Universit`
a degli Studi di Pisa
FACOLT `A DI SCIENZE MATEMATICHE, FISICHE E NATURALI Corso di Laurea in Fisica
Tesi di laurea
Higher derivative NLSM and the construction of
a critical 4-dimensional brane
Candidato: Matteo Romoli Matricola 535780 Relatore: Dr. Omar Zanusso Anno Accademico 2019-20
Contents
1 Introduction 7
1.1 String theory: derivation of Polyakov action . . . 7
1.2 String theory: fields and critical dimension . . . 10
1.3 Conformal field theories . . . 12
1.4 Conformal anomaly . . . 16
1.5 Higher dimensional objects: Dp-brane . . . 18
1.6 Fourth order Non-Linear σ Model and 4-branes . . . 20
2 Two-dimensional string 23 2.1 Bosonic string action . . . 23
2.2 Weyl anomaly . . . 24
2.3 Covariant background expansion . . . 24
2.4 Heat kernel method . . . 27
2.5 Dynamical metric . . . 27
2.6 Low-energy effective action . . . 30
3 Higher derivative nonlinear σ model 33 3.1 Covariant background field expansion . . . 34
3.2 Heat Kernel method . . . 37
3.3 Fixed points analysis . . . 41
3.4 Adding a dimension 2 term . . . 42
4 4-brane theory 47 4.1 4-brane action . . . 47
4.2 Dynamical metric . . . 50
4.3 Conformal invariance and critical dimension . . . 55
4.4 Discussion of β(T ) and the case A 6= 0 . . . . 56
5 Conclusions 59 A Covariant background field expansion 65 A.1 Generating functionals and background field method . . . 65
A.2 Covariant background field method . . . 66
A.3 Geodesic covariant derivative . . . 67 3
4 CONTENTS
B Heat Kernel method 69
B.1 Heat kernel introduction . . . 69 B.2 Heat kernel in curved space . . . 73 B.3 Higher derivative heat kernel . . . 74
Summary
String theory is one of the most ambitious and fascinating projects in the history of theoretical physics. It purports to be an all-encompassing theory of the universe, unify-ing the forces of Nature, includunify-ing gravity, in a sunify-ingle quantum mechanical framework. Using the only assumption that all particles are, at a fundamental level, excitations of one-dimensional strings, the theory tries to create a model in which general relativity, electromagnetism and Yang-Mills gauge theories emerge.
This revolutionary strong postulate, on which string theory is developed, could provide solutions to many of the problems that arise from traditional particle physics, such as the incompatibility between quantum mechanics and general relativity, giving thus the basis for a theory of quantum gravity. A particularly successful feature of the string theory framework is to consistently reproduce the dynamics of spacetime in the form of Einstein’s equations. In fact, they emerge at low energies, through the requirement of internal consistency of string theory, if seen as a field theory. Together with Einstein’s equations, string theory also predicts the number of spacetime dimen-sions, known as critical dimension, which is a rather unique achievement if compared to other approaches. In fact, the string is believed to be the unique theory which is capable of such goals.
The critical dimension comes about from the requirement that the theory is con-formal (or Weyl) invariant. This requirement translates to the fact that the trace of the stress-energy tensor, Tµ
µ, must be zero, even after taking into account the quanti-zation process that, in general, could break the symmetry through the quantum Weyl anomaly. The trace of the stress-energy tensor is directly related to the β functions coming from the process of renormalization of the underlying field theory.
It is worth noting that the string theory framework often requires the inclusion of higher dimensional generalizations of the string itself, known as Dp-branes, which are (p + 1)-dimensional objects. Branes are often used to fix boundary conditions for open strings and provide basic degrees of freedom to construct gauge theories. However, in this framework, branes are also treated in a rather passive way, since the action by which they are described presents conceptual problems in relation to quantization and renormalization.
In this thesis, we introduce an entirely new four dimensional object that we call the 4-brane. It could naively be compared to a D3-brane, being both four dimensional object, but perhaps that would be misleading and certainly incomplete. In fact, we show that the 4-brane can play the very active role of producing Einstein’s equations at low energies. This happens because we construct the 4-brane by paying close attention to the same requirement of Weyl invariance that is enforced on the critical string.
6 CONTENTS From the field-theoretical point of view, the 4-brane is regarded as a higher-derivative nonlinear sigma model (NLSM), whose fields have the natural interpretation of coordinates of the four dimensional brane. When treating it as a field theory, and discussing the necessary steps to eliminate its conformal anomaly, we follow closely the traditional steps of string theory, looking for both a space-time low-energy description and a critical dimension. In practice, most of the original work of this thesis is on the renormalization of a generalized higher-derivative NLSM on a curved background, which is the brane itself, and on the determination of the main contributions to the Weyl anomaly. Among various things, we find that scale invariance of the NLSM implies that the 4-brane is governed by the so-called Fradkin-Tseytlin-Paneitz-Riegert operator, which is the unique fourth order conformal operator. Furthermore, the breaking of conformal symmetry can be caused by the insertion of a term which has an interaction of mass dimension two and therefore could play the role of a mass for the model.
We conclude the thesis by applying the results found for the NLSM to the devel-opment of the consistent 4-brane theory. As anticipated, a particularly interesting result is that the 4-brane seems to work as well as the string in producing a low-energy limit that contains the dynamics of general relativity. Even more surprisingly, the consistency of the 4-brane determines a critical value for the number of spacetime dimensions. Even though these conclusions must be taken with a grain of salt, espe-cially because much more work is needed to truly establish the 4-brane as a completely consistent object, we could speculate that the new four dimensional object is a serious contender to the uniqueness of the string.
Chapter 1
Introduction
String theory is one of the most ambitious and fascinating project in the history of physics. It purports to be an all-encompassing theory of the universe, unifying the forces of Nature, including gravity, in a single quantum mechanical framework. With the only assumption that matter is, at a fundamental level, composed by one-dimensional objects instead of point particles, it is possible to create a model in which general relativity, electromagnetism and Yang-Mills gauge theories emerge. The special role of this model is precisely due to the fact that it is possible to consistently reproduce the geometry of the spacetime. In fact, after introducing in this chapter theoretical aspects on the models we are going to work on, we show in chapter 2 how to derive general relativity equations from the closed bosonic string action. Then, in chapter 3, we focus our attention on the fourth order non linear sigma model (NLSM) with emphasis on conformal invariance aspects. In chapter 4 we use the NLSM to describe a theory based on a 4-dimensional brane (4-brane), that we will be soon introduced. We formulate our model in complete analogy to string theory. We show that it is possible to derive a coherent theory based on a different fundamental entity that reproduce, at least at one-loop approximation, Einstein field equations.
1.1
String theory: derivation of Polyakov action
A convenient analogy to introduce the Polyakov’s action begins with the relativistic point particle. In this introduction we follow the presentation given in the most of the works on string theory, as in [1], [2] and [3]. We are considering a D-dimensional Minkowski spacetime R1,D−1 working with signature η
AB = (−1, +1, ..., +1). The relativistic action of a point particle is
S = −m Z
dτ q
−ηABX˙AX˙B, (1.1)
where A = 0, 1, ..., D − 1. The parameter τ , is an arbitrary parameter that labels the position of the particle along the worldline. This action is nothing else than the proper time R ds. This formulation is manifestly Poincar´e invariant and has also a gauge symmetry: it is invariant under any reparametrization ˜τ(τ ) (monotonic function). This implies that not all D degrees of freedom XAare dynamical, because the solutions
8 CHAPTER 1. INTRODUCTION are function of τ and τ itself is not meaningful. Like all gauge symmetries, it is not really a symmetry at all. Rather, it is a redundancy in our description, that is evident in the presence of an extra condition of the momenta pA:
pApA+ m2 = 0. (1.2)
This is a the mass-shell constraint for a relativistic particle of mass m. From the worldline perspective, it tells us that the particle is not allowed to sit still in Minkowski space: at the very least, it had better keep moving in a timelike direction with (p0)2 ≥ m2. To treat better our theory a fictitious degree of freedom can be added by inserting an independent metric on the worldline. We are coupling with 1d gravity (“einbein”), calling the field e(τ ). The action become
S = 1 2
Z
dτ e−1X˙2− em2. (1.3)
This extra variable does not change the physical degrees of freedom because e is determined by its equation of motion, that, upon insertion in (1.3), reduces the action to (1.1). However the formulation (1.3) is better for two reasons: firstly, it works for massless particles and secondly, the absence of the annoying square root means that it is easier to quantize in a path integral framework.
