thus, to find the BRST invariant action it is imposed that S is a local polynomial in the fields and their derivatives (with no higher order derivatives), with dimension bounded by 4, vanishing ghost number, Euclidean spacetime isometry, obeying sS = 0 . (63) Condition (63) is actually the BRST equivalent of the gauge principle. Therefore, following (63), the solution is then given by

−1 S = S0 + s∆ sS0 = 0 ,S0 6= s(something) . (64)

where ∆−1 is a local integrated polynomial in the ﬁelds and their derivatives, with dimension bounded by 4 and ghost number −1. The fact that BRST doublets always belong to the trivial sector combined to the Faddeev-Popov 17 symmetry yield that c, b and c are not allowed at S0, i.e. S0 = S0(A). Hence, it can be shown that (Exercise 1.34) (see [43])

S0 = SYM , Z α ∆−1 = d4x cA ∂ AA + bA + βf ABC cBcC . (65) µ µ 2 where α and β are dimensionless gauge parameters. For the particular case β = 0, (64) reduces to the Faddeev-Popov result(56) (Exercise 1.35). In fact, (57) can be written as a BRST exact object Z α S = s d4x cA ∂ AA + bA . (66) gf µ µ 2 which means that the gauge fixing does not interfere with the physical observables. An example of a non-physical quantity is the gauge field propagator (Exercise 1.36), 1 hAA(−p)AB(p)i = δAB [T + αL ] , (67) µ µ p2 µν µν where the transverse and longitudinal projectors read p p T = δ − µ ν , µν µν p2 p p L = µ ν . (68) µν p2 The gauge field propagator is then clearly gauge dependent and describes massless non-physical states. C.DRAFT Renormalization, physical states and unitarity There are three very important consequences of BRST symmetry: renormalizability, definition of physical states and unitarity. Renormalizability states that all divergences can be consistently eliminated by a suitable redefinition of the fields and parameters. This can be performed through the algebraic renormalization techniques [43]. Essentially, BRST symmetry is imposed for the quantum action Γ, defined through e−Γ = Z. (69)

17 This is not a trivial exercise. For the non-trivial part, write down a linear combination of all possible integrated polynomial terms using A, ∂, κ, and f ABC . Impose the requirements above listed and then apply s. The result must vanish, a property that will provide specific values for the coefficients of the linear combination. The final result should be the Yang-Mills action. The trivial part is a bit more complicated due to the higher number of fields (which includes A as well). 17

The quantum action can be perturbatively expanded around the classical action,

2 Γ = Γ0 + Γ1 + Γ2 + ...... (70)

where Γ0 = SYMFP , is a small expansion parameter and Γn, for n > 0, are integrated local polynomials in the fields and their derivatives bounded by dimension four, vanishing ghost number and Euclidean Lorentz invariance. 18 Imposing sΓ = 0 and using sΓ0 = 0, it is possible to show that, at first order, Γ1 can be reabsorbed in Γ0 by means (1) of multiplicative redefinition of the fields and parameters, ensuring that Γ = Γ0 + Γ1 has the same form as Γ0. The method is recursive and can be applied to all orders. Thus, Yang-Mills theories are renormalizable to all orders in perturbation theory, at least in a large class of covariant gauges. The gauge principle is used to determine physical observables at the classical level. In non-Abelian theories, the BRST symmetry must be taken in favor of gauge symmetry because of the presence of the Faddeev-Popov ghosts. Thus, (56) can also be considered at classical level. Therefore, since the BRST symmetry is postulated to be the fundamental symmetry of Yang-Mills theories (it generalizes the gauge symmetry), it should be used to define the physical states of the quantum theory. For that, it is used the nilpotent BRST charge Qbrst which is the generator of BRST symmetry. From (Exercise19 1.36), Z brst 3 brst Q = d j0 , (71)

we can deﬁne the physical quantum states as BRST invariants,

Qbrst|ψi = 0 , (72)

where |ψi is required to have vanishing ghost number [64, 65] and to respect all symmetries displayed by SYMFP . However, following the cohomological prescription (60) and (61), |ψi will have a non-trivial piece |ψiphys and a trivial redundancy |ψinphys, namely

brst brst Q |ψiphys = 0 |ψiphys 6= Q |somethingi ,

brst brst Q |ψinphys = 0 |ψinphys = Q |somethingi . (73)

It is clear that |ψinphys has a vanishing norm (Exercise 1.38). Thus, the actual physical states are those belonging to the non-trivial BRST cohomology in the space of vanishing ghost number states. Unitarity of the S-matrix ensures the probability conservation in physical processes [6, 64, 65]. BRST symmetry plays a fundamental role in that issue, guaranteeing that states with negative norm do not appear as asymptotic states, i.e. they are not observables. The demonstration relies on the deﬁnition of physical states, Ward identities20 and commutation relations involving Qbrst.

D. Asymptotic freedom and conﬁnement

The renormalization of the coupling parameter κ leads to the concept of asymptotic freedom [6, 66–68], which means that the coupling parameter is higher as lower is the energy scale. In fact, at ﬁrst order in perturbation theory, it is found that [66, 67] DRAFT 16π2 κ2 = , (74) 11N µ2 ln 2 3 Λqcd

18 All symmetries must be imposed as well. 19 Noether‘s theorem [6] establishes that X ∂L A jµ = δΦ . ∂ (∂ ΦA) allfields µ

20 Ward-Takahashi identities are functional equations describing the symmetries of an action. At quantum level, their role is to provide recurrence relations between Green funtions. 18

where N is the Casimir of the gauge group (This is the dimension of the vector space of the fundamental representation), µ is the energy scale and Λqcd is the renormalization group cutoff [68]. The renormalization group cutoff is a phenomelogical parameter which is associated with the pion mass. From (74) it is clear that as µ → ∞, κ → 0. Thus, as higher the energy, more weakly coupled to each other are the gauge fields and perturbation theory is very consistent. At this level, the gauge fields are almost free. On the other hand, as µ → Λqcd, µ → ∞. In this case, the gauge fields become strongly coupled and perturbation theory does not hold anymore, i.e. the theory become highly non-perturbative. The point µ = Λqcd is known as the Landau pole and indicates a phase transition of the theory which is recognized as the hadronization phenomenon [69]. Below µ = Λqcd the Yang-Mills action must be replaced by a suitable theory where the excitations are hadrons and glueballs21. The concept of confinement emerges from the behavior (74) together with the global G symmetry22. The essence of the global symmetry is to forbid any algebra-valued to be a physical state, Qbrst|ψA >6= 0. Only states carrying no group indexes would allowed at the physical sector. This property should hold at all energy scales23. Even at high energies, where the coupling is very small and the gauge excitations barely interact among each other, is also impossible to observe them due to its algebra-valued character. At this regime, considering also quarks, the theory describes the so-called quark-gluon plasma, a phase in which quarks and gluons are virtually free, even though it is impossible to isolate a single quark or gluon. By decreasing the energy, the coupling increases and the interactions between gluons (and quarks) become stronger and stronger til the point of hadronization at the Landau pole. Then, hadrons and glueballs, which are experimentally observables, are in fact states with no gauge charge.

III. GRIBOV AMBIGUITIES

In this section, we will introduce the problem pointed out by V. N. Gribov in his seminal paper [14]. The essence of the problem lies on the fact that the Faddeev-Popov method, described in Sect. II, is not sufficient to fix the gauge redundancy completely. It means that, after applying the Faddeev-Popov gauge fixing procedure, a residual symmetry remains: there still are configurations obeying the same gauge fixing and related through gauge transofrmations. Such spourious configurations are called Gribov copies and, although present in a wide class of gauges, their elimination is not fully understood. The first (partial) attempt to deal with such copies was realized by Gribov himself, and it was worked out in more generality by Zwanziger later, [70]. A local and renormalizable action, free from infinitesimal Gribov copies (copies that are related to another configuration through an infinitesimal gauge transformation) was established, the so-called Gribov-Zwanziger action. This action was firstly constructed in the Landau gauge and it has a very interesting feature: the term which is responsible to guarantee the elimination of copies breaks the BRST symmetry (58a-58d) in a soft manner24. Further non-perturbative effects as the presence of condensates can be taken into account in this scenario giving rise to the Refined Gribov-Zwanziger action, [71], which provides gluon and ghost propagators in very good agreement with lattice results. In this section, we provide an introduction to these developments in a very concise form. For further details, we refer to more complete treatments which are listed along the text.

A. Statement of the problem

As discussed in Sect. II, the quantization of gauge theories requires the imposition of a condition over the gauge fields. Although the Faddeev-Popov framework (or BRST quantization) provides a very efficient way to fix a gauge in perturbation theory, it fails at the non-perturbative level. The reason is the following: The application of the FP procedure tacitly assumes that the gauge fixing condition has a unique solution for each gauge orbit (44). This assumption does not hold whenDRAFT we enter the non-perturbative sector and spurious configurations are not eliminated from the path integral quantization. As a matter of illustration, we consider pure Yang-Mills action in four-dimensional Euclidean space with SU(N) gauge group25 and we impose the Landau gauge condition over the gauge fields, which

21 This is actually an open problem in theoretical physics. 22 In QCD this symmetry is the color symmetry associated with the SU(3) global group. In electroweak interactions the SU(2) × U(1) local and global symmetries are broken to the U(1) group due to spontaneous symmetry breaking. Hence, confinement is not a feature of electroweak interactions. 23 There are no experimental evidence of confinement violation whatsoever. 24 A breaking is called soft if it is an operator with dimension lower than spacetime dimension. Otherwise, it is a hard breaking. 25 The group SU(N) is chosen to fit with the existing literature. The generalization to a generic semi-simple Lie group G can be straightforwardly done. 19

is the case of α = 0 in (57)

A ∂µAµ = 0 . (75) An equivalent gauge configuration A0 is obtained from A through a gauge transformation, given by eq.(33). If we assume that A satisfies the condition (75), the gauge fixing is consistent if eq. 75 is not satisfied by a configuration A0, which belongs to the same gauge orbit of A. To simplify the question, let us consider an infinitesimal gauge transformation26 with gauge parameter ξ as in (34). Therefore,

0A A AB B Aµ = Aµ + Dµ ξ , (76) satisﬁes the Landau gauge condition if

0A A AB B AB B ∂µAµ = ∂µAµ + ∂µDµ ξ ⇒ ∂µDµ ξ = 0 . (77)

ab It implies that a Gribov copy exists if the operator −∂µDµ , which is the Faddeev-Popov operator in the Landau gauge, has zero-modes. Although this is not a trivial task, we can start our analysis as an eigenvalue problem, i.e.,

AB AB B AB ABC C B A M = −∂µDµ ξ = −∂µ(δ ∂µ − κf Aµ )ξ = (A)ξ , (78) where is the correspondent eigenvalue associated to the gauge configuration A. This equation can be faced as a Schrödinger-like equation, where A plays the role of the potential. An extra item to this analysis is that the operator −∂D, with the condition ∂A = 0, is Hermitian (Exercise 1.39) and, therefore, has a real spectrum. From eq.(78), it is possible to realize that for small values of A, just positive are allowed due to the positivity of −∂2 in Euclidean space. However, for a suitable value of A, as it becomes larger, the operator −∂D admits a zero-mode and, then, a negative eigenvalue. This allows the following picture: Around A ≈ 0, i.e., at the perturbative level, the FP operator does not have any zero-modes and the usual gauge-fixing procedure is safe. Nevertheless, as far as we get from the perturbative sector, the FP operator reaches zero-modes and the gauge fixing is ill-defined. From this qualitative analysis, it is possible to see that the appearance of Gribov copies is a genuine non-perturbative effect. This poses a non-trivial issue: The Faddeev-Popov gauge fixing procedure does not take into account the existence of Gribov copies at the non-perturbative regime. Therefore, the path integral quantization is not free of spurious configurations, which must be eliminated anyway. At the present moment, we have discussed just the existence of infinitesimal copies. However, dealing with such class of copies is already highly non-trivial and very few is known about the general problem with finite gauge transformations. Before ending this general discussion, we must highlight a very important remark: In the way we presented the existence of copies, might seem to the reader that this is a peculiarity of the Landau gauge. However, it was shown by I. M. Singer [15] that the existence of such configurations is a very general feature, and, in particular, all Lorentz covariant gauge conditions suffer from this problem. This enforces the reasoning that this problem is much more delicate than a simple technical issue. For a more formal justification, see appendix B.

B. Gribov’s solution

In his seminal work, [14], Gribov proposed the first attempt to eliminate gauge copies. The idea, at a first glance, might seem very naive and many drawbacks can be pointed out. However, many essential properties which were not proved by Gribov in his paper were proved later on, providing a firm ground to his proposal. Therefore, although it is not the final answer we haveDRAFT so far, the solution proposed by Gribov is simple and elegant enough to be presented before the more recent attempts, namely, the Gribov-Zwanziger framework. We reenforce, however, that we will just present the main ideas and we refer to appropriate references for more details. In particular, our exposition is based on [45, 72, 73]. Furthermore, unless the contrary is said, we will restrict ourselves to the Landau gauge. As mentioned before, the FP operator is Hermitian in the Landau gauge. Hence, its spectrum is real and, therefore, it is possible to define positivitness and a hierarchy ordering for the eigenvalues . In particular, we argued that there is a region where this operator is positive. So, inside this region, the operator does not contain zero-modes and no infinitesimal Gribov copies are present27. Gribov’s proposal was simply to restrict the domain of integration of the

26 We will only consider infinitesimal gauge transformations and, hence, only infinitesimal Gribov copies. 27 This does not prevent the appearance of finite Gribov copies. 20

path integral to this region. With this, we eliminate a large set of copies, namely, the inﬁnitesimal ones. But, before implementing such idea, we must highlight some important properties of this region. AB AB The region Ω where M ≡ −∂µDµ is strictly positive is known as the ﬁrst Gribov region, or simply Gribov region. Therefore,

A A AB Ω = Aµ , ∂µAµ = 0, M > 0 . (79) This region enjoys some very important properties. In particular, • it is bounded; • it is convex; • all gauge orbits cross Ω;

A • the configuration Aµ = 0 belongs to Ω. The first two properties ensures that configurations very close to the so-called Gribov horizon, the boundary which contains the first zero-modes of the FP operator, have a finite distance to the origin in configuration space. The third property is fundamental: It guarantees that the restriction is consistent, i.e., the restriction of the path integral to Ω does not eliminate any physical configuration, which is essential. The proof of such properties can be found in [74, 75]. Finally, the fourth property ensures that the perturbative region is inside the Gribov region and its proof is trivial. These properties make the restriction a consistent proposal. A very important comment is the following: The restriction to Ω, although satisfies those very important properties does not eliminate all Gribov copies. As mentioned before, Gribov copies related by finite gauge transformations may belong to Ω, [76]. The region which is truly free from copies is known as the fundamental modular region (FMR). Nevertheless, we do not have a systematic way to implement the restriction of the path integral to this region so far. So, the best we can do is to restrict to Ω. Formally, for the restriction to Ω, we write for the path integral the following modified measure, Z A −SYMFP Z = N DADc¯Dc δ(∂µAµ )e V(Ω) , (80)

where N stands for a possible normalization factor and V(Ω) is the factor responsible for the restriction to Ω. Essentially, the factor V(Ω) is a Heaviside function limiting the functional integral up to the Gribov horizon. To characterize V(Ω) we will ﬁrst notice that, in the Landau gauge, the operator which generates inﬁnitesimal Gribov copies, namely, the FP operator, is intimately related to the ghost sector of the theory. In particular,

