Strings, Branes, Sigma Models, and AdS Hints

Particles, strings, and branes can all be described by summing over their histories in a path integral. A particle sweeps out a worldline, a string sweeps out a worldsheet, and a $p$-brane sweeps out a $(p+1)$-dimensional worldvolume. This common viewpoint connects confining flux tubes to the Polyakov string, worldsheet anomalies to target-space field equations, and the string’s Liouville or radial direction to scale.

The same questions reappear at every dimension: What is the geometric action? What are the gauge redundancies on the worldvolume? Which background fields couple to the object? What does renormalization on the worldvolume mean for spacetime physics?

This page is a synthesis, not a complete string theory course in miniature. The goal is to connect the course’s earlier ideas—Wilson loops, defects, conformal invariance, anomalies, sigma models, and RG—to the language of strings, branes, and holographic geometry.

Required background. Wilson loops, worldlines, monopole plasma, and confinement supplies the surface representation of a confining Wilson loop. Worldlines, worldsheets, and reparametrization gauge supplies the geometric actions and gauge redundancies used here.

Helpful background. Conformal anomalies, Liouville action, and nonlocal effective actions develops the anomaly that makes the conformal factor dynamical.

From worldlines to worldsheets

Polyakov normalization and sigma-model beta function

The worldsheet coordinates are $\xi^a$, $a=1,2$. Target-space coordinates are $Y^m(\xi)$. When the target includes a Liouville or radial coordinate, we often write

$$Y^m=(X^\mu,\phi).$$

The Euclidean Polyakov action for a bosonic string in flat target space is normalized as

$$S_P[X,g]={1\over4\pi\alpha'}\int_\Sigma d^2\xi\sqrt g\,g^{ab}\partial_aX^\mu\partial_bX_\mu.$$

The string tension is

$$T_s={1\over2\pi\alpha'}.$$

Numerical coefficients in Liouville and sigma-model beta functions depend on conventions for $\alpha'$, the worldsheet metric, the Liouville field, and the dilaton. We define $\beta^G_{mn}=\mu\,dG_{mn}/d\mu$ below. The invariant message is that quantum Weyl invariance constrains the target-space background fields.

The particle path integral gives the prototype. A scalar particle propagator may be represented schematically as

$$G(x,y)\sim \sum_{\gamma:x\to y} e^{-mL(\gamma)},$$

where $L(\gamma)$ is the length of the path. This is not just a calculational trick. It says that the quantum object is described by summing over all possible histories, with a geometric action suppressing long histories.

A confining Wilson loop suggests the next step. If a loop $C$ bounds an electric flux tube, the low-energy contribution is not a path but a surface $\Sigma$ with boundary $\partial\Sigma=C$:

$$W(C)\sim \sum_{\Sigma:\,\partial\Sigma=C}e^{-T_sA(\Sigma)}.$$

Here $A(\Sigma)$ is the area and $T_s$ is an effective string tension. In a strongly coupled lattice gauge theory this formula appears directly: the Wilson loop is saturated by plaquette surfaces. In a continuum confining theory it becomes an effective description of the flux tube.

A particle propagator sums over paths, while a confining Wilson loop suggests a sum over surfaces ending on the loop. This is the conceptual bridge from worldlines to worldsheets.

There is an immediate energy–entropy lesson, familiar from the Ising model at the beginning of the course. Suppose the number of lattice surfaces of area $A$ grows like

$$N(A)\sim e^{s_0A}.$$

If every surface has bare weight $e^{-T_0A}$, then the combined statistical weight behaves like

$$N(A)e^{-T_0A}\sim e^{-(T_0-s_0)A}.$$

Thus the physical or renormalized string tension is roughly

$$T_{\rm eff}=T_0-s_0.$$

When $T_{\rm eff}$ is large, surfaces are microscopic and rigid. When $T_{\rm eff}$ is tuned toward zero, large fluctuating surfaces dominate. That tuning is the string analog of approaching a critical point. The continuum string is not a single smooth surface chosen by hand; it is a critical limit of a sum over fluctuating surfaces.