Now we are ready to move to strings. We consider only closed strings, even if a brief introduction of the open strings is made on section (1.3), and, firstly, string on a flat D-dimensional spacetime. As a point-particle could be represented as a world line in Minkowski spacetime, a string sweeps out a worldsheet. To parametrized it we use a timelike coordinate x0 and a spacelike coordinate x1 that has a range [0,2π). it is possible to define a map from the worldsheet to the Minkowski spacetime, that we call XA(x0, x1). For closed strings we require that XA(x0, x1) = XA(x0+ 2π, x1). Let us call xµ = (x0, x1); µ = 0, 1: if we want to describe the dynamics of the string we need an action and, as is natural to expect, we require that it does not depend on the coordinate we choose on the worldsheet. We are asking for reparametrization invariance. As the action for a pointlike particle is proportional to the length of the worldline, we may suppose that the action is proportional to the area of the worldsheet, that is a curved surface embedded in spacetime. The induced metric on this surface, gµν, is the pull-back of the metric on Minkowski space:
gµν = ∂XA
∂xµ ∂XB
∂xν GAB. (1.4)
So we can write the Nambu-Goto action for a relativistic string: S = −T
Z
d2xp− det (g). (1.5)
By the definition in (1.4), calling ˙XA= ∂X A
∂x0 and (X
A)0 = ∂X A
∂x1 , the action could be written S = −T Z d2x q −( ˙X)2(X0)2+ ( ˙X · X0)2, (1.6)
1.1. STRING THEORY: DERIVATION OF POLYAKOV ACTION 9 where T is a proportionality constant with an interpretation of the string tension (mass per unit length). To show it we can work in a gauge with X0 ≡ t = Rx0, where R is a constant that is needed to balance up dimensions. Consider a snapshot of a string configuration at a time when d~x/dx0 = 0 so that the instantaneous kinetic energy vanishes. Evaluating the action for a time dt gives
S = −T Z
dx0dx1Rp(d~x/dx) = −T Z
dt ×(spatial length of string), (1.7) but, when the kinetic energy vanishes, the action is proportional to the time integral of the potential energy,
potential energy = T × (spatial length of string). (1.8) So T turns out to be an energy per unit length as claimed. It is sometimes convenient to introduce α0 as:
T = 1
2πα0. (1.9)
It is easy to verify that α0 has the dimension of a length squared and we can introduce a natural length scale ls, called string scale:
α0 = ls2. (1.10)
The Nambu-Goto action has two symmetries: Poincar´e and reparametrization in-variance. To derive the equations of motion for the Nambu-Goto string, we first introduce the momenta
Π(xA0) = ∂L ∂ ˙XA = −T (X·X˙ 0)X0 A−(X 02)X˙A q (X·X˙ 0)2− ˙X2X02 Π(xA1) = ∂L ∂X0A = −T (X·X˙ 0)X˙ A−(X˙2)XA0 q (X·X˙ 0)2− ˙X2X02 (1.11)
and then we obtain
dΠ(xA0) dx0 +
Π(xA1)
dx1 = 0. (1.12)
Recalling action (1.5) and varying the determinant (δ√−g = 1 2
√
−ggαβδg
αβ) we can write equation (1.12) in a manifestly covariant way:
∂µ( √
−ggµν∂νXA) = 0, (1.13)
Following the ideas that led us to write action (1.3) we can introduce the Polyakov action: SP = 1 4πα0 Z d2x√−ggµν∂ µXA∂νXBGAB. (1.14) The field gµν corresponds to the metric on the world sheet, we have now coupled the Nambu-Goto action with 2d gravity. The equation of motion for XA is
∂µ( √
−ggµν∂
10 CHAPTER 1. INTRODUCTION but now gµν is treated as an independent variable. Therefore we can vary the action respect to the metric:
δSP = T 2 Z d2xδgµν √ −g∂µXA∂νXB− 1 2 √ −ggµνgρσ∂ρXA∂σXB GAB = 0. (1.16) Note that for the moment we have always considered flat spacetime, so we could have written GAB = ηAB, but, as we are interested in curved spacetime, changing notation it is not convenient. We obtain
gµν = 2f (x)∂µX · ∂νX, (1.17)
where the function f (x) is given by
f−1 = gρσ∂ρX · ∂σX. (1.18)
The definition of gµν differs from (1.4) for the conformal factor f, but for the Weyl invariance of the Polyakov action
SP(X, g) → SP(X, Ω2g)
the equation of motion are unaffected. So the action (1.14) and (1.6) are the same, but, similarly to the point particle analysis, the first is simpler on the quantization point of view.
There is one condition: we want the variation of the Polyakov action with respect to the metric to be zero. This is analogous to ask the stress-energy tensor Tµν to vanish. In practice, this leads to two constraints:
T01= ˙X · X0 = 0, T00= T01 = 1 2( ˙X 2+ X02 ) = 0. (1.19)
The first constraint tells us that the motion of the string must be perpendicular to the string itself. In other words, the physical modes of the string are transverse oscillations. There is no longitudinal mode.
1.2
String theory: fields and critical dimension
Without going in too much detail we want to present the fundamental fields on which string theory is built. The process of quantization pass trough the construction of a Fock space and the most popular way to do it is using the so called lightcone quantization, based on the use of lightcone coordinates x± = x0± x1. This choice of coordinates drastically simplifies the equations of motion, that become
∂+∂−XA= 0 (1.20)
and then the general solution could be written as XA(x0, x1) = XA
L(x+) + XRA(x −
1.2. STRING THEORY: FIELDS AND CRITICAL DIMENSION 11 for arbitrary functions XA
L(x+) and XRA(x
−). These describe left-moving and right-moving waves respectively.
Expanding in modes the functions considering all constraints as symmetries and periodical conditions and imposing canonical commutation relations we can derive some fundamental results:
The physical solutions of the string are described by 2(D−2) functions, D−2 for right-moving and D − 2 for left-moving. This counting has a nice interpretation: the degrees of freedom describe the transverse fluctuations of the string (see section 1.1).
A definition of a vacuum state that is different from the usual vacuum of the quantum field theories because carries another quantum number (position or momentum). For example, if we work in momentum space, we label the fun-damental state also with the eigenvalue of the momentum operator |0, pi. It is associate to the tachyon, that has the peculiarity to posses negative mass. We do not take it into consideration, anyway the extension to the superstring resolve this problem.
A set of creation and annihilation operators αi
n,α˜jm with n, m ∈ Z \ {0} and i, j = 1, ..., D − 2 . The negative values of n, m are associated to creation op-erators and positive values to annihilation opop-erators. The αi
n are used to the right-moving modes, while ˜αj
m for the left-moving ones.
In this formalism the first excited states are given by αi−1α˜
j
−1|0, pi. (1.22)
They are (D − 2)2 states with mass M2 = 4 α0 1 − D −2 24 . (1.23)
However a problem arises: the operators αi and ˜αi each transform in the vector representation of SO(D − 2) ⊂ SO(1, D − 1). However we want these states to fit into some representation of the Poincar´e group. Recalling Wigner’s classification of representations of the Poincar´e group, we start by looking at massive particles in R1,D−1. After going to the rest frame of the particle by setting pA= (p, 0, . . . , 0), we can watch how any internal indices transform under SO(D − 1) and the conclusion of this is that any massive particle must form a representation of SO(D−1). Nevertheless we have (D − 2)2 states and we cannot pack these states into a representation of SO(D − 1). It seems that the first excited states of the string cannot form a massive representation of the D-dimensional Poincar´e group. And this is why we must impose to have massless excitations. This does not allow us to go in rest frame. The best that we can do is choose a spacetime momentum for the particle of the form pA = (p, 0, . . . , 0, p). In this case, the particles fill out a representation of the little group SO(D − 2). This means that massless particles get away with having fewer internal
12 CHAPTER 1. INTRODUCTION states than massive particles and so (1.22) sit in a representation of SO(D − 2). We learn that if we want the quantum theory to preserve the Lorentz symmetry that we started with, then the states have to be massless. Looking at (1.23) we see that there is just one possible dimension of the spacetime that agrees with our assumption:
D= 26. (1.24)
Before commenting this result let us make the observation that the states (1.22) transforms in the 24 ⊗ 24 representation of SO(24). These decompose into three ir-reducible representations:
traceless symmetric ⊕ anti-symmetric ⊕ singlet (= trace ).
To each of these modes, we associate a massless field in spacetime such that the string oscillation can be identified with a quantum of these fields. The fields are:
GAB(X) , BAB(X) , Φ(X). (1.25)
The first is a massless spin 2 particle, identified as the graviton, the second is an anti-symmetric tensor field which is usually called “Kalb-Ramond field” or, in the language of differential geometry, the “2-form”, and the scalar field is called dilaton. Some comments on these fields are added in chapter 2 where we show how they appear in the complete bosonic string action.
