Strings, Branes, Sigma Models, and AdS Hints

The quantum field theory on a string worldsheet constrains the spacetime through which the string moves. At leading order in slowly varying backgrounds, cancellation of the worldsheet Weyl anomaly gives equations for the target metric and dilaton. In a flat critical bosonic string, the conformal weights of physical vertex operators constrain spacetime masses. These are concrete consistency tests; neither a sum over confining surfaces nor a resemblance between a radial coordinate and scale establishes a gauge/string duality.

This lesson develops those tests and finishes with a separate geometric calculation: the dilation isometry of AdS and the allowed boundary powers of a free bulk scalar. The assumptions linking a worldsheet theory, a spacetime background, and a boundary QFT remain visible at each step.

Required background. Worldlines, worldsheets, and reparametrization gauge supplies auxiliary metrics and constraints. Wilson loops, worldlines, monopole plasma, and confinement supplies the regulated surface expansion and its confinement limits.

Helpful background. Conformal anomalies and Liouville theory explains why the quantum Weyl factor need not decouple.

Geometric actions and sums over surfaces

A particle sweeps out a worldline, a string a worldsheet, and a spatial $p$-brane a $(p+1)$-dimensional worldvolume. For this geometric discussion, both the worldvolume metric and target metric $G_{mn}$ are positive Euclidean metrics. A smooth immersion $X^m(\xi)$ induces

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

The volume action is

$$S_p=T_p\int d^{p+1}\xi\,\sqrt{\det h}, \qquad T_p>0.$$

Thus $p=0$ gives $m\int ds$, while $p=1$ gives the Nambu–Goto area $T_sA$. Here $T_s=1/(2\pi\alpha')$ and $\alpha'$ has dimensions of length squared.

The geometric action does not by itself define a quantum sum. In the particle case, the well-defined regulated construction used in the previous lesson is a heat-kernel path integral followed by a proper-time integral. Its typical continuum paths are Brownian and do not have finite ordinary arclength. Likewise, the plaquette surfaces in a strong-coupling lattice expansion come with a regulator, constraints, and nontrivial weights. Replacing them by an integral over smooth embeddings with weight $e^{-T_sA}$ requires a measure and a controlled long-distance approximation.

A simple counting model explains the possible competition. At fixed regulator and boundary loop, suppose the number of surfaces of physical area $A$ has leading exponential growth $N(A)\sim e^{s_0A}$. With bare weight $e^{-T_0A}$,

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

Both $s_0$ and $T_0$ have units of inverse area. Their difference is the effective exponential suppression rate in this model. Sending it toward zero removes that suppression; it does not prove that a continuum limit exists or identify its universality class. Area tension also does not supply a bending-rigidity term. A derivation of an effective flux-tube string must establish which further operators and configurations matter.

The auxiliary worldsheet metric

Introduce an independent positive metric $g_{ab}$ on the two-dimensional worldsheet. For the pure kinetic action,

$$S_P[X,g]=\frac{T_s}{2}\int_\Sigma d^2\xi\,\sqrt g\,g^{ab}h_{ab}.$$

Varying its inverse metric gives

$$\delta_gS_P=\frac{T_s}{2}\int_\Sigma d^2\xi\,\sqrt g\, \Theta_{ab}\delta g^{ab}, \qquad \Theta_{ab}=h_{ab}-\frac12g_{ab}g^{cd}h_{cd}.$$

The constraint is $\Theta_{ab}=0$. In the positive covariant-metric stress convention of lesson 31, the lower-index stress is $-T_s\Theta_{ab}$. For positive nondegenerate $h$, the constraint makes $g$ proportional to $h$. Choosing the Weyl representative $g=h$ gives $g^{ab}h_{ab}=2$ and hence

$$S_P=T_s\int_\Sigma d^2\xi\,\sqrt{\det h}=S_{NG}.$$

This eliminates an auxiliary metric at fixed smooth embedding. It neither minimizes over embeddings nor proves equality of quantum measures. The calculation also concerns the pure kinetic action; the nonconstant dilaton coupling introduced below changes its metric equation. The classical construction and its local gauge fixing are developed in Polyakov 1987, §9.5, pp. 176–177.

