Scale invariant theories & conformal field theories

dx

dy dw

ÝÑ dv

zÑz² ÝÝÝÝÝÝÝÝÑ

Figure 2.6. Holomorphic complex functions describe conformal maps between their domain complex planez“x`iyand the image complex planefpzq “Refpzq`i Imfpzq “ w`iv. In the above figure the analytic functionfpzq “ z² is shown. As one can see the angles of the intersection of the coordinate functionsxandyare preserved, but not the area enclosed by contour lines. Moreover, the plots indicate that the transformation of an infinitesimal area elementdxdy Ñdwdvcan be composed of a scaling and a rotation as shown in the inset.

since the metric fieldgpxqdoes not change when “flowing” alongv. For a conformal transforma-tion of the metric (2.145) the Killing equatransforma-tion (2.146) is modified to

Lvg“2σpxqg ÝÝÝÝÝÝÑ^local

coordinates ∇_µv_ν`∇_νv_µ“2σgµν,⁴⁰ (2.147) whereΩpxq “1´σpxqfollows from the linearized relationg¹“Ω²g«g´Lvg. Exponentiating the infinitesimal transformation defined by σpxqshows that a conformal transformations is in fact a Weyl rescaling

g¹_µνpx¹q “ e^´2σpxqg_µνpxq. (2.148) In local coordinates the trace of the conformal Killing equation (2.147) yields

2∇µv^µ“2σd ñ σ“ 1

d∇µv^µ. (2.149)

Inserting (2.149) into the conformal Killing equation allows to construct various constraint equa-tions depending on the dimensiond, and respective solutions for the vector fieldv. For a com-prehensive and exhaustive treatment of the conformal invariance/group seee.g. [69]. We will be content with understanding the different solutions ford “1,2 andd ą 2. We see that for d“1(2.147) is trivially satisfied, so any vector fieldvdescribes a conformal transformation, due to the fact that the concept of angles between tangent vectors is meaningless in one dimension.

Ford“2 (2.147) reduces to the Cauchy-Riemann equations, so the vector fieldvcan be con-structed by all holomorphic functions. Thus, the two dimensional conformal group possesses an infinite number of generators and an infinite number of associated conserved currents/charges.

In dimensionsdą2, the vector fieldvcan in general be expressed as follows

v^µ“a^µ`cx<sup>µ</sup>`ω^µ_νx^ν´x_νx^νb^µ`2bνx<sup>ν</sup>x<sup>µ</sup>, (2.150) where a<sup>µ</sup> describes infinitesimal translations, the antisymmetric ωµν infinitesimal rotations, c infinitesimal scaling transformations or dilation, andb<sup>µ</sup>so-called special conformal transforma-tions. In general the Poincaré symmetry and scale invariance is not fully equivalent to conformal symmetry due to the existence of the special conformal transformations. Constructing the asso-ciated generators, one can show that the Poincaré group including scale transformations form a subgroup of the conformal group. Futhermore, by constructing appropriate generators analo-gous to the Poincaré generators it can be shown [69] that the conformal group is isomorphic to SOpd`1,1qin Euclidean spacetime andSOpd,2qin Minkowski spacetime, respectively.

Conformal Field Theory

Operators⁴¹transforming under the conformal group can be constructed by representing the Lie derivative in local coordinates of the conformal transformation

O

¹ “

O

´Lv

O

in the following way

Lv“v^µ∇_µ`∆σ´1

2p∇_µv_ν´∇_νv_µqS^µν, (2.151)

40The local coordinate representation of the metric Lie derivative in terms of simple covariant derivatives is only valid for a metric connection,i.e.∇g“0.

41Note that the CFT community developed a slightly different jargon as usually used in QFT: fields are called operators, and to add to the confusion there are two special classes of operators which are called primary and quasi-primary fields. Primary fields transform in a simple “tensorial” manner under local and global conformal transformations, whereas quasi-primary fields transform only under global conformal transformations. In two dimensions primary fields have the simplest energy-momentum operator product expansion with at most second order poles.

where∆denotes the scaling dimension of the operator

O

as defined in (2.117) and explained in Section2.3.5, whereasS^µν denotes theOpdqgenerators. A scalar conformal covariant operator transforming in a simple “tensorial” way under global conformal transformations is called quasi-primary. This transformation law can be derived from the conformal generators following (2.90) and is given by

ϕpxq ÝÑϕ¹px¹q “

dx¹ dx

´^∆ϕ{^d

ϕpxq “Ωpxq^∆ϕϕpxq, (2.152) where we have inserted the Jacobian of the coordinate transformationxÝÑx¹“Ω^´1x. Due to this additional transformation behavior, the two-point and three-point correlation functions are highly constrained

@ϕ¹₁px¹₁qϕ¹₂px¹₂qD

“Ω^∆1`∆2@

ϕ1pΩx¹₁qϕ1pΩx¹₂qD ñ xϕ1px1qϕ2px2q y “ C12

|x1´x2|^∆1`∆2, (2.153) where C12 is a constant coefficient and we implicitly used translation and rotation invariance.