Now that all the fields are presented it is possible to come back to (1.24). This is the well-known critical dimension and it turns out to be the only possible dimension of spacetime where a theory of string can be consistently formulated with the inclusion of the dilaton background field. This is a fundamental result in the string theory, because it is the only theory that tells us the dimensions of the spacetime. Neither general relativity nor quantum field theories provide a model in which the structure of spacetime emerges. This is one of the most important point of this framework. In the next chapter we show an alternative way to calculate the critical dimension, which can be generalized to our 4-brane model.
1.3
Conformal field theories
A crucial role in all the calculations that we are going to do is represented by the conformal symmetry of the theory and, in particular, by the conformal anomaly that appears in the quantization. Therefore is important to introduce the argument. We follow the presentation given by most of the string books. See, as example, [1]. A conformal transformation is a change of coordinates x → ˜x(x) such that the metric trasforms as
gµν(x) → Ω(x)2gµν(x). (1.26)
A conformal field theory (CFT) is a field theory invariant under these transformations. This means that the physics of the theory looks the same at all length scales. We could think the conformal invariance as a local change of scale which preserves the angles
1.3. CONFORMAL FIELD THEORIES 13
Figure 1.1: Example of conformal invariance: the two worldsheet metrics shown in the figure are seen by the Polyakov string as equivalents.
between all lines. For example the two worldsheet metrics shown in the figure 1.1 are seen by the Polyakov string as equivalents.
We could interpret the transformation of the metric in different ways depending on whether we are considering a fixed background metric or a dynamical one. In the first case the conformal symmetry is a real global symmetry, with corresponding currents. In the second case we can interpret the conformal invariance as local, that is gauge symmetry. CFTs find a large application in several physical problems, for example in the study of ferromagnetic transition. A beautiful image in [4] can be used as an example to visualize implications of conformal invariance, we report it in figure 1.2.
Figure 1.2: Example of conformal transformation on configurations of the ferromag-netic Ising model. The statistical minimum of the model, equivalent to the ground state in a quantum field theory, can be seen as a superposition of configurations of which the left and right images are representatives. On the left, the model is above the critical temperature and spins are uncorrelated because of thermal fluctuations, therefore a conformal transformation f changes the quantitative features of the repre-sentative, as can be seen from the stretching and contracting of lengths. On the right, the model is at the critical temperature and the correlation length is infinite (assum-ing negligible effects from the finite size of the configuration). In this second case, all spins are correlated and there are fluctuations of arbitrary size. The application of a conformal transformation f changes the configuration to a physically equivalent one, and therefore leaves the thermal superposition invariant. [4]
14 CHAPTER 1. INTRODUCTION transformations:
δgµν(x) ' 2φ(x)gµν(x). (1.27)
CFTs provide important results, the first is that, on a classical point of view, the stress-energy tensor is traceless. To see it we can choose to vary the action under a scale transformation, δgµν → gµν, that is a special case of conformal transformation. Then, for a CFT, we can write
0 = δS = Z d2x δS δgµν δgµν = − 1 4π Z d2x√g Tµµ. (1.28) Even if Tµ
µ = 0 classically, this might not be the case quantum mechanically. We present the main features of quantization of a CFT on a Euclidean flat space and using complex coordinates. The generalization on a non-trivial background is discussed along the section. We move first to Euclidean coordinates (x1, x2) = (x1, ix0) and then to complex coordinates (z, ¯z) ≡ (x1 + ix2, x1 − ix2) = (x1 − x0, x1 + x0). The most general conformal transformation is given by
z → z0 = f (z) z →¯ z¯0 = ¯f(¯z). (1.29) The holomorphic derivatives are
∂z ≡ ∂ = 1
2(∂1− i∂2) ∂z¯≡ ¯∂ = 1
2(∂1+ i∂2). (1.30) We have ds2 = (dx1)2+ (dx2)2 = d¯zdz and the flat metric becomes
gz ¯z = 1 2 0 1 1 0. (1.31) Note that under this construction vectors with up indices are of the form vz = v1+ iv2 and vz¯ = v1 − iv2, when the indices are down we write v
z = 12(v1 − iv2) and v¯z = 1
2(v1 + iv2). The stress-energy tensor traceless property is now Tz ¯z = 0 and, as it is conserved1, we have ¯∂T
zz = 0 and ∂T¯z ¯z = 0, so Tzz(z, ¯z) = Tzz(z) ≡ T (z) and Tz ¯¯z(z, ¯z) = T¯z ¯z(¯z) ≡ ¯T(¯z). It is easy to demonstrate also that the conserved Noether currents J, ¯J under the generic infinitesimal conformal transformations z0 = z + (z) and ¯z0 = ¯z+ ¯(¯z) are
Jz = 0 J¯z = ¯T(¯z)¯(¯z)
Jz¯ = T (z)(z) J¯¯z = 0. (1.32)
To take into account quantum aspects we have to introduce the operator product expansion (OPE). In CFTs for “local operator”, that we simply call field, we intend any local expression that we can write down, instead of the usual meaning in a generic QFT. The OPE is a statement that tell us what happens as local operators approach each other. The idea is that two local operators inserted at nearby points can be closely approximated by a string of operators at one of these points:
Oi(z, ¯z)Oj(w, ¯w) = X
k
Cijk(z − w, ¯z −w)O¯ k(w, ¯w). (1.33)
1Note that on a general background we do not have ∂
µTµν =, but ∇µtµν = 0, with ∇ covariant
1.3. CONFORMAL FIELD THEORIES 15 An operator equation like (1.33) is always thought as statement which hold as operator insertions inside time-ordered correlation functions:
hOi(z, ¯z)Oj(w, ¯w)...i = X
k
Cijk(z − w, ¯z −w)hO¯ k(w, ¯w)...i. (1.34)
We do not always denote the h·i but we always know how to think about these quan-tities. It is fundamental that the correlation functions are always assumed to be time-ordered, so we can write Oi(z, ¯z)Oj(w, ¯w) = Oj(w, ¯w)Oi(z, ¯z). In (1.34) for “...” we intend every local operator that is at a distance from others much greater than |z − w|. Considering path-integral formalism it is easy to write the Ward identity
− 1 2π
Z
∂αhJα(x)O1(x1) . . .i = hδO1(x1) . . .i , (1.35) that, for our case, becomes
i 2π I ∂ dz hJz(z, ¯z)O1(x1) . . .i− i 2π I ∂ dz hJ¯ z¯(z, ¯z)O1(x1) . . .i = hδO1(x1) . . .i . (1.36) Considering results (1.32) we can simplify drastically because Jz is holomorphic while J¯z is anti-holomorphic and this means that the contour integral simply picks up the residue i 2π I ∂ dzJz(z)O1(x1) = − Res [JzO1] . (1.37) This is the reason why we have used complex coordinates formalism. The residue in the OPE between the two operators
Jz(z)O1(w, ¯w) = . . . +
Res [JzO1(w, ¯w)]
z − w + . . . (1.38)
We can now write
δO1(x1) = − Res [(z)T (z)O1(x1)] δO1(x1) = − Res¯(¯z) ¯T(¯z)O1(x1) , (1.39) where the minus sign comes from the fact that the H d¯z boundary integral is taken in the opposite direction.
This result means that if we know the OPE between an operator and T (z) and ¯
T(¯z), then we immediately know how the operator transforms under conformal sym-metry. Or, flipping the previous statement over its head, if we know how an operator transforms then we know at least some part of its OPE with T and ¯T. Looking at translations, where we have O(z − ) = O(z) − ∂O(z), since for translations the Noether current is the stress-energy tensor, we can write
T(z)O(w, ¯w) = . . . +∂O(w, ¯w) z − w + . . . T¯(¯z)O(w, ¯w) = . . . + ¯ ∂O(w, ¯w) ¯ z −w¯ + . . . (1.40) An operator O is said to have weight(h, ˜h) if, under δz = z and δ¯z = ¯¯z, O transforms as
16 CHAPTER 1. INTRODUCTION The terms ∂O in this expression would be there for any operator. They simply come from expanding O(z − z, ¯z −¯¯z). The terms hO and ˜hO are special to operators which are eigenstates of dilatations and rotations. We can now introduce the spin s
s = h − ˜h (1.42)
and the scale dimension ∆
∆ = h + ˜h. (1.43)
Now we can improve (1.40) for the T O OPE
T(z)O(w, ¯w) = . . . + hO(w, ¯(z−w)w)2 + ∂O(w, ¯w) z−w + . . . ¯ T(¯z)O(w, ¯w) = . . . + ˜hO(w, ¯(¯z− ¯w)w)2 + ¯ ∂O(w, ¯w) ¯ z− ¯w + . . . (1.44)
An operator is primary if in its OPE with T and ¯T we have no other singularities than the (z − w) and (z − w)−2.
Now we focus our attention to the T T OPE, that gives rise to the most important definition of CFTs. T(z)T (w) = (z−w)c/2 4 + 2T (w) (z−w)2 + ∂T (w) z−w + . . . ¯ T(¯z) ¯T( ¯w) = (¯z− ¯c/2˜w)4 + 2 ¯T ( ¯w) (¯z− ¯w)2 + ¯ ∂ ¯T ( ¯w) ¯ z− ¯w + . . . (1.45) A (z − w)−3 term would violate z ↔ w invariance, and we cannot have terms over (z − w)−4. The constants c and ˜c are known as central charges2. They measure the degrees of freedom of a CFT. This fact, proved by Zamolodchikov [5], is known as the c-theorem. For a theory on a curved background we require, for consistency, that c= ˜c.
1.4
Conformal anomaly
Two-dimensional case
The concept of central charge is crucial, because enters in our theory in a way that we must approach cautiously. We have said that we want to build a conformal invariant theory. At classical level it implies that the trace of stress-energy tensor must vanish. The problem that arises in general is that, after quantized a CFT on a curved two-dimensional background, which means that we have constructed the path integral formulation, the (expectation value) of stress-energy tensor trace do not result to be zero. In fact, altough we are working with an invariant action, nothing assures that the measure of integration is and, rather, in general it is not. This is the problem of the quantum anomaly, that emerges in various areas of physics, such as in QCD when looking at axial U(1) symmetry. For our interests we obtain
hTµ µi =
c 12π
(2)R, (1.46)
1.4. CONFORMAL ANOMALY 17 where we have denoted with (2)R the two-dimensional worldsheet Ricci scalar. This is the conformal (or Weyl) anomaly and it is a crucial point for the future analysis. To prove it we have to look at the Tz ¯zTw ¯w OPE. We can write
∂zTz ¯z(z, ¯z)∂wTw ¯w(w, ¯w) = ¯∂z¯Tzz(z, ¯z) ¯∂w¯Tww(w, ¯w) = ¯∂¯z∂¯w¯ c/2 (z − w)4 + . . . , (1.47)
where we have used the energy conservation ∂Tz ¯z = − ¯∂Tzz. The r.h.s. of (1.47) has a singularity for z = w. This implies that
¯ ∂z¯∂¯w¯ 1 (z − w)4 = 1 6∂¯z¯∂¯w¯ ∂z2∂w 1 z − w = π 3∂ 2 z∂w∂¯w¯δ(z − w, ¯z −w).¯ (1.48)
Inserting this into the correlation function (1.47) and stripping off the ∂z∂w derivatives on both sides, we end up with
Tz ¯z(z, ¯z)Tw ¯w(w, ¯w) = cπ
6 ∂z∂¯w¯δ(z − w, ¯z −w).¯ (1.49) So the OPE of Tz ¯z and Tw ¯w almost vanishes, but there is some strange singular be-haviour going on as z → w. This is usually referred to as a contact term between operators and, as we have shown, it is needed to ensure the conservation of energy-momentum. We now see that this contact term is responsible for the Weyl anomaly. We assume thatTµ
µ = 0 in flat space. Our goal is to derive an expression for Tµµ
close to flat space. Firstly, consider the change of Tµ
µ under a general shift of the metric δgµν. δTµµ(x) = δ Z Dφe−STµµ(x) = 1 4π Z Dφe−S Tµµ(x) Z d2x0√g δgρσTρσ(x0) . (1.50)
Considering a Weyl transformation δgµν = 2ωδµν : δTµ µ(x) = − 1 2πR Dφe −S Tµ µ(x)R d2x 0ω(x0) Tν ν (x 0) . (1.51)
Now we understand why Tz ¯z(z, ¯z)Tw ¯w OPE determines the Weyl anomaly. Having Tµµ(x)Tν ν (x0) = 16Tzz(z, ¯z)Tw ¯w(w, ¯w) and 8∂z∂¯wδ(z − w, z − ¯w) = −∂2δ(x − x0) we obtain Tµµ(x)Tνν (x 0 ) = −cπ 3 ∂ 2 δ(x − x0) . (1.52)
We now plug this into (1.51) and integrate by parts to move the two derivatives onto the conformal factor ω. We are left with
δTµ µ = c 6∂ 2ω ⇒ Tµ µ = − c 12π (2)R. (1.53)
18 CHAPTER 1. INTRODUCTION Four dimensional case
The concept of conformal anomaly could be generalized for a dimensional CFT on a 4d curved background, since we want to try describe our 4-brane . In [6] was shown that the trace anomaly has to be of the form
hT µ
µ i = aC
2+ bE + cR2
+ dR. (1.54)
Where C2 is the square of Weyl tensor:
C2 = RµνρσRµνρσ − 2RµνRµν+ 1 3R
2, (1.55)
and E is the Euler characteristic integrand, known also as Gauss-Bonnet term
E = RµνρσRµνρσ − 4RµνRµν+ R2. (1.56) The R can always be removed by adding a finite local counterterm to the action [5], so we can ignore it. Instead, it is not totally clear the role of R2 term. In [7] there are considerations about the fact that we cannot have it, while in [8] we can read that the coefficient of R2 could appear on higher loops calculations and, in general, is gauge dependent. Since we are interested in one loop calculations we will not go any further with this discussion, agreeing with the fact that we expect the trace anomaly to be of the form3 hT µ µ i = c (4π)2C 2+ a (4π)2E. (1.57)
Where the (4π)2 is just a convention factor.
1.5
Higher dimensional objects: Dp-brane
Although our work in chapter 2 focuses only on closed strings, the theory provides a description of the open ones and, in this context, the concept of brane appears. The dynamics of a generic point on a string is governed by local physics. This means that a generic point has no idea if it is part of a closed string or an open string. The dynamics of an open string must therefore still be described by the Polyakov action. The difference is in considering boundary conditions. We can check it taking Polyakov action in conformal gauge4
S = − 1 4πα0
Z
d2x ∂µX · ∂µX. (1.58)
Varying the action we obtain5 δS = − 1 2πα0 Z τf τi dx0 Z π 0 dx1∂µX · ∂µδX = 1 2πα0 Z d2x(∂µ∂ µX) · δX + total derivative . (1.59)
3For historical reasons the standard notation of the coefficients in the anomaly is c for the C2
term and a for the E. This is because when the problem of 4d conformal anomaly was started to be studied physicists thought that the coefficient of C2was the 4d generalization of the central charge.
4An open string is parametrized by spatial coordinate in interval [0,π].
5Let us consider the string evolving from some initial configuration at x0 = τ
i to some final
configuration at x0= τ f.
1.5. HIGHER DIMENSIONAL OBJECTS: DP -BRANE 19 Total derivative picks up the boundary contributions
1 2πα0 Z π 0 dx1X · δX˙ τf τi − 1 2πα0 Z τf τi dx0X0 · δX x=π x=0 . (1.60)
The first term is the kind that we always get when using the principle of least action. The equations of motion are derived by requiring that δXµ = 0 at τ = τ
i and τf and so it vanishes. However, the second term is new. In order for it too to vanish, we require
∂µXAδXA = 0 at x = 0, π. (1.61)
This results in two different boundary conditions: Neumann boundary conditions
∂µXA= 0 at x = 0, π. Dirichlet boundary conditions
δXA = 0 at x = 0, π.
Considering Dirichlet boundary conditions for some coordinates and Neumann for the others we have
∂µXA= 0 for A = 0, . . . , p
XB = cB for B = p + 1, . . . , D − 1. (1.62) This fixes the end-points of the string to lie in a (p + 1)-dimensional hypersurface in spacetime such that the SO(1, D − 1) Lorentz group is broken to
SO(1, D − 1) → SO(1, p) × SO(D − p − 1). (1.63) This hypersurface is called a D-brane or, when we want to specify its spatial dimension, a Dp-brane6. So we have seen how strings framework includes higher dimensional elements. However, the role of these entities is, although crucial, rather “passive” in the theory. They are dynamical object and could be described by an analogous of the Nambu-Goto action
SDp = −Tp Z
dp+1xp− det(g) (1.64)
where Tp is the tension of the Dp-brane and xµ, µ = 0, . . . , p, are the worldvolume coordinates of the brane. gµν is the pull back of the spacetime metric onto the world-volume. This is the Dirac action. A theory like (1.64), besides presenting several problems in quantization process [1], is not renormalizable in four dimensions.
In Chapter 4 we focus our attention to an higher dimensional object that could be considered an analogous of a D3-brane, but, to emphasize the fact that we are working with a four dimensional entity, we call it just 4-brane. So we are looking higher dimensional objects on a more active point of view.