Z AB A B A −1 −SYM hc¯ (x)c (y)i = DAδ(∂µAµ )V(Ω)det(M) M (x, y)e (81)

where we have integrated over the ghost ﬁelds. Thus, performing the one-loop computation of the ghost two-point function within the usual Faddeev-Popov theory (55), we obtain the following result [14, 45] (Exercise 1.40)

A B AB AB 1 1 hc¯ (−p)c (p)i = δ G(p) = δ 2 9 , (82) p 11κ2N Λ2 44 1 − 48π2 ln p2

2 with Λ being the ultraviolet cutoﬀ. The function G(p) has two singularities: p2 = 0 and p2 = Λ2 − 1 48π . For DRAFTκ2 11N 2 Λ2 large momenta, G(p) ≈ 1/(p ln p2 ) which is the perturbative regime and we are inside the Gribov region as we 2 2 1 48π2 discussed before. Also, it is possible to note that for p < Λ exp − κ2 11N , G becomes complex, which means that we are outside the Gribov region28 Ω. On the other hand, the pole p2 = 0 corresponds to the fact we have reached the Gribov horizon, since p2 is a positive function. This implies that the restriction to the Gribov region avoids singularities diﬀerent from p2 = 0. In fact, this is the away we can impose the restriction of the path integral, i.e., let us impose that no poles but p2 = 0 are allowed in the ghost two-point function. This is the so-called no-pole condition, which was the original proposal by Gribov and allows the characterization of V(Ω). We must highlight that

28 A similar discussion will take place at Sec. X C 21 this proposal was worked out up to second order in perturbation theory originally. The generalization to all orders will be discussed in the next section. For the time being, we compute the connected ghost two-point function up to second order using the usual FP path integral (55). We will omit the details, but they can be found in [45, 72, 73]. The idea of using the (55) is because we want to find a condition for the ghost propagator in the usual theory that identifies the Gribov horizon, i.e. we want to identify the FP zero modes with the usual ghost propagator. Then, this identification will be used to define V(Ω). Moreover, A must be considered as external fields in such a way that V(Ω) will be a function of A itself. The G function is then

1 1 1 Nκ2 Z d4q (p − q) p G(p, A) = + AA(−q)AA(q) µ ν , (83) p2 V p4 N 2 − 1 (2π)4 µ ν (p − q)2 and V is the spacetime volume. We rewrite (83) as 1 G(p, A) = (1 + σ(p, A)) , (84) p2 with 1 1 Nκ2 Z d4q (p − q) p σ(p, A) = AA(−q)AA(q) µ ν . (85) V p2 N 2 − 1 (2π)4 µ ν (p − q)2

Then, (84) is rewritten in Born approximation as 1 1 G(p, A) ≈ , (86) k2 1 − σ(p, A) and the no-pole condition can be simply implemented by σ(p, A) < 1. Although we will not give the proof [72, 73], σ(p, A) decreases as p2 increases. So, the no-pole condition can be rephrased as σ(0,A) < 1 where [45] (Exercise 1.41)

1 1 Nκ2 Z d4q 1 σ(0,A) = AA(−q)AA(q) . (87) V 4 N 2 − 1 (2π)4 κ κ q2

Hence, an explicit expression for V(Ω), in terms of σ, can be written:

V(Ω) = θ(1 − σ(0,A)) . (88)

Even though we have determined an expression for V(Ω), in its current form expression (88) is not suitable for practical purposes. Nevertheless, we can incorporate V(Ω) into the classical action using an integral representation for the Heaviside function, i.e.

Z +i∞+ dβ V(Ω) = θ(1 − σ(0,A)) = eβ(1−σ(0,A)) . (89) −i∞+ 2πiβ Thus,

Z +i∞+ dβ Z A −SYMFP β−ln β−βσ(0,A)) Z = N DADc¯Dc δ(∂µAµ )e e , (90) DRAFT−i∞+ 2πi The trick here is to set β to its minimizing value (the saddle point method [6, 14]), i.e. δZ/δβ = 0, providing the so called gap equation (Exercise 1.42),

3 Z d4q 1 1 = Nκ2 , (91) 4 (2π)4 (q4 + γ4) where the so called Gribov parameter γ is deﬁned as β N 1 γ4 = κ2 0 , (92) N 2 − 1 2V 22

and β0 is the solution of the saddle point. Two observations are in order: First, the actual value of γ is the one determined by (91). Second, in (92), the rate β0/V is assumed to be ﬁnite as V 7−→ ∞. Hence, after a suitable normalization, we ﬁnally have29 (Exercise 1.43)

Z R 4 2 4 4 A −SYMFP d x4(N −1)γ −γ σ(0,A)) Z = DADc¯Dc δ(∂µAµ )e e , (93)

where the thermodynamic limit V 7−→ ∞ was taken. the term γ4σ is clearly momentum-dependent and, thus, a non-local term while the V γ4 is a vacuum term30. However, since the new term is quadratic in the gauge ﬁeld, it aﬀects the tree level propagator which is now given by (Exercise 1.44)

p2 hAA(p)AB(−p)i = δAB T . (94) µ ν p4 + γ4 µν

where, as we have established, the Landau gauge is assumed (α = 0). Some comments are (focusing on QCD), however, important: The restriction of the path integral to the Gribov region brings deep modifications to the gluon propagator. In particular, it is suppressed in the infrared region and displays two complex poles. This makes the interpretation of gluons as part of the physical spectrum impossible, which can be faced as a manifestation of confinement. It is a quite remarkable feature that the partial solution of a technical problem provides such physical behavior. The computation of the ghost propagator also has novelties, but we will omit it here because it is not relevant to our present purposes: its behavior is enhanced in the infrared regime at one-loop order. Its behavior is ∼ 1/k4. Finally, the integral in the gap equation (91) is divergent. Thus, to solve it, renormalization is required. A local and renormalizable action which effectively implements the restriction to the Gribov region Ω is then needed. This is achieved by the so-called Gribov-Zwanziger action, which will be exposed in the next section.

C. The Gribov-Zwanziger action and BRST soft breaking

In the last section, we discussed the solution proposed by Gribov for the elimination of infinitesimal gauge copies. Nevertheless, this solution was implemented up to second order in perturbation theory and we might wonder if an all order proposal could be developed. The answer is affirmative and was first worked out by D. Zwanziger in [70]. Inhere, again,we will not expose all the reasoning followed by Zwanziger, but just mention some important features. Instead of dealing with an all order no-pole condition, Zwanziger studied the spectrum of the Faddeev-Popov operator in the Landau gauge, which is actually the same thing [77, 78]. In particular, the restriction to the Gribov region is implemented through the imposition that the functional integral should be performed in the region of configuration space where the smallest eigenvalue of M = MAA/(N 2 − 1), i.e. (A), is non-negative. We must notice that the eigenvalue is a function of the gauge field. Zwanziger argued that this condition is well implemented, in the thermodynamic limit, by requiring the trace of M is non-negative. The condition found by Zwanziger is given by

4V (N 2 − 1) − H(A) ≥ 0 , (95)

with Z 2 4 4 ABC B −1AD DEC E H(A)DRAFT = g d xd y f Aµ (x) M (x, y)f Aµ (y) , (96) being the so-called horizon function. Also, Zwanziger was able to effectively implement (95) by a modification in the classical action. The result for the modified path integral which takes into account the restriction to the Gribov region Ω is given by

Z 4 R 4 2 4 A −[SYM+γ H(A)− d x4(N −1)γ ] Z = DAδ(∂µAµ )det(M) e , (97)

29 An important step here is that the term ln β can be neglected at the thermodynamic limit because the Heaviside function, at this limit, converges to the Dirac delta function. See [45]. 30 This term is negligible at classical level. At quantum level, however, it is essential to obtain the gap equation (91) 23

where γ - the Gribov parameter - is not free, but determined in a self-consistent way through the so-called horizon condition,

2 hH(A)iGZ = 4V (N − 1) , (98)

with the expectation value being evaluated with the path integral modiﬁed measure. It is possible to show that eq. (98) reduces to the gap equation given by (91) at lowest order, i.e., in the Gribov approximation. From (97), it is possible to read the modiﬁed classical action, known as the Gribov-Zwanziger action, which implements the restriction to the Gribov region Ω (to all order no-pole condition), i.e.

1 Z Z S = d4x F A F A + d4x (¯cA∂ DABcB + bA∂ AA) + γ4H(A) − 4γ4V (N 2 − 1) , (99) GZ 4 µν µν µ µ µ µ

which, due to the presence of H(A), is a non-local action. Nevertheless, it can be casted in a local form through the AB AB introduction of a set of auxiliary fields, namely, a pair of commuting (ϕ ¯µ , ϕµ ) and a pair of anti-commuting fields AB AB 31 (¯ωµ , ωµ ). With this new set of fields, the Gribov-Zwanziger action is expressed as (Exercise 1.45) 1 Z Z Z S = d4x F A F A + d4x (¯cA∂ DABcB + bA∂ AA) − d4x ϕ¯AC MABϕBC − ω¯AC MABωBC GZ 4 µν µν µ µ µ µ µ µ µ µ Z ABC AD BE E CD 2 4 ABC A BC 4 2 − gf ∂µω¯ν Dµ c Eϕν − gγ d x f Aµ (ϕ ¯ + ϕ)µ − 4V γ (N − 1) . (100)

By integrating over the auxiliary fields, we regain expression (99) (Exercise 1.46). The auxiliary fields, also known as Zwanziger fields, form a BRST quartet (two BRST doublets),

AB AB AB sϕµ = ωµ , sωµ = 0 , AB AB AB sω¯µ =ϕ ¯µ , sϕ¯µ = 0 . (101) As a first test, it can be shown that the action (101) reduces to the usual gauge fixed Yang-Mills action in the limit γ → 0 (Exercise 1.47). The Gribov-Zwanziger action has a very peculiar property: It breaks explicitly the BRST symmetry (Exercise 1.48). Albeit explicit, the breaking is soft because it is proportional to the Gribov parameter. It means that, in the ultra-violet regime, the Gribov parameter becomes negligible and the BRST symmetry is recovered (together with all well established properties of perturbative quantum theory). Hence, the explicit breaking of the BRST symmetry is a direct consequence of the restriction of the domain of the path integral to the Gribov region. Since the restriction to the Gribov region allows the interpretation of confinement through the modification of the gluon propagator, we immediately realize that there is a deep connection between confinement and BRST soft breaking. It is important to mention that the Gribov-Zwanziger action is renormalizable to all orders in perturbation theory. The proof is beyond the scope of these notes, but we refer to [70, 72, 73] for detail.

D. The Reﬁned Gribov-Zwanziger action

Besides the restriction to the Gribov region, which is a non-perturbative effect in the sense that in the perturbative regime Gribov copies play no role, other non-perturbative features might be taken into account. Doing so, a suitable modification of the Gribov-ZwanzigerDRAFT framework is achieved and its final form fits very well with the most recent lattice data [79–81]. The action which leads to those results is the so-called Refined Gribov-Zwanziger action [71]. a a Essentially, the Gribov-Zwanziger action displays non-vanishing values for condensates, in particular, for hAµAµi and ab ab ab ab hϕ¯µ ϕµ −ω¯µ ωµ i. The non-vanishing values at one-loop are proportional to the Gribov paramete [71, 82]. Therefore, the restriction to the Gribov region brings more non-trivial features at the non-perturbative level. The proposal is to take into account these instabilities of the Gribov-Zwanziger action from the very beginning by introducing those

31 The demonstration is performed in the full path integral (97). 24 condensates. It implies a modiﬁcation to the Gribov-Zwanziger action as follows

1 Z Z Z S = − d4x F A F A + d4x (¯cA∂ DABcB + bA∂ AA) − d4x ϕ¯AC MABϕBC − ω¯AC MABωBC , RGZ 4 µν µν µ µ µ µ µ µ µ µ Z m2 Z − κf ABC ∂ ω¯ADDBEcEϕCD − κγ2 d4x f ABC AA (ϕ ¯ + ϕ)BC + d4x AAAA+ , µ ν µ ν µ µ 2 µ µ Z Z 2 4 AB AB AB AB 4 4 2 −M d x (ϕ ¯µ ϕµ − ω¯µ ωµ ) − d x4γ (N − 1) ,

(102a)

which is the Reﬁned Gribov-Zwanziger (RGZ) action. This action, as the Gribov-Zwanziger action, breaks the BRST symmetry in a soft manner. The gluon propagator computed from the RGZ action is (Exercise 1.49)

p2 + M 2 hAA(p)AB(−p)i = δAB T , (103) µ ν (p2 + m2)(p2 + M 2) + 2g2Nγ4 µν which, as previously discussed, does not vanish at the origin p → 0. An important remark is that for m2 = 0, it still goes to a non-vanishing value, due to the presence of the condensates of auxiliary fields which naturally emerge from the localization of the horizon function. The ghost propagator is not enhanced anymore in this framework, i.e., as G(p2) ≈ 1/p2. Some considerations are in order: The Refined Gribov-Zwanziger scenario has passed through several non-trivial tests in the last few years. In particular, besides the very good agreement with lattice results for the gluon and ghost propagators, [79–81], it also provides a good qualitative agreement with glueball spectrum [83, 84], the correct sign to the Casimir energy in the MIT bag model [85] and a coupling to scalar and fermionic matter that also fits lattice simulations [86]. As presented here, the (Refined) Gribov-Zwanziger action might seem to be restricted to the Landau gauge. Indeed, its construction is based on very peculiarities of Landau gauge and it was the first gauge condition which was worked in details. However, we must mention that we already have at our disposal the extension (or, at least, the partial extension) to other gauges, e.g., the maximal abelian, linear covariant and Landau - maximal Abelian interpolating gauges, see [49, 82, 87, 88]. Also, many aspects of the soft BRST breaking are being studied not only from the analytical point of view, but also from lattice simulations [50, 89]. This scenario seems to be a very suitable way to study non-trivial effects from the non-perturbative regime of Yang-Mills theories in an analytical fashion. The issue of the BRST symmetry breaking is highly non-trivial and a matter of current research, its final status still lacks. It is worth mention, however, that a non-perturbative version of the BRST operator has been found in [50] where its action on the RGZ fields are highly non-local. Nevertheless, this new BRST operator is nilpotent and reduces to the ordinary BRST operatorDRAFT at the perturbative level. 25

PART II: GRAVITY AS A GAUGE THEORY

In this part we focus on the construction of gravity in the first order formalism. Some definitions will be made in order to clarify the main differences between first and second order formalisms, also known as vierbein and metric formalisms, respectively. The main references on general relativity in the first and second order formalisms are [10, 12, 90–98].