Defects, strings, and branes

The same ladder can be organized by the spatial dimension $p$ of the object whose history is summed over. A pointlike instanton is inserted at a Euclidean event and is sometimes called a $p=-1$ brane. A particle is a $p=0$ object and sweeps out a line; a string is a $p=1$ object and sweeps out a surface; a general $p$-brane sweeps out a $(p+1)$-dimensional worldvolume.

The geometric action for a relativistic $p$-brane is the volume of its worldvolume:

$$S_p[X]=T_p\int d^{p+1}\xi\sqrt{\det h_{ab}},$$

where

$$h_{ab}=\partial_aX^m\partial_bX^nG_{mn}(X)$$

is the induced metric. The cases $p=0$ and $p=1$ give

$$S_0=m\int ds, \qquad S_1=T_s\int d^2\xi\sqrt{\det h}.$$

The $p=1$ case is the Nambu–Goto action for a string. It measures the area of the embedded worldsheet.

The same path-integral idea repeats in increasing dimension: local events, worldlines, worldsheets, and higher branes are weighted by geometric actions and by their couplings to background gauge fields.

Extended objects also couple naturally to differential-form gauge fields. A charged particle couples to a one-form $A$ through

$$S_{E,\rm int}=-iq\int_\gamma A.$$

A string can couple to a two-form $B$ through

$$S_{E,B}=-iq\int_\Sigma B,$$

and a $p$-brane couples to a $(p+1)$-form potential $C_{p+1}$ through

$$S_{E,\rm WZ}=-iq\int_{W_{p+1}}C_{p+1}.$$

This is the natural generalization of the Wilson-line phase. An electrically charged $p$-brane couples to a $(p+1)$-form potential, while the Hodge-dual field strength measures the corresponding magnetic charge. The topology of these electric and magnetic couplings underlies generalized Dirac quantization and the relation among monopoles, vortices, and branes.

The Polyakov action and conformal gauge

The Nambu–Goto action is geometrically transparent but technically inconvenient because of the square root. The Polyakov trick introduces an independent worldsheet metric $g_{ab}$:

$$S_P[X,g]={T_s\over2}\int d^2\xi\sqrt g\,g^{ab}\partial_aX^m\partial_bX^nG_{mn}(X).$$

For a flat target and $T_s=1/(2\pi\alpha')$, this becomes

$$S_P[X,g]={1\over4\pi\alpha'}\int d^2\xi\sqrt g\,g^{ab}\partial_aX^\mu\partial_bX_\mu.$$

Classically, varying $g_{ab}$ gives the worldsheet stress-tensor constraint

$$T_{ab}=\partial_aX\cdot\partial_bX-{1\over2}g_{ab}g^{cd}\partial_cX\cdot\partial_dX=0.$$

When the induced metric is nondegenerate, solving this constraint sets $g_{ab}$ equal to the induced metric up to a Weyl factor and returns the Nambu–Goto area action.

The Polyakov form has a larger gauge redundancy:

$$\xi^a\mapsto \xi'^a(\xi), \qquad g_{ab}\mapsto e^{2\omega(\xi)}g_{ab}.$$

The first is worldsheet diffeomorphism invariance; the second is Weyl invariance. Locally in two dimensions one can use them to set

$$g_{ab}=e^{2\omega}\hat g_{ab}.$$

Classically the conformal factor $\omega$ drops out of the kinetic action. Quantum mechanically it need not drop out, because the functional measure is anomalous. The matter fields $X^\mu$ contribute central charge $c=D$, while the diffeomorphism ghosts contribute

$$c_{\rm gh}=-26.$$

Thus a bosonic string in flat space is Weyl invariant without an extra Liouville dynamics only when

$$D-26=0.$$

Away from criticality, the conformal factor becomes a genuine field. After a field redefinition to a canonically normalized Liouville field $\chi$, its action has the schematic form

$$S_L[\chi]={1\over4\pi}\int d^2\xi\sqrt{\hat g}\, \left[(\hat\nabla\chi)^2+Q\hat R\chi +4\pi\mu e^{2b\chi}\right].$$

The sign and the parameters $Q$ and $b$ depend on the central-charge deficit and on conventions; in the common spacelike normalization, $c_L=1+6Q^2$. Quantum consistency requires the full matter, ghost, and Liouville central charges to cancel,

$$c_{\rm matter}+c_{\rm gh}+c_L=0,$$

not merely $D-26=0$ after Liouville dynamics is included. The structural point is that the scale of the worldsheet metric becomes an additional dynamical coordinate. In noncritical string theory, one often interprets this Liouville coordinate as an emergent target-space direction.