6Here D stands for Dirichlet, while p is the number of spatial dimensions of the brane. So, in this
20 CHAPTER 1. INTRODUCTION
1.6
Fourth order Non-Linear σ Model and 4-branes
In the previous section we have generalized the Nambu-Goto action for a brane, but this procedure does not seem to lead to a well-defined object. Then we might think about trying to extend the Polyakov action, whose generalization is the fourth order non-linear sigma model (NLSM). In general a NLSM is a theory whose field takes its values in a Riemannian manifold M. The NLSM field values can therefore be considered as a set of coordinates in the internal manifold M whose metric is field-dependent. This analogy is the reason why we always call the fields X. The standard action in flat space is [9]
S = 1 2 Z d4x XAXBGAB+∇µ∂νXA∂µXB∂νXCAABC+∂µXA∂µXB∂νXC∂νXDTABCD , (1.65) where XA= ∇
µ∂µXA, andGAB, AABC and TABCD are all coupling tensors that are X-dependents.
It is possible to give a conformal invariant version on a curved background of the (1.65): S = 1 2 Z d4x√g XAXBGAB + ∇µ∂νXA∂µXB∂νXCAABC + ∂µXA∂µXB∂νXC∂νXDTABCD− 2(4)R µν ∂µXA∂νXA+ 2 3 (4)R∂ µXA∂µXA . (1.66) We have excluded a Wess-Zumino-Witten term
c Z
d4x µνρσBABCD∂µXA∂νXB∂ρXC∂σXD (1.67) since it is parity-violating, but could also be introduced at any time. It is natural to interpret the higher derivative NLSM as a theory which describes a four dimensional brane.
Since we do not have a Nambu-Goto formulation, but we still want to attempt a geometrical interpretation, we take a hint from the theory of physical membranes [10]. We focus on crystalline membranes theory and, in this context, we write the simplest isotropic effective Hamiltonian [11]:
H = Z ddx κ 2 ∂2X~ 2 + u∂µX∂~ νX~ 2 + v∂µX∂~ µX~ 2 + t 2 ∂µX∂~ µX~ (1.68) Where the notation7 X~ = XA(x) A = 1, .., D is due to the fact that we are looking at the case of a flat Eucledian D-dimensional target spacetime, while the membrane coordinates are xµ µ = 0, .., d − 1. We note that (1.68) is similar to the Landau-Ginsburg-Wilson Hamiltonian when identifying ∂µX~ with the usual ~φorder parameter. In mean field, for t > 0, ∂µX∂~ νX~ = 0 and the surface is crumpled. For t < 0, ∂µX∂~ νX~ has a spontaneous expectation value
∂iX∂~ jX~ = − t
4(u + dv)δij (1.69)
1.6. FOURTH ORDER NON-LINEAR σ MODEL AND 4-BRANES 21 and the surface is flat. Rescaling the coordinates xµ in such a way that ∂
µX∂~ νX~ = δµν we can write H as H= Z ddx κ 2 ∂2X~2+ µ (uµν)2+ λ 2(uµµ) 2 (1.70) where uµν = 1 2 ∂µX∂~ νX − δ~ µν (1.71) κ is a parameter called “bending rigidity” and µ and λ are the “Lam´e elastic coeffi-cients”. We can confront (1.65) and (1.70): there is a correspondence between κ and GAB and the tensor TABCD could be thought as a generalization of the Lam´e coef-ficients. This is our starting point, which convinces us to study our 4-brane with a generalized higher derivative NLSM on a curved background. Although the parallel between physical membranes and our objects may seem not completely convincing, we would like to emphasize that Polyakov action also emerges in this context. This gives us another reason to try to develop our model.
Chapter 2
Two-dimensional string
2.1
Bosonic string action
We have already presented in section 1.2 the fields on which the string theory is built. Now let us try to see their role in the action. First of all we have to deal with GAB, which we could geometrically interpret as the metric on the target manifold, so we see a simple way to include it in our bosonic string action
SG= 1 4πα0
Z
d2x√g GAB(X)∂µXA∂µXB), (2.1) intended as a natural generalization of (1.14). Now we want to insert the Kalb-Ramond antisymmetric field BAB(X): we look for the most general renormalizable action that is conformally invariant and Weyl invariant. The solution is to write
SB = 1 4πα0
Z
d2x√g µνBAB(X)∂µXA∂νXB, (2.2) where, as we have already said, XA(x) (A = 0, .., D − 1) maps the string into a D-dimensional manifold M. The combination of (2.1) and (2.2), after a redefinition of the tensors GAB and BAB to absorb a factor 2πα0, has the form of a generalized NLSM in curved space. We therefore can use this analogy to find the counterterms coming from the quantization of the spacetime coordinates X while the metric is fixed, which is properly the case of NLSM and postpone the considerations coming from the inclusion of a dynamical metric.
Therefore we have seen how it is possible to describe a bosonic string propagating in a non-trivial background with a generalized NLSM defined on a two-dimensional surface with intrinsic metric gµν. However the closed string admits another massless excitation: the dilaton. One possible solution, first presented by Fradkin and Tseytlin [12], is to add the renormalizable, but non Weyl invariant, term
Sφ = 1 4π
Z
d2x√g Φ(X) ·(2)R, (2.3)
where (2)R is the two-dimensional curvature scalar and Φ is the background dilaton field defined in M. Note that it enters in the action with an extra α0 in front. This
24 CHAPTER 2. TWO-DIMENSIONAL STRING fact implies that, even if the inclusion of a term as (2.3) would breaks explicitly the (classical) Weyl symmetry, the extra power of α0 tell us that for |α0| 1 the theory maintain the invariance. We can say that the breaking of symmetry is at order α0, considering the dilaton term as a quantum correction. Surprisingly we verify that, after taking into account quantum corrections, it is possible to restore the symmetry and no inconsistencies appear. The complete action is therefore
S = 1 4πα0 Z d2x√g GAB(X)∂µXA∂µXB+ µνBAB(X)∂µXA∂νXB+ α0Φ(X) ·(2)R. (2.4)
2.2
Weyl anomaly
To be a consistent theory we must have Weyl invariance. This results in the requirment that the expectation value of energy-momentum tensor must be traceless:
hTµ
µi = 0. (2.5)
It is possible to write the quantity in (2.5) as 2πhTµ µi = ¯β (G) AB √ ggµν∂µXA∂νXB+ ¯β (B) AB µν∂ µXA∂νXB+ ¯β(Φ) √ g(2)R, (2.6) where the ¯β are called Weyl anomaly coefficients and are related to the β functionals of the theory. We show later in the chapter how, because we must first introduce some concepts that enter the ¯β definitions. The β functionals enter directly in the global scale invariance, which is related to R Tµ
µ: 2π Z Tµµ = βAB(G)√ggµν∂µXA∂νXB+ βAB(B)µν∂µXA∂νXB+ β(Φ) √ g(2)R. (2.7) After determined the β we can turn to ¯β discussion. We first work considering the metric fixed, which means we want to renormalize the NLSM. Succesively we include the dynamical metric.
2.3
Covariant background expansion
We look for one-loop corrections coming from the quantization of the coordinates X which, in terms of functional generators associated to the path integral formulation, means that we want to determine the contributions coming from the DX integration. We include the Dg integration later in the chapter.
The first idea to determine one-loop expansion would be to work with the standard background field method, of which a brief description has been given in Appendix A, which provides a parametrizations of quantum fluctuations in a linear way around a fixed field, which is called background field:
2.3. COVARIANT BACKGROUND EXPANSION 25 After this parametrization the one loop effective action, which gives us the coun-terterms, could be in function of the Hessian of the action respect to the quantum corrections: Γ1−loop = 1 2Tr ln( δS δηAδηB) . (2.9)
This technique,with the parametrization (2.8), would lead to a problem in the formu-lation, because we would parametrize the action in terms of a non-covariant quantity, ηA, since it is defined as a difference of coordinates. To avoid this problem we use the so called geodesic expansion, which uses a non-linear parametrization of the quantum fluctuations in terms of normal coordinates, defined through the use of geodesics on the manifold M. The Hessian in eq.(2.9) is substituted with the Hessian, respect to the normal coordinate ξA, of the quadratic part of action S(2), which can be written as S(2) = 1
2∇ 2
sS. For the definition of the geodesic covariant derivative we refer to the Appendix A. For the moment we just perform the calculations. It is easier to work with one piece at time. The first one gives
SG(2) = 1 2∇ 2 sSG= 1 8πα0 Z d2x√g∇2s(∂µXA∂µXA) = = 1 8πα0 Z d2x√g(∇s(2∇µξA∂µXA)) = = 1 4πα0 Z d2x√g(∇µξA∇µξBGAB− RACBDξAξB∂µXC∂µXD). (2.10)
Where the notation SG(2) means that we are considering the quadratic action of just the GAB part in (2.4).