IV. EQUIVALENCE PRINCIPLE AND GEOMETRODYNAMICS

Following Einstein’s ideas on the construction of general relativity (GR) [99], gravitational effects can be locally eliminated for an object in free falling. By “locally”, it is meant a small spacetime region where gravity does not vary significantly. A famous example is the free falling elevator experiment. This property is the essence of the modern interpretation of the equivalence principle. Let us quickly discuss two equivalence principle enunciates: • Inertial and gravitational mass are equivalent. This enunciate is known as Weak Equivalence Principle (WEP), which states that the inertial mass of a body is equal to its the gravitational mass32. The consequence is that all test particles locally experiment the same acceleration under gravity. This principle is known since Galileo Galilei. • For local spacetime regions, special relativity laws hold. This version is known as the Strong Equivalence Principle (SEP). It explicitly states that gravitational fields cannot be detected by local experiments. However, for sufficient large spacetime regions, gravitational field inhomogeneities cannot be neglected since tidal forces take place. The WEP states that gravity effects can be locally eliminated by a suitable non-inertial reference frame. The SEP aditionally states that, locally, spacetime is a Minkowski flat spacetime. Hence, it is intuitive to try to describe gravitational effects as a dynamical geometric formulation of spacetime which, locally, is reduced to spetial relativity. In this formulation, spacetime is deformed by matter while matter experiments a deformed spacetime due to the presence of other matter distributions. Thus, the spacetime geometry is dynamically generated by the presence of matter. This type of theories are called geometrodynamical theories or, simply, geometrodynamics. These ideas lead Einstein to formulate a special set of equations describing spacetime dynamics due to matter distributions, the so called Einstein equations, 1 R − R − 2Λ2 g = kT . (104) µν 2 µν µν

33 µν where Rµν is the Ricci tensor (related to spacetime curvature), R = Rµν g is the curvature scalar, gµν is the metric tensor, Λ is the cosmological constant, k is related to Newton’s constant (it can be determined by taking the Newtonian limit of (104)), and Tµν is the energy-momentum tensor of the matter distribution. The l.h.s of (104) is purely geometric34 while the r.h.s is related to pure physical properties of the matter distribution. Essentially, given a matter distribution, (104) determines the geometric properties of spacetime. In this scenario, the metric tensor is the fundamental ﬁeld. 2 µ It is worth mention that the trace of (104) leads to (Exercise 2.1) R = 4Λ − kT where T = T µ. Thus, even in the absence of matter distributions, a curvature can be generated by the cosmological constant, R = 4Λ2, which is known to exist with an extremely small observational value (∼ 10−44peV 2). In fact, the cosmological constant in (104) is recognized as the dark energy componentDRAFT of the Universe.

A. General relativity in the metric formalism

The metric formalism of GR, or second order formalism, is based on the principle that gµν is the only fundamental ﬁeld of gravity. This is actually the original idea adopted by A. Einstein in the construction of (104). All other geometric quantities are obtained from the metric tensor.

32 The gravitational mass is the equivalent of the electric charge in Coulomb force, which cannot be associated to inertia. 33 We will dissect the geometric quantities of a diﬀerential manifold in the next section. 34 Perhaps Λ can have a diﬀerent origin other than geometry. 26

Let us start our definitions with the metric tensor gµν , which is the fundamental field of GR. Inherent to this geometrical object are all information about lengths, angles, areas, and volumes on a differentiable manifold M 4. It is responsible to describe and to characterize the spacetime metric properties. For simplicity we will omit the position µ µ µ dependence of most quantities. With gµν , the distance between two near points, x and x + dx , is easily computed,

2 µ ν ds = gµν dx dx . (105)

Clearly, the metric tensor is a symmetric second order tensor, i.e., gµν = gνµ. An inverse metric tensor is assumed, gµν , through

λµ µ gνλg = δν , (106)

The SEP states that, locally, gµν ≈ ηµν , where ηµν is the Minkowski metric tensor. In the absence of a gravitational field, ηµν becomes global in the spacetime. The first object we can construct from the metric tensor is the affine connection, which characterizes the parallelism properties of spacetime. The affine connection rises from the study of parallel transport in M 4 of a generic vector field. It can also be obtained from the imposition that a covariant derivative of a vector must transform as a second rank tensor. Hence (Exercise 2.2), lit can be defined the action of the covariant derivative as

∇ T µ1µ2···µk = ∂ T µ1µ2···µk β ν1ν2···ν` β ν1ν2···ν` + Γµ1 T λµ2···µk + Γµ2 T µ1λ···µk + ··· βλ ν1ν2···ν` βλ ν1ν2···ν` − Γλ T µ1µ2···µk − Γλ T µ1µ2···µk − · · · . (107) βν1 λν2···ν` βν2 ν1λ···ν`

µ where the aﬃne connection Γβν is deﬁned under two strong requirements:

• Metric compatible: The metric tensor is covarantly constant, ∇βgµν = 0.

α α α α α • Torsion free: T µν = Γµν − Γνµ = 0, which implies Γµν = Γνµ. In general, the affine connection is independent of the metric. In the metric formalism, however, it is only a function of gµν . These requirements state that non-metricity and torsion vanishes at all spacetime points. Thus, parallel transport preserves lengths and angles and also forbids twists in M 4. Under these conditions, the affine connection is reduced to the so called Christoffel symbols (Exercise 2.3) 1 Γα = gαρ (−∂ g + ∂ g + ∂ g ) . (108) µν 2 ρ µν µ ρν ν µρ This is the connection adopted by Einstein in the original formulation of the GR. The Riemann-Christoffel tensor, or simply Riemann tensor, corresponds to the spacetime curvature. This tensor is defined through the commutation of covariant derivatives over a vector, i.e. (Exercise 2.4)

α α β [∇µ, ∇ν ] V = R βµν V , (109) where

α α α α λ α λ R βµν = ∂µΓνβ − ∂ν Γµβ + ΓµλΓνβ − ΓνλΓµβ . (110) The main properties of the RiemannDRAFT tensor in the metric formalism are

• It is antisymmetric in the ﬁrst and last pair of indexes: Rαβµν = −Rαβνµ and Rαβµν = −Rβανµ.

• It is symmetric by interchanging the first and second pairs of indexes: Rαβµν = Rµναβ. • Because the covariant derivative defines parallel transport, the curvature measures the failure of the parallelo- gram law. In other words, parallel transport of a vector along two different directions will differ by interchanging the order of the directions. The Riemann tensor can be used to calculate two important objects present at Einstein equations (104). These αβ α µν quantities are the Ricci tensor and the curvature scalar, namely, Rµν = g Rβµαν = R µαν and R = g Rµν , respectively. We notice however that, ultimately, the Riemann tensor depends exclusively on the metric tensor. 27

It is possible to obtain the Einstein equations (104) from a variational principle. This is achieved by deﬁning the Einstein-Hilbert action,

Z √ 1 S = d4x g (R − 2Λ) + T µ , (111) EH 16πG µ

where g = | det gµν | and G is the universal gravitational constant (Newton’s constant). It is a straightforward computation to obtain (104) from (111) (Exercise 2.5). It is worth mention that (104) encondes Newtonian gravity. In fact, by taking the weak ﬁeld limit we recover the Newton’s theory of gravitation where g00 is related to the Newtonian potential. In this way, gµν would play the role α of a relativistic potential while Γµν would be the associated gravitational force. This interpretation is also evident from the geodesic equation

d2xµ dxα dxβ = −Γµ . (112) ds2 αβ ds ds where s is the line element of the geodesic and xµ the position of the particle along the geodesic curve. This interpretation is consistent. However, an alternative interpretation can be deﬁned in order to make gravity closer to gauge theories. This is the task of the next section.

B. First order formalism of gravity

The most direct generalization of the metric formalism is due to A. Palatini and E. Cartan [100–102], which considers µ gµν and Γαβ as independent dynamical variables, i.e. there are two independent fields for gravity. In this way, new degrees of freedom rises. This formalism is called Palatini formalism and brings torsion into the game. The effect is a new set of field equations δSEH /δΓ = 0. Nevertheless, no torsion shows up unless the energy-momentum tensor of the matter distribution carries spin-density components (Exercise 2.6). Nevertheless, non-metricity remains a vanishing quantity. Theories which nonmetricity is included are usually called metric-affine gravity [32]. The so called first order formalism [94] is equivalent to the Palatini formalism, but it is written in terms of different 35 a ab 4 variables, namely, the vierbein eµ(x) and the spin-connection ωµ (x) where small Latin indexes refer to the M tangent space coordinates. In particular, for economical purposes, we will now employ the differential form notation36 a a and use the vierbein one-form e (x) and the spin-connection one-form ω b(x). Let us quickly construct these objects.

1. Vierbein

Let spacetime be a smooth37 four-dimensional manifold M 4. At each point x ∈ M 4 we define the tangent space 4 Tx(M ) with flat metric given by the Minkowski metric, η = diag(−1, 1, 1, 1). Due to the SEP, tangent spaces of this form are equivalent to free falling local reference frames. In this way, we start a description of spacetime in full 4 4 explicit accordance with the SEP. By definition of tangent spaces, there is an isomorphism between TxM and M , which is a linear map relating coordinates y of the tangent space and coordinates x of M 4. This map is local, is called vierbein, and is given by

a a µ DRAFTdy = eµdx , (113) where, due to dim M = dim T (M), we also have a, b, c . . . ∈ {0, 1, 2, 3}. Sometimes, the tangent indexes are called frame indexes while spacetime indexes are called world indexes. Because a distance measure must be invariant under this isomorphism, we must have the following relation (Exercise 2.7)

a b gµν ≡ ηabeµeν . (114)

35 a In d dimensions eµ is called vielbein and for specific dimensions we have zweibein, dreibein, vierbein... for 2, 3, 4... dimensions. 36 A brief exposition about differential forms can be found in Appendix (A). For detailed texts we refer to []. In our notation, all wedge products are implicit understood, and, hence, will be omitted in most equations. 37 By smoth, it is meant a C∞ class manifold, i.e. an infinitely differentiable manifold. 28

4 a Thus, a metric gµν on M is induced by the vierbein eµ and the tangent space metric ηab. The inverse vierbein is deﬁned through the set of relations:

a µ a eµeb = δb , a ν ν eµea = δµ . (115) The flat Minkowski metric is characterized by the isometries of the tangent space. These isometries are described by the Lorentz group SO(1, 3), but the local version of the Lorentz group, i.e. Lorentz transformations which depend on a the spacetime position. Thus, eµ is not unique because there are infinity equivalent virbeins related by the elements of the local group SO(1, 3). In other words, locally, there are infinity equivalent tangent spaces related SO(1, 3). Under local Lorentz transformations, the vierbein behaves as

a a b eµ 7−→ L beµ , (116)

where L = L(x) ∈ SO(1, 3). It is worth mention that the vierbein has n2 = 16 independent components whereas the metric has n(n + 1)/2 = 10 independent ones. The diﬀerence is exactly the n(n − 1)/2 = 6 independent n = 4- dimensional local rotations characterized by the Lorentz group. ∗ 4 Ultimately, we deﬁne the vierbein one-form, which refers to a soldering form in the cotangent space Tx (M ), which 4 is dual to Tx(M ), by

a a µ e = eµdx , (117)

µ ∗ 4 where dx is a local basis of Tx (M ). Clearly, (118) turns to

a a b e 7−→ L be , (118)

which, for inﬁnitesimal transformations reduce to (Exercise 2.8)

a a a b e 7−→ e + ξ be , (119)

a a a where ξ b is the infinitesimal group parameter. Obviously, due to its algebra-valued character, ξ b = −ξb . An important observation is that the action of the Lorentz group on the vierbein leaves unaltered the spacetime coordinates xµ but affects the vierbein. Thus, the local Lorentz group can be visualized as a gauge group for gravity. Under this assumption, through (114), the metric tensor is a gauge invariant (Exercise 2.9). On the other hand, (118) does not describe the gauge transformation of a connection but of a matter field! Thus, it remains to determine what quantity is the connection in this scenario.

2. Spin-connection

Once again, the concept of parallel transport concept can be used to deﬁne the connection. Inhere, we are interested in tangent and cotangent spaces. Let x and x + dx two near points on the spacetime manifold M 4. The parallel 4 4 transport of Lorentz tensors between tangent spaces on these points, i.e. Tx(M ) 7−→ Tx+dx(M ), is realized through a 38 the spin-connection ω bµ(x), sometimes called Lorentz connection . However, we are interested in the diﬀerential forms formalism. Thus, the parallel transport between two near cotangent spaces T ∗(M 4) 7−→ T ∗ (M 4) is realized DRAFTx x+dx by the Lorentz algebra-valued spin-connection one-form

a µ b ω = ω bµdx Ta . (120)

b b a a Hence, due to Ta = −T a, we have ω b = −ωb . The corresponding exterior covariant derivative is deﬁned in the usual way,

D = d + ω , (121)

38 The word “spin” rather than “Lorentz” was introduced because the spin-connection is responsible for the coupling with fermions. 29

acting on tangent space indexes. Moreover, the Minkowski local metric ηab and the Levi-Civita tensor abcd are 39 covariant constant quantities (Exercise 2.10) , i.e. Dηab = dηab = 0 and Dabcd = dabcd = 0. Under Lorentz local transformations, ω behaves as a typical gauge connection,

ω 7−→ L−1(ω + d)L. (122)

Expression (122) makes the covariant derivative (121) an actual covariant quantity (Exercise 2.11). For inﬁnitesimal transformations we have (Exercise 2.11)

ω 7−→ ω + Dξ . (123)

Or, in components

a a a ω b 7−→ ω b + Dξ b . (124)

where

a a a c c a Dξ b = dξ b + ω cξ b − ω bξ c . (125) Transformation (122) is a typical gauge transformation. Hence, the spin-connection can be interpreted as the actual gravity gauge ﬁeld. The aﬃne connection, having no free frame indexes, is a gauge invariant quantity and is computed by

α α a a b Γµν = ea ∂µeν + ωµbeν . (126)

Expressions (126) and (114) deﬁne the change of variables from the ﬁrst order variables to Palatini variables. In particular, expression (126) is called consistency condition and is obtained by demanding the vierbein to be fully covariant constant, i.e.

(full) a a a b α a ∇µ eν = ∂µeν + ω beν − Γµν eα = 0 . (127) We reenforce the fact that, in the Palatini and first order formalisms, the affine connection is not symmetric anymore. This should be evident from (126). In fact, the anti-symetric contribution is the definition of the torsion α α α tensor 2Tµν = Γµν − Γνµ.