The Nambu–Goto action may be rewritten using an independent worldsheet metric. After conformal gauge fixing, the Weyl factor is pure gauge classically but becomes the Liouville mode when the matter-plus-ghost central charge does not vanish.

This is one of the deepest uses of two-dimensional CFT in physics. The same anomaly that produced the Schwarzian term and the Virasoro central charge now controls the consistency of a fluctuating geometry.

Sigma models as spacetime backgrounds

The flat-space string action is only the first term in a much broader two-dimensional field theory. A general bosonic sigma model has target-space fields as couplings:

$$S_\Sigma={1\over4\pi\alpha'}\int d^2\xi\sqrt g\, g^{ab}G_{mn}(Y)\partial_aY^m\partial_bY^n +{i\over4\pi\alpha'}\int B_{mn}(Y)dY^m\wedge dY^n$$ $$\hspace{3.2cm} +{1\over4\pi}\int d^2\xi\sqrt g\,\Phi(Y)R^{(2)} +\int d^2\xi\sqrt g\,T(Y)+\cdots.$$

The target-space metric $G_{mn}$, two-form $B_{mn}$, dilaton $\Phi$, and tachyon or scalar source $T$ are not merely external decorations. From the worldsheet perspective they are coupling functions. Therefore they run under the two-dimensional RG.

A string moving in a background sees target-space fields as worldsheet couplings. Requiring Weyl invariance constrains the allowed backgrounds.

This is a stunning reversal of viewpoint. In ordinary QFT, beta functions tell us how couplings depend on scale. In string theory, the beta functions of a two-dimensional QFT become equations of motion for spacetime fields.

To see the seed of this idea, recall the nonlinear sigma model with $n^2=1$:

$$S={1\over2\alpha}\int d^2\xi\,\partial_an\cdot\partial_an, \qquad n\in S^{N-1}.$$

With $\ell$ a length scale, the one-loop running coupling is

$${d\alpha\over d\log\ell}={N-2\over2\pi}\alpha^2+O(\alpha^3).$$

Equivalently, with $\mu=1/\ell$,

$$\mu{d\alpha\over d\mu}=-{N-2\over2\pi}\alpha^2+O(\alpha^3).$$

For $N>2$, the model is weakly coupled at short distances and strongly coupled at long distances. The one-loop coefficient is proportional to the Ricci curvature of the target sphere. More generally, define the metric beta function with respect to the energy scale $\mu$ by

$$\beta^G_{mn}\equiv\mu{dG_{mn}\over d\mu} =\alpha' R_{mn}+O(\alpha'^2)$$

in a common convention. In terms of the length scale $\ell=1/\mu$, this is $dG_{mn}/d\log\ell=-\alpha'R_{mn}+\cdots$, the Ricci-flow sign often used in geometry. Including the dilaton and setting $B=0$, the leading condition for Weyl invariance is schematically

$$\boxed{ R_{mn}+2\nabla_m\nabla_n\Phi+O(\alpha')=0. }$$

The same equation follows from varying the target-space effective action

$$S_{\rm eff}\sim\int d^DY\sqrt G\,e^{-2\Phi} \left(R+4(\nabla\Phi)^2+\cdots\right).$$

Thus a two-dimensional quantum anomaly has become a target-space gravitational equation.

The leading metric beta function of a sigma model is controlled by target-space curvature. Worldsheet Weyl invariance therefore imposes equations on the target geometry.

A common pitfall is to identify this equation with the ordinary Einstein equation too quickly. The sigma-model beta function gives $R_{mn}$, not directly $R_{mn}-{1\over2}G_{mn}R$. The Einstein-like form appears after deriving the spacetime effective action and choosing the appropriate frame and dilaton variables.

Vertex operators and string scattering

In a first-quantized particle path integral, local insertions create or absorb external particles. In a string path integral, vertex operators play the same role.