Worldsheet diffeomorphisms and Weyl rescalings $g_{ab}\mapsto e^{2\omega}g_{ab}$ allow conformal gauge locally. Global moduli, boundary conditions and residual gauge transformations still have to be treated. With $D$ free target-coordinate fields, the matter central charge is $D$; the reparametrization ghosts contribute $-26$. The flat bosonic matter-plus-ghost theory therefore cancels this anomaly at $D=26$; see Tong 2009, §5.3, p. 119, PDF.

For a noncritical theory, cancelling the anomaly requires additional dynamics rather than treating the conformal factor as a harmless extra coordinate. The real spacelike Liouville construction in lesson 33 has its own central-charge and reality restrictions. We will not identify that field with an arbitrary holographic radial coordinate. The following calculation uses a critical bosonic background and a different question: how do nonconstant target couplings affect Weyl invariance?

Metric running and Weyl consistency

Let $Y^m$ be target coordinates. On a closed worldsheet, set the antisymmetric two-form and tachyon backgrounds to zero and retain the metric and dilaton:

$$\begin{aligned} S_\Sigma={}&\frac{1}{4\pi\alpha'}\int d^2\xi\,\sqrt g\, g^{ab}G_{mn}(Y)\partial_aY^m\partial_bY^n\\ &+\frac{1}{4\pi}\int d^2\xi\,\sqrt g\,\Phi(Y)R^{(2)}. \end{aligned}$$

$R^{(2)}$ is worldsheet curvature; $R_{mn}$ below is target curvature. The dimensionless dilaton $\Phi$ and the metric are coupling functions of a two-dimensional QFT. The normalization of the curvature coupling is given in Tong 2009, §7.2.2, p. 167, Eq. (7.12), PDF. Worldsheets with boundary require the corresponding boundary completion; the equations here concern the bulk couplings.

Work in a covariant perturbative prescription at fixed target coordinates. If curvature radii and length scales of target-field variation are large compared with $\sqrt{\alpha'}$, the leading metric running with energy scale $\mu$ is

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

The remainder denotes higher-derivative curvature terms with their required dimensions. The leading tensor coefficient is a worldsheet loop result, not something obtained by dimensional analysis alone; its background-field derivation is given in Tong 2009, §7.1.1, pp. 161–164, Eqs. (7.4)–(7.6), PDF. In terms of a length scale $\ell=1/\mu$, the sign reverses: $dG_{mn}/d\log\ell=-\alpha'R_{mn}+\cdots$.

The sphere gives a useful normalization check. For $n\in S^{N-1}$,

$$S=\frac{1}{2\alpha}\int d^2\xi\,(\partial_an)^2, \qquad G_{mn}=\frac{2\pi\alpha'}{\alpha}\,\bar G_{mn}, \qquad R_{mn}=(N-2)\bar G_{mn},$$

where $\bar G$ is the unit-sphere metric. Differentiating the metric coefficient and equating it to $\alpha'R_{mn}$ yields

$$-\frac{2\pi\alpha'}{\alpha^2}\, \mu\frac{d\alpha}{d\mu}\,\bar G_{mn} =\alpha'(N-2)\bar G_{mn},$$

and therefore

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

For $N>2$ the weakly coupled model is asymptotically free. Exercise 3 integrates its leading running and identifies where that approximation fails.

Ordinary metric running is not the complete Weyl condition. The dilaton curvature coupling improves the trace response, even on a worldsheet that is flat before variation. Denote the improved coefficients by a bar. In the critical $D=26$, zero-two-form truncation,

$$\begin{aligned} \bar\beta^G_{mn} &=\alpha'\bigl(R_{mn}+2\nabla_m\nabla_n\Phi\bigr) +O(\alpha'^2),\\ \bar\beta^\Phi &=\alpha'\left[(\nabla\Phi)^2-\frac12\nabla^2\Phi\right] +O(\alpha'^2). \end{aligned}$$

Both must vanish at the order being retained. With $D$ free coordinate fields and no compensating sector, the dilaton coefficient also contains $(D-26)/6$; it cannot be discarded away from criticality. These coefficients and their relation to the trace anomaly are given in Tong 2009, §7.2.3, p. 169, and §7.4.4, p. 182, PDF and Callan et al. 1985, §2, pp. 595–597, Open PDF.

Why the metric and dilaton equations must be used together

The leading string-frame functional for this critical truncation is, up to an overall normalization,

$$I[G,\Phi]=\int d^DY\,\sqrt G\,e^{-2\Phi} \left[R+4(\nabla\Phi)^2\right].$$

It is the tree-level target-space action, with boundary terms suppressed here by compactly supported variations. Its relation to the worldsheet conditions is discussed in Callan et al. 1985, §2, p. 597, Open PDF. To see the relation explicitly, define

$$\mathcal D=R+4\nabla^2\Phi-4(\nabla\Phi)^2.$$

Varying the prefactor and the kinetic term and integrating derivatives of the variations by parts gives

$$\begin{aligned} \delta I=\int d^DY\,\sqrt G\,e^{-2\Phi}\Big[ &\left(R_{mn}+2\nabla_m\nabla_n\Phi -\frac12G_{mn}\mathcal D\right)\delta G^{mn}\\ &-2\mathcal D\,\delta\Phi\Big]. \end{aligned}$$

The dilaton equation sets $\mathcal D=0$. Only after using it does the metric equation reduce to $R_{mn}+2\nabla_m\nabla_n\Phi=0$. At this order,

$$\mathcal D= \frac{G^{mn}\bar\beta^G_{mn}-4\bar\beta^\Phi}{\alpha'}.$$

This is the promised consistency test: the two-dimensional Weyl equations agree with the joint spacetime field equations in this specified truncation. Small $\alpha'$ corrections require small target curvature and slow variation; neglecting string loops separately requires an appropriate small string coupling $g_s=e^{\Phi_0}$ about the chosen constant reference value. Neither control guarantees a stable bosonic vacuum: the bosonic spectrum includes a tachyon.

Vertex weights and spacetime masses

Now specialize to the flat critical bosonic string with constant dilaton. Use a Euclidean target for the free-field calculation, then continue target momentum to the site’s Lorentzian convention. A complex worldsheet coordinate is denoted by $w$, to keep it distinct from the AdS radial coordinate below. The local singular covariance, with the noncompact zero mode treated separately and an additive infrared reference understood, is

$$\langle X^m(w,\bar w)X^n(0)\rangle =-\frac{\alpha'}2\delta^{mn}\log|w|^2.$$

Define the normal-ordered plane-wave operator $V_p={:}e^{ip\cdot X}{:}$. The holomorphic stress is $T(w)=-(1/\alpha'){:}\partial X\cdot\partial X{:}$. Contracting one derivative with the exponential gives $-i\alpha'p^m/(2w)$; the double contraction therefore gives the double pole

$$T(w)V_p(0)\sim \frac{\alpha'p_E^2/4}{w^2}V_p(0) +\frac1w\,\partial V_p(0).$$

The antiholomorphic calculation is identical, so

$$h=\bar h=\frac{\alpha'p_E^2}{4}.$$

The normal-ordering and stress-OPE calculation is developed in Tong 2009, §4.3.3, pp. 80–82, PDF. An integrated matter insertion $\int d^2w\,V_p$ must have weights $(1,1)$ to be invariant under conformal changes of coordinates. Thus $p_E^2=4/\alpha'$. After analytic continuation $p_E^2=-p_L^2$, the mass is

$$m^2=-\frac4{\alpha'}.$$

This is the closed bosonic tachyon. It is a property of the specified bosonic theory, not a universal prediction for every effective string.