3. Curvature, torsion and hierarchy identities

The curvature two-form is easely deﬁned by the double action of the covariant derivative (Exercise 2.12)

D2 = R = dω + ωω , (128)

where, in components, the curvature two-form reads

a a a c R b = dω b + ω cω b . (129) The curvature tensor and theDRAFT curvature 2-form are related by (Exercise 2.13) 1 Rab = ea eb Rαβ dxµdxν , (130) 2 α β µν

αβ where R µν generalizes the Riemann tensor (110) for the case of non-vanishing torsion. It is important to emphasize that (Exercise 2.14) R 6= Dω, which should be clear from (125). The torsion two-form is deﬁned as the covariant exterior derivative of the vierbein,

a a a a b T = De = de + ω be , (131)

39 Hint: Use (125) below. 30 and it is is related to torsion tensor as (Exercise 2.15) 1 T a ≡ ea T α dxµdxν . (132) 2 α µν In the case of vanishing torsion, (131) can be solved for ω, namely a b a ω be = −de . (133) α By combining (133) and (126) it is possible to show that Γµν will be Christoffel symbols, as expected (Exercise 2.16). By the vanishing torsion condition, we see that is possible to describe GR in the first order formalism, in despite of the fact that gµν is not considered the fundamental field in favor of the vierbein. The geometrical interpretation of Rab and T a are quite simple. The curvature carries information about the paral- lelogram rule of vector addition. Hence, by taking the two consecutive parallel transports along different directions, namely nµ and then tµ. Then take the same vector and parallel transport it again, but inverting the direction order, namely tµ and then nµ. The resultant vectors will end up at the same point, but will not coincide because the manifold is curved, i.e. Rab 6= 0. On the other hand, if T a 6= 0, the two resulting vectors may coincide under parallelism, but will not lie at the same point because the manifold is twisted. The curvature and torsion obey an hierarchy chain of equations given by the Cartan descent structure equations (Exercise 2.17):

T a = Dea , (134a) a a b DT = R be , (134b) a DR b = 0 , (134c) where the last identity one is the Bianchi identities for the curvature. We emphasize here that the set of equations (134a-134c) are natural consequences of the geometry we are constructing.

4. Einstein-Hilbert action

The change of variables (114) and (126), when applied to the Einstein-Hilbert action (111), leads to (Exercise 2.18)

Z 1 S = d4x e Rab eµeν − 2Λ2 + T µ ea , (135) EH 16πG µν a b a µ

a √ µ where e = det(eµ) = g and T a is the projected energy-momentum tensor. The variation of (135) with respect to a a eµ and ω bµ leads to (Exercise 2.19), 1 Rab eν − ea Rbc eν eα − 2Λ2 = 8πGT a , µν b 2 µ να b c µ a a b Tµν = 8πGS bν e µ , (136) respectively. In (136), we have deﬁned the spin density as α a δT c c S bµ = a e α . (137) δω bµ In the form notation, the Einstein-HilbertDRAFT action (135) turns out be an integrated local four-form, polynomial in the ﬁelds and their derivatives, given by (Exercise 2.20)

Z 1 Λ2 S = Rabeced + eaebeced + Θ ea + S ωab , (138) EH 32πG abcd 6 a ab

a a ab a where the energy-momentum tensor was split according to Tae = Θae + Sabω . Clearly, Θ does not depend on the spin-connection and Sab does not depend on the vierbein. The corresponding ﬁeld equations (Exercise 2.21) read now Λ2 Rbced + ebeced = 16πGΘ . abcd 3 a c d abcdT e = 16πGSab , (139) 31

Needless to say, the ﬁrst equations in (136) and (139) are equivalent and coincide with the Einstein equations (104) for the vanishing torsion case. Moreover, since the second equations in (136) and (139) are algebraic, the vanishing torsion case is always achieved for vanishing spin densities.

C. Gravity as a gauge theory

The similarity between Yang-Mills connection and spin connection should be clear from their gauge transformations, for instance (33) and (122). Both of them are connections with respect to a gauge group. Moreover, recalling that the Yang-Mills curvature, in differential form notation, is F = dA + κAA, another similarity appears with respect to the curvature (128). However, a few important differences are in order: • The main difference is related to the geometric structures of both theories: The gauge group of gravity has an intimate relation with spacetime, i.e., the gauge group is identified with the local isometries of the spacetime tangent space. In Yang-Mills theories, on the other hand, the gauge group describes an internal space with no relation with spacetime geometry whatsoever. The vierbein is the actual object which connects the internal group space with spacetime. Hence, the vierbein is the field responsible to address the SEP to the theory. The consequence is that the corresponding gauge field describes the parallelism properties between near cotangent spaces. • Torsion seems to have no analogue with respect to Yang-Mills theories. In fact, a vierbein field is not present at Yang-Mills theories. Nevertheless, under Lorentz gauge transformations, the vierbein has an explicit matter character, in despite of its geometrical appeal. Thus, in order to have a parallel, a kind of matter field should be introduced in Yang-Mills theories. • The Einstein-Hilbert action (138) has no relation with the Yang-Mills action, which, in differential form notation reads 1 Z S = F A ∗ F A . (140) YM 2

• Another point is that the Yang-Mills theory, at least at perturbative level, is massless. The Einstein-Hilbert action, however, has at least two mass parameters right from the beginning, namely, Newton and cosmological constants. This feature is not only related to the fact that we have very diﬀerent actions, but also because the vierbein is dimensionless with respect to canonical dimensions, i.e. [e] = 0. • It is also important to notice that the Hodge dual ∗ is not required to construct the EH action. In fact, the absence of the Hodge operator opens the possibility to interpret gµν as an induced quantity. See next section.

a a • Finally, by thinking about the Einstein-Hilbert action (138) as a genuine ﬁeld theory for e and ω b, it lacks quadratic terms in the action, i.e. there are no kinematical therms. Hence, it seems that a ﬁeld theoretical perturbative approach is out of question, at least in the usual way.

V. GENERALIZED THEORIES

In the first order formalism of gravity there is a limited number of Lagrangians which have physical consistency. a a a a The basic bricks to build a theoryDRAFT of gravity in the first order formulation are e , ω b, R b, T and the gauge invariant tensors ηab and abcd. Under the restriction to first order derivatives, there are two main approaches to obtain generalized actions, namely, the Lovelock action [103] and the Mardones-Zanelli action [93].

A. Lovelock theory

The theory of gravity formulated by D. Lovelock in 1971 consists in to describe the gravitational interaction as an extension of the Einstein’s GR [103] by adding to Einstein-Hilbert action (138) all possible extra terms but still with the requirements of locality, vanishing torsion and derivatives order not higher than 1. We notice that, vanishing 32 torsion and ﬁrst order derivatives imply on second order derivatives for the vierbein, see (133). Lovelock was able to ﬁnd a generic action in D dimensions given by

D Z 2 X k SLovelock = akLD , (141) k=0 where ak are generic constants and

k a1a2 a2k−1a2k a2k+1 aD LD = a1a2···aN R ··· R e ··· e . (142) For any spacetime dimension, the ﬁeld equations generated by the action (141) will be second order diﬀerential equations for the metric. Let us consider, for instance, D = 2. In this case we have (Exercise 2.22)

1 Z S = a eaeb + a Rab , (143) L2 32πG ab 0 1 which is the usual EH action with cosmological constant in two dimensions. This action is known to be topological, i.e. it provides no ﬁeld equations at all (Exercise 2.23). For D = 4 we obtain (Exercise 2.24)

1 Z S = a eaebeced + a Rabeced + a RabRcd , (144) L4 32πG abcd 0 1 2 The ﬁrst term is recognized as the cosmological constant term while the second term is the well known EH term. The novelty in (144) is the last term. However, in four dimensions, this term is actually topological and, hence, provides no contribution to the ﬁeld equations. This term is known in the literature as the Gauss-Bonnet term.

B. Mardones-Zanelli theory

The Mardones-Zanelli formulation [93, 94] is a generalization of the Lovelock action with the relaxation of the torsionless condition. A generic expression for the D-dimensional case is not at our disposal. The main new features at this construction are the inclusion of torsional series and Chern-Simons terms. Let us write a few examples, starting with the two dimensional case. In two dimensions, no extra terms are added to (143) because all two-form terms that can be construct are already there. For three dimensions, on the other hand, we have

1 Z 2 S = a eaebec + a Rabec + b T ae + b ωa dωb + ωa ωb ωc . (145) MZ3 32πG abc 0 1 1 a 2 b a 3 b c a

Again, the first term is the cosmological constant term and the second is the EH term. The third term is the Chern- Simons three-form for the vierbein and the fourth term is the usual topological Chern-Simons three-form for the connection. It is known that, for certain situations, the last term is equivalent to the first two terms [104]. The four-dimensional case is obtained as Z 1 a b c d ab c d ab cd ab a b a SMZ4 = abcd aDRAFT0e e e e + a1R e e + a2R R + b1R Rab + b2Rabe e + b3T Ta . (146) 32πG The first three terms coincide with the Lovelock action. The fourth term is the Pontryagin topological term and is related to the connection CS term through

2 RabR = d ωa dωb + ωa ωb ωc . (147) ab b a 3 b c a

The last two terms are, in fact, equivalent up to surface terms. Moreover, for b2 = −b3, the last two terms become the Nieh-Yang topological term which is related to the vierbein CS term [105],

a b a a Rabe e − T Ta = d(T ea) . (148) 33

VI. QUANTIZATION ATTEMPTS

It is widely known that GR has deep problems in what concerns its quantum consistency. In fact, the ﬁrst attempts to quantize GR were found to be inconsistent [130–132]. The idea is to perturb the metric around a ﬁxed background40, for instance, the Minkowski background,

gµν = ηµν + khµν , (149)

where k is a constant with negative mass dimension [k] = −1 related to Newton‘s constant, and quantize the gravitational waves modes hµν . The linearized version of the EH action (111) is known as the Pauli-Fierz action and enjoys an Abelian gauge symmetry for the field hµν . The quantum field hµν , being symmetric in its indexes, describes spin-2 excitations associated to the graviton field. However, the standard quantization methods for gauge theories fail due to the fact that the theory is not renormalizable, i.e. extra couplings are required. The extra counterterm indicates that quadratic terms on the curvature should be included in the original theory. If this is correctly done, then the linearized theory is actually renormalizable. However, the extra terms provide non-physical modes associated to higher order derivatives [133]. Consequently, although renormalizable, the theory with higher order derivatives is not unitary. Many other alternatives to quantize GR have been proposed. Perhaps, the closer approach to the background linearization method above described is the asymptotic safety method [22, 23]. In this method, the renormalization group method is employed in the looking for fixed points, in despite of the non-renormalizability of EH action. The claim is that EH action is non-perturbative, hence, perturbative expansions and counterterms absorption may not make sense. In this sense, the full non-linearized EH action is considered in a path integral approach. Of course, to find these fixed points is a highly non-trivial task and the final solution still lacks. Another popular technique is the loop quantum gravity theory [20, 21] where a different set of gravity variables is quantized as fundamental quantities. These variables are related to the spacetime holonomies and, hence, automatically predicts spacetime granulation. the method is background independent and it seems to possess evidence of a nice classical limit [136]. However, huge obstacles remain to be overcame such as the solution of the so called Hamiltonian constraint and the understanding of how time is encoded in this theory. More recently, a power counting renormalizable proposal have emerged, the Horava-Lifshitz gravity [24–26]. In this approach, spacetime diffeomorphisms are explicitly broken and power counting renormalizable actions are constructed with space and time independent terms. The diffeomorphisms are assumed to be a macroscopic phenomenon, i.e. all parameters of the theory should converge to macroscopic values for which diffeomorphism is gained. Of course, the formal renormalizability proof lacks and a consistent classical limit is not known. A different approach to treat quantum gravity is the large class of theories known as emergent gravities. In these theories, gravity as geometrodynamical theory is an emergent phenomenon produced by all kinds of quantum effects which have nothing to do with spacetime at some energy scale. In other words, gravity as geometry is a classical limit of a more fundamental theory which is not related to spacetime dynamics. Hence, one could say that quantum gravity and geometrodynamics would be different sides of the same coin. some examples we will not discuss can be found in [27–30] and references therein. Perhaps the most popular method to describe gravity in quantum level is superstrig theory [16–19] which concerns the quantization of string systems instead of particles. In this approach, all spin excitations are contained in the string vibration modes, including the spin-2 graviton modes. Hence, gravity as geometrodynamics emerges as an effective effect. Moreover, superstring theory is known to have the property of unifying all four interactions. Nevertheless, it predicts extra particles still not observed by experiments such as the superpartner particles. It also predicts the spacetime dimension to be ten, i.e. six more than the four known dimensions. Just like the extra particles, the extra dimensions have not been observed yet. Still in effective gravity theories,DRAFT there is a whole bunch of gauge theories which could lead to geometrodynamics, see for instance [31–42]. Generally speaking, the idea is to define a metric independent action enjoying gauge symmetry. In most approaches, a topological action is defined and a spontaneous symmetry breaking is enforced, freeing the degrees of freedom which will be interpreted as gravity. An example in which the starting point is not a topological action and there is no need of Higgs fields is explored with some detail in the next part.

40 The same idea can be employed in the ﬁrst order variables. 34

PART III: EFFECTIVE GRAVITY FROM YANG-MILLS THEORIES

In this part we focus on a speciﬁc mechanism that generates a generalized geometrodynamical gravity theory from a Yang-Mills action based on the group SO(5). A deep analogy with QCD is discussed. As so, the Yang-Mills action is taken as a proposal for a quantum gravity theory while the geometrodynamical classical limit would be a generalization of GR. These and similar ideas can be extensively found in [31–42] and references therein. In most of this part we will remain at classical level, at least for most of the technical discussions. Nevertheless, the idea is that everything we do with SYM should be extended to its renormalized quantum version Γ.

VII. GENERAL IDEAS

As discussed in Sec. VI, geometrodynamical theories such as gravity have many problems in what concerns their quantization. In fact, the principles of QFT and GR seem to carry fundamental incompatibilities. The main obstacle in quantizing a system is to find, if any, the correct quantization method because the classical theory is always a limit of another quantum theory. In this sense, the quantization techniques can not be taken as fundamental approaches to obtain the quantum version of a classical system: The most fundamental theory is the quantum one while the classical theory is a particular macroscopic case, i.e. an effective version of the quantum theory. The Faddeev-Popov quantization and Gribov ambiguities are striking examples of that idea. Remarkably, in some cases, the quantum and classical theories seem to have no relation whatsoever. This is the case of the SM, which is fully described by Yang-Mills theories coupled to spinorial (quarks and leptons) and scalar (Higgs) fields (needless to say, Yang-Mills theories displays quantum consistency, renormalizability and unitarity). In fact, at very high energies, the SM is a gauge theory based on the group SU(3) × SU(2) × U(1) where the corresponding gauge fields are massless and mediate the interactions between all matter massless fields. As the energy decreases, the three sectors split, with different behavior each. The QCD sector is confined and develop several mass parameters (Gribov mass, condensates...) and suffers hadronization while the electroweak sector suffers spontaneous symmetry breaking to the subsector U(1): roughly speaking, the vector bosons become massive while the photon remains massless. Hence, at low energies, three completely different theories emerge.

The question we can formulate is: Can gravity also be an exotic limit of a typical gauge theory? Or, equivalently: Can quantum gravity be a standard gauge theory in four dimensions?

The answer is not trivial... at all: In all three fundamental interactions of the SM, the path between the perfect Yang-Mills theories at high energies and the cumbersome low energy regime is quite difficult, even obscure. In the case of QCD, the hadronization phenomenon, intimately linked to confinement, lacks a theoretical description. In other words, how to start from the Yang-Mills action and end up at an action in terms of hadrons and glueballs? The essence of this question almost perfectly fits to our gravity problem. In QCD we know the quantum theory and the final low energy states. In gravity, we know the effective low energy theory, i.e. GR. However, we do not know the starting fundamental theory, nor the mechanism leading it to geometrodynamics. Therefore, as a starting point, because Yang-Mills gauge theories work so nicely in flat spacetime for three of the four fundamental quantum interactions, we postulate that:

Quantum gravity is a Yang-Mills gauge theory in four-dimensional Euclidean spacetime R4. This choice, instead of a MinkowskiDRAFT spacetime, has three main reasons: • First, because QFT is only solvable in Euclidean spaces. Reliable computations require a Wick rotation from the Minkowkian to the Euclidean space [6]. However, a Wick rotation is known to be valid only at perturbative level. At non-perturbative scales, there are no indications that a Wick rotation can be performed. Thus, since we are interested in non-perturbative eﬀects, it is necessary to start with a Euclidean spacetime. • The second reason, and perhaps the most important, is that the Euclidean space is the simplest geometry. Since we are constructing a theory that will determine spacetime geometry, starting with the simplest geometry seems to be the right choice. • Third, in four dimensions, the coupling parameter of a Yang-Mills theory is dimensionless. As a consequence, 35

pure Yang-Mills action is massless41. This fact is important because it prevents the gauge ﬁeld, which has dimension 1, to be directly associated with the vierbein, which has vanishing dimension. This means that, at quantum level, the gauge theory and the spacetime are completely unlinked.

Hence, from the third motivation above, to identify the vierbein at low energies, it seems that mass parameters must appear at some level. Moreover, since Newton’s constant carries dimension −2, it becomes evident that at least one mass parameter must appear in order to obtain gravity at low energies. Therefore, since Yang-Mills theories naturally generate mass parameters as the energy decreases, our choice sounds promising. In fact, we pick the Gribov parameter as the mass that will be responsible for this generalization. It remains to decide which group (or class of groups) is the most suitable group to describe our mechanism. The gauge field is an algebra-valued 1-form and thus, carries dim G components with respect to the algebra of the gauge group. The gauge group G has to fulfill a few requirements [39, 107]: • The group must have dim G ≥ dim ISO(4), in such a way that the degrees of freedom of gravity are covered. • The group must decompose at least42 as G = H + Q where Q = G/H must be a symmetric space and H, obviously, a stability group. The stable group must share a morphism with Lorentz-type groups and Q must define a vector representation of H. Thus, H can be identified with local Lorentz transformations and Q with a sector which expands the vierbein. As a consequence, the gauge components can be identified with the spin-connection and the vierbein. • It can be very convenient to have a symmetry breaking G → H, in such a way that the field at the Q sector acquires a matter-type transformation with respect to H. That is why Q must be a symmetric space. It is important to mention that, depending on the gauge group, a symmetry breaking might not be needed. This occurs if the gauge transformations could be partially identified with local Lorentz transformations and partially identified with diffeomorphisms. A renowned three dimensional example can be found in [104, 108]. Perhaps the simplest group satisfying these conditions is the (Exercise 3.1) N = 5 special orthogonal group. Then, the choice is that G = SO(5) is the symmetry group of our Yang-Mills gauge theory describing quantum gravity. In summary, gravity will be described at quantum and classical level as follows:

The theory has two sectors, the UV, which is a massless gauge theory in Euclidean four-dimensional space, and the IR sector, which presents soft BRST symmetry breaking and dynamically generated mass parameters. The UV sector is a standard, non-Abelian, asymptotic free, gauge theory of spin-1 excitations. Although the degrees of freedom coincide in number with a first order gravity, this identification is forbidden unless a mass parameter arises, so that a vierbein can be defined. Once the energy starts to decrease, the soft BRST symmetry breaking takes place, the Gribov parameter and other possible masses appear. At this stage, propagators of the fundamental fields develop complex poles, a fact that is interpreted as an evidence that these excitations are ruled out from the physical spectrum of the theory (in QCD, this is recognized as confinement). Physical observables must be defined at this point. In QCD, the observables are hadrons and glueballs. In gravity, the low energy observables must be geometry. Thus, one possibility is to identify the H gauge field with the spin-connection and the Q field with the vierbein. The consequence is a first order gravity where geometry is determined by the usual relations (114) and (126).

If all above requirements are fulfilled, we gain a deep analogy between gravity and quantum chromodynamics. At high energies, both theories are well defined quantum gauge theories in a four-dimensional flat space. Both theories present soft BRST breaking and have to be redefined in order to establish the physical content at the infrared regime. In the case of QCD, the physical content are confined states identified with hadrons and glueballs. In gravity, the physical contentDRAFT are identified with geometric properties of spacetime. We now turn to the specifics of these ideas.

41 In Yang-Mills theories, the four-dimensional case is the only case where the coupling parameter is dimensionless. 42 Larger groups are also allowed. The extra decompositions will generate a matter sector in the resulting gravity theory, [40]. 36

VIII. SO(5) YANG-MILLS THEORY

As discussed, one possible realization of the above ideas occurs in a gauge theory based on the group SO(5) in a four-dimensional Euclidean spacetime, R4. The algebra of the group is given by 1 J AB,J CD = − δAC J BD + δBDJ AC − δADJ BC + δBC J AD , (150) 2 where J AB are the 10 anti-hermitian generators of the gauge group, antisymmetric in their indexes. Capital Latin 43 5 indexes are chosen to run as {5, 0, 1, 2, 3}. The SO(5) group defines a five-dimensional flat space, RS, with invariant AB 4 5 Killing metric given by δ ≡ diag(1, 1, 1, 1). The spaces R and RS have no dynamical relation whatsoever. The SO(5) Yang-Mills action is renormalizable and unitary. Thus, Gribov ambiguities and BRST soft breaking takes place. Moreover, a few extra mass parameters may emerge. The presence of these masses will be forwardly used. However, we will fix our attention only at the Yang-Mills action and the gauge field one-form, here denoted by Y .

A. Projection and rescaling

The five-dimensional orthogonal group may be decomposed as a direct product, SO(5) ≡ SO(4) × S(4) where S(4) ≡ SO(5)/SO(4) is a symmetric coset space with four degrees of freedom. This decomposition is carried out by projecting the group space in the fifth coordinate A = 5. Defining then J 5a = J a, where small Latin indexes vary as {0, 1, 2, 3}, the algebra (150) decomposes as (Exercise 3.2) 1 J ab,J cd = − δacJ bd + δbdJ ac − δadJ bc + δbcJ ad , 2 1 J a,J b = − J ab , 2 1 J ab,J c = δacJ b − δbcJ a , (151) 2 where δab ≡ diag(1, 1, 1, 1), J ab ∈ algSO(4), and J a ∈ algS(4). The gauge connection follows the same decomposition, A B a b a Y = Y BJA = A bJa + θ Ja where, obviously, A ∈ algSO(4), and θ ∈ algS(4). The gauge transformations in Eq. (34) are decomposed, at infinitesimal level, as (Exercise 3.3) κ Aa 7−→ Aa + Dαa − (θaζ − θ ζa) , b b b 4 b b a a a a b θ 7−→ θ + Dζ + κα bθ . (152) a b a where the full gauge parameter splits as ξ = α bJa + ζ Ja and D = d + κA is the covariant derivative with respect to the sector SO(4) of the algebra. The field strength also decomposes (Exercise 3.4), κ F = Ωa − θaθ J b + KaJ , (153) b 4 b a a where

a a a c Ω b = dA b + κA cA b , a a a a b DRAFTK = Dθ = dθ + κA bθ . (154) The starting action is, as postulated, the Yang-Mills action (39). In form notation, it reads (Exercise 3.5) 1 Z S = F A ∗ F B , (155) YM 2 B A which, from (153), decomposes as (Exercise 3.6) 1 Z 1 1 1 S = Ωa ∗Ω b + Ka∗K − Ωa ∗(θ θb) + θaθ ∗(θ θb) . (156) 2 b a 2 a 2 b a 16 b a

43 In comparison with the notation used in PART I, the correspondence is given by A (PART I) 7−→ AB = −BA (PART III). 37

Once the energy starts to decrease the set of mass parameters, together with BRST soft breaking, dynamically arises. At this point, it is convenient to perform the following redefinitions A 7−→ κ−1A, θ 7−→ κ−1mθ , (157) where m is a mass scale which depends on the mass parameters of the theory and still has to be determined. The transformations (157) are not accidental. Both sectors are rescaled with κ−1 in order to factor out the coupling parameter outside the action, a standard procedure in Yang-Mills theories [6]. On the other hand, the mass parameter affects only the θ-sector, transforming it in a field with dimensionless components. It turns out that this is the unique possibility if one wishes to identify θ with a vierbein field. If also A is rescaled with a mass factor, then it would never be possible to identify it with the spin-connection. The action (156) is then rescaled to (Exercise 3.7) 2 2 4 1 Z a b m a m a m S = Ω ∗Ω + K ∗K − Ω ∗(θ θb) + θaθ ∗(θ θb) , (158) 2κ2 b a 2 a 2 b a 16 b a

a a a c a a where Ω b = dA b + A cA b, K = Dθ , and the covariant derivative is now given by D = d + A. Moreover, a reparameterization of the SO(5) generators is required due to the existence of a mass scale, i.e., a stereographic 5 projection is now allowed if one identifies the mass parameter with the radius of the gauge manifold RS, see [39]. a a In essence, the whole θ field must be invariant under the rescaling (157), i.e., under (157), θ Ja 7−→ θ Ja. The consequence for the algebra (151) is (Exercise 3.8) 1 J ab,J cd = − δacJ bd + δbdJ ac − δacJ bc + δbcJ ad , 2 m2 J a,J b = − J ab , 2κ2 1 J ab,J c = δacJ b − δbcJ a . (159) 2 So far, we have done nothing. Action (158) is exactly the original action (155) with a specific decomposition of the fields and a very simple rescaling of the fields.

B. Contraction and symmetry breaking

In order to obtain gravity, the main step is to consider that the rate m2/κ2 tends to vanish at a specific low energy scale44. At this point, this is only an assumption, however, due to asymptotic freedom, it is not an improbable hypothesis45. Taking this limit at the rescaled algebra (159), an Inönü-Wigner contraction46 [109] takes place, enforcing the SO(5) group to be contracted down to the Euclidean Poincarégroup (Exercise 3.10). The gauge symmetry is then dynamically deformed to the Euclidean Poincarégroup, SO(5) −→ ISO(4), for some values κ in the strong coupling regime. The details of this deformation can be understood as a stereographic projection (Exercise 3.11), see the appendix in [42]. An Inönü-Wignercontraction is a deformation of the group to another group which is not a subgroup of the former. Hence, since the Poincarégroup is not a symmetry of the action (158), a dynamical symmetry breaking takes place. Nevertheless, the Lorentz-type group SO(4) is not only a common subgroup, ISO(4) ⊃ SO(4) ⊂ SO(4), but is also a stability subgroup for both groups. Thus, the theory suffers a symmetry breaking SO(5) −→ SO(4). Under the SO(4) gauge symmetry, the transformations (152) reduce to (Exercise 3.12) a a a DRAFTA b 7−→ A b + Dα b , a a a b θ 7−→ θ − α bθ , (160) where (157) was assumed. Thus, after the contraction, the field A becomes the only gauge field with respect to the Lorentz group while θ has migrated to the matter sector (it suffers only group rotations under infinitesimal Lorentz transformations, i.e. it is a vector representation of the group). Now, the theory is ready to be identified with gravity.

44 We remark that, due to the hierarchy of fundamental interactions, for gravity, a low energy scale would be much higher when compared to all other fundamental interactions. 45 In Sec. X C we will show that this is actually what happens when we take m to be the Gribov mass parameter. 46 Perhaps the most known example of an In¨onu-Wigner contraction in Physics is the deformation of the Lorentz group to the Galileo group for v/c 7−→ 0 (Exercise 3.9). 38

IX. INDUCED GRAVITY

The broken theory described in Sect. VIII B can generate an effective geometry if [39, 40, 107]: (i) every configuration (A, θ) defines an effective geometry (ω, e); (ii) there exists a mapping from each point x ∈ R4 to a point X ∈ M4 of the deformed space. In order to preserve the algebraic structure already defined in R4, it is demanded that this mapping is an isomorphism; (iii) The local gauge group SO(4) defines, at each point of the mapping, the isometries of the tangent space TX (M). Thus, θ and A can be identified with the vierbein e and spin connection ω, respectively, through

ab µ a b ab µ ωµ (X)dX = δaδb Aµ (x)dx , a µ a a µ eµ(X)dX = δaθµ(x)dx . (161)

In expressions (161), Latin indexes {a, b ...} belong to the tangent space TX (M). Moreover, it is always possible to impose that the space of all p-forms in R4 is mapped into the space of p-forms in M4, namely, Πp 7−→ Πe p, and the same for the Hodge spaces, ∗Πp 7−→ ?Πe p, where ? is the Hodge dual in M4. This mapping47, together with the identiﬁcations in Eq. (161), provides (Exercise 3.12)

1 Z 1 1 Λ2 S = Ra ?R b + T a ?T − Rabeced + eaebeced , (164) 16πG 2Λ2 b a a 2 abcd 4 abcd

a a a c a a a b where R b = dω b +ω cω b and T = de +ω be are the curvature and torsion, respectively. Newton and cosmological constants are determined by

m2 = κ2/4πG = 4Λ2 . (165)

Action (164) is a gravity action in the first order formalism presenting (in order of appearance) a quadratic Yang- Mills-type curvature term, a quadratic torsion term, the Einstein-Hilbert term, and the cosmological constant term. In summary, an Yang-Mills theory based on the SO(5) group can be identified with gravity if a mass parameter emerges at some energy scale and if the 2-step deformation SO(5) 7−→ ISO(4) 7−→ SO(4) takes place. In our hypothesis, the deformation is driven by a dynamical behavior of the ratio m2/κ2 7−→ 0. In a way, the degrees of freedom of the gauge fields are absorbed by spacetime which acquires a deformed character. Hence, all other matter fields would feel this deformation in the way, independently of their character. This phenomenon can be visualized as the rising of the equivalence principle. Now, we start to explore the physical contents of this model at quantum and classical levels.

X. QUANTUM ASPECTS

A. Weinberg-Witten theorems, emergent gravity and spin-1 gravitons

In [110], Weinberg and Witten established two very powerful theorems in QFT. Essentially, they forbid: (i) massless charged states with helicity j > 1/2 which have a conserved Lorentz-covariant current and (ii) massless states with helicity j > 1 which have conservedDRAFT Lorentz-covariant energy-momentum tensor.

47 The formal aspects of the mapping between a gauge theory in Euclidean spacetime and a gravity theory can be discussed in terms of fiber bundle theory. The details can be found in [39, 40, 107]. The result was discussed for general manifolds with d arbitrary dimensions. Let the original manifold have metric tensor gµν (x) while the effective metric tensor is defined as geµν (X). Thus, the matrix that defines the map x 7−→ X for all points in the effective manifold is given by 1/2d ν g˜ να L = g˜ gαµ . (162) µ g Its inverse is given by 1/2d −1ν g να L = g g˜αµ . (163) µ g˜ In (162) and (163), g = det gµν and ge = det geµν are taken as non-vanishing quantities. Thus, ambiguities are absent in this mapping. 39

At first sight, the first theorem would forbid gauge theories to exist because vector bosons and gluons are charged massless states which carry helicity j = 1. However, the conserved gauge current associated with these states are not Lorentz covariant [110]. Similarly, the second theorem forbids spin-2 states with conserved Lorentz-covariant energy-momentum tensor to be defined. In traditional GR the metric tensor gµν is not a Lorentz covariant tensor since it is constructed over a Riemannian manifold. The linearized version of GR, on the other hand, is also a gauge theory and the same principles of the first case apply [110]. Moreover, the energy-momentum tensor of the graviton field identically vanishes. On the other hand, the Weinberg-Witten theorem (ii) applies to emergent gravity theories for which massless spin-2 states are generated as effective or composite states that could be associated with graviton excitations. Nevertheless, one may argue that the Weinberg-Witten theorem (ii) applies to the present mechanism. However, this is not the case here. First, the theory has a few mass parameters and the theorem holds only for massless states. Second, and more important, there are no spin-2 states in this model. The fields are identified with geometry and not with spin-2 composite fields. Gravity emerges as geometrodynamics and not as a field theory for spin-2 particles in flat space. Nevertheless, renormalizability establishes that the quantum action has the same form of its classical version. The difference between the classical and quantum actions lies on the fields and parameters, which are their respective renormalized versions. Hence, although each configuration (A, θ) can define a geometry (ω, e), the mapping should not be applied to each configuration in the path integral, but at the quantum action itself. In this way, the resulting geometrodynamical theory is obtained from the full dynamical content of the original gauge theory. The conclusion is that there is no violation of the Weinberg-Witten theorem. Another question may raise at this point: Where are the spin-2 excitations of the graviton? And it is an important question. The main point here was already discussed: The present approach is not a quantization of gravity geometric variables. Inhere, gravity is an effective manifestation of another theory, i.e. geometrodynamics is a classical limit of a quantum gauge theory of spin-1 excitations. On the other hand, if our SO(5) gauge theory is in fact quantum gravity and the graviton is the mediator of gravity at quantum level, then, in the present theory, the graviton has spin-1. Hence, this definition differs from the usual graviton definition which is associated with massless spin-2 states. Nevertheless, after the emergence of gravity as geometrodynamics, linearization is allowed for weak gravitation and the spin-2 states might also be considered. In that case, these states are classical states associated with propagating fields and not quantum states. Another way to think of the spin counting difference lies on the map R4 7−→ M4 where the local gauge group is identified with the isometries of the tangent spaces of M4. In this process, the group indexes are interpreted as spacetime tangent indexes (an association prevent at high energies due to the massless character of Yang-Mills action). Thus, after the map, each field gains an extra index and thus, extra spin degrees of freedom.

B. Cosmological constant problem

A remarkable feature of the present theory is that Newton and cosmological constants are related through Λ2 = κ2/16πG. From asymptotic freedom, κ2 is a big quantity at low energies. And, by assumption, G is small. Thus, Λ should be very big. In fact, if this is true, we can make two important remarks: (i) The first term in (164) can be safely neglected. Moreover, due to the absence of matter fields, torsion can be taken as very small. The resulting theory is then, the usual Einstein-Hilbert theory with (a very big) cosmological constant (see Sec. XI A); (ii) although 2 astrophysical predictions [111, 112] determine that Λobs is very small, quantum field theory predicts [113–115] a very 2 large Λqft. Thus, the contribution of a pure gravitational cosmological constant, which is big in our case, can drive the cosmological puzzle to a final consistent answer. In fact, following [39, 116], the renormalized cosmological constant 2 2 2 2 of our model could be determined through Λobs = Λren + Λqft ≡ Λe . This means that, by summing all vacuum contributions to the last term inDRAFT (164), it should be replaced by Λe2. We will in fact assume this at Secs. XI C and XI B.

C. Quantum predictions

It is possible to perform actual explicit computations within a semi-perturbative approach, see [40] for the first 1-loop estimates and [117] for 1 and 2-loop improved computations. In these notes, without loss of generality, we confine ourselves to 1-loop results. The important thing at this point is to wisely find the correct mass parameters to account for m. It is intuitive that the Gribov parameter is the most probable candidate because it is a genuine non-perturbative and inevitable parameter and it is gauge invariant [49, 50]. 40

1. Running parameters

In perturbative QFT, the running of the parameters can be determined by the renormalization factors of fields and parameters. In the case of the Gribov parameter, it is determined by a gap equation obtained from the minimization of the quantum action with respect to γ2, see Sec. III. At 1-loop, the gap equation is48 [14, 46] 3Nκ2 Z d4p 1 = 1 . (166) 4 (2π)4 p4 + γ4 where N = 5 and p is the internal momentum. Following [46], by employing the MS renormalization scheme at the gap equation (166), it is not difficult to find (Exercise 3.13) Nκ2 5 3 Nγ4 − ln = 1 , (167) 16π2 8 8 µ4 where µ is the energy scale and the trivial solution γ2 = 0 was excluded. Equation (167) provides (Exercise 3.14)

5 4 16π2 2 e 6 2 − γ = √ µ e 3 Nκ2 . (168) N On the other hand, recalling that the coupling parameter is determined through the renormalization group equations [66, 67], we can write

Nκ2 1 = , (169) 16π2 11 ln µ2 3 Λ2

where Λ is the renormalization group cutoﬀ. Hence, combining (168) and (169), it is found (Exercise 3.15)

5 2 −35/9 2 e 6 2 µ γ = √ Λ 2 . (170) 5 Λ We can see from (170) and Fig. 4 that, as higher the energy is, as smaller the Gribov parameter is. This is the expected behavior [14, 46, 48] of γ2 because at high energies perturbative massless Yang-Mills theory must be recovered. At the deep infrared region, however, it seems to diverge. This behavior is an evidence that the semi-perturbative approximation needs improvements. In fact, it is known from lattice simulations for unitary groups that the coupling parameter is finite at the origin [118, 119]. This kind of improvement could affect the infrared sector of the Gribov parameter in such a way that it could also be finite. DRAFT

5 2 FIG. 1: Gribov parameter as function of energy scale. The energy is in units of Λ and the Gribov parameter in units of e√6 Λ . 5

48 It is important to notice that, because we are dealing with Yang-Mills theories, the following computations can be derived in the same manner that of [46], only keeping in mind that the fundamental Casimir here is 5 and the group dimension is 10. 41

Now, the ratio γ2/κ2 is easily determined from (169) and (170),

2 5 ! 2 −35/9 2 γ 55 e 6 2 µ µ = √ Λ 2 ln 2 . (171) κ2 48π2 5 Λ Λ

The very interesting behavior of γ2/κ2 is plotted in Fig. 2 and the inverse ratio is displayed in Fig. 5. As expected, at high energies, γ2/κ2 asymptotically vanishes. Then, in a scale right above Λ, γ2/κ2 achieves a maximum. At this region, BRST soft breaking dominates and the rescaling of the fields (157) is allowed. After that, it fastly drops to zero at µ = Λ. Is exactly at this point that the theory suffers the Inönü-Wignerdeformation which induces the breaking to the SO(4) theory and the geometric phase starts over. This point is also recognized as the point of phase transition in non-Abelian gauge theories. Below this point, another theory takes place. In the case of QCD, ΛQCD ≈ 237MeV , the Yang-Mills action should be replaced by an action based on hadrons and glueballs excitations. In the case of gravity, the Yang-Mills action must be substituted by a geometrodynamical action and the phase transition scale is expected to be around Planck scale.

FIG. 2: The ratio γ2/κ2 as function of energy scale. The energy is in units of Λ and the Gribov parameter in units of 5 2 55 e√6 Λ . 48π2 5

Below the transition point Λ, where the coupling parameter diverges, the squared coupling parameter acquires negative values. This indicates that below Λ the perturbative predictions are actually meaningless. However, from lattice predictions [119, 120], there are strong evidences that the non-perturbative coupling is actually finite at the origin and presents no divergence at the transition scale. On the other hand, we can argue that the divergence is a strong signal that there is a phase transition at that point. Thus, neither way, the coupling parameter should drop out in favor of an effective coupling. In the case of QCD, it is not known how to obtain this parameter from the dynamics of Yang-Mills theories. However, in the present case, we can interpret the ratio γ2/κ2 as the effective coupling, which is related to Newton‘s constant. The analysis for the cosmological constant is easier because Λ ∼ γ2. Thus, it should be very large indeed. As discussed in Sec. X B, this is actually a very welcome feature because it may compensate the QFT predictions and, eventually, it can provide a smallDRAFT effective value [116].

2. Numerical estimates

There are two main ways to calculate the parameters of interest. The usual way [6, 68] is: i) to fix the renormalization group parameter at the transition energy. Its value is a phenomenological quantity and cannot be derived by theory. In 2 our case is Planck energy; ii) chose a energy scale in such a way that Nκ2/16π2 and ln(µ2/Λ ) are as small as possible; iii) compute the physical parameters you want. Another way is: i) Fix a physical parameter to its experimental value; 2 ii) chose a energy scale in such a way that Nκ2/16π2 and ln(µ2/Λ ) are as small as possible; iii) Compute the other parameters as well as the transition scale. We choose to discuss here the second way (both ways are discussed in detail in [117] and there are no significant difference between them). 42

FIG. 3: The ratio κ2/γ2 as function of energy scale. The energy is in units of Λ and the Gribov parameter in units of √ 48π2 5 −2 55 5 Λ 2. e 6

Following the second approach, the scales µ and Λ can be estimated by ﬁxing the current value of Newton’s constant to G−1 ≈ 1.491 × 1032T eV 2. Thus, combining (165) and (171), one easily achieves

−70/9 2 31 2 2 µ µ 7.323 × 10 T eV = Λ ln 2 . (172) Λ Λ

2 We have little freedom to choose µ as long as Nκ2/16π2 < 1. Let us work at µ2 = 2Λ . Thus, Nκ2/16π2 ≈ 0.393. 2 This provides ln(µ2/Λ ) ≈ 0.693. Accepting these as reasonable values, we achieve for the renormalization group scale

2 Λ ≈ 2.122 × 1033T eV 2 , (173)

2 32 2 which is quite close (for a 1-loop semi-perturbative estimate) to Planck energy Ep = 1.491×10 T eV . Although some approximations and extrapolations have been considered, we can interpret these values as a good result, indicating that the geometric phase of gravity appeared right before Planck scale. This means that right above Planck scale, where quantum mechanics starts to make sense, gravity is already in its geometric phase. We can also estimate the value of the cosmological constant for the chosen scales. From (165) and (170), it provides

Λ2 ≈ 1.106 × 1032 T eV 2 . (174)

This is a huge amount of energy and is three orders of magnitude greater than quantum ﬁeld predictions [121] of 2 28 2 2 −68 2 Λqft ∼ −3.71 × 10 T eV and would not cancel it to provide the observational data value Λobs ∼ 1.686 × 10 eV . However, for 1-loop approximation at zero temperature, it is quite remarkable that a solution accommodating Newton constant and Planck energy scale could be found.

DRAFTXI. CLASSICAL ASPECTS Let us now study some classical aspects of the theory.

A. Classical ﬁeld equations

The variation of the action (164) with respect to ω and e leads to (Exercise 3.16)

3 Λ2 Rbc ? (R e ) + T b ? (T e ) + D?T − ε Rbced − ebeced = 0 , (175) 2Λ2 bc a b a a abcd 3 43

for the vierbein, and (Exercise 3.17)

3 D?R + 2e ?T − ε T ced = 0 , (176) Λ2 ab b a abcd

for the spin-connection. The last term in (175) is recognized as the Einstein term with cosmological constant. Nevertheless, as discussed in Sect. X B, the cosmological constant term in (175) will sum with all vacuum terms of the matter ﬁelds action. Therefore, by this phenomenological argument, we should replace Λ 7−→ Λ,e only at this term. Thus,

! 3 Λe2 Rbc ? (R e ) + T b ? (T e ) + D?T − ε Rbced − ebeced = 0 , (177) 2Λ2 bc a b a a abcd 3

where Λe is the observational cosmological constant, which is extremely small. Equations (175) and (176) are the full ﬁeld equations, valid at all sectors of the geometric phase. On the other hand, for a suﬃcient low energy scale where all other theories achieve their vacuum states, equations (177) and (176) must be employed. The Einstein limit can be obtained with a very subtle argument: Since Λ2 is a very large quantity, all terms proportional to Λ−2 are small perturbations as long as the curvatures are not too strong. Hence, at zeroth order approximation, equations (177) and (176) reduce to ! Λe2 T b ? (T e ) + D?T − ε Rbced − ebeced = 0 . b a a abcd 3 c d 2eb ?Ta − εabcdT e = 0 . (178)

The second of (178) is algebraic in the torsion and, hence, a solution T = 0 is allowed49. Thus, (178) reduce to ! Λe2 ε Rbced − ebeced = 0 . (179) abcd 3

which is the well know Einstein equations, i.e. the ﬁrst of (139) in vacuum.