The second one requires a bit more hard work. The simplest way to do it is to find SB(1), rearrange the terms through the anti-symmetry property of tensors BAB and and only at that point calculate SB(2).
SB(1) = ∇sSB = 1 4πα0 Z d2x√g∇s(µνBAB∂µXA∂νXB) = = 1 4πα0 Z d2x√g µνξC∇CBAB∂µXA∂νXB+ 2µνBAB∇µξA∂νXB. (2.11)
The second term in (2.11) can be integrated by parts, obtaining: 1 4πα0 Z d2x√g 2µνB AB(X)∇µξA∂νXB = = 1 4πα0 Z d2x√g −2µν∇CBAB∂µXC∂νXBξA− 2µνBAB∇µ∂νXBξA µ↔ν = = 1 4πα0 Z d2x√g −2µν∇CBAB∂µXC∂νXBξA = = 1 4πα0 Z d2x√g − µν∇CBAB∂µXC∂νXBξA+ µν∇BBAC∂µXC∂νXBξA (2.12)
26 CHAPTER 2. TWO-DIMENSIONAL STRING where, in the last step, we just splitted in two terms that allow us to introduce the totally antisymmetric field strength HABC ≡
1
2(∇ABBC+ ∇BBCA+ ∇CBAB). At the end the expression is
SB(1) = 1 4πα0
Z
d2x√gµν 2HABC∂µXB∂νXCξA. (2.13) Now we can obtain SB(2) = 1
2∇sS (1) B : SB(2) = 1 4πα0 Z d2x√gµν − ∇AHBCD∂µXC∂νXDξAξB+ HABC∇µξA∂νXBξC + HABC∂µXA∇νξBξC. (2.14) The inclusion of dilaton it is now quite simple:
Sφ(2) = 1 8π
Z
d2x√g ∇A∇BΦ(X) · R(2)ξAξB. (2.15) The quadratic part of the complete action is now of the form hξ, ∆ξi where h·, ·i denote the inner product, not to be confused with the expectation value, and ∆ is the hessian which we want to use to compute Γ1−loop. As one can read in Appendix B, where the Heat-Kernel method is properly introduced, we need to find a new connection that allow us to write ∆ in the form −(gµν∇
µ∇νI + E), where I and E(x) are operators of the internal space. In particular, E is an endomorphism.
The last two terms of (2.14) are linear in ∇, so ∆ is not compatible with form desired, but the solution is to define a new connection
ˆ
∇µξA = ∇µξA+ νµ(∂νXB)HABCξ
C. (2.16)
This transformation allows us to absorb the “wrong” pieces of equation (2.14) in GAB∇µξA∇νξB at the cost of inserting a new term of endomorphism. Because of the way H enters in the covariant derivative it may be naturally considered as a torsion. The complete quadratic action is
S(2) = 1 4πα0 Z d2x√g ˆ ∇µξA∇ˆµξBGAB− RACBDξAξB∂µXC∂µXD − µν∇AHBCD∂µXC∂νXDξAξB+ ∂µXC∂µXEHACDHDEBξ A ξB +α 0 2∇A∇BΦ(X) · R (2)ξAξB . (2.17)
From now on until the chapter we avoid to use the hat to indicate the complete covari-ant derivative of (2.16), since it is always clear what we are using. Note that (2.17) coincide with equation (2.2) in [9], where a generalized Riemann tensor is introduce. In [13] (eq. (3.16)) there is a difference in the definition of HABC: it does not include the factor 1
2 and this reflects in a factor 1
2 in front of the torsion part and so the coefficients in the quadratic part are different, but all the results are compatible with each other. Our Hessian is in the end of the form desired:
∆AB = − gµν∇µ∇νGAB+ RACBD∂µXC∂µXD + µν∇AHBCD∂µXC∂νXD − ∂µXC∂µXEHACDHDEB −
α0
2∇A∇BΦ(X) · R
2.4. HEAT KERNEL METHOD 27
2.4
Heat kernel method
We can derive the expression of the one-loop effective action in terms of the heat kernel and so, trough the Seeley-de Witt expansion, of the coefficients ak. Since we are looking for local divergences we take the coincidence limit, denoted with square brackets [ak] = ak(x, x). The expression is
Γ1-loop = − 1 2 X k≥0 Tr Z ddx √ g[ak] (4π)d/2 Z ∞ 0 ds sd/2+1−k. (2.19)
We are interested in UV divergences, that appears in the limit s → 0, because s is a variable conjugate to the momentum. Therefore it is possible to add a (little) mass term η2 to regulate IR divergnces for s → ∞
Γ1-loop= − 1 2 X k≥0 Tr Z ddx √ g[ak] (4π)d/2 Z ∞ 0 ds e −sη2 sd/2+1−k = −1 2 X k≥0 Tr Z ddx √ g[ak] (4π)d/2 (η 2)d/2−kΓ(k − d/2) . (2.20)
In two dimensions there are poles just for k = 0, 1, but after analitycal continuing and taking η → 0 only k = 1 contribute survives, so the only coefficient that appear is [a1].
Γ(d=2−)1-loop |div = − 1 2Tr Z d2x √ g 4π (2)R 6 δAB− EAB 2 = = 1 4π Z d2x√g − D 6R (2)+ R AB∂µXA∂µXB + µν∇CH CAB∂µXA∂νXB− ∂µXA∂µXBHAB2 −α 0 2∇ 2Φ(X) · R(2) , (2.21) where H2
AB = HACDHBCD. From the expression (2.21) it is possible to derive the β functionals at one loop approximation. We obtain:
βAB(G)= α0(RAB− HAB2 ) βAB(B) = α0∇CH ABC β(Φ) = D 6 − α0 2∇ 2 φ (2.22)
The usual form of the β includes other terms, coming from 2-loops calculations [14].
2.5
Dynamical metric
We have found the contribution coming from the DX integration. Given the relation between the β-functionals and the trace anomaly we can understand that βΦ is related
28 CHAPTER 2. TWO-DIMENSIONAL STRING to the central charge, implying that every scalar in the theory weighs 1/6. This is related to the fact that under conformal transformations the measure DX transforms as[15]
DX → DXe48πD SL, (2.23)
where the 48π is for convention and SLis the Liouville action, defined as the difference between the Polyakov action1,
SP[g] = − 1 96π Z d2x√gR1 ∆R (2.24)
evaluated before and after a Weyl rescaling: SPge2σ − SP[g] = −
1
24πSL[σ; g]. (2.25)
So the Liouville action is
SL[σ; g] = Z
d2x√g[σ∆σ + σR], (2.26)
and it provides a dynamic for the conformal mode, which we want to eliminate because we are requiring conformal invariance. We now need to find the contribution from Dg integration, that, as we just said, does not integrate the conformal mode. This process passes trough the standard definition of path integral
Z = 1 V ol
Z
DgDXe−S. (2.27)
The “Vol” term is all-important. It refers to the fact that we should not be integrating over all field configurations, but only those physically distinct. The usual way of doing this is through the gauge-fixing process, which is followed by the discussion of ghosts. This is what is always found in string theory books: the so called bc-ghost system. We have two gauge symmetries: diffeomorphisms and Weyl transformations. The change of the metric under a general gauge transformation is
gµν(x) −→ gµνζ (x 0 ) = e2ω(x)∂xρ ∂x0µ ∂xσ ∂x0νgρσ(x). (2.28) In two dimensions these gauge symmetries allow us to put locally the metric into any form ˆg, which is called fiducial metric and represent our choice of gauge fixing. Our goal is to only integrate over physically inequivalent configurations, using the standard Faddev-Popov method. This involves the insertion Z of the identity
1 = ∆F P(g) Z
Dζ δ(g − ˆgζ), (2.29)
1Same name, but different action. There is no need to find a way to tell them apart because we
2.5. DYNAMICAL METRIC 29 where ∆F P is the Faddeev-Popov determinant. Note that
∆−1F P gζ = Z Dζ0δgζ− ˆgζ0 = Z Dζ0δg −gˆζ−1ζ0 = Z Dζ00δ g −ˆgζ00 = ∆−1F P[g] (2.30)
where, in the second line, we have used the fact that Dζ, which is Haar measure, is invariant. The resulting path integral expression is called Z[ˆg] since it depends on the choice of fiducial metric ˆg. The first thing we do is use the δ g − ˆgζ delta-function to do the integral over metrics:
Z[ˆg] = 1 Vol Z DζDXDg∆F P[g]δ g − ˆgζ e−S[X,g] = 1 Vol Z DζDX∆F Pˆgζ e −S[X,ˆgζ] = 1 Vol Z DζDX∆F P[ˆg]e−S[X,ˆg], (2.31)
where in the last line we have used the fact that the action is gauge invariant as the Faddev Popov determinant, proved in (2.30). Equation 2.31 is therefore independent from ζ, so the Dζ integration could be performed cancelling the “Vol ” term. Now the problem is the calculation of ∆F P, which leads as usual to the ghost action. The final expression is
Z[ˆg] = Z
DXDbDc exp (−S[X, ˆg] − Sghost[b, c, ˆg]) , (2.32) where the Sghost[b, c, ˆg] is
Sghost[b, c, ˆg] = 1 2π
Z
d2x√gbµν∇µcν (2.33) The role of these ghost fields is to cancel the unphysical gauge degrees of freedom, that are non-conformal modes. This is related to the fact that in lightcone quantization, which breaks the Lorentz invariance, the ghosts leave only the D − 2 transverse modes of Xµ. The ghost action forms a completely decoupled system, so it is possible to evaluate the contributions it brings to the trace anomaly using complex coordinates and TT OPE. We do not perform the calculations, first of all because they are widely known steps and also we want to try to use another method. The conclusion is that the Dg integration leads to a change in βΦ, that becomes:
β(Φ) = D −26
6 −
α0 2∇
2Φ. (2.34)
Changing the normalization of the central charge so that the DX integration leads to a central charge cX = D and the Dg contributes to a cg = −26 we have the central charge of the system
30 CHAPTER 2. TWO-DIMENSIONAL STRING However, we can think the above combination from another perspective. It is possi-ble to introduce another measure D0g which integrates also the conformal mode. Now the problem is analogous to two-dimensional quantum gravity [16] and we can use its solutions to derive the expression of anomaly. We do not repeat the calculation, the fundamental result is that the integration over all metric degrees of freedom would bring a −25 in the expression of βΦ. Of course this is not correct, because we have in-tegrated over the conformal mode σ. The latter, as a scalar quantum field, contributes with a cσ = +1 to the anomaly. We have
− 25 = c0g = cg+ cσ =⇒ cg = c0g− cσ = −26. (2.36) This, as expected, brings to the correct estimate for the central charge of the system and so for the anomaly. In the end we have found the same result, but following a completely different path. It is easy to understand why we have used it: it is simply generalizable to the four-dimensional case, using the already resolved calculation of higher-derivative gravity.
