U
NIVERISTY OF
P
ISA
D
EPARTEMENT OF
P
HYSICS
C
OURSE OFT
HEORETICALP
HYSICSEffective description of
fluid membranes and quantum
gravity
Author
Simon Kanka
n.520178
Supervisor
Dr. Omar Zanusso
April 6, 2021
Contents
1 Introduction 3
2 Nonlinear Realization 7
2.1 Inverse Higgs Mechanism . . . 7
2.2 Coset construction . . . 9
2.2.1 Internal symetries . . . 10
2.2.2 Space-time symmetries . . . 11
2.2.3 Effective action . . . 12
2.2.4 Fluid membrane and Thetered membrane relation . . . 14
3 Covariant variations of a fluid membrane 17 3.1 Covariant variations along the normal . . . 19
3.1.1 First variation of intrinsic geometry . . . 20
3.1.2 First variation of extrinsic geometry . . . 21
3.1.3 Euler-Lagrange equations . . . 22
3.2 Second order variations . . . 23
3.3 Energy bound and Monge representation . . . 25
4 Renormalization and crumpling transition 28 4.1 Partition function and gauge fixing . . . 28
4.2 Wilsonian Renormalization . . . 31
4.2.1 Crumpling transition . . . 34
4.2.2 Fixed Area prescription . . . 35
5 Relation to gravitational theories 41 5.1 Polyakov and Nambu-Goto actions . . . 42
5.2 Dynamical diffeomorphisms . . . 44
5.3 Analytical continuations . . . 44
5.3.1 Two-dimensional membrane . . . 46
5.3.2 N >2 and asymptotic safety . . . 47
5.3.3 Dynamical diffeomorphisms and curvature terms . . . 49
6 Conclusions 51 Appendices 54 A Expanded second order variations . . . 54
B Heat-kernel method . . . 57 1
CONTENTS 2 B.1 General construction . . . 57 B.2 Traces with non-minimal differential operator . . . 58 B.3 Application to the fluid membrane case . . . 59
Chapter 1
Introduction
Effective descriptions of membranes have found applications in many areas of physics. It comes naturally to describe solid-state physics like the graphene as a crystalline membrane and the description of lipid bilayers as a fluid membrane (figure 1.1a). The general phase-diagram of a membrane is complicated [2], and, in many cases, we do not know what order transition exist between these phases. We will therefore specialize our research on a specific thermodynamical phase of the membrane and analyze its mechanical properties that also can depend on the model one uses like in figure 1.1b. The second aforementioned example, for the applications of a membrane model in solid-state physics, that corresponds to a fluid membrane model built up by Helfrich [3], is the one we will analyze here in this thesis, having this a relation to gravitational theories, on which we will give later a brief explanation. Turning back to the membrane discussion, we know that by using the Renormalization Group (RG) action on a statistical model, the fixed
(a) The Helfrich model, extended to ar-bitrary dimension in this work, was originally used to describe, in the continuous limit, the properties of a membrane that is composed, in the microscopic scale, by a lipidic bi-layer.(Illustration by Frank Boumphrey, MD)
(b) Here we show one of the possi-bles diagram mecanichal phases. In our model we will have the distinction be-tween a crumpled and an extended me-chanical phase [1].
Figure 1.1
CHAPTER 1. INTRODUCTION 4 points, if they exist, divide distinct phases when one flows back to the infrared region. That is what we see when we take the Wilsonian RG in momentum shell regularization applied to a fluid membrane model and study the different situa-tions with arbitrary fluid membrane dimension, N, and general bulk dimension, D. The found transitions are going to be of the second type and the associated citical exponents are universal for all fluid membranes, independently of their microscopic origin.
The most general description of a membrane embedded in an arbitrary space is defined by a set of functions such that. :
r : M→ N, (1.1)
in which M and N are arbitrary differentiable manifold. Naturally, one would have that the target space N has a bigger dimension of M. The target space has an arbitrary metric, and the presence of the membrane does break the group of isometries associated with it. The broken generators are associated with space-time coordinates and must be treated carefully. In fact, this procedure does gen-erate Goldstone bosons modes, like the case of internal symmetries, but such modes are related to each other and will give rise to an over numerated Gold-stone modes. We need then to introduce an inverse Higgs constraint to have the real degrees of freedom for our description. Such argument will be studied in the following chapter where we are going to describe the construction of mod-els by using the Coset method. We will find the elementary objects to build a Lagrangian density that is invariant under the action of the full isometry group, whose basic blocks transform linearly under the action of the subgroup of un-broken generators. We will also consider the case of a crystalline membrane in which no translations tangential generator to the membrane let the vacuum state invariant. This will bring then to a different number of Goldstone modes with re-spect to the fluid membrane case. In fact, the crystalline model is composed from different monomers that have a fixed distance, thus breaking all the tangential deformations.
To describe a real fluid membrane, one would need to insert a self-avoiding constraint on it so that, in a possible crumpled phase, the membrane does not in-tersect itself and does still have some bending energy that would go against such a trend. This constraint would unfortunately be a nonlocal interaction, which is makes computations very difficult, so we neglect the effects of self-avoidance and concentrate on the simpler mode’s RG flow to analyze second orders phase-transition. Although our fluid membrane will not describe real objects it is still interesting to study its behavior to obtain the critical exponential indices with arbitrary bulk and membrane dimensions.
A fluid membrane model, with specific limits concerning both bulk and mem-brane dimension, has a good relationship with possible new gravitational and gravity-like theories. In fact, we can see the embedding functions in a string the-ory optic. We know that the Nambu-Goto action does describe the action of a string in flat space-time and, if we make a Wick rotation to describe the theory
CHAPTER 1. INTRODUCTION 5 with a Euclidean metric, then the action is
SNG =T
Z
d2ζ√g. (1.2)
This action is related to the Polyakov action by Lagrange multipliers SP =
T 2
Z
d2ζ√ggab∂a~r·∂b~r. (1.3)
When working in a two-dimensional space, the integration of the Ricci scalar, which is associated with the auxiliary metric gab, on the surface is a topological invariant, usually known as the Euler characteristic and can be added to the ac-tion. This newly added term does not modify the equations of motion (EOM) of the embedding functions. Polyakov action can then be seen as an action de-scribing scalar matter fields interacting via a gravitational field living on the same Manifold. Then the matterless limit is equivalent to the limit D →0 and this can be seen as a new gravitation theory. The membrane model, when restricted only to no curvatures’ terms, that is to the lowest order in the derivative expansion, is identical to an extended version of the Nambu-Goto action. Basing ourselves on this relation we can then take the same limit for our model. This will then bring a new gravitation theory with its specific characteristic terms. We can then extend the above reasoning to dimensions higher than two and interpret the limit of zero bulk dimension as a quantum gravity theory of higher dimension.
There is another special limit that can be related to a gravity-like model. This limit corresponds to the case in which the embedding space dimension corre-sponds to the membrane’s one. We can look at said correspondence by taking the case of an embedding function that goes to the same initial manifold, with or without the same metric, and this corresponds to an active point of view of the gauge associated with change of coordinates. A model that has diffeomorphisms as dynamical variables can have applications in the problem of dark energy in cosmology and the problem of observables in General Relativity [4]. The main building block of an action, in covariant models, is the matrix
Bµν := g µρ(
ϕ∗h)ρν =gµρ∂ρϕα∂νϕβhαβ(ϕ). (1.4)
Here ϕαare scalar field, g
µν the space-time metric, hαβ(ϕ)is the target space
met-ric and ϕ∗h is the pullback of h by ϕ. The transformations rules of the defined tensor Bµνare the same as one would have in an ordinary nonlinear sigma model
and a diffeomorphism-invariant action can be constructed by taking the traces of it. The simplest action is the action for harmonic maps
S= −f
2
2
Z
d4x√g Tr(B). (1.5) If we assume a flat target space metric, we obtain the extended Polyakov action and we know that this is related to Nambu-Goto action by Lagrange multipliers. This is equal to the reduced effective fluid membrane model with no curvatures. We can then argue that in the aforementioned limit, taken after the RG step, this
CHAPTER 1. INTRODUCTION 6 results in a theory of dynamical diffeomorphism with dimensions higher or lower than four.