B. Spherical symmetry static solutions

Let us consider the ﬁeld equations (176) and (177) for vanishing torsion and very high values of Λ2, ! 3 Λ˜ 2 Rbc ? (R e ) − ε Rbced − ebeced = 0 . (180) 2Λ2 bc a abcd 3

We remark here that the ﬁrst term in (180) will be considered as a very small perturbation. In, (176), for vanishing torsion, the remaing term is virtually zero and can be safelly neglected since it does not aﬀect any other dynamical term. We take the usual ansatz [10–13]DRAFT for static and spherically symmetric metric, namely e0 = −eα(r)dt , e1 = eβ(r)dr , e2 = rdθ , e3 = r sin θdφ . (181)

49 It is important to understand that this limit can only be taken outside a matter distribution of for suﬃcient small spin densities, otherwise, T 6= 0 in general. 44

Hence, equation (180) decomposes as

" 2 2# e−2β∂ β 1 − e−2β e−2β∂ β 1 − e−2β σ 2 r + + 2 r + + 3λ = 0 , r r2 r r2 " 2 2# e−2β∂ α 1 − e−2β e−2β∂ α 1 − e−2β σ 2 r + − 2 r + + 3λ = 0 , r r2 r r2 ( 2 2) h i2 e−2β∂ α e−2β∂ β σ e−(α+β)∂ e−β∂ eα + r + r + r r r r e−2β∂ α e−2β∂ β −e−(α+β)∂ e−β∂ eα − r + r + 3λ = 0 , r r r r (182)

for a = 0 , a = 1 and a = 2, respectively. The constants in (182) are 3 σ = − , 2Λ2 Λ˜ 2 λ = − , (183) 3 We notice that the diﬀerential equations for a = 2 and a = 3 are identical. The remaining equations can be solve by perturbation theory around the well-known de Sitter Schwarzchild solution []. A long computation yields, at fourth order in perturbation theory,

a b b c c c e−2β = 1 + 1 + a r2 − η 1 + b r2 + 3 − η2 1 + c r2 + 3 + 4 + r 2 r 2 r4 r 2 r4 r7 d d d d e e e e e − η3 1 + d r2 + 3 + 4 + 5 − η4 1 + e r2 + 3 + 4 + 5 + 6 , r 2 r4 r7 r10 r 2 r4 r7 r10 r13 (184)

1 Λ˜ 2 with η = σλ ≡ 2 Λ2 being the small perturbation parameter. At zeroth order we have a1 = −2GM, a2 = λ. On the other hand, b1, c1, d1 and e1 are constants of integration while the rest of the constants have dependence in these ﬁrst ones. Clearly, for r 2GM, the asymptotic solution is the de Sitter spacetime, as expected,

2 ˜ 2 2 1 2 2 ds = 1 + Λpr dt + + r dΩ , (185) 2 1 + Λ˜ pr

2 3 4 with Λ˜ p = a2 − b2η − c2η − d2η − e2η . DRAFTC. Cosmology The details of this section can be found in [135]. Again, let us start by considering only the subtle vanishing torsion case. Considering a homogeneous and isotropic metric means that there exists a special spacetime foliation where each spatial section is maximally symmetric50. Therefore, the metric must be of a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) type [134]

dr2 ds2 = −dt2 + a2(t) + r2dΩ2 , (186) 1 − kr2

50 A maximally symmetric space is a space that has the same number of symmetries as the Euclidean space. In other words, a space is maximally symmetric if it has n(n + 1)/2 linearly independent Killing vectors for an n-dimensional space 45

where a(t) is the scale factor, dΩ2 is the solid angle and the constant k ∈ {−1, 0, 1} deﬁnes the curvature of the spatial sections. As usually used in the literature, it is convenient the following variables

a¨ a˙ 2 k l ≡ , h ≡ + . (187) a a a2

−1 2 −1 ˜ 2 51 We also deﬁne, ρΛ ≡ χ Λ and ρΛ˜ ≡ χ Λ . Hence, substituting the metric (186) in the ﬁeld equations (176) and (177), it is found

3 2 χ h − h + (ρ + ρΛ˜ ) = 0 , 2χρΛ 3 3 2 χ l − l − (ρ + 3p − 2ρΛ˜ ) = 0 , 2χρΛ 6 1 2 ∂t(l + h) = 0 . (188) (χρΛ) These are the complete equations governing the evolution of the cosmological scenario. We will explore them in diﬀerent energy scale sectors. We notice that the third equation in (188) is a novelty information. In typical theories this equation is not present. It arises from the fact that we have a curvature squared term and the fact that we are considering the ﬁrst order formalism.

1. The Λ-CDM model

The infrared sector is characterized by a low energy regime for gravity, i.e. weak curvature regime. The characteristic 2 energy scale of the model is given by Λ or, equivalently, through ρΛ. Thus, we have to compare the linear terms h and l with ρΛ. If h and l ρΛ, then the quadratic terms in (188) become negligible. Eﬀectively, one can reach the IR regime by taking the limit Λ2 → +∞ in (176) and (177). Hence52, χ h = (ρ + ρ ), (189) 3 Λ˜ χ l = − (ρ + 3p − 2ρ ), (190) 6 Λ˜ which are exactly the Einstein ﬁeld equations with a cosmological term. Therefore, the present stage of the Universe is correctly described by the model.

2. High curvature regime

Let us now consider a strong curvature regime in which the curvature square term is comparable with the other terms. In contrast to the IR regime, now the quadratic curvature becomes comparable to ρΛ and can no longer be neglected in (188). In what follows we shall outline separately the evolution of a vacuum from a matter ﬁlled spacetime. • Vacuum model The dynamical equationsDRAFT for a vacuum universe in the UV sector follow from (188) by setting ρ = p = 0,

3 2 χ h − h + ρΛ˜ = 0 , 2χρΛ 3 3 2 χ l − l + ρΛ˜ = 0 , 2χρΛ 3 ∂t(l + h) = 0 . (191)

51 Perfect ﬂuit energy-momentum tensor has to be added in (177). 52 See Sect. XI A for the argumentation to drop the third equation (188). 46

The ﬁrst two equations above are algebraic relations for h and l respectively. In fact, both equations have the same structure showing that h and l share the same spectrum. It is straightforward to identify the roots as

2 r 2 Λ ρΛ˜ (h, l) = ΛdS± = 1 ± 1 − 2 , (192) 3 ρΛ

Since both h and l are constant, the third equation in (191) is automatically satisﬁed. The Ricci scalar reads 2 2 R = 6(l + h) which for l = h gives R = 12 ΛdS± and for l 6= h we have R = 4Λ . The evolution of the scale 2 factor can automatically be integrated from the equation h = ΛdS±. The possible solutions read  cosh (Λ t); k = −1 , 1  dS± a(t) = 2 exp (ΛdS±t); k = 0 , (193) ΛdS±  sinh (ΛdS±t); k = 1 ,

q 2 where ΛdS± ≡ ΛdS±. One can immediately recognize these solutions being three different foliations of a de 2 Sitter universe with a cosmological constant given by ΛdS±. aa¨ 2 −1 −2 −2 The deceleration parameter is defined as q = − a˙ 2 = −l(h − k/a ) = −1/ 1 − k ΛdS±a , which is always negative. Hence, an accelerated expanding phase is manifest. ˜ 2 2 The effective cosmological constant depends both on ρΛ˜ and ρΛ. In the limit Λ /Λ 1, we can expand up to first order and obtain " !# Λ2 Λ˜ 2 Λ2 ≈ 1 ± 1 − . (194) dS± 3 Λ2

2 2 2 The root ΛdS+ is approximately proportional to Λ while the root ΛdS− is approximately proportional to Λ˜ 2. Given the enormous value of Λ2, the first root represents an universe with a violent de Sitter phase. Therefore, this phase can be associated with an inflationary expansion. On the other hand, the second root would correspond to a smooth accelerating phase similar to the late time expansion in the ΛCDM model. This result may be a Hint that a general solution connecting the inflationary era with the ΛCDM model could be developed. • Perfect fluid model In the second order formalism one has to provide an equation of state that in this case reduces to give a functional dependence of the pressure in terms of the energy density, i.e. p = p(ρ). In this case, the equation of state of the fluid is of major importance to completely specific the dynamic system. In contrast, the first order gravity analyzed in this paper has an extra set of dynamical field equations (188). They are a consequence of the dynamically independent character of the spin connection field. In principle we have three variables to determine the evolution, namely, the scale factor, the energy density and the pressure. However, with the thermodynamic equation of state we would have four equation for three variables and the system would be overdetermined. Indeed, the set of equations (188) is sufficient to establish the time evolution of all the variables. A possible way to reconcile this situation with a thermodynamic description of matter is to interpretation the UV sector as a regime where the gravitational field does not distinguish the nature of the matter fields, i.e. all perfect fluids gravitate in the same mannerDRAFT in the UV regime. For the two thermodynamic quantities it is found (see fig. 4) 2 2 3ξ0(4χρΛ − R0) −4 9ξ0 −8 R0 R0 ρ = 2 a − 2 a − ρΛ˜ − + 2 , (195) 4χ ρΛ 2χ ρΛ 4χ 32χ ρΛ 4χ − R0 −4 p = − ρ + 2 ξ0a , (196) χ ρΛ

The particular case ξ0 = 0 freezes the value of the energy density and the pressure becomes p = −ρ. Therefore, we have to assume the general case where ξ0 6= 0. Equations (195) and (196) show the universal behavior of the pressure and energy density independent of the equation of state of the ﬂuid. Indeed, if one assumes a barotropic equation of state p = ωρ with constant ω, equations (195) and (196) implies ω = −1. Thus, the only consistent barotropic equation of state in the UV sector is p = −ρ. It is clear then that the energy density and 47

FIG. 4: Behaviour of pressure and energy density according to scale factor. pressure given by (195) and (196) must be associated with a non-conservation of the energy-momentum tensor. Indeed, equations (188) are consistent with a˙ ρ˙2 ρ˙ + 3 (ρ + p) + = 0, (197) a 8(˙a/a)(ρ + ρΛ˜ − ρΛ/2) which verifies the non-conservation of the usual energy-momentum tensor. The first two terms are the usual components while the last non-linear term comes from the high curvature corrections. The scale factor can also be determined as s 2 ±αt 9k − 3R0ξ0 ∓αt 6k a(t) = x0e + 2 e + , (198) R0x0 R0 p where α = R0/3 and x0 > 0 is a constant of integration. The above solution has non-trivial R0 = 0 and k = 0 limits. Three different simple situations of (198) are displayed in Fig. 5. First, we take vanishing Ricci scalar but nonzero spatial section, providing q p 2 a(t) = ξ0k − k(t ± |ξ0|) . (199)

The ± sign within the square term does not change qualitatively the evolution. For k = 1 we have a big bang-big√ crunch solution with√ an initial and a final singularity. The scale factor reaches its maximal value at tmax = ξ0 where a(tmax) = ξ0. The k = −1 has to disjoint branches. Hence, both branches have an initial singularity and expands or it is a collapsing universe with a future singularity. For the plus√ sign the initial singularity is located at t = 0 while theDRAFT final singularity in the collapsing phase is at t = −2 ξ0. For flat spatial section the

FIG. 5: Time evolution of the scale factor. constant ξ0 must be positive deﬁnite and we have q p a(t) = ±2 ξ0t, (200) 48

where the constant of integration was chosen to locate the singularity at t = 0. Once again we have a collapsing phase for t < 0 that reaches the singularity and an expanding phase with initial singularity at t = 0. Finally, let us only comment that the characteristic of the general solution (198) depends on the interplay of the constants k, R0 and ξ0. In particular, the evolution can describe a bounce if the second term has the same sign as the ﬁrst one, namely, if R0 < 3/ξ0.

D. Dark stuﬀ

Nowadays, dark matter and dark energy could be expected features from a good fundamental gravity theory. They can have a completely different origin. However, the fact that dark matter only interacts only with gravity could be a clue about its gravity origin. Dark energy is not a problem if Λe is at our disposal. Dark matter, on the other hand, is a bit more complicated. There are two possible ways that the present approach could account for dark matter. The first one is to enlarge the gauge group to a bigger group encoding the SO(5). A known example is the SL(5, R) group, as in [36, 40]. In this case, the extra sector of the gauge group manifests itself in the geometric sector as a matter field which interacts only with gravity. Moreover, the mass of this field is very large since it is proportional to Λ2. This means that the range of this field is very very small. Hence, this field should very difficult to be detected, as expected from dark matter. The other possibility is to remain with the SO(5) group and consider the contribution of the Gribov-Zwanziger fields. This contribution is also small because these terms are proportional to ~2, since it comes from 1-loop contributions. Moreover, they would also interact only with gravity while a very small equivalence principle violation is expected due to the non-covariance of the Faddeev-Popov operator. this last property is not discarded from a possible dark matter behavior, see for instance [122–124].

Appendix A: Diﬀerential forms

This appendix is not a rigorous exposition of exterior differential forms. We provide here a brief survey about the main definitions and results only for operational matters. In other words, we provide here a kind of practical tutorial about differential forms. For details we refer to the vast existing literature on the topic, see for instance [62, 125–128].

1. Exterior product and p-forms

For generality purposes, let us consider D-dimensional manifold M D as the starting scenario. For each differentiable D D curve passing at a generic point x ∈ M , there is a tangent vector field. The tangent space Tx(M ) is the collection of all tangent vectors at x. The differential operators {∂µ}, with µ ∈ {1, ··· ,D}, are the components for which a D ∗ D basis in Tx(M ) is constructed. The dual space Tx (M ) is called cotangent space and its basis is defined by the objects dxµ. Such duality is imposed through the inner product rule

ν ν (∂µ, dx ) = δµ . (A1)

∗ D The exterior product is typically represented by the symbol ∧. For two components of the basis for Tx (M ), the exterior product is defined as the anti-commuting product DRAFTdxµ ∧ dxν = dxµ ⊗ dxν − dxν ⊗ dxµ = −dxν ∧ dxµ . (A2) In the case of µ = ν we have a null exterior product, obviously. It is easy to see that the exterior product is associative. For the sake of simplicity, from now on, we omit the ∧ symbol from the notation. An exterior differential form ω, or simply a p-form, is defined by 1 ω = ω dxµ1 dxµ2 ··· dxµp , (A3) p! µ1µ2···µp where ωµ1µ2···µp are covariant tensor components of order p 6 d. Due to (A2), these components are completely anti-symmetric on their indexes. Moreover, the restriction p 6 d follows from the obvious fac The exterior product between a p-form α and a q-form β results in a r-form γ such that r = p + q, which is non-vanishing if r 6 d, given by γ = αβ = (−1)pqβα , (A4) 49 where the factor ±1 comes from the anti-commuting nature of the exterior product: two generic forms will anti- commute only if both forms are odd in their ranks.

2. Exterior derivative

The diﬀerential operator d, which is called exterior derivative, is deﬁned by

ν d = ∂ν dx . (A5) It acts on p-form as follows 1 1 dω = (∂ ) dxν ω dxµ1 dxµ2 ··· dxµp = ∂ ω dxν dxµ1 dxµ2 ··· dxµp . (A6) p! ν µ1µ2···µp p! ν µ1µ2···µp It is evident that the exterior derivative is a map from the space of p-forms to the space of (p + 1)-form. The exterior derivative has two important properties. The ﬁrst of them concerns its nilpotency, d2 = 0. The second is about its application on a product of forms and the Leibniz rule. To illustrate this second point, let us take a p-form α and a q-form β. The action of the exterior derivative on the product is explicitly shown below,

d(αβ) = dαβ + (−1)pαdβ . (A7)

If the exterior derivative acts on a sum of forms, then

d(c1α + c2β) = c1dα + c2dβ , (A8) where c1 and c2 are 0-forms.