For the simplest closed-string scalar mode, a plane-wave vertex has the schematic form

$$V_p=\int d^2\xi\sqrt g\,e^{ip\cdot X(\xi)}.$$

For a graviton-like excitation one uses

$$V_{\epsilon,p}=\int d^2\xi\, \epsilon_{mn}\partial X^m\bar\partial X^n e^{ip\cdot X}.$$

The condition that the integrated vertex operator be marginal is the worldsheet version of the spacetime mass-shell condition. For a free boson normalized by

$$\langle X^m(z)X^n(0)\rangle=-{\alpha'\over2}\delta^{mn}\log|z|^2,$$

the operator $e^{ip\cdot X}$ has weights

$$h=\bar h={\alpha'p_E^2\over4}$$

in Euclidean target signature. Under Wick rotation $p_E^2=-p_L^2$, while the global (+---) convention gives $p_L^2=m^2$. The closed-string tachyon condition $h=\bar h=1$ therefore corresponds to

$$m^2=-{4\over\alpha'}.$$

Likewise, the operator

$$\epsilon_{mn}\partial X^m\bar\partial X^n e^{ip\cdot X}$$

is marginal when $p_L^2=0$ after continuation, with transverse and gauge-equivalence conditions on $\epsilon_{mn}$. This gives the graviton, antisymmetric tensor, and dilaton in the closed-string spectrum.

Closed-string scattering amplitudes are worldsheet CFT correlators of vertex operators, integrated over insertion points and divided by conformal Killing transformations.

At tree level for closed strings, the worldsheet is a sphere. Fixing three punctures by the global conformal group leaves one complex modulus $z$ in a four-point amplitude. The amplitude contains integrals of the schematic form

$$A_4\sim\int d^2z\,|z|^{\alpha'p_1\cdot p_2} |1-z|^{\alpha'p_2\cdot p_3}\times(\text{polarization factor}).$$

This one formula contains several earlier themes at once. The limits $z\to0$, $z\to1$, and $z\to\infty$ are OPE limits on the worldsheet. In spacetime, they become factorization channels of the scattering amplitude. The poles are the infinite tower of string states.

Open strings and boundary operators

For an open string, the worldsheet has a boundary. An integrated physical boundary vertex must be a boundary primary of conformal weight one, so that its integral respects boundary reparametrization and Weyl invariance. The gauge-field vertex is

$$V_A=\int_{\partial\Sigma} ds\,\dot X^\mu A_\mu(X),$$

which for a plane wave $A_\mu(X)=\epsilon_\mu e^{ip\cdot X}$ becomes

$$V_{\epsilon,p}=\int ds\,\epsilon_\mu\dot X^\mu e^{ip\cdot X(s)}.$$

This is the stringy descendant of the Wilson-line coupling $\int A_\mu dX^\mu$. The boundary of the worldsheet behaves like the worldline of a charged object.

Open-string vertex operators live on the boundary. The massless vector vertex is the boundary Wilson-line coupling expanded in modes.

Excited boundary operators contain products of higher derivatives of $X$ and give an infinite tower of open-string states, including higher-spin states. A general matter operator has the schematic form

$$V\sim\int ds\, \epsilon_{\mu_1\cdots\mu_r} \prod_{j=1}^{r}\partial_s^{n_j}X^{\mu_j} e^{ip\cdot X}, \qquad n_j\ge 1.$$

The precise basis is organized by the oscillator modes of the open string, but the physical idea is already visible: a string has internal vibration modes, and these appear as particles of increasing spin and mass.

There is also an important duality of interpretation. The same cylinder worldsheet can be sliced in two ways: as a loop of an open string, or as a closed string propagating between two boundaries.

A cylinder admits an open-string channel and a closed-string channel. This open–closed duality is the worldsheet origin of many gauge/string correspondences.

This is a good place to remember the earlier Wilson-loop discussion. A large Wilson loop can be represented by a string worldsheet ending on the loop. In an open-string channel, the boundary degrees of freedom look like charged particles and gauge fields. In a closed-string channel, the same diagram looks like exchange of closed-string modes, including the graviton and dilaton.