Unfortunately, both aforementioned limits do not have clear physical inter-pretations from the point of view of the membrane model (they are perfectly le-gitimate models on their own). In fact, the case of the analytic continuation to a zero-target space dimension brings an unclear unbroken symmetry group. Our subgroup, that lets the void invariant, is composed of the rotations around the normal directions of the fluid membrane, SO(D−N), and this would be an ill-defined group for D < N. This situation is similar to the case for Parisi-Sourlas model [34], or the limit N →0 of O(N)model that describes self-avoiding walks. Something similar happens when we consider the case of equal space-time di-mension of the target and of the membrane. In this case, outside the problem of having the group SO(D−N) going to the identity element we do not have any clear physical explanation of what role does the extrinsic and intrinsic curva-tures play in the dinamycal diffeomorphisms model. We will readdress to such problems in the conclusions, where we will try to give ideas for a sounder geo-metrical interpretation of the aforementioned limits, by searching for a bulk space with the correct isometries, that are spontaneously broken by the presence of the membrane, to reproduce the two limits.
This thesis work is organized as follow. In the first chapter we will resume the Inverse Higgs mechanism and its relation to the known Higgs mechanism. We will than show how to obtain the main elements to build up an effective model to describe fluid and crystalline membranes by using the nonlinear realization with the Coset method. This method is fundamentally based on the use of the Maurer-Cartan one-form. In the following chapter we will write down the fun-damental objects that characterize the embedding of an object in a target space both with arbitrary dimensions. After that we will compute the Hessian of the action constructed before up to O(K3). We will also show what the EOM for the embedding function up to the aforesaid order are and show that for an open dif-ferentiable manifold our covariant construction gives the same results that the ones obtained for specific cases. This will be needed to put some constraints on the physical region of our coupling constants. The fourth chapter will be focused on the definition of the Wilsonian Renormalization procedure in a momentum-shell cutoff regularization and the principal result of the thesis: the trend of the dimensionless coupling constants with both bulk and membrane’s arbitrary di-mension. We will also try to give a fixed area prescription to the model, which is interesting for technical reasons, because it would give us statistical observables that directly depend on it, rather than on the tension. The final chapter will con-sist of the analytic continuations and a more sophisticated study of the results we gain.
Chapter 2
Nonlinear Realization
We are going to formulate our effective action for a N-dimensional membrane embedded in a D-dimensional space with a flat bulk metric by using the coset construction. The generators of our symmetry group are space-time dependent, and this fact will give rise to the inverse Higgs mechanism. The broken inverse Higgs mechanism is the complementary of the Higgs mechanism that gives mass to the Goldstone boson fields when the broken symmetries are gauged [5]. In our case we will have an overcounting of the real number of Goldstone modes. This will be solved by imposing to zero the covariant derivative along the membrane of the Goldstone boson associated with the broken translation.
2.1
Inverse Higgs Mechanism
Let us take a general case in which we have a group G that breaks spontaneously to a subgroup H. By analogy with our cases let us mark with a the unbroken translational generators and with i the broken ones. We are here going to follow the discussion of such a problem in [6]. Take φ to be our parameter order of the spontaneous symmetry breaking. By acting on its mean value on the vacuum we generate the massless Goldstone bosons,
δφ=ciTihφi. (2.1)
If we have non-trivial solutions, imposing to zero (2.1), then we have a relation between our Goldstone modes for the long wavelength ci(x). This brings a
reduc-tion in the number of independent Goldstone fields. We know that the number of independent solutions to (2.1), nx, is always ≥ 0. In a general formulation
we have that the independent Goldstone bosons are dim(G) −dim(H) −nx. In
our case we have, for fluid membranes, we have that nxand such a phenomenon
happens whenever we are working with generators that depend on space-time directly. In the case the generators are all from internal symmetries than (2.1) has not non-trivial solutions. We are now going to analyze the case in which there are some broken translation generators and some non-broken. If all the transla-tion generators are broken than we would have the normal case for the counting. To understand the previous sentence, we must remark that the only propagating
CHAPTER 2. NONLINEAR REALIZATION 8 directions for the Goldstone bosons are the ones along the surface and their asso-ciated generators are the unbroken translation. Acting with Pa we show this, in
fact we have
0= PaciTihφi = [Pa, ciTi]hφi = −i(∂aci− f ij
a ciTj)hφi, (2.2)
[Pa, Ti] =i fa ji Tj+i fa bi Tb. (2.3)
If the Ti are all internal generators then [Pa, Ti] = 0 and the only solutions to
(2.1) are trivial. If that is not the case, we have a relation between Goldstone bosons and, in particular, we can impose some of the covariant derivatives, of the Goldstone bosons, to zero.
Like we mentioned before there is an analogy between the Higgs mechanism and the inverse one. Specifically, this becomes clear when we consider the Higgs mechanism with the unitarity gauge:
ζi =0, (2.4) Zia|ζ=0= 1 f∇aζ i = 1 f∂aζ i+Zi a|ζ6=0+O(ζ), (2.5)
in which ζiare the Goldstone bosons and Ziaare the gauge fields associated with
the broken generators. The above equation (2.5), is obtained by taking the in-finitesimal transformation of the gauge field under the broken group transforma-tion, that has as parameters the Goldstone fields. Now, we see explicitly that the inverse Higgs mechanism corresponds to requesting the opposite condition, like we said at the beginning of the chapter. That is, we are imposing our gauge fields to zero and when building the action this one will depend only on the covariant derivatives of the Goldstone fields.
This can also be seen by only considering the algebra of the full group un-der analysis. If we take the Poincaré group, that has the following commutation relations for the generators,
[Jµν, Jρσ] =i(ηνρJµσ−ηµρjνσ−ηνσJµρ +ηµσJνρ), (2.6)
[Jµν, Pσ] =i(ηνσPµ−ηµσPν). (2.7)
Then we see, that by taking the commutator value in the presence of a differen-tiable manifold, a N-dimensional membrane, the commutator between the un-broken tangential translations and the un-broken rotations are
[Pa, Jbi]hφi = iηabPihφi. (2.8)
This confirms to us that the broken rotations and translations do not generate independent Goldstone bosons. Such process can be represented graphicaly for the case of a string that breaks the three-dimensional Poincaré group down to the two-dimensional one in 2.1.
We are now going to give an explicit example of the application, via the alge-bra method, to have the right count of the number of Goldstone bosons in case the broken group is a full conformal group. We must add to the Poincaré algebra,
CHAPTER 2. NONLINEAR REALIZATION 9
Figure 2.1: Here it is represented a normal deformation of the string cause by local perpendicular translations. Such effects can be nullified by a local rotation [6].
the commutation relation that exists between these generators with the generator for dilatation, D, the generators for special conformal transformations, Kiand the one between these two groups of generators.
[Jµν, Pρ] =i(ηµρPµ−ηµρPν), (2.9)
[Jµν, Kρ] =i(ηνρKµ−ηµρKµ), (2.10)
[Pµ, Kν] =2iJµν −2iηµνD, (2.11)
[D, Kµ] =iKµ, (2.12)
[D, Pµ] = −iPµ. (2.13)
If the Conformal group spontaneously breaks down to the Poincaré group, we have that only one Goldstone boson brings the information of such procedure and we know that the said boson is the dilaton.
2.2
Coset construction
We are ready to analyze the general construction for effective Lagrangian action. This method is described in various paper and a detalied argument about it is given by Coleman, Wess and Zumino in the following two articles [7], [8], in which they also treated the problem with external broken symmetries groups. Its fundamental ingredient is the one-form Maurer-Cartan from which we obtain the basic elements to build any theory who breaks spontaneously G, is invariant under H and its element transform linearly under the latter group action.
We write down explicitly the commutators between the generators of the bro-ken and unbrobro-ken Poincaré symmetry. This will be of fundamental importance when we are going to choose the representative for the group element of G. We emphasize, that if we restrict ourselves to a transformation of H we want it to have a linear realization. What we need are the relations with Jij and Jab. They
CHAPTER 2. NONLINEAR REALIZATION 10 are the Lorentz transformation or Euler rotations that do not mix distinct types of indices. [Jij, Pa] =i[ηjbPi−ηibPj] = 0, (2.14) [Jab, Pc] =i[ηbcPa−ηacPb], (2.15) [Jij, Ph] =i[ηjhPi−ηihPj], (2.16) [Jab, Pi] =i[ηbiPa−ηaiPb] = 0, (2.17) [Jij, Jah] =i[ηjaJih−ηiaJjh−ηjhJia +ηihJja] = i[ηihJja−ηjhJia], (2.18) [Jab, Jic] =i[ηbiJac−ηaiJbc−ηbcJai +ηacJbi] = i[ηacJbi −ηbcJai]. (2.19)
The fundamental thing here is that the commutator does not give generators that are not present in the commutator. This will be needed for linear realization and for the Maurer-Cartan form. We begin by taking the simplest case possi-ble, that corresponds to only internal symmetries without gauge fields, to show the method to build up the action.
2.2.1
Internal symetries
Let us change the notation. We take a group G and a subgroup H. We call Zathe generators of broken symmetries and Vi of the unbroken ones. We assume they have the following relations,
[Vi, Vj] =i fajhVh, (2.20)
[Za, Vi] =i faicZc, (2.21)
[Za, Zb] =i fabcZc+i fabiVi. (2.22) A generic element g∈ G near the identity can be parametrized like
Ω=eiuiVieiθaZa. (2.23) We now introduce the Goldstone bosons, θa(x), associated to the broken genera-tors and define the transformation under a generic element of the group, g, as
geiθa(x)Za =eiθ0aZah(θ(x), g), (2.24)
h(θ(x), g) =eiu(θ,g)
iVi
. (2.25)
Let us define the transformation of θ like
gθa(x) = θ0a(θ(x), g). (2.26)
Under a generic element group of G the Goldstone bosons do not transform lin-early: g1g2eiθ aZa =g1eiθ a 2(θ,g2)Zah 2(θ, g2) = eiθ a 1(θ2(θ,g2),g1)Zah 1(θ1, g1)h2(θ2, g2) = (2.27) =eiθ12a (θ,g1g2)Zah( θ, g1g2)
CHAPTER 2. NONLINEAR REALIZATION 11 If g belongs to the subgroup H we have a linear realization,
geiθaZa =eiuiVieiθaZae−iujVjeiujVj =eiθaζabZbeiujVj. (2.28)
This is possible thanks to the commutator property (2.21). We take the Maurer-Cartan differential form, that is defined as Ω−1dΩ and parametrize Ω = eiθaZa. We also parameterize the one form as
Ω−1dΩ=iωa
ZZa+iωVi Vi. (2.29)
The transformation properties under a generic element g are e−iθ0aZadeiθ0bZb =eiuiVie−iθaZag−1d[geiθbZbe−iujVj] =
=h(θ, g)e−iθ
aZa
d[eiθbZbh−1(θ, g)] = (2.30)
=h(θ, g)[iωaZZa+iωVj Vj]h−1(θ, g) +h(θ, g)dh−1(θ, g).
From this we see how our Maurer-Cartan differential form change,
ω0ZaZa =h(θ, g)ωaZZah
−1(
θ, g), (2.31)
ω0ViVi =h(θ, g)ωVi Vi−ih(θ, g)dh−1(θ, g). (2.32)
Taking a single element of the form ωZa,
ω0aZ =h(θ, g)ωZa h−1(θ, g), (2.33)
we obtain the covariant derivative of the Goldstone bosons. This transforms un-der a linear representation of the group H. The covariant un-derivative is defined as
∇µθa =
∂ωaZ
∂xµ. (2.34)
It is also possible to define a covariant derivative for matter fields, but it is not of our interest and we omit it.
2.2.2
Space-time symmetries
Now that we have seen the general construction in the case of only internal sym-metries breaking, we are going to take the problem in the presence of broken space-time symmetries. We take the case of the spontaneous symmetry break-ing of the Poincaré group due to the presence of the membrane [9, 10, 11]. The presence of such an object breaks spontaneously the compact group ISO(D)of the Poincaré group to the direct product of SO(D−N) ×SO(N). The normal transla-tions and the Lorentz transformation that involve indices from two diverse basis, tangential and normal, are broken. We have seen that these symmetries give rise to not independent Goldstone bosons and therefore we have to apply the inverse Higgs mechanism.
CHAPTER 2. NONLINEAR REALIZATION 12 We consider only space-time symmetries and our choice of parametrization near identity is
Ω=eiYa(x)PaeiYi(x)Pieiθia(x)Jia. (2.35)
Everytime, it is possible to take a gauge such that Ya(x) = xa. Acting with a general g∈ G gives
gΩ=eix0aPaeiY0i(x0)Pieiθ0ia(x0)Jia. (2.36)
By taking g = eieaPa then the application of such element group gives us the
transformation rules
x0a =xa+ea, Y0i(x0) =Yi(x), θ0ia(x0) =θia(x). (2.37)
That gives us a nonhomogeneous transformation and we need to include it in the representation of the broken symmetries. Another point of view could be that if we did not include it, then our linear realization is not possible. Following the same procedure used for the internal symmetry case we parametrize
Ω−1dΩ=i[
ωaPPa+ω⊥hPh+ωjbJJjb +ωabJJab+ωijJJij]. (2.38)
We take the transformed version of the above equation Ω0−1dΩ0 =
hΩ g−1d[gΩ h−1] =hΩ−1(dΩ)h−1+hd(h−1). (2.39) From which we have the following tranformation rules for the covariant deriva-tives ω0aPPa =hωaPPah−1, (2.40) ω0h⊥Ph =hωh⊥Phh−1, (2.41) ω0jbJJjb =hωjbJJjbh −1, (2.42) ω0abJab +ω0ijJij =h(ωabJab+ωijJij)h−1−ih d(h−1). (2.43)
With this we conclude the proof of the searched transformation rules for our ele-ments obtained via the Maurer-Cartan one-form.
2.2.3
Effective action
We must now give to the elements that we obtained some geometrical interpreta-tion. First, we rewrite the obtained form in terms of a term that plays the role of a vielbein, and the covariant derivatives of the Goldstone bosons. That is by taking
ωaP =eabdxb, ωh⊥ = ∇aYhdxa, ωjbJ = ∇aθjbdxa, (2.44)
and for the elements associated with H
CHAPTER 2. NONLINEAR REALIZATION 13 By writing down explicitly the Maurer-Cartan form, we have that
Ω−1dΩ=iea
b[Pa+ ∇aYhPh+ ∇aθjcJjc]dxb+ [ωabcJab+ωijcJij]dxc. (2.46)
The same form written with the chosen parametrization of the element group near the identity gives
Ω−1dΩ=i∂ aYµΛµρPρdx a+ i 2(Λ −1)µ ρ∂aΛ ρβJ µβdx a, (2.47)
in which the tensor Λµρ is the Lorentz bulk transformation depending on the
Goldstones bosons θia.
The tensor multiplying all the Goldstone bosons covariant derivatives eabplays, like we said, the role of a vielbein for the membrane. This analogy is clear when we build the volume term, that corresponds to the lowest derivative order ele-ment of our action. We cannot naively use the membrane parametrizationVN
a=0dxa
because it has bad transformation properties. The elements with the chosen prop-erties under the group action G and H are eabdxb from which we obtain
N ^ a=0 eabdxb =det(e) N
∏
a=0 dxa. (2.48)We want to have an effective action that is described by geometrical objects that characterize our membrane. To achieve this, let us rewrite the Volume element in terms of the covariant derivatives of our Goldstone bosons and take as the target space metric a flat one,
Z det(e) N
∏
a=0 dxa = Z q s det(e eT)det(η) N∏
a=0 dxa, (2.49) where s is the sign representative of η and can be 1 or −1. From the Maurer-Cartan form we have that eab =∂bYαΛαaand substitution of it into (2.49) givesZ q s det(∂bYαΛαd∂cYβΛβd) N
∏
a=0 dxa. (2.50) Finally, we are going to apply our inverse Higgs constraint, which gives us a relation between the Goldstone bosons. The relation between these fields derives, like we saw, from the nonzero commutators[Jai, Ph] = iηihPa. The usual inverseHiggs constraint, whose physical origin has been discussed at the beginning of this chapter, is∇aYh=0. The equation written, with the use of the relations, is
∇aYj =∂aYαΛαj =0 (2.51)
and putting these in the Volume term
Z q s det(∂bYα∂cYα)d Nx =Z √s γ bcd Nx. (2.52)
CHAPTER 2. NONLINEAR REALIZATION 14 In the final equivalence we introduced the induced metric on the manifold. At the lowest order we obtain, from our nonlinear realization of the spontaneous breaking of the Poincaré group, the Nambu-Goto action, that is the action of an embedded bosonic brane. We want to build an action of higher order that con-tains the first non-trivial geometrical objects in function of the Goldstone bosons covariant derivatives. By requesting the invariance under diffeomorphisms and working in an arbitrary dimension, we request that the first non-trivial term will be made of contraction of at least two covariant derivatives.
If we identify Yα(x) to be our embedding function, then it comes naturally
to define the bulk Lorentz transformations, that mixes different kind of indices, with a normal basis,
Λαj:=n
j
α. (2.53)
In fact, we have that
nαj·n h β =Λ j α Λ αh = ηjh. (2.54)
We first need to identify∇aθic. By using the antisymmetric property of the Lorentz
generators, we have
∇aθic =eab(Λ−1)iρ∂bΛρa, (2.55)
where we introduced eab = (e−1)ab. We can express the external curvature in function eaband∇aθib. Kiab =∂aYµ∂bniµ =e c aΛ µ c(−Λµdeba∇aθid) = −edaecb∇cθid. (2.56)
Then, the general form of an effective action not coupled to matter is S= Z dNx√γ[µf + kf 2 K 2+ k 2R] (2.57)
where we used the Gauss-Codazzi relation (3.18) that gives a relation between the composition of the extrinisc curvature and the intrinsic one.
2.2.4
Fluid membrane and Thetered membrane relation
It is also interesting to see what happens in the case our membrane is not in the fluid phase but in a crystalline one. We will have a different tangential transla-tional group that lets the vacuum state invariant and the number of generators of said group is smaller with respect to the total number of generators of tangential translations. The membrane will be described as a reticle that has a defined space interval separating the monomers on it and this will result in a minor number of inverse Higgs constraints, that means we will have to work with a bigger number of Goldstone fields. In fact, in this case, under mechanical stress, each monomer fluctuates as a pointlike object. A pointlike object has a definite position in space
CHAPTER 2. NONLINEAR REALIZATION 15 and therefore its configurations break all translations, leaving the Euclidean rota-tions invariant (2.2). The coset is thus
ISO(D)/SO(D), (2.58) And we can choose as representative
Ω=eirµPµ. (2.59)
The Maurer-Cartan one-form associated with it is Ω−1
∂aΩ =I∂arµPµ, (2.60)
in which we introduced a set of coordinates ζa for our membrane. The most general Hamiltionian, truncated to the second order of the derivative expansion, that has as fundamental objects the covariant derivative is
Ht = Z dNζ[ε0+µt 2 (∂ar µ)2+kt 2(∂ 2rµ)2+u( ∂arµ∂brµ)2+ (2.61) +v(∂arµ∂arµ)2],
which is known as the tethered membrane model and the couplings introduced are in order the chemical potential, the surface tension, the extrinsic rigidity and the Lamé coefficients, that describe the elastic properties of the membrane. The model we obtained is not reparametrization invariant and this is the main dif-ference from the fluid model we have. Nevertheless, it is possible to give a fluid model description of a tethered membrane and we will outline here the steps nec-essarily to do it without explicitly doing such calculations. We are going to see it in the case of the Hamiltonian of the tethered model with neglected rigidity and elasticity properties. To obtain such a description we need to perform a coordi-nate change from the crystalline patch ζa to an arbitrary coordinate patch ya and rewrite the above action in terms of the fields in (2.46). The last task is realized by performing the following symmetry transformation
∂arµ →R(ζ)µν∂ar
ν, (2.62)
while the coordinate change gives rise to the determinant
∂ζa
∂yb =e
a
b . (2.63)
After this manipulations we obtain Ht = Z dNy√h ε0+ N 2µt+ µt 2 (∇aφ i)2 . (2.64)
Then by performing a path integration of the Goldstone field πi, which here plays the role of the order parameter of the breaking of the symmetry SO(D) →
SO(N) ×SO(D−N), we obtain an effective model to describe thetered mem-branes with a fluid membrane model. [9]
CHAPTER 2. NONLINEAR REALIZATION 16
(a) This is the fluid membrane config-uration where there are rappresented the monomers, that are free to diffuse. This system is, in the macroscopic scale, invariant under any tangential transla-tion.[9]
(b) The monomers, seen as pointlike objects that breaks fully the transla-tional invariance of the embedding. In the crystalline phase the binding(lines) are intact. A displacement of any monomer does change the configura-tion of the membrane in the macro-scopic scale too.[9]
Figure 2.2: The crystalline and fluid mechanical phase of a membrane. The dots represent the monomers.
Chapter 3
Covariant variations of a fluid
membrane
In this section we are going to work out the construction of the two fundamental forms of an embedded N-dimensional membrane in a D-dimensional space that has an arbitrary metric gµν. We are going to use Greek letters for bulk space and Latin ones for local coordinates indices on the membrane. Take the embedding function rµ(
ζa) with µ = 0, ..., D−1 and a = 0, ..., N−1. The new basis will be
(ea, ni)defined from :
ea : =∂arµ∂µ, (3.1)
g(ni, nj) =δij, (3.2)
g(ea, ni) =0, (3.3)
where we used the metric operator applied to vectorial fields. The indices i represent the elements orthonormal to the membrane and takes the values in 1, ..., D−N. A representative image of the defined basis is given in figure 3.1. We have an induced metric on the manifold γab := g(ea, eb) that arises from the
pull back action of the embedding on the target space metric. The generalized
Figure 3.1: We gave here a visual representation of the embedding of the fluid membrane in a flat target space.