3. The Hodge dual operator

The Hodge duality consists in an operation that maps a p-form into a dual (D − p)-form. The Hodge dual operator is a map as follows, √ g µ ···µ ∗ dxµdxµ ··· dxµ = 1 p dxµp+1 ··· dxµD , (A9) 1 2 p (D − p)! µp+1···µd where g = | det gµν |. Hence, the Hodge dual associated to a p-form ω is √ g µ ···µ ∗ ω = ω 1 p dxµp+1 ··· dxµD . (A10) p!(D − p)! µ1···µp µp+1···µd

The Hodge duality is an isomorphism that maps the space of all p-forms Ep in the space of all D − p-forms ED−p as

p ? D−p E 7−−→ E . (A11) Since the dimension of the space p-form is given by D D! DRAFTp dim = ≡ , (A12) E p p!(D − p)! it is straightforward to see that D dim D−p = , (A13) E D − p i.e., the dimension of the dual space has the same dimension of the space of p-forms. The Hodge dual operator satisﬁes the linear property,

∗ (c1α + c2β) = c1 ∗ α + c2 ∗ β , (A14) where c1 and c2 are 0-forms. When the Hodge dual operator is twice employed, we obtain 50

( (−1)p(D−p) if D is Euclidean, ?? = M . (−1)p(D−p)+1 if MD is Minkowskian. Hence, the double Hodge dual operation brings back the p-form to the original shape up to a signal. Another interesting mathematical relation is about Levi-Civita tensor. In a D-dimensional Euclidean space the volume form is related to that tensor as 1 dx1 ··· dxD = dxµ1 ··· dxµD ⇒ dxµ1 ··· dxµD = µ1···µD dx1 ··· dxD , (A15) D! µ1···µD while in a D-dimensional Minkowskian space we have 1 dx0 ··· dxD−1 = dxµ1 ··· dxµD ⇒ dxµ1 ··· dxµD = −µ1···µD dx0 ··· dxD−1 , (A16) D! µ1···µD since the full contraction between the Levi-Civita tensor is given as follow, ( D! for Euclidean space µ1···µD = µ1···µD −D! for Minkowskian space . It is also possible to employ the vielbein to project the Levi-Civita tensor indexes,

ea1 ea2 ··· eak = , µ1 µ2 µk a1a2···akµk+1µk+2···µk+` µ1µ2···µkµk+1µk+2···µk+` e µ1 e µ2 ··· e µk a1a2···akµk+1µk+2···µk+` = µ1µ2···µkµk+1µk+2···µk+` . (A17) a1 a2 ak The inverse relations are easily found

ea1 ea2 ··· eak µ1µ2···µkak+1ak+2···ak+` = a1a2···akak+1ak+2···ak+` , µ1 µ2 µk e µ1 e µ2 ··· e µk = . (A18) a1 a2 ak µ1µ2···µkak+1ak+2···ak+` a1a2···akak+1ak+2···ak+`

4. Integration of diﬀerential forms

The integral of a p-form ω over an oriented region U ∈ Rp is defined as Z Z 1 Z 1 µ1 µp 1 p ω = ωµ1···µp dx ··· dx = o(x) ωµ1···µp dx ··· dx , (A19) U U p! U p! where o(x) = ±1 depending on the orientation of the coordinate basis. The result is, obviously, a (p − 1)-form. An important property that uses the integration of p-forms is contained in the definition of the inner product. Let α and β be generic p-forms. If α ∗ β is an r-form, then we define the inner product on a p-dimensional manifold M p as Z Z (α, β) = α ∗ β = β ∗ α . (A20) d M U Finally, the Gauss-Stokes’ theorem in this formalism can be enunciated: Z I dω = ω . (A21) DRAFTU ∂U

Appendix B: Geometrical aspects of gauge theories

In this appendix we summarize some formal concepts about the geometrical structure of gauge theories. A parallel with the gravity geometrical structure is also presented. Notions about topological spaces, maps and manifolds may be required. Proofs are omitted. Standard references are [2, 62, 125]. See also [63] for a short discussion and [129] for an advanced approach. For simplicity we employ diﬀerential form notation and set κ = 1. 51

1. Principal bundles and connections

We have learned about Lee groups at Sec. I C 1. Together with a differential manifold (spacetime, in this case), A Lie group form the basic structure for gauge theories. Such structure is a sophisticated product of topological spaces called principal bundles, which is a particular case of the more general concept of fibre bundles. Formally, a Fibre bundle is a space B(G, M, F ), called total space, obeying the following conditions [62] • B is a differential manifold;

• M, called the base space, is a differential manifold with an open covering {Ui}; • F , called fibre, is a differential manifold; • G is a Lie group, called structure group, with left action on F ; •∃ a surjective map53 π : B 7−→ M, called projection map.

54 −1 55 •∃ a diﬀeomorphism φi : {Ui} × F 7−→ π (Ui) such that π ◦ φi(p, f) = p where p ∈ M and f ∈ F . Equivalently, φi(p, f): F 7−→ Fp is a diﬀeomorphism.

−1 • Whenever Ui ∩ Uj 6= ∅ we have tij(p) = φi (p, f) ◦ φj(p, f): F 7−→ F ∈ G. Moreover, tij(p) must respect

– For p ∈ Ui, tii(p) = id; −1 – For p ∈ Ui ∩ Uj, tij(p) = tji (p);

– For p ∈ Ui ∩ Uj ∩ Uk, tij(p)tjk(p) = tik(p);

−1 The inverse of the projection defines the fiber at p ∈ M, π :(p) = Fp, which is a “copy” of F at p. The diffeomorphism φi is called the local trivialization and is analogous to the concept of local Cartesian product. The maps tij is a smooth map between φi and φj and are called transition functions. Further important definitions are:

−1 • A section on B is a smooth map s : M 7−→ B π ◦ s = id. Thus, s(p) = π (p) ∈ Fp. A global section is a section defined independently of Ui while a local section is only defined in Ui. • A fibre bundle is a trivial bundle if all transition functions can be taken as identity maps, i.e. a trivial bundle is a direct (Cartesian) product M × F . Equivalently, a trivial bundle is a fibre bundle which admits a global section. Nevertheless, a fibre bundle is also trivial if the base space M is contractible to a point. A principal bundle is simply a fibre bundle where the fibre coincides with the structure group, F = G, and is denoted by P (G, M). Besides the left action of G on F = G, the right action of G on the fibre can be defined by −1 means of: Let u ∈ G and q = (p, fi) ∈ P then φi (qu) = (p, fiu). Applying φi(q) from the left in this last expression, we find qu = φi(p, fiu). The right action is map Fp 7−→ Fp and is possible because of the closure property of groups. A connection on a principal bundle is the fibre bundle analogue of affine connections in Riemannian manifolds. It allows to compare vector in the tangent space of P (G, M), providing the notion of parallelism in the total space P . To do this, we split the tangent space of P , Tq(P ), at a point q = (p, g) ∈ P into its vertical and horizontal subspaces, namely Vq and Hq, respectively. The vertical space is tangent to a fiber Gp. As a consequence, as discussed in Sec. I C 1, it is equivalent to the algebra of G. The horizontal space is a complement of Vq, which is not unique. Hence, a connection is a uniqueDRAFT split of Tq(P ) obeying • Tq(P ) = Hq ⊕ Vq

H Q H V • A vector ﬁeld X ∈ P split as X = X + X , where X ∈ Hq and X ∈ Vq;

• At the same ﬁbre, horizontal spaces are related by the right action of G, i.e. Rg : Hq 7−→ Hqg.

53 A surjective map between two topological spaces is a map whose image is entirely covered by the map. 54 A diﬀeomorphism between two topological spaces is a smooth and continuous map whose inverse is also continuous and smooth, i.e. a diﬀeomorphism is a smooth homeomorphism. 55 Obviously, q = (p, f) is a point of the total space B. 52

Thus, a connection is a set of conditions that picks only one specific Hq amongst all possible horizontal subspaces. In accordance with the above definitions (see [62]), it is possible to define the connection one-form as an algebra- ∗ valued one-form ω ∈ Tq (P ) × AlgG, which is a projection of Tq(P ) in Vq. The corresponding local connection form ∗ 1 ∗ is the pullback of ω to Ui, i.e. Ai = σi ω ∈ Ω (Ui) × AlgG where σi is the pullback from q to p associated to a local 1 section σi and Ω (U)i) is the space of one-forms in Ui ∈ M. To ω be uniquely defined on Ui ∩ Uj, it is required that

−1 −1 −1 Ai 7−→ t ijAitij + tij dtij , (B1) where d is the exterior derivative. To make contact with the known concept of affine connections, we have to define the horizontal lift of a curve: Given two curves c : [0, 1] 7−→ M and ec : [0, 1] 7−→ P , ec is a horizontal lift of c if π ◦ ec 7−→ c and the tangent vector of c belongs to H . It can be shown [62] that there exists a map π−1(c(0)) 7−→ π−1(c(1)) while q 7−→ q which defines e ec 0 1 the parallel transport of q0 through the curve ec in such a way that Z µ q1 = σi(c(1))O exp − Aiµdx , (B2) c where σi(c(1)) is the local section at c(1) ∈ Ui, O is the path ordering operator and t ∈ [0, 1] is the parametrization of the curve. Eq. (B2) clearly states that the components of Ai are responsible for the parallel transport along c. The covariant derivative rises in a natural way as the horizontal exterior derivative in P [62, 63], namely dP = D. In Ui, the covariant derivative components are

Di = d + Ai . (B3)

The commutation between covariant derivatives measures the commutativity of the parallel transport of vector in P 2 along a curves ec1 and ec2 which have the same starting point ec1(0) = ec2(0). In fact, D = Dω leads to components of the form

Fi = dAi + AiAi . (B4)

The basic structure of the gauge theories we are considering in this lectures is the principal bundle P (G, R4) where the base space is the four-dimensional Euclidean spacetime, G is the gauge group. In a sense, P (GR4) is a formal manner to localize the rigid Lie group G by making a copy of it at each point of the spacetime. Moreover, since R4 can be contracted down to a point, P (G, R4) is a trivial bundle. The local connection form in P (G, R4) is then recognized as the gauge field and (B1) is the corresponding gauge transformation in the notation of fibre bundles. Thus, the gauge field is a natural geometrical quantity in the abstract space P (G, R4). The covariant derivative components (B3) are recognized as the covariant derivative in gauge theories (32) while the curvature components (B4) is directly identified with the field strength (36). Hence, a pure geometrical scenario is set to explore gauge theories. Given a gauge invariant classical action for A and the respective field equations, the solution of the theory with appropriate boundary conditions is equivalent to find the appropriate unique horizontal space Hq. For a quantum theory however, we need another principal bundle which takes into account all possible gauge configurations, i.e. the set of all possible horizontal spaces in P .

2. Universal bundles

The most suitable scenario forDRAFT a quantum gauge theory is a universal bundle, denoted by U(G, A/G) where the total space is the space of all possible gauge connections configurations U = A while the fibre and structure group are the Lie group G. The base space is the space off all independent gauge connections A/G, i.e. the space of gauge connections not related through gauge transformations. In other words, one defines equivalence classes in A and take one representative A of each equivalence class, the set of all A is the base space and is called moduli space. The fibre is constructed by π−1 which manifest itself through the gauge orbit (44),

Au = u−1(A + d)u . (B5)

In this context, the gauge fixing is an attempt to define a global section in U, which is not possible because U is not trivial since A/G is not contractible to a point. In fact, this is the mathematical kernel of Gribov ambiguities [15] which states that is not possible to globally fix the gauge of a gauge theory, i.e. it is not possible to pick only one representative of each equivalence class [A]. 53

Besides Au and F u, there are two more natural structures in U, namely, the BRST operator and the Faddeev-Popov ghost ﬁeld. If one deﬁnes the functional exterior derivative in U by56

Z δ δ = d4x δAu , (B6) u µ δAµ

u µ where δAµ is the functional analogous of dx , it is possible to show that

δAu = −D(Au)(u−1δu) + u−1δAu . (B7)

Clearly, the first term is the derivative along the fibre G while the second term is the derivative along the base space A/G. Let us consider the anti-commuting argument of the covariant derivative of the first term, c = u−1δu, its derivative reads57

δc = −u−1δuu−1δu = −cc . (B8)

We remark that, although c is an anti-commuting quantity, cc = c2 6= 0 because it is an algebra-valued quantity. If we consider δ along the ﬁbre, we end up with

u u δ fibreAµ = −D(A )c ,

δ fibrec = −cc . (B9)

Thus, the BRST operator can be interpreted as the functional derivative in U along the ﬁbre58, s = δ . Moreover, fibre c is recognized as the Faddeev-Popov ghost ﬁeld. Geometrically, c is the Maurer-Cartan form on group space while the second equation in (B9) is known as the Maurer-Cartan structure equation.

3. Gravity geometrical structure

Gravity can also be described in terms of principal bundles and, hence, as a gauge theory. However, extra features arise. 4 4 4 First, let us consider a diﬀerential manifold M and the tangent spaces TX (M ) at points X ∈ M . The set of all tangent spaces,

\ 4 T = TX (M ) , (B10) X∈M 4

4 4 is called tangent bundle. The tangent bundle has M as the base space, the fibre is TX (M ) and the structure group is the GL(4, R) group, characterizing the transition functions as local coordinate transformations. It should be clear µ 4 that a point q ∈ T is characterized by q = (X, v) where V is a vector v = v ∂µ while π : T 7−→ M provides −1 4 ∗ π (X) = TX (M ). Analogously, the cotangent bundle T can be defined as the set of all cotangent spaces in M by µ changing vectors by forms and the basis ∂µ by the dual basis dx . The fundamental structure for gravity is the frame bundle or, equivalently, the coframe bundle. The coframe bundle C is the equivalent of the principal bundle P (G, R4) of gauge theories. The coframe bundle is associated to 4 a a µ the cotangent bundle where theDRAFT fibre at X ∈ M is the set of all local coframes e = eµdx that can be defined 4 4 in TX (M ) and the base space is M . An element of C is then q = (X, e). Since all frames are related through GL(4, R4) transformations (right action), the fibre coincides with the GL(4, R4) group. Hence, since the transition 4 functions are the same as the cotangent bundle, tij ∈ GL(4, R ), the coframe bundle actually is a principal bundle C = (GL(4, R4),M). Since the right action of the group leaves the coordinates in p intact, it is associated with gauge transformations while the left action is associated with coordinate transformations. Thus, C is the perfect basic structure to define gravity. In fact, since GL(4, R4) is contractible to O(4), the reduced coframe bundle O =

56 Just like the usual exterior derivative, δ is an anti-commuting operator. 57 We have used δ)(u−1u) = 0. 58 It is an easy exercise to show that, in algebra components, (B9) reduce to (58a-58b) 54

P (O(4),M 4) coincides with the mathematical structure of a Einstein-Cartan gravity. Moreover, the natural connection in O is recognized as the spin-connection ωab. In essence, gravity is a gauge theory where the gauge group is identiﬁed with the local isometries of spacetime. Although a background independent quantum version of any gravity theory appear to be inconsistent, its mathematical structure is not, yet it is a bit more complicated than U, deﬁned at the previous section... ♠

Acknowledgments

We are thankful to the organizing committee for the opportunity in giving this lectures at the X Escola do Centro Brasileiro de Pesquisas F´ısicas, 13-24 July 2015. ADP acknowledges SISSA and the Theoretical Particle Physics division for hospitality and support. The Conselho Nacional de Desenvolvimento Cient´ıficoe Tecnológico59 (CNPq- Brazil) and the Pró-Reitoriade Pesquisa, Pós-Gradua¸cãoe Inova¸cão(PROPPI-UFF) are acknowledged for financial support.

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