Using the special conformal transformation, one can show that the correlator is only non-zero if

∆1“∆2,i.e. only quasi-primary fields with the same scaling dimension are correlated xϕ1px1qϕ2px2q y “ C12

|x1´x2|^2∆. (2.154)

Comparing with the definition of the critical exponent for correlators η in Table 2.5 we find 2∆ϕ “ d´2`η or η “ 2´d`2∆ϕ. This is in agreement with the hyperscaling relation η “ 2`d´2yJ (as expected!) where the scaling dimension of the operator is related to the scaling of the relevant source fieldJ byy_J “d´∆ϕ. In the same way, the three point function is fixed up to a constantC123as

xϕ1px1qϕ2px2qϕ3px3q y “ C123

|x1´x2|^{∆1`∆2´∆3</sup>|x2´x3|<sup>∆2`∆3´∆1}|x1´x3|^{∆3`∆1´∆2}. (2.155) Higher order correlation functions cannot be constrained by the scaling and special conformal transformation alone because here we can construct conformally invariant dimensionless cross ratios such as ^|x_|x¹₁^´_´x^x2₃^|_|^|x_|x³₂^´_´x^x4₄^|_|. The non-existence of conformal invariants for two and three point functions, due to the special conformal transformation, allows for the totally fixed simple forms in (2.154) and (2.155). Apart from the constraints imposed on the correlation functions, there are additional constraints for the conserved currents related to the Poincaré symmetry, in partic-ular the energy-momentum tensor.

Polyakov’s Theorem

Using Noether’s theorem we can define a tensor field by varying the action under symmetry transformations as

δvS“ ż

d^dx T^µν∇µv_ν “^! 0.

For translational invariancevν “aνwe find the conservation of charge or divergence condi-tion

∇µT^µν“0,

whereas the rotational invariance implies a symmetric energy-momentum tensor since its generator is totally antisymmetric

T^µνωµν “0 ^ ωµν “ ´ωνµ ñ T^µν “T^νµ.

Finally, conformal symmetry together with the conservation law and the symmetry condition gives rise to a traceless condition

T^µ_µ “0.

This follows from demanding that the conservation law must hold for conformal transforma-tions, see AppendixDof [70].

Formally, the traceless condition for the energy-momentum tensor can be derived directly from the general source term of a quantum field theory. Lorentz invariance (or translational and ro-tational invariance) imposes already strong constraints on the possible source terms. In general, there is no consistent interacting theory for massless fields with higher spin excitations than spin two in flat spacetime.⁴² Heuristically, a higher spin field possess more independent components that cannot be fixed by symmetries which in turn yields unphysical degrees of freedom,e.g. in d“ 4 a massless field has only two physical degrees of freedom. Therefore, the most general source term is given by

Ssource“ ż

d^dxpφλ`j<sub>µ</sub>A<sup>µ</sup>`T_µνg^µνq. (2.156) Physically, we can view the scalar source term λas a Lagrange multiplier fixing the value of the scalar, the vector source terms are usually related to gauge connections such as the elec-tromagnetic Up1q gauge field A^µ or, conversely, a charged current jµ induces an interaction mediated by the gauge field (see the discussion in Section2.3.2 below (2.96)). Considering Poincaré invariant theories, the related massless excitations to the energy-momentum tensor are spin two excitations, so we naturally expect the metric as the source term. But even for non-relativistic theories the deformation of a physical object due to external stresses is described by a displacement field changing distances compared to the free reference metric. In this casesT_µν is usually called stress-energy tensor related to the strain tensor of the deformed object. Applying a conformal coordinate transformation to (2.156) and inserting the generating functional for the energy-momentum tensor yields

δvlnZrgµνs “ ż

d^dxδlnZrg_µνs

δg^µν δvg^µν “ ż

d^dx TµνpLvg^µνq “2 ż

d^dx σTµνg^µν “^! 0, (2.157) which directly implies the traceless condition for arbitrary conformal deformationsσ

g^µνTµν “T^µ_µ “0. (2.158)

This also holds true for quantum field operators, where the corresponding Ward identities can be derived for the translation, rotation (Lorentz) and scaling symmetry. Nonetheless, conformal symmetry can be broken by quantum effects leading to a conformal anomaly. Since we learned from the RG approach that the critical fluctuations are correlated on length scales of the inverse reduced temperature (2.138) or the mass operator, conformal anomalies introduce a characteris-tic length scale signaling the breakdown of the effective theory description in the chosen degrees