CHAPTER 3. COVARIANT VARIATIONS OF A FLUID MEMBRANE 18 Gauss-Weingarten equations are :
Daeb =γabcec−Kiabni, Kiab :=g(Daeb, ni) = Kiba, (3.4)
Dani =Kiabeb+ω ij
a nj, ωaij := g(Dani, nj) = −ωaji, (3.5)
where we introduced the two one fundamental forms: γabc the Levi-Civita con-nection and Kiab the extrinsic curvature. We introduced in a different form the projection of the covariant derivative of the normal base along the normal direc-tion. Usually this would have been introduced directly in the definition of the derivative. We choose not to do so, having this peculiar significance. It is the connection of the remaining O(D−N) symmetry group. If we take a general transformation O of such a group:
ni →Oijnj, (3.6)
Da˜ni = Da(Oijnj) =OijKjabeb+ [(DaOij) +Oikωa jk ]nj = (3.7)
=K˜iabeb+ [(DaOij) +Oikωa jk ](O−1) j
h˜nh. (3.8)
From this we obtain the transformation properties:
Kiab →OijKjab ωa →O ωaO−1+O,aO−1 (3.9)
Let us observe how we can redefine the following objects in a covariant way:
∇aeb :=Daeb−γabcec =Kiabni, (3.10)
˜
∇aφib := ∇aφbi −ωa jiφjb. (3.11)
We have the usual Levi-Civita connection for the indices that are on the mem-brane and another one for normal ones. If we worked in the case of a hypersur-face than this new transformation would not have taken place. The same holds for D= N−2, two-dimensional membrane, in which it corresponds to an abelian gauge group and can be nullified everywhere. We can define its curvature as fol-low:
Ωabij = ∇bωaij− ∇aωbij+ωb ki ωakj−ωa ki ωbkj. (3.12)
We still must obtain the relations between the extrinsic and internal curvature. These are obtained by using the Ricci’s identity and acting on our base with the covariant bulk derivative twice.
DfDaec = [∂fγacb+γaceγf eb−KaciKfbi]eb+ (3.13) + −[γacbKf bj+∂fKacj +Kaciωf ji ]nj, DfDani = [∂fKabi+Kadiγf db+ωa ji K bj f ]eb+ (3.14) + [∂fωa ji −KabiKf bj+ωa jiωf jh]nj,
CHAPTER 3. COVARIANT VARIATIONS OF A FLUID MEMBRANE 19 [Df, Da]ec = [∂fγacb−∂aγf cb+γaceγf eb−γf ceγaeb−KaciKfbi+ (3.15) +Kf ciKabi]eb − [γacbKf bj−γf cbKabj+∂fKacj−∂aKf cj+ +Kaciωf ji −Kf ciωa ji ]nj = = [Rbc f a−KaciKfbi+Kf ciKabi]eb− [∇˜ fKacj −∇˜aKf cj]nj, [Df, Da]ni = [∇˜ fKabi−∇˜ aKfbi]eb+ [∂fω ij a +∂aωfij+ωa hi ωfhj+ (3.16) + −ωf hi ωahj−KabiK j f b +K bi f K j ab]nj = = [∇˜ fKabi−∇˜ aKfbi]eb+ [Ωa fij−KabiK j f b +K bi f K j ab ]nj. (3.17)
From the above equations, in which we are still working with an arbitrary bulk metric, we obtain the integrability conditions:
g(R(ea, ef)ec, ed) =Rf acd−Kf ciKadi+KaciKf di, Gauss-Codazzi, (3.18)
g(R(ea, ef)ec, ni) =∇˜ fKaci−∇˜aKf ci, Codazzi-Mainardi, (3.19) g(R(ea, ef)ni, nj) =Ω ij f a −K bi f K j ab +K bi a K j f b , Ricci. (3.20)
From the Ricci integrability condition we see that it is always possible, indepen-dently from the bulk metric, to have a zero-torsion curvature, Ωabij, locally. If our target space is a flat one than all the above equations are identically null. For our interest, 3.18, gives us a relation between combinations of the extrinsic cur-vature and the Riemannian tensor of our membrane. Let us write explicitly this connection,
Rfacd =KfciKadi−KaciKf id (3.21) Rad :=Rfa f d =KiKadi−Ka f iKf id (3.22) R :=Radγad =KiKi−Ka f iKf ai (3.23)
3.1
Covariant variations along the normal
We are now going to take the variations of our embedding function rµ up to the
second order on the parameter along the normal deformation. Why we only take the normal ones will be clear when we are going to talk about the second order one. The procedure will be equal to the one used by Capovilla in [12] and [13]. We will proceed with the background field method, that is we will take our vari-ation on a background configurvari-ation of the membrane. A general varivari-ation of the embedding function involves both normal and tangential directions,
δrµ =φini+φaea. (3.24)
We are interested only in the variations along the normal base. The local tan-gential deformations have no physical properties and can be seen as reparame-terizations of our membrane. This works only if the membrane does not have a
CHAPTER 3. COVARIANT VARIATIONS OF A FLUID MEMBRANE 20 boundary. To show this we take the infinitesimal reparameterization
ζa →ζa+ea(ζ). (3.25)
This transforms our embedding function as rµ(
ζ) → rµ(ζ) +ea ∂
∂ζar
µ(
ζ) (3.26)
and by taking back our definition of the tangential base (3.1) we see that by an appropriate choice of such infinitesimal variation, we can always absorb such terms. This will correspond with our choice of gauge-fixing, that gives an ultra-local condition and will thus not generate any ghost mode. We will express our variations like derivatives along the vector field δ :=φiniand use the fact that for
infinitesimal variations Dδea =Daδ.
3.1.1
First variation of intrinsic geometry
We begin by decomposing the general variation of eaalong the base,
Dδea =βabeb+Jaini, (3.27)
βab=g(Dδea, eb) = βba, (3.28)
Jaj =g(Dδea, nj), (3.29)
and using the properties of our base we obtain
βab =g(Daδ, eb) =φig(Dani, eb) = φiKabi, (3.30)
Jai =g(Daδ, ni) = φkωa ik + ∇aφi =∇˜aφi. (3.31)
The first variation of the induced metric gives
Dδγab =Dδg(ea, eb) =2g(Dδea, eb) = 2K
i
ab φi, (3.32)
and by requesting that Dδ(γabγbc) = 0
Dδγ
ab = −
γac(Dδγcd)γdb = (3.33)
= −2Kdciφi.
Here we introduced the inverse of the metric and denoted it as usual by γab. We directly write down the first variation of the are element√sγ
Dδ √ sγ = 1 2 √ sγγabDδγab = √ sγγabKabiφi = √ sγKiφi, (3.34)
where s represents the signature of our metric, that can be (−,+, ...,+), Minkowskian, or(+, ...,+), Euclidean.
CHAPTER 3. COVARIANT VARIATIONS OF A FLUID MEMBRANE 21 We request that our tangential derivations are torsionless so that we can ex-press our Levi-Civita connection as
γabc = 1 2γ c f(D aγbd+Dbγad−Ddγab). (3.35) By noticing that −(Dδγed)γecγabd = (Dδγ ec) γedγabd = (3.36) = 1 2Dδγ f c(D aγb f +Dbγa f −Dfγab),
we can express the variation along the normal direction as Dδγ c ab = 1 2γ cd(∇ aDδγbd+ ∇bDδγad− ∇dDδγab) = (3.37) = (∇aKbci+ ∇bKaci− ∇cKabi)φi+ + (Kbciδaf +Kaciδ f b −K i ab γf c)∇fφi.
The variations of the Riemannian curvature and its contraction are DδR a bcd =DcDδγ a bd −DaDδγ a bc +Dδ(γ a ce γbde) −Dδ(γ a de γbce) (3.38) = ∇cDδγ a bd − ∇dDδγ a bc , DδRbd = ∇aDδγ a bd − ∇dDδγ a ba , (3.39) DδR=DδRbdγbd+RbdDδγ bd = (3.40) = ∇f∇dDδγf d−∆(γbdDδγbd) +RbdDδγ bd.
From this we have our first important result for two-dimensional membranes, Dδ( √ −γR) = −2 √ −γKabiφiGab+ √ −γ[∇f∇dDδγf d+ (3.41) + −∇a∇a(γbdDδγbd)] = = −2√−γKabiφiGab+total derivatives.
This remarks the fact that for a 2 dimensional manifold, theR d2ζ√γRis a
topo-logical invariant, and like one would expect our deformations, that do not change the topology of the membrane, result in a null value. To see this we must only remember that Gab =Rab−12γabR and in 2 dimensions we have Rab = 12γabR.
3.1.2
First variation of extrinsic geometry
We begin like before by calculating the first variation of the base normal vectors, Dδni = −Jaiea+ηijnj, Jai = −g(Dδni, ea) = g(ni, Dδea) = ∇˜ aφi, (3.42)
CHAPTER 3. COVARIANT VARIATIONS OF A FLUID MEMBRANE 22 The tensor ηij does transform like a connection under the group O(D−N),
η →OηO−1+ (DδO)O
−1. (3.43)
Therefor we introduce a covariant variation defined as ˜
Dδψi =Dδψi−η
j
i ψj. (3.44)
We could also work directly by imposing ηij = 0 everywhere, being our action covariant and in other words we do not have free indices. In this new notation we have ˜Dδni = −(∇˜aφi)ea. The first variation of the extrinsic curvature in a general
bulk metric is ˜ DδK i ab = −D˜δg(Daeb, n i) = −g(D δDaeb, n i) +g(D aeb, Jdied) (3.45) g(Daeb, Jdied) =g(γabcec, Jdied) =γabc∇˜cφi (3.46) g(DδDaeb, n i) =g(R( δ, ea)eb, ni) +g(DaDδeb, n i) = (3.47) =g(R(nj, ea)eb, ni)φj+Dag(Dδeb, n i) − g(Dδeb, Dan i) = =g(R(nj, ea)eb, ni)φj+Da(∇˜ bφi)+ (3.48) + −[KbdjKadiφj+∇˜bφkωaik] ˜ DδK i ab = −∇˜ a∇˜ bφi+ [g(R(ea, nj)eb, ni) +Kb jdKadi]φj (3.49)
We know that our extrinsic curvature is symmetric in the tangential indices. That suggests that there is another integrability condition that could be obtained from the above variation. This is not our case, because it is related to the Ricci integra-bility one (3.20). (∇˜a∇˜b−∇˜b∇˜a)φi = [g(R(ea, nj)eb, ni) −g(R(eb, nj)ea, ni) +KaciKcbj+ (3.50) + −KbciKcaj]φj (∇˜a∇˜b−∇˜b∇˜a)φi =Ωab ji φj (3.51) g(R(ea, nj)eb, ni) = −g(R(nj, eb)ea, ni) −g(R(eb, ea)nj, ni) (3.52) Ωabijφj = [−g(R(eb, ea)nj, ni) +KaciK cj b −K i bc K cj a ]φj (3.53)
At least we calculate the variation of the connection ωaij.
˜ Dδω ij a =Dδω ij a −ηikωakj−ηjkωaik (3.54) Dδω ij a =g(R(δ, ea)ni, nj) +g(DaDδn i, nj) +g(D ani, Dδn j) = (3.55) =g(R(nk, ea)ni, nj)φk+ (∇˜bφi)K bj a − (∇˜bφj)Kabi+∇˜aηij ˜ Dδω ij a − ∇aηij =Dδω ij a −∇˜aηij (3.56)
3.1.3
Euler-Lagrange equations
At this point of the discussion, we are going to take a digression about the first variation of our effective theory and obtain the Euler-Lagrange actions to the or-der K2. We remark that all our computation is made up to the terms of that order
CHAPTER 3. COVARIANT VARIATIONS OF A FLUID MEMBRANE 23 because we are working on a phenomenological action. The first variation reads
˜ DδHf = Z dNy√γ[(µf + kf 2 K 2)K j−kfKj∆˜ −kfKiKabiKabj+ (3.57) + −kKabjGab]φj = = Z dNy√γ[(µf + kf 2 K 2)K j−kf∆K˜ j−kfKiKabiKabj−kKabjGab]φj
and our equations of motion to the order of interests are
µfKj =0. (3.58)
This term derives only from the variation of √γ, our metric measure, and
con-firms the well-known result that extremal surfaces have a vanishing trace of the second fundamental form.
3.2
Second order variations
Before we directly apply the variations to our action, we must make some re-marks on the general variation. We are first taking an infinitesimal variation of the embedding and seeing what this brings to our effective field theory mem-brane with the objective to calculate the Hessian, the perturbative expansion and then confront the results with the initial action. At the second order if we take only the elements, we need to consider the variations of our attached fields φi. Luckily, that is not our case, and the evaluation is the naive computation of nor-mal variations on geometric objects. The general deformation up to the second order of our action will be equal to the normal variation plus a total derivative and a piece proportional to the Euler-Lagrange equations. The tangential varia-tion of ea, of γaband of √ γare Dδkea =Da(φ be b) = (Daφb)eb+φb(γabcec−Kabini) = (3.59) = (∇aφb)eb−Kabiniφb, Dδkγab = ∇aφb+ ∇bφa, (3.60) Dδk √ γ=∇aφa (3.61)
The total deformation along δ=δ⊥+δkof our action rewritten as S= R dNζ√γε, where we choose s=1, is DδS= Z dNζ[Dδ⊥( √ γε) +Dδk( √ γε)] = (3.62) = Z dNζ√γ[εKiφi+ (Dδ⊥ε) + ∇a(εφ a)].
We have that up to a total derivative the Euler-Lagrange eq. depend only on the normal deformation. The above equation gives us again an ulterior explication of why we can neglect the parallel deformation for a membrane without a border.
CHAPTER 3. COVARIANT VARIATIONS OF A FLUID MEMBRANE 24 Let us parametrize Dδ⊥(
√
γe) = Aiφiand rewrite the above variation as variation
of density elements, DδS= Z dNζ[Aiφi+Da( √ γεφa)]. (3.63)
We know that for infinitesimal variations we can change the order between Dδ and Da. By acting with a second variation on the action and excluding the total
derivative part, we gain D2δS= Z dNζ[AiDδ⊥φ i+ (D δ⊥Ai)φ i+D a(Aiφiφa)]. (3.64)
That is the anticipated result. When we consider the total value and not only its density, then we have up to modulus Euler-Lagrange eqs. and total derivatives that the important pieces came from the perpendicular variations of the densities. We are going to write down the results of the second variations up to order O(K4)and modulo total divergencies for our density terms. The extended calcu-lations including the proportional terms to K4 can be found in appendix A. We will also impose directly the constrain ηij = 0 being the new defined directional derivation applied to a scalar equal to the not covariant one. The variation of the induced metric is D2δγab =2(KaciK cj b φiφj−φ i∇˜ a∇˜bφi). (3.65)
For the inverse of the metric tensor we use Dδ2M−1= −Dδ[M −1(D δM)M −1] = (3.66) =2M−1(DδM)M −1(D δM)M −1−M−1(D2 δM)M −1,
and from this we obtain
Dδ2γa f =2[3KadiKd jf φiφj+γc fφi∇˜ a(∇˜ cφi)]. (3.67)
The variation of the elements densities are D2δ(µf √ γ) = −µf √ γφi∆δijφj+ (µf √ γφi{KiKj−KabiKabj}φj) (3.68) D2δ(1 2 √ γkfK2) =kf √ γφi{∆2δij +2KabiKabj∆−KjKi∆− K 2 2 ∆ δij+ (3.69) +2KhKach∇a∇cδij}φj D2δ(1 2 √ γkR) =k√γφi[Gabδij +2KabiKj−2KadiKd jb + (3.70) + (KcdiKcdj−KiKj)γab]∇a∇bφj (3.71)
Here we can take an easy check on the variations that we made. We know that in the case of a two-dimensional membrane the integrated density intrinsic cur-vature corresponds to a topological invariant. That means that every continu-ous deformation, that does not change its topology, should be null. To do so we rewrite the relation between the Ricci curvature and the extrinsic ones
Rab δij
D−N =K
ab
CHAPTER 3. COVARIANT VARIATIONS OF A FLUID MEMBRANE 25 It is interesting to notice that up to this order we have an operator Lij that is self-adjoint. This derives from the peculiarity of having as a result only terms that are combination of even derivatives and symmetric tensors in normal indices.
3.3
Energy bound and Monge representation
In this final part we are going to analyze the bounds for our coupling constants in the Monge representation[14] with a precise gauge condition for the normal base. We will see that these results are the same we would have obtained in our covariant formalism with a flat background. We will assume that our constraints on the physical validity region of our coupling constants are still valid when we will analyze they RG flow in the D → 0 case. This makes no sense when we are talking about N-dimension physical membranes embedded in a D-dimensional flat space but will have one when we discuss the relation between the SO group and the SP group.
For a flat membrane we take the following parametrization rµ(~
ζ) = (ζ1, ..., ζN, h1(~ζ), ...., hD−N(~ζ)). (3.73)
From the definition 3 we have that our induced metric is
γab =δab+ ∂h
i
∂ζa
∂hi
∂ζb. (3.74)
The inverse up to order O(h4)is
γab ' δab−∂ah·∂bh (3.75)
and from this we can build all our intrinsic geometrical objects:
γabc '∂ch·∂a∂bh, (3.76) R '∂a[∂ah·∂2h] −∂b[∂ah·∂b∂ah], (3.77) q det(γab) = q exp(Tr ln(δab+∂ah·∂bh)) '1+ 1 2∂ah·∂ ah = (3.78) =1+1 2|∂ah| 2.
Our normal base vectors are defined by two properties : they must be ortohog-onal to every tangent base vector of the membrane and to each other. We can parametrize the nias(fai, gij)and the aforementioned conditions are
(δab, ∂ahk) ·ni =0= fai+ (∂ahk)gik, (3.79)
ni·nj =δij = faif j
a+gikgik. (3.80)
We do not have to worry whatever the indices are up or down in these formulas. Up to our order of interest the respective tensor who lowers or raises them is the
CHAPTER 3. COVARIANT VARIATIONS OF A FLUID MEMBRANE 26 identity in Euclidean space. We must pay attention that with δabwe also indicate the Kronecker delta. These conditions do not completely determine parameteri-zation. In fact, we have N(D−N)dof fixed by [3.79] and(D−N)(D−N+1)/2 from [3.80]. We still have to determine(D−N)(D−N−1)/2 conditions and these are exactly the number of generators for the unbroken subgroup SO(D−
N). Evaluating the conditions up to order O(h2)gives us for the tangential com-ponents
(fai)0+ (fai)1 ' −∂ahk[(gik)0+ (gik)1], (3.81)
(fai)0 =0, (3.82)
(fai)1 = −∂ahk(gik)0. (3.83)
The perpendicular components satisfy the following equations
(gik)0(gjl)0 =δij, (3.84)
(gik)0(gjl)1δkl = − (gik)1(gjl)0δkl, (3.85)
(gik)0(gjl)0∂ahk∂ahk = − (gik)1(gjl)1δkl. (3.86)
The natural choice for(gik)0 is to put it equal to δik, the Kronecker delta. In fact,
if we had all null heights than the usual choice for the linear independent normal vectors would be of this form. From this choice we obtain for the remaining two conditions.
(gij)1 = − (gji)1 (3.87)
δikδjl(∂ahk)(∂ahl) = − (gik)1(gjl)1δkl (3.88)
Combining both the conditions, we have that the only solution is the trivial one. This gives us too the notion that, in this case, we have ∂ahi∂ahj=0. We have an
in-sight that for the Helfrich model applied to N-dimensional fluid membranes the first kinetical term will be a quartic divergent one. There is a subtle thing that can be seen in our passages: we used our dof for the normal vector parametrization by imposing(gij)0. After imposing this conditions we have our normal vectors,
up to O(h),
niµ = (−∂ahi, δik). (3.89)
Now we can construct our extrinsic curvatures using their definition Kiab = − ∂arµ,bniµ = −(0, ∂a∂bh
k) · (
∂dhi, δij) = −∂a∂bhi (3.90)
Ki = − ∂2hi, (3.91)
and verify that (3.18) still holds order to order
CHAPTER 3. COVARIANT VARIATIONS OF A FLUID MEMBRANE 27 Finally, we have our action written in the Monge representation with small nor-mal fluctuations H ' Z dNζ{µf + kf 2 ∂ 2 hi∂2hi+ k 2[∂a(∂ a hi∂2hi) −∂b[∂ahj∂b∂ahj]} = (3.93) =µf A+ kf 2 Z dNy hi∇2hi+ k 2B.
In conclusion we see that our√γRterm do only contribute giving a border term
and for a membrane without borders gives no result. Thus, by setting B= 0, we have no constraints on the intrinsic geometrical coupling. We can make similar arguments, based solely on the topology of our membrane, for the other two. That is by requesting the energy positivity we have both µf and kf positive.
Now we can see the relation between our covariant deformations and this result. If we take a flat background metric and having shown that the first con-tributing variations of our Hamiltonian/Action are with the background metric method, the ones to the second order, we see that our defined small heights, hi(ζ),
are exactly our attached fields φi(ζ). To evaluate the same Hamiltonian, we need
to put in the total variation the geometric objects calculated with γab = δab. This
has been done in this section and we have that both curvatures are null in a flat background. We have
H'
Z
dNζ[µf(1−φi∆φi) +kfφi∆2φi], (3.94)
In which the second term multiplying the surface tension can be rewritten in the form of a total derivative and ∂aφi∂aφi. When we impose the same gauge
conditions on the above variations, we have that this term is null. This shows that we have the same result obtained in the Monge representation. They would differ if the membrane had a boundary.
Chapter 4
Renormalization and crumpling
transition
After having computed the variation of our effective action up to the second order let us first formalize the construction of our renormalization procedure. We be-gin by writing down the partition function in the euclidean formalism, imposing a gauge fixing for the parametrization invariance [15]. The discussion is followed by a review of the Wilsonian renormalization procedure ( [16], [17], [18]) and the calculation of the β-functions of our coupling constants. We chose to use a UV cut-off regularization so that we mantain the explicit power divegercencies and have not to fixe neither the bulk dimension and the membrane dimension.It is im-portant here to remark on the difference between the phase diagram, that we will obtain for our coupling constants, and the RG diagram. The way we are going to use the RG step is seen as the derivation from microscopic properties described at an ultraviolet scaleΛ and parametrically on the values of interactions. This ap-proach is rendered explicit in figure 4.1 where the arrows point to the IR region. We are interested in the critical surface for which, if there exists a non-trivial FP( it has also flow directions going outside it), then this surface differentiates between distinct phases of the model. It is now important to notice that the critical points are related to scale invariance but do depend on the scale. However, by looking at the system for increasingly large scales, we can systematically wash out the dependence on the UV cutoff (the inverse of the length scale at which our initial model gives a good description) by tuning to criticality and then fall into the RG fixed point (RGFP) that does not depend on scale.
4.1
Partition function and gauge fixing
We define the partition function of the embedded brane likeZ
D[r]e−βH. (4.1)
The functional measure D[r]corresponds to the integration over all possible em-bedding functions of the N-dimensional fluid membrane. We request that it sat-isfies the following two conditions :
CHAPTER 4. RENORMALIZATION AND CRUMPLING TRANSITION 29
Figure 4.1: We show the phase-diagram in the case of two distinguished phases, in which the red line denotes is the parametrized curve of our model at a fixed scale with different values of the interactions. The intersecation of such curve with the direction that brings us to the RGFP is the critical surface.
• Invariance under reparametrization;
• Locality : small deformations at distant points should be treated indepen-dent and their measure should factorize.
Written in such a way the partition function would give us a divergent term. We need to consider the Faddev-Popov process, that is to fix a gauge. Usualy, such a procedure gives rise to ghost fields. In our case, it is possible to forget about them because they will have an associated ultra local propagator. Let us briefly review the Faddev-Popov mechanism [19]. We want to compute the value of an integral whose components are invariant under the action of a group G. Like we said before, this integral, without a gauge fixing condition, would be divergent because we are integrating on equivalent configurations for G. Let us call Fasome gauge fixings and dg the invariant dimension of the group. Our partition function becames Z= Z [dX] { 4G[X] Z [dg]δ(Fa[X]) }e−βH. (4.2)
That is the same as the inital one in which we have inserted the element between curly braces that is equal to identity. Being our group measure invariant under the action of the group, we have that
4G[X] :=
Z
[dg]δ(Fa[X])
−1
(4.3) is gauge invariant and can be computed on any configuration we want. This give us the possibility to isolate the integral over the gauge group and the choice to perform our calculations on a gauge slice. Parametrizing the gauge fixing
condi-CHAPTER 4. RENORMALIZATION AND CRUMPLING TRANSITION 30 tion in (4.3) gives rise to a determinant.
4G[X]δ(Fa[X]) =det(MG a b) J[θ =0] δ(F a[X]) (4.4) MGab := δ(Fa[X]g(θ)) δθb |θ=0 (4.5)
We represented with J[θ] the Jacobian determinant of the group measure and θ
the parametrization variables of the group. MGabis the Faddev-Popov matrix. By
neglecting all the terms that can be put in the normalization costant we have Z=
Z
[dX] (det(MGab)δ(Fa[X])e
−βH. (4.6)
Usualy one introduces some auxiliary fields Ba to exponentiate the gauge fixing condition and the ghost fields ca,ca to exponentiate the Faddev-Popov
determi-nant.
Let us go back to our Membrane configurations invariant under diffeomor-phisms. We need not to take in consideration the other symmetry gauges, be-cause they will not give rise to null modes after we impose the physical gauge fixing. Under an inifnitesimal reparametrization transformation we have
ζa →ζ0a =ζa+ea(ζ), Xµ(ζ) → X0eµ(ζ) = X
µ(
ζ) +ea∂X
µ
∂ζa (ζ). (4.7)
We chose a background configuration Xµ0 and take small fluctuations, x, around it. Our gauge fixing will be, like we saw for our variational calculations, to set such variations only along the normal directions. The variation of the gauge fix-ing is δFa(ζ) = δ[X(ζ) −X0(ζ)] · ∂ ∂ζaX0(ζ) = (4.8) =δX(ζ) · ∂ ∂ζaX0(ζ) = =eb(ζ)[ ∂ ∂ζbX0 (ζ) + ∂ ∂ζbx (ζ)] · ∂ ∂ζaX0(ζ) = =eb(ζ)[(γ0)ab(ζ) + ∂ ∂ζbx(ζ) · ∂ ∂ζaX0(ζ)].
The kernel of the Faddov-Popov determinant reads
δFa(ζ) δeb(ζ0) =δ(ζ−ζ 0)[( γ0)ab(ζ) + ∂ ∂ζbx(ζ) · ∂ ∂ζaX0(ζ)]. (4.9)
and by expressing it with the ghost fields we have an ultralocal propagator that does not interact with the other objects present in the theory. We will also not introduce the auxiliary field for the gauge-fixing and will directly apply it on our action. It is important to notice that we did not considered the other symmetries of our action. In fact we had constructed an effective action that is invariant under ISO(D) ×diff. Once we did this choice they do not have anymore degenerates solutions.
CHAPTER 4. RENORMALIZATION AND CRUMPLING TRANSITION 31
4.2
Wilsonian Renormalization
We remark that our gauge-fixing only works for small perturbations that have not long wavelength correlations. The apparatus of the Wilson renormalization than works perfect for our case being also our theory only effective. That is we are not interested in completing the theory after a UV scale but work its properties from a fixed scaleΛUV. This justifies our order expansion up to O(K4)and will be later
discussed following Codello argument in [20].
The basic idea of the Wilson Renormalization group is to distinguish the par-tition function in two sectors: one over high frequencies modes and one over the lower ones. That is in a general point of view we take the fields φ like
˜ φ+(p) = 0 |p| > ΛUV ˜ φ+(p) Λ< |p| < ΛUV 0 |p| < Λ (4.10) and ˜ φ−(p) = ( 0 |p| >Λ ˜ φ−(p) |p| <Λ. (4.11)
That is our partition function is defined on these fields as Z[φ] =
Z
[dφ+] [dφ−]e−SΛUV(φ++φ−) =
Z
[dφ−]e−SΛ(φ−), (4.12)
where in the last equality we integrated over the high frequencies modes. That is we can define the effective action atΛ
e−SΛ(φ−) =
Z
[dφ+]e−SΛUV(φ++φ−)
. (4.13)
Comparing it with the usual partition function used in QFT for the computations of the Feynman diagrams, we see that the fields with momentum smaller than some cutoff, work like external currents. We still want to match our original action with the one obtained after the integration of the high modes. To make such a match up we need to rescale our quantities. We define Λ = bΛUV with
b ≤1 and obtain for our action in terms of the slow modes
Z dNy(µf + kf 2 K 2+ k 2R) → Z dNy0(b−Nµf + kf 2 b −N+2K2+ (4.14) +k 2b −N+2R).
Notice that the geometrical object have a b−2being both function of two deriva-tives in the tangential base. It is useful to notice here that the powers of b corre-spond exactly to the opposite of the mass dimension of our coupling constants. In the case there were no interactions then we have a trivial scaling properties