A level-one tensor insertion has matter part

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

Its weights are $(1+\alpha'p_E^2/4,1+\alpha'p_E^2/4)$. The mass-shell condition is then $p_L^2=0$, but the operator must also be primary: the higher stress-OPE poles require transverse polarizations. Nonzero massless kinematics use the analytic continuation to Lorentzian or complex momenta; a real Euclidean vector of zero norm is zero. The physical spectrum further identifies null or gauge-equivalent states and includes the ghost sector. Weight counting is a necessary test, not a replacement for these constraints or a proof that a finite background deformation is exactly marginal. See Tong 2009, §5.4, pp. 122–126, PDF.

For an open string with Neumann boundary conditions in these directions, the image contribution doubles the boundary covariance to $-2\alpha'\delta^{mn}\log|\tau|$. A normal-ordered boundary exponential has weight $\alpha'p_E^2$. The vector insertion

$$\int_{\partial\Sigma}d\tau\, \epsilon_m{:}\partial_\tau X^m e^{ip\cdot X}{:}$$

has weight one on the mass shell $p_L^2=0$, with the corresponding primary and physical-state conditions. Its form is the plane-wave expansion of the boundary coupling $\int A_m\,dX^m$. The covariance and physical boundary insertion are given in Tong 2009, §4.7, p. 109, Eq. (4.57), PDF and Tong 2009, §5.4.2, pp. 125–126, PDF. Dirichlet boundary conditions and general brane spectra need their own treatment; this Neumann calculation does not establish them.

The two channels of an annulus

One precise relation between open and closed descriptions is already visible in boundary CFT. Take a rectangle of spatial width $\ell$ and Euclidean-time circumference $\beta$, identify the time edges, and impose boundary conditions $a,b$ at the two spatial ends. At fixed modulus its partition function is

$$\begin{aligned} Z_{ab}(\beta,\ell) &=\operatorname{Tr}_{\mathcal H_{ab}(\ell)} e^{-\beta H_{\rm open}(\ell)}\\ &=\langle B_a|e^{-\ell H_{\rm closed}(\beta)}|B_b\rangle. \end{aligned}$$

The first slicing propagates interval states around periodic time; the second propagates circle states between the two boundary states. The Hamiltonians include their Casimir energies. This is a change of slicing of a specified two-boundary path integral; see Cardy 2008, §3, pp. 9–11, Eqs. (27)–(30), PDF. A string amplitude additionally requires the matter/ghost theory and integration over its modulus.

A disk ending on one Wilson loop has a different number of boundary components. Changing its slicing does not make it this annulus, and the channel identity alone supplies no massless gravitational spectrum for a generic confining flux tube. The further ingredients belong to D-branes, open–closed duality, and gauge sectors.

AdS dilations and scalar boundary powers

Consider a separate Euclidean target geometry, with radius $R>0$ and $d$ boundary coordinates:

$$ds^2=d\phi^2+e^{2\phi/R}d\vec x^{\,2} =R^2\frac{dz^2+d\vec x^{\,2}}{z^2}, \qquad z=Re^{-\phi/R}>0.$$

At fixed $\phi$, a coordinate separation $|d\vec x|$ has proper length $e^{\phi/R}|d\vec x|$. The simultaneous scaling $(z,\vec x)\mapsto(\lambda z,\lambda\vec x)$, $\lambda>0$, leaves the metric invariant. In the first coordinates it shifts $\phi\mapsto\phi-R\log\lambda$. This dilation isometry relates the radial position to boundary length scale.

It does not show that this metric solves the preceding string equations. Its Ricci tensor is $R_{mn}=-dG_{mn}/R^2$, so a constant dilaton with vanishing two-form fails $\bar\beta^G=0$ at finite $R$. A supported string background requires additional fields or sectors and their equations; the geometric calculation alone does not construct one.

Deriving the scalar exponