Bosonic string theory

been many attempts to formulate a quantum theory of gravity, such as loop quantum gravity [46,92–94], string theory [95–99], non-commutative geometry [100], string-net condensates [101–104], just to name a few. To my knowledge, only string theory has been successful to incorporate the holographic principle and allows for explicit realizations, for example

• Matrix Theory: M-Theory¹⁰is conjectured to be dual to D0-branes (particles) in ten di-mensions, providing a second quantization scheme of M-Theory in flat space in a certain set of limits (the infinite momentum frame^N{^RÑ 8[105]). Thus, the eleven dimensional membranes (extended objects of M-theory) emerge from point like fundamental degrees of freedom in ten dimensions described in terms of supersymmetric (NˆNmatrix) quantum mechanics withN Ñ 8. The additional dimension is removed by compactification on a circle with radiusR “gsℓs related to the string couplinggs and the string lengthℓs. The eleven dimensional M-theory is recovered by the strong coupling limitRÑ 8. For more details the reader is urged to consult the comprehensive review [106]. A major difference to the AdS/CFT correspondence lies in the fact that matrix theory deals with an infinite vol-ume Minkowski space, whereas the AdS spacetime possess a boundary with a well defined Cauchy problem,i.e. the boundary conditions of the “AdS box” introduce sources for the interacting bulk fields. An overview of the connection between matrix theory and AdS/CFT can be found in [107].

• AdS/CFT Correspondence: The AdS5/CFT4 correspondence connects holographically a conformalN “4supersymmetricSUpN_cqYang-Mills theory in four dimensions to type IIB string theory on AdS5ˆS⁵. This holographic duality will be explained in the subsequent sections and extended in a way to geometrize the renormalization group method.

t t

x x

point particle

trajectoryds string

worldsheetdA

Figure 3.2. A relativistic point particle is described by the actionS “ ´mş

ds, whereds² “ g_µνdx^µdx^ν describes the relativistic worldline. The particles trajectory is deter-mined by the coordinate functionsx^µpτqparametrized by the worldline parameter τ. A one dimensional string traces out a worldsheet in two dimensional spacetime with the corresponding Nambu-Goto actionS“ ´Tsş

dA, where the particles mass is replaced by the string tensionTs, defined as (potential) energy over the spatial string length.

This is a gauge symmetry on the worldsheet where the coordinate functions transform as scalars and the induced metric transforms as a rank two cotensor

h¹_αβpσ¹q “ Bσ^γ Bσ^1α

Bσ^δ

Bσ^1βh_γδpσq, (3.11)

or for infinitesimal transformationsf^αptσuq “σ^α`ǫ^αptσuq,

x^a ÝÑx^a´ǫ^αBαx^a, h_αβÝÑh_αβ´ p∇αǫ_β`∇βǫ_αq. (3.12)

• Weyl rescalingas defined in (2.148)

h_αβÝÑ e^´2wpτ,σqh_αβ, (3.13) which allows to conformally deform the worldsheet and to describe the worldsheet by a two dimensional CFT. The different conformally equivalent worldsheets can be viewed as different gauges describing the same physical state.

The classical Polyakov (string sigma model) action Ss“ ´Ts

2 ż

d²σ?

´hh^αβgabpxqBαx^aBβx^b, (3.14) is invariant under the above mentioned symmetries. The Polyakov action can be derived from the Nambu-Goto action explained in Figure3.2by eliminating the square root of the area term.

This yields an additional auxiliary fieldh^αβ which can be viewed as the induced metric on the worldsheet up to a conformal factor. Thus, we can unleash the full power of two dimensional conformal symmetry described by holomorphic functions. The two parametrization transforma-tions arising from the local diffeomorphism invariance may be used to write a conformally flat metrichαβ “ Ωpτ, σq²ηαβ which can be viewed as a certain gauge choice called the conformal gauge. Together with the Weyl rescaling we may fix the three independent components of the two dimensional induced metric such that the worldsheet can be written in flat Minkowski co-ordinatesi.e. h_αβ “ η_αβ. Moreover, let us first work with a flat target spacetime g_abpxq “ η_ab

x¹

x² x³

τ

σ

x^µpτ, σq

Figure 3.3. The worldsheet of a string moving in the target spacetime is parametrized by two parametersτ andσand the area in the Nambu-Goto actionS “ ´Tsş

dAcan be written in terms of the induced metric on the worldsheet given by the pullback of the coordinate functions,i.e.PrGsαβ“gabBx^a

Bσ^α Bx^b

Bσ^β, asdA“a

´detpPrGsαβqd²σ with d²σ “ dτdσ. This idea is easily extended to higher dimensional objects such as membranes tracing out a worldvolume and the corresponding hypersurface enters the action with a volume element. Note that the embedding of the string is not restricted to three dimensions as suggested by the labels of the target space coordinate system.

and later generalize to a curved spacetime. The equations of motion in the conformal gauge corresponding to (3.14) are simply given by the free wave equation

BαB^αx^a “` Bτ²´ Bσ²

˘x^apτ, σq “0, (3.15)

with the additional constraint that the conformal energy-momentum tensor vanish

Bτx^aBσxa“0, pBτxq²` pBσxq²“0. (3.16) Solutions of (3.15) in light cone coordinatesσ^˘“τ˘σyield left and right moving waves

x^apτ, σq “x^a_Lpσ<sup>`</sup>q `x^a_Rpσ^´q. (3.17) The constraints (3.16) relate the Fourier modes of the left and right moving waves to the ef-fective mass of the string. Additionally, we need to impose the periodicity conditionx^apτ, σq “ x^apτ, σ`2πqfor closed strings and boundary conditions for open strings. There are two types of boundary conditions, called Neumann and Dirichlet, as shown in Figure 3.4. The Dirichlet boundary conditions define a hypersurface in space, a so calledD_p-brane, wherepdetermines the number of spatial directions, i.e. a D0-brane is a particle, a D1-brane a string, and D_pą2 -branes are higher dimensional mem-branes. Interestingly, theD_p-branes are dynamical charged solitonic objects (where D_´1-branes can be considered as instantons). A soliton is a kink-like solution interpolating between two different (vacuum) states and hence are related to unstable vacua and spontaneous symmetry breaking. In higher dimensions solitonic solutions are related to vortices determined by their winding numbers which can be viewed as the classification of maps from circles to circles or for arbitrary dimensions the homotopy groups of spheres. Indeed, DP-branes are dynamical solutions to the supergravity equations of motion. The connection of supergravity to string theory will be elucidated at the end of the section,c.f. Figure3.8. Detailed accounts of D-branes and their role in string theory can be found in [109,110].

After quantizing (3.17) physical states arise from the oscillatory states of the string according to

Neumann directions σ

Dirichlet directions

Figure 3.4. Open string solutions must be supplemented with two types of boundary condi-tions, Dirichlet boundary conditions defined byx^a|σ“0,π “ const. and Neumann boundary conditions given byBσx^a|σ“0,π “ 0. Dirichlet boundary conditions de-fine a p`1 dimensional hypersurface in d dimensional spacetime positioned at x<sup>b</sup> “c<sup>b</sup> withb“p`1, . . . , d´1 (Dirichlet directions) where along thep`1 hy-persurface directions Neumann conditions apply,i.e. Bσx<sup>a</sup> “ 0 witha “ 0, . . . , p (Neumann directions). These so-calledDp-branes break the global Lorentz group SOp1, d´1qintoSOp1, pq ˆSOpd´1´pq. In [108] the true meaning of a Dp -brane apart from defining the Dirichlet boundary conditions was discovered. In the context of superstring theoriesDP-branes carry Ramond-Ramond charges. From a supergravity point of viewD<sub>p</sub>-branes are solitonic charged objects preserving half of the supersymmetric generators with tension scaling as the inverse string cou-plingg<sup>´</sup><sub>s</sub> <sup>1</sup>(see also Section3.2). For closed strings the boundary conditions reduce to the periodicity conditionx<sup>a</sup>pτ, σq “x<sup>a</sup>pτ, σ`2πq.

the decomposition ofSOpd´1qfor massive andSOpd´2qfor massless single particle states. To quantize the classical string theory we can go about in various wayse.g. in a specific gauge, such as the lightcone gauge or covariant quantization imposing the constraints as operator equations, or the gauge invariant path integral method following the BRST¹³ quantization scheme. In all three cases the consistency of the theory fixes the spacetime dimensions¹⁴ i.e. the requirement of massless (massive) particles to transform under the little groupSOpd´2q(SOpd´1q) of the Lorentz symmetry or the absence of the conformal anomaly

xT^α_αy “ ´ c

12R“ ´cGhosts`cCFT

12 R“ ´´26`d R

“! 0 ñ d“26, (3.18)

assuming the d dimensional coordinate functions to be free scalar fields with central charge c “ 1. The critical dimension for bosonic string theory is d “26. The spectrum of the closed strings is formed by a tachyonic ground state, indicating the instability of the bosonic string theory, three massless field arising from the first excited state which can be described in terms of irreducible representations,i.e. the trace representation, describing the spin zero dilaton Φ,˜ the antisymmetric representation, describing the spin one Neveu-Schwarz two formB, and the symmetric traceless representation, describing the metric/spin two gravitonG. Higher excita-tions are massive with masses scaling asα^1´1 “ ℓ²_s which are quite large for small strings. For example, for a fundamental theory including quantum gravity the string length will be close to

13Named after Becchi, Rouet, Stora and Tyutin see [111,112]

14In the BRST quantization scheme this is not completely true since any CFT with the opposite central charge of the ghosts,i.e.c“26removes the conformal anomaly,i.e. there is no need to choosedfree scalar fields.

Figure 3.5.

Modes of open strings contains states that transforms as “photons” under the Lorentz group, whereas closed strings admit states that transform under symmetric traceless representations of the Lorentz group and hence can be identified as “gravitons”. The graviton arises as the quadrupole fluctua-tion of a closed string.

Photon

Graviton

Figure 3.6.

The replacement of point particles with ex-tended strings yields a purely topological smooth Feynman diagram. Locally, every point of the worldsheet diagram looks like a free propagating string, interactions are only affect-ing global properties of the worldsheet. Note that conformal invariance allows only to com-pute on-shell correlation functions.

the Planck lengthℓP and the higher excited states carry masses close to the Planck mass MP. The open string sector spectrum consists of a tachyonic ground state confined to the brane and for the first excited state we find two types of massless excitations, longitudinal and transverse to the brane. The longitudinal excitation describes a spin one gauge boson on the brane trans-forming under the brane Lorentz groupSOp1, pq, which can be identified as a “photon”, whereas the transverse excitations transform as scalars under theSOp1, pqand as vectors under the re-mainingSOpd´p´1q. A simple visualization of physical particles corresponding to the excited string states is displayed in Figure3.5. In the open string sector the tachyonic state signals the instability of the brane decaying into a state of closed strings. This state is only a local minimum of the potential, however, there exists an additional global minimum with value´8. Consider-ing interactions as shown in Figure 3.6, the topology of the worldsheet is modified depending on the particular string emission or absorption process. Since two dimensional gravity is purely topological, we include a topological term encoding the type of interaction process which, ac-cording to the Gauss-Bonnet theorem, yields a weighting factor given by the Euler numberχof the worldsheet

Ss“S_s^p0q` λ 4π

ż d²σ?

´hR“S_s^p0q`λχ. (3.19)

An expansion of the above action allows us to identifygs“ e^λ as the closed string coupling since an emission and subsequent re-absorption changes the genus by one — switching from the sphere topology to the torum topology — and hence the Euler number by two via,χ“2p1´#handlesq “ 2p1´genusq. Clearly, an open string emission and re-absorption process will change the Euler number only by one due to the effect of boundariesχ“2p1´#handlesq ´#boundaries, so the open string coupling is given by?gs. As we will see in the effective low-energy action in curved spacetime (3.21), the parameterλis dynamical and not a free parameter. Turning to strings in curved backgrounds, we can write the generalized action including all three closed string fields

Ss “ ´ 1 4πα¹

ż d²σ?

´h´

h^αβG_abpxqBαx^aBβx^b`iǫ<sup>αβ</sup>B<sub>ab</sub>pxqBαx<sup>a</sup>Bβx<sup>b</sup>`α¹ΦR˜ ^phq¯

. (3.20)

Conformal invariance is broken after quantization, so we have to determine the corresponding β-functions in the energy-momentum tensor and set them to zero. This is done by expanding the action (3.20) in the string length parameterℓs“?

α¹to obtain an interacting quantum field theory and calculate corrections to the naïve scaling dimension up toOpα¹²q, which amounts to a one loopβ-function. Then, the Weyl invariance/traceless condition of the energy-momentum tensor can be generated from a low-energy effective action in the curved spacetime fields

Seff-closed “ ´ 1 2κ0

ż d²⁶x?

´Ge^{´2 ˜Φ} ˆ

R´ 1

12HabcH^abc`4∇aΦ∇˜ <sup>a</sup>Φ˜ `Opα¹q

˙

, (3.21) where the three form is the exterior derivativeH “dBand the Ricci scalarRis calculated from the curved spacetime background metricG. Comparing with (3.19), we see that the expectation value of the dilaton field determines the string coupling viags “exppxΦ˜yq, so the string length κ²₀ „ℓ²⁴_s is the only free parameter in the theory. For the open string sector the effective action of tree-level physics to leading order inα¹is given by

Seff-open“ ´ 1 4q

ż

d²⁶xe^´Φ˜ “ tr`

FabF^ab˘

`Opα¹q‰

. (3.22)

The low-energy effective action is of Yang-Mills type (2.101), where the trace is taken over the representation of the non-Abelian gauge group. The gauge fieldsAcouple to the boundary of

the worldsheet ż

C

A“ ż

dx^aA_a “ ż

dτBτx^aA_a, (3.23)

which can be interpreted as open string ends carrying gauge field charges. This is an important ingredient when connecting open strings toDp-branes. We will utilize this fact in the AdS/CFT correspondence,c.f. Section3.2.

Let us conclude the (hopefully quite intriguing to the reader) bosonic string theory section by taking stock of the impact of extended fundamental objects to quantum theory:

• Interaction and free propagation is “unified” by removing the “singular points” in inter-action diagrams. The properties of the interinter-action diagram are invariant under Lorentz transformations and hence the “disappearance” of the fixed interaction point cures the UV divergences in perturbative quantum gravity. Technically, this arises from the expansion of the low-energy effective action in curved spacetime (3.21) in orders ofα¹ about the point particle limitα¹ Ñ0. Each additional factor ofα¹ adds another “loop” in the sense of the semi-classical expansion explained in (2.22) (α¹Ø¯h) and an extra factor of Ricci curvature e.g. at two loopsR^p2q“Rp1`α¹Rqwhich is reminiscent of the perturbative expansion in (3.5). In this sense, the controlled expansion inα¹andgsallows to compute the coefficient of the finite counter terms and therefore cures the UV divergences. This is known up to two loops and for supersymmetric string theories in the pure-spinor formalism up to five loops.

• Different known fundamental forces are “unified”,e.g. forces transmitted by spin one and spin two bosons. To be more precise, the high-dimensional critical spacetime should be reduced to our “perceived” four dimensional spacetime by a compactification of the addi-tional spacetime directions in the sense of the old Kaluza-Klein theory unifying electromag-netism and gravity. The unification of gauge interactions and gravity can also arise from D-branes, where closed strings (describing gravity) exchanged between D-branes are viewed as loops of open strings (describing gauge interactions). Moreover, from a reductionist point of view, fundamental particles with different intrinsic properties arise dynamically from a single entity, the fundamental string.

Im Dokument Gauge/Gravity duality (Seite 95-101)