Warped targets and the radial scale

The notes now point toward a geometric idea that later became central in holography. Suppose the target-space metric has the warped form

$$ds^2=d\phi^2+a^2(\phi)d\vec x^{\,2}.$$

The coordinate $\phi$ changes the proper length scale of the $\vec x$ directions. A local excitation with fixed proper mass is seen with a different coordinate energy at different values of $\phi$. Thus motion in the extra direction resembles motion in energy scale.

The most important special case is

$$ds^2=d\phi^2+e^{2\phi/R}d\vec x^{\,2}.$$

With the dimensionful Poincaré coordinate

$$z=R e^{-\phi/R},$$

this becomes the Poincaré metric on Euclidean anti-de Sitter space:

$$ds^2=R^2{dz^2+d\vec x^{\,2}\over z^2}.$$

The transformation

$$\vec x\mapsto\lambda\vec x, \qquad \phi\mapsto\phi-R\log\lambda$$

leaves $d\phi^2+e^{2\phi/R}d\vec x^{\,2}$ invariant. Equivalently,

$$\vec x\mapsto\lambda\vec x, \qquad z\mapsto\lambda z$$

leaves $R^2(dz^2+d\vec x^{\,2})/z^2$ invariant. The conformal boundary lies at $z\to0$, or $\phi\to+\infty$. This is the clean geometric reason the radial coordinate is tied to scale transformations.

In Poincaré AdS, boundary scale transformations are bulk isometries. A boundary operator insertion may be represented by a bulk field or string vertex operator whose radial profile records the RG scale.

The conformal group of flat $d$-dimensional space is realized as the isometry group of $AdS_{d+1}$. Translations and rotations act on the boundary coordinates $\vec x$. Dilatations act as the simultaneous scaling above. Special conformal transformations arise from AdS isometries that mix $z$ and $\vec x$; equivalently, they are generated by inversion, translation, and inversion on the boundary.

This is the geometric hint behind the AdS/CFT correspondence. A conformal gauge theory, such as four-dimensional $\mathcal N=4$ supersymmetric Yang–Mills with schematic Lagrangian

$$\mathcal L={1\over g_{\rm YM}^2}\operatorname{Tr}\left(F_{\mu\nu}F^{\mu\nu}+\cdots\right),$$

has local gauge-invariant operators

$$\mathcal O_n(x)=\operatorname{Tr}\bigl(F^2(x)\bigr),\quad \operatorname{Tr}\bigl(FD\cdots F(x)\bigr),\quad\ldots$$

A dual string or gravity description packages these operators into bulk fields or string vertex operators. In momentum space one writes schematically

$$V_n(p)=\int d^2\xi\sqrt g\,\Omega_n(Y(\xi))e^{ip\cdot X(\xi)},$$

where $\Omega_n$ is a wavefunction in the radial and internal directions. The bulk wave equation relates its mass to the scaling dimension of the boundary operator:

$$m_n^2R^2=\Delta_n(\Delta_n-d)$$

for a scalar field in $AdS_{d+1}$.

What is and is not being derived here

The course has built the ingredients that make holography plausible: Wilson loops as worldsheets, conformal symmetry, stress tensors, sigma-model beta functions, Liouville/radial scale, and open–closed string duality. A full derivation of AdS/CFT requires much more: D-branes, supersymmetry, large-N gauge theory, and the decoupling limit. The present page should be read as a structural map, not as a proof.

The course in one loop

The last page loops back to the first page of the course. The Ising model taught us that a statistical sum can be reorganized as a sum over loops or domain walls. Kramers–Wannier duality taught us that nonlocal variables may be the right local variables in the dual description. The continuum limit taught us that criticality produces fields. The CFT pages taught us that two-dimensional scale invariance, stress tensors, and anomalies are powerful enough to determine spectra and correlators. The gauge-theory pages taught us that Wilson loops diagnose confinement and that compactness creates topological defects. The string pages reinterpret those Wilson-loop surfaces as dynamical worldsheets.

So the conceptual chain is

$$\text{spins} \longrightarrow \text{loops and defects} \longrightarrow \text{fields and CFT} \longrightarrow \text{flux-tube worldsheets} \longrightarrow \text{strings and geometry}.$$

The punchline is not that every QFT is secretly a simple string theory. The punchline is sharper: whenever a QFT admits a useful description in terms of fluctuating extended objects, the tools developed in this course—duality, RG, conformal invariance, anomalies, and gauge-invariant observables—are the natural language for finding it.

Exercises

Exercise 1: Eliminating the worldsheet metric

Start from the Euclidean Polyakov action

$$S_P={T_s\over2}\int d^2\xi\sqrt g\,g^{ab}h_{ab}, \qquad h_{ab}=\partial_aX\cdot\partial_bX.$$

Assume the induced metric $h_{ab}$ is nondegenerate. Show that varying with respect to $g^{ab}$ implies that $g_{ab}$ is proportional to $h_{ab}$, and that substituting this result gives the Nambu–Goto action

$$S_{NG}=T_s\int d^2\xi\sqrt{\det h}.$$

Solution

The variation is

$$\delta S_P={T_s\over2}\int d^2\xi\sqrt g \left(h_{ab}-{1\over2}g_{ab}g^{cd}h_{cd}\right)\delta g^{ab}.$$

Thus

$$h_{ab}-{1\over2}g_{ab}g^{cd}h_{cd}=0.$$

This equation says that $h_{ab}$ is proportional to $g_{ab}$:

$$h_{ab}=\lambda g_{ab}, \qquad \lambda={1\over2}g^{cd}h_{cd}.$$

In two dimensions the proportionality factor is not fixed because of Weyl invariance. Choose the Weyl gauge $g_{ab}=h_{ab}$. Then

$$\sqrt g\,g^{ab}h_{ab}=\sqrt h\,h^{ab}h_{ab}=2\sqrt h,$$

so

$$S_P={T_s\over2}\int 2\sqrt h=T_s\int d^2\xi\sqrt{\det h}.$$

This is the Nambu–Goto action.

Exercise 2: Entropy renormalizes the string tension

Assume that the number of lattice surfaces of area $A$ bounded by a large loop behaves as $N(A)\sim e^{s_0A}$. If each surface has bare Boltzmann weight $e^{-T_0A}$, find the effective tension and state the condition for large surfaces to proliferate.

Solution

The contribution of surfaces with area $A$ is approximately

$$N(A)e^{-T_0A}\sim e^{s_0A}e^{-T_0A}=e^{-(T_0-s_0)A}.$$

Thus

$$T_{\rm eff}=T_0-s_0.$$

If $T_{\rm eff}>0$, large surfaces are exponentially suppressed. If $T_{\rm eff}\to0^+$, large surfaces are no longer strongly suppressed, and a continuum fluctuating-surface description can emerge. If $T_{\rm eff}<0$, the naive surface sum is unstable and must be regulated by additional physics.

Exercise 3: The O(N) sigma-model correlation length

Let

$${d\alpha\over d\log\ell}={N-2\over2\pi}\alpha^2$$

for the two-dimensional $O(N)$ sigma model with $N>2$. Solve for $\alpha(\ell)$ in terms of $\alpha_0=\alpha(\ell_0)$ and estimate the scale $\xi$ at which perturbation theory breaks down.

Solution

Separate variables:

$${d\alpha\over\alpha^2}={N-2\over2\pi}d\log\ell.$$

Integrating from $\ell_0$ to $\ell$ gives

$$-{1\over\alpha(\ell)}+{1\over\alpha_0} ={N-2\over2\pi}\log{\ell\over\ell_0}.$$

Therefore

$$\alpha(\ell)= {\alpha_0\over1-{N-2\over2\pi}\alpha_0\log(\ell/\ell_0)}.$$

The denominator vanishes at

$$\xi\sim \ell_0\exp\left({2\pi\over(N-2)\alpha_0}\right).$$

This is the dynamically generated long-distance scale. In energy variables $\mu=1/\ell$, the coupling decreases logarithmically in the ultraviolet.

Exercise 4: Marginality of the closed-string tachyon vertex

For a free boson with

$$\langle X^m(z)X^n(0)\rangle=-{\alpha'\over2}\delta^{mn}\log|z|^2,$$

the operator $e^{ip\cdot X}$ has weights

$$h=\bar h={\alpha'p_E^2\over4}.$$

Find the condition for the integrated closed-string vertex

$$V_p=\int d^2z\,e^{ip\cdot X(z,\bar z)}$$

to be marginal, and translate it into the Lorentzian mass of the state.

Solution

The measure $d^2z$ has weights $(-1,-1)$, so the integrand must have weights $(1,1)$ for the integrated operator to be invariant. Therefore

$$h=\bar h=1.$$

Using $h=\bar h=\alpha'p_E^2/4$ gives

$$p_E^2={4\over\alpha'}$$

in Euclidean target signature. Wick rotation gives $p_E^2=-p_L^2$, and the global (+---) convention identifies $p_L^2=m^2$. Therefore the corresponding state has

$$m^2=-{4\over\alpha'}.$$

This is the closed bosonic string tachyon.

Exercise 5: AdS scaling as an isometry

Show that

$$ds^2=R^2{dz^2+d\vec x^{\,2}\over z^2}$$

is invariant under

$$z\mapsto\lambda z, \qquad \vec x\mapsto\lambda\vec x.$$

Then rewrite the metric using $z=R e^{-\phi/R}$.

Solution

Under the scaling,

$$dz\mapsto\lambda dz, \qquad d\vec x\mapsto\lambda d\vec x, \qquad z^2\mapsto\lambda^2z^2.$$

Thus

$${dz^2+d\vec x^{\,2}\over z^2} \mapsto {\lambda^2(dz^2+d\vec x^{\,2})\over\lambda^2z^2} ={dz^2+d\vec x^{\,2}\over z^2}.$$

Now set $z=R e^{-\phi/R}$. Then

$${dz\over z}=-{d\phi\over R}, \qquad R^2{dz^2\over z^2}=d\phi^2,$$

and

$$R^2{d\vec x^{\,2}\over z^2}=e^{2\phi/R}d\vec x^{\,2}.$$

Therefore

$$ds^2=d\phi^2+e^{2\phi/R}d\vec x^{\,2}.$$

A rescaling of the boundary coordinates is compensated by a shift of $\phi$.

Exercise 6: Bulk scalar mass and boundary dimension

For a scalar field in $AdS_{d+1}$, the near-boundary relation between the bulk mass and the scaling dimension is

$$m^2R^2=\Delta(\Delta-d).$$

Solve for $\Delta$ and state which branch is usually chosen for standard quantization.

Solution

The quadratic equation is

$$\Delta^2-d\Delta-m^2R^2=0.$$

Thus

$$\Delta_\pm={d\over2}\pm\sqrt{{d^2\over4}+m^2R^2}.$$

In standard quantization one usually chooses

$$\Delta=\Delta_+={d\over2}+\sqrt{{d^2\over4}+m^2R^2}.$$

The roots are real above the Breitenlohner–Freedman bound $m^2R^2\ge-d^2/4$. In the window

$$-{d^2\over4}<m^2R^2<-{d^2\over4}+1,$$

both falloffs are normalizable and alternate quantization with $\Delta=\Delta_-$ may be possible; standard quantization uses the larger root $\Delta_+$.

Further reading

• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer-Verlag, 1997), chapters on free bosons, vertex operators, Virasoro symmetry, and worldsheet CFT.
• M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, vols. 1–2 (Cambridge University Press, 1987).
• S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from non-critical string theory,” Physics Letters B 428 (1998), 105–114.
• J. M. Maldacena, “The large-N limit of superconformal field theories and supergravity,” Advances in Theoretical and Mathematical Physics 2 (1998), 231–252.
• J. Polchinski, String Theory, vols. 1–2 (Cambridge University Press, 1998).
• A. M. Polyakov, Gauge Fields and Strings, Contemporary Concepts in Physics, vol. 3 (Harwood Academic Publishers, 1987), Chapters 9–10.
• E. Witten, “Anti-de Sitter space and holography,” Advances in Theoretical and Mathematical Physics 2 (1998), 253–291.
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed. (Oxford University Press, 2002), chapters on nonlinear sigma models and renormalization-group flow.