42An interesting possible interacting higher spin theory with massless excitations is Vasiliev’s higher spin gravity in AdS4

spacetime, see [71]

of freedom. For example in chiral QCD the conformal anomaly sets the scale for color confine-ment and the acquired mass of the confined quarks/composite hadrons (since the quark masses

„Op1 MeVqare much smaller than the confinement scale„Op1 GeVq). As we will see in Sec-tion3.1.2, for a consistent string theory the conformal anomaly of the local Weyl symmetry must be removed which leads to a constraint on the numbers of spacetime dimensions. For purely bosonic string theory the critical dimension isd“26, whereas the supersymmetric string theo-ries are consistent in ten dimensional spacetime. For condensed matter applications some of the symmetries might not be present in the system. Usually, the Poincaré symmetry is broken due to anisotropies (no rotation invariance), lattices (no translation invariance) or non-relativistic de-grees of freedom (no boosts). This allow for different scaling behavior of space and time which is classified by yet another exponent the so-called dynamical scaling exponentzdefined as

tÝÑ t

c^z, xÝÑ x c, ωÝÑc^zω, kÝÑck.

(2.159)

Forz “1 we recover relativistic scaling symmetries found in one-dimensional electron liquids or in two-dimensional Graphene sheets. These systems can be described by a chiral quantum field theory and show a linear massless dispersion relation when tuned to their critical point.

A non-relativistic analogue of the conformal group is the Schrödinger group [72] where the Schrödinger scaling (see Schrödinger equation) yields z “ 2. More complicated scaling sym-metries, depending on the effective dispersion relation, might involve non-integer values for the dynamical scaling exponent. In certain gravity duals one finds a particular scaling behavior called Lifshitz scaling as well as more complicated scaling symmetries of AdS-spaces. Additionally, there exists a spacetime metric that allows for hyperscaling violation, parametrized by another scaling exponent [73].

To conclude the section about renormalization, I hope the reader is convinced that renormal-ization elegantly connects the microscopic world with the macroscopic world and in the same way different scale invariant theories that describe distinct physical phenomena. It is one of the most powerful methods to extract relevant degrees of freedom/information of complicated mi-croscopic theories and relates them to universal effective theories. In Chapter3we show that the Gauge/Gravity duality geometrize this wonderful method in a fascinating way and at the same time allows us to deal with new classes of strongly coupled, scale invariant theories and extend the applicability of the RG method.

3

Gauge/Gravity Duality

In this chapter we will establish the “original” AdS5/CFT4 correspondence devised in [74] and somewhat refined in [75,76]. In this case AdS5 stands for the five dimensional Anti-de Sitter spacetime, a hyperbolic spacetime with a boundary, which is a solution to Einstein’s classical equations of gravity with negative cosmological constant. On the other side of the duality we have a supersymmetric conformal field theory in four dimensions with vanishingβ-function at all energy/length scales even with regard to perturbative and non-perturbative quantum correc-tions. The duality belongs to the very powerful class of weak/strong dualities,i.e. we can map a strongly coupled conformal field theory with some additional properties to a weakly curved classical gravity. Furthermore, the AdS5/CFT4correspondence is an example of the holographic nature of (quantum) gravity as is exemplified by its unusually thermodynamic behavior near black holes. Therefore, we will start with a small survey of interesting features of gravity, then move on to explain a possible way to quantize gravity, which gives rise to a vast mathemati-cal web of theories (see Figure3.8) and are known under the name of (super) string theories.

Again dualities play an important role to map between these different string theories. In order to understand the AdS5/CFT4correspondence, we will only explain the crucial objects originating from a certain type of string theory. Clearly, this exposition of string theory will not do justice to the huge and still intricate subject, but I hope the more stringy inclined reader does not mind the writer’s lack of expertise on the subject. As a next step, we will employ the equivalence of the partition function to construct generating functionals and incorporate symmetry break-ing mechanism [77], as explained in Section 2.3. Most importantly, we will utilize the recipe of calculating response and Green functions [78] — a real time fluctuation-dissipation theorem connected to the Schwinger-Keldysh approach to Green functions [79] — to compile a dictionary relating equilibrium and dynamical properties of strongly correlated systems to a simple gravita-tional computation. Finally, we focus on generalizations of the original correspondence and show its deep and intriguing connection to renormalization group flows and to extend the dictionary to finite temperature/density calculations designed to tackle problems within condensed matter theory. As we will see, the generalized gauge/gravity duality, understood as a weak/strong dual-ity, is a powerful tool for extending field theoretic methods into previously inaccessible regimes.

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Im Dokument Gauge/Gravity duality (Seite 83-90)