It is important to mention the fact that the procedure of including the calculation from gravity is a well-known analogy in string theory. Some articles [17] use another perspective: we can just include the gravity and then the critical dimension would be D = 25, which means 25 Euclidean dimensions with O(25) symmetry. But we would have a dynamics, with the minus sign, for the conformal mode. Then it could be thought as a temporal coordinate and included in the X, that now become 26 scalars with symmetry O(1, 25). Strictly speaking we would have D = D0 − 1 → D0 = 26. This point of view has the merit that explicits the Minkowski coordinates.
2.6
Low-energy effective action
Let us turn back to the discussion of section 2.2. The link between ¯β and the β is [18]: ¯ βAB(G)= βAB(G)+ ∇(AWB)+ ∇A∇BΦ, ¯ βAB(B)= βAB(B)+ H C AB WC + ∂[ALB]− 2HABC∇CΦ, ¯ β(Φ)= β(Φ)+α 0 2∇ A Φ∇AΦ + α0 2W A Φ∇AΦ, (2.37)
where WA = WA(G) and LA = LA(B). These quantities are zero in one loop ap-proximation2. As we have said, the complete first order in α0 is given by two-loops calculation, so we include the missing terms. We do it just because the discussion results more interesting and more complete, but all considerations we are going to do remain valid in one loop approximation. The Weyl anomaly coefficients are:
¯ βAB(G)= RAB − HAB2 + 2∇A∇BΦ + O(α0), ¯ βAB(B)= ∇CH ABC− 2HABC∇CΦ + O(α0), ¯ β(Φ)= D −26 6 + α0 2(∇Φ) 2 − (∇2Φ) − R + 1 3H 2 + O((α0 )2). (2.38)
2.6. LOW-ENERGY EFFECTIVE ACTION 31 We want them equal to zero to have Weyl invariance at quantum level. First of all we understand that we need to put
D= 26, (2.39)
since there is no other terms that can cancel the leading order part in ¯β(Φ). We have obtained the critical dimension again, and we have understood in a deeper way its meaning: it is the only dimension of spacetime in which is possible to formulate a conformal invariant string theory.
Now there is a problem that must be taken into consideration: ¯β(Φ) plays the role of the central charge of the system. It is clear that we want it to be a c−number, but we see in its definition how it results to be an operator depending on spacetime. String theory solves the problem in an interesting way: if we want NLSM to be conformal invariant we first must impose ¯βAB(G)and ¯βAB(B) equal to zero, otherwise there is a confor-mal anoconfor-maly even on a flat space-time. The remarkable fact is that these conditions implies that ¯β(Φ) is a constant. This is true at any order [19].
We can rearrange Weyl-anomaly equations into a more suggestive form: RAB− 1 2GABR= HAB2 − 1 6GABH 2 − 2∇A∇BΦ + 2GAB∇2Φ ≡ ΘAB, ∇CH ABC = 2(∇CΦ)HABC, ∇2Φ − 2(∇Φ)2 = −1 2H 2. (2.40)
ΘAB is symmetric tensor and, if the equations are satisfied, we want it to be conserved. However, (2.40) can also be considered as equations of motion for the background in which the string propagates. We now change our perspective: we look for a D = 26 dimensional spacetime action which reproduces them as the equations of motion. This is the low-energy effective action of the bosonic string,
S = 1 2κ2 0 Z d26X√−Ge−2Φ R − 1 3HABCH ABC+ 4∂ AΦ∂AΦ , (2.41)
where the overall constant involving κ0 is not fixed by the field equations, but can be determined by coupling these equations to a suitable source as described, for example, in [1]. On dimensional grounds alone, it scales as κ2
0 ∼ ls24 where α 0 = l2
s.
Action (2.41) governs the low-energy dynamics of the spacetime fields. The caveat “low-energy” refers to the fact that we only worked with the first order of expansion in α0.
Something rather remarkable has happened here: we started with an action defined on the worldsheet and just looking for coherence we are led to an action governing how spacetime and other fields fluctuate in D = 26 dimensions.
Chapter 3
Higher derivative nonlinear σ
model
Since we want to describe a four-brane theory we first need to work with generalized fourth order NLSM [20] in curved space. We work with the general action
S = 1 2 Z d4x√g XAXBh(4)AB+ ∇µ∂νXA∂µXB∂νXCAABC + ∂µXA∂µXB∂νXC∂νXDTABCD+ (4)R µν ∂µXA∂νXBHAB +(4)R∂µXA∂µXBQAB + Φ(C)C2+ Φ(E)E+ Φ(R)(4)R2 , (3.1)
where we remind that XA = ∇
µ∂µXA. The inclusion of other terms is discussed later, but we do not consider Wess-Zumino type terms, containing completely antisym-metric tensor , to simplify the calculation. Every coupling tensor has to be interpret as X−dependent. h(4)AB is the generalization of GAB of the two-dimensional case and therefore represents the role of the metric. The “(4)” apex is to distinguish it from the h(2) term, whose insertion is discussed later in the chapter. The tensor A can be assumed to be totally symmetric without loss of generality. The tensor T must have the following symmetry properties
TABCD = TBACD = TABDC = TCDAB (3.2)
and tensors HAB and QAB are symmetric. We have also introduced curvature terms whose coupling we have denoted with Φ in analogy with the string case. In our work we work with A = 0, because it drastically simplifies our calculation without changing the most interesting results. However, the whole discussion could in general be repeated considering A, nothing in the method used would change.
Much of this chapter focuses on calculations. In this sense it can be cumbersome, but we believe necessary to show explicitly the steps as we have not had complete references on this topic to follow.
34 CHAPTER 3. HIGHER DERIVATIVE NONLINEAR σ MODEL
3.1
Covariant background field expansion
Again we need to use covariant background field expansion. The procedure is analogous to the previous chapter. We start from XA
XBh(4)AB term, where = ∇µ∂µ. ∇s XAXBh (4) AB = 2(∇sXA)XBh (4) AB = = 2 ∇µ∇s∂µXA+ [∇s, ∇µ]∂µXAXBh (4) AB = = 2 ∇µ∇s∂µXA+ RCD EA ξ C ∂µXD∂µXEXBh(4)AB = = 2 ξA+ R A CD Eξ C∂ µXD∂µXEXBh (4) AB. (3.3)
Since we want to find the one loop corrections we need the expression for the fourth order Hessian that enters in the expression of the effective action. To find it we need to derive the quadratic part of the action. Hence the quantity of interest is :
Sh4(2) = 1 2∇ 2 s X A XBh(4)AB = ∇s ξA+ R A CD Eξ C∂ µXD∂µXEXBh (4) AB , (3.4)
which is composed of two terms. Applying ∇s to XBh (4)
AB we repeat the previous calculation, obtaining (ξA+ R A CD Eξ C∂ µXD∂µXE(ξB+ RCD EB ξ C∂ µXD∂µXEh (4) AB = = ξAξBh(4)AB+ 2ξARCDAEξC∂µXD∂µXE + RCD EA RF GAHξ C ∂µXD∂µXEξF∂µXG∂µXH, (3.5)
while for the rest we need to compute. The paradigm, as explained in Appendix B, is to move the geodesic covariant derivative to the right, through continuous use of commutators, until it finds a ξ on which to act, because ∇sξA vanishes. Therefore for the other part of (3.4), which is
∇s ξA+ RCD EA ξ C∂ µXD∂µXEXBh (4) AB, (3.6) we have ∇sξA= ∇s∇µ∇µξA = ∇µ∇s∇µξA+ [∇s, ∇µ]∇µξA = = ∇µ∇s∇µξA+ RCD EA ∇ µξEξC∂ µXD = = ∇µ RCD EA ξ EξC∂µXD + R A CD E∇ µξEξC∂ µXD = = ∂µXF∇FRCD EA ξ EξC∂µXD+ R A CD E 2∇µξEξC∂µXD + ξE∇ µξC∂µXD + ξEξCXD (3.7) and ∇s RCD EA ξ C∂ µXD∂µXE = ξF∇FRCD EA ξ C∂ µXD∂µXE + R A CD Eξ C∇ µξD∂µXE + RCD EA ξ C∂ µXD∇µξE. (3.8)
3.1. COVARIANT BACKGROUND FIELD EXPANSION 35 We now want to collect everything together, reminding ourselves that we want to find the Hessian of the action, which means we want to write the action in a form hξ, ∆ξi. We obtain Sh4(2) = 1 2 Z d4x√g ξAξBh(4)AB+ 2ξC ξARCDAE∂µXD∂µXE + ξCξFR A CD ERF GAH∂µXD∂µXE∂νXG∂νXH + ∂µXF∇FRCDAEξEξC∂µXDXA+ 2ξC∇µξERCDAE∂µXDXA + ξE∇ µξCRCDAE∂µXDXA+ ξEξCRCDAEXDXA + ξCξF∇ FRCDAE∂µXD∂µXEXA+ ξC∇µξDRCDAE∂µXEXA+ + ξC∇ µξERCDAE∂ µXD XA = = 1 2 Z d4x√g ξA h(4)AB2+ 2RACBD∂µXC∂µXD + R C AD ERBF CG∂µXD∂µXE∂νXF∂νXG− ∇ERADBC∂µXE∂µXDXC − 3RADBC∂µXDXC∇µ− RACBD∂ µXD XC∇µ − RACBDXC XD + ∇BRACDE∂µXC∂µXEXD + RABCD∂ µ XDXC∇µ ξB = = 1 2 Z d4x√g ξA h(4)AB2+ 2RACBD∂µXC∂µXD − (3RADBC+ RACBD − RABCD)∂µXD
XC∇µ
+ (∇BRACDE − ∇ERACBD)∂µXE∂µXCXD− RACBDX C XD+ + RAD EC RBF CG∂µXD∂µXE∂νXF∂νXG ξB = = 1 2 Z d4x√g ξA h(4)AB2+ 2RACBD∂µXC∂µXD − 4RADBC∂ µ XDXC∇µ + (∇BRACDE − ∇ERACBD)∂µXE∂µXCXD− RACBDX
C XD+ + R C AD ERBF CG∂µXD∂µXE∂νXF∂νXG ξB, (3.9) where in the last step we have collected some terms through Ricci identity. This result agrees with [21] and [9], where is discussed the higher derivative NLSM in flat space.
Let us move to ∂µXA∂µXB∂νXC∂νXDTABCD(X) geodesics expansion. This is simpler since we do not have a ∇s acting on a , which is the most complicated
36 CHAPTER 3. HIGHER DERIVATIVE NONLINEAR σ MODEL computation. 1 2∇ 2 s ∂µXA∂µXB∂νXC∂νXDTABCD = = 1 2∇s 4∇µξ A∂µXB∂ νXC∂νXDTABCD+ ∂µXA∂µXB∂νXC∂νXDξF∇FTABCD = = 2R A EF GTABCDξEξG∂µXF∂µXB∂νXC∂νXD+ 2∇µξA∇µξB∂νXC∂νXDTABCD + 4∇µξA∂µXB∇νξC∂νXDTABCD+ 4∇µξA∂µXB∂νXC∂νXDξF∇FTABCD+ + 1 2∂µX A∂µXB∂ νXC∂νXDξFξG∇G∇FTABCD. (3.10) So, in the end, we obtain:
ST(2) = 1 2 Z d4x√g ξA 2R D AC BTDEF G∂µXC∂µXE∂νXF∂νXG − 2∂νXC∂νXDTABCD − 4∂µXC∂νXDTACBD∇µ∇ν − 4∂µXB∂ νXC∂νXD∇ETAECD∇µ+ +1 2∂µX G∂µXF∂ νXC∂νXD∇A∇BTGF CD ξB. (3.11)
Now we want to evaluate (4)Rµν∂
µXA∂νXBHAB. The first variation gives SH(1) ∼ ∇s((4)R µν ∂µXA∂νXBHAB) = =(4)Rµν(2∇µξA∂νXBHAB+ ∂µXA∂νXBξC∇CHAB) (3.12) and then SH(2) = 1 2 Z d4x√g 1 2∇s (4)Rµν(2∇ µξA∂νXBHAB+ ∂µXA∂νXBξC∇CHAB) = = 1 2 Z d4x√g (4)Rµν RCD EA ξEξC∂µXD∂νXBHAB + ∇µξA∇νξBHAB + ∇µξA∂νXBξC∇CHAB + ∇µξA∂νXBξC∇CHAB + ∂µXA∂νXBξCξD∇C∇DHAB = = 1 2 Z d4x√g (4)Rµν RCD EA ξEξC∂µXD∂νXBHAB + ∇µξA∇νξBHAB + 2∇µξA∂νXBξC∇CHAB+ ∂µXA∂νXBξCξD∇C∇DHAB = = 1 2 Z d4x√g (4)RµνξA RBD AE ∂µXD∂νXCHEC− HAB∇µ∇ν − 2∂νXC∇BHAC∇µ+ ∂µXC∂νXD∇A∇BHCDξB. (3.13)
The calculation for Q piece is analogous to the previous for H. We have SQ(2) = 1 2 Z d4x√g (4)R RCD EA ξEξC∂µXD∂µXBQAB+ ∇µξA∇µξBQAB + 2∇µξA∂µXBξC∇CQAB + ∂µXA∂µXBξCξD∇C∇DQAB. (3.14)
3.2. HEAT KERNEL METHOD 37 Finally we can treat curvature terms, whose expansion is quite simple since it is identical to the string case. So the contribution are trivial, simply of the form
1
2∇A∇Bφξ
AξB· ..., which add just endomorphism terms:
SΦ(2) (C) = 1 2 Z dx√g 1 2∇A∇BΦ(C)ξ AξBC2, SΦ(2) (E) = 1 2 Z dx√g 1 2∇A∇BΦ(E)ξ AξBE, SΦ(2) (R) = 1 2 Z dx√g 1 2∇A∇BΦ(R)ξ A ξB (4)R2. (3.15)
We have computed the quadratic action, which is composed by many terms that, as a matter of order, we have presented separately. Now we want to extract the fourth order Hessian which, analogously to the previous chapter, we call ∆.
3.2
Heat Kernel method
Because we have fixed A = 0, the general form of the ∆ operator is
∆AB = h (4) AB
2+ Bµν
AB∇µ∇ν + CABµ ∇µ+ DAB. (3.16)
We want (3.16) to be self-adjoint. This request means that these identities must be verified:
BµνAB = BνµBA,
CABµ = −CBAµ + ∇νBBAµν + ∇νBνµBA, DAB = DBA+ ∇νCBAµ + ∇ν∇νBBAνµ.
(3.17)
We can also put BµνAB = BABνµ at the cost of adding another endomorphism term. To arrive at the complete form of ∆ is not an easy problem. We can solve it finding first a non self-adjoint ˜∆ integrating by parts the covariant derivatives on the same on the same ξ (this is, for example, exactly what we have written in the final part of (3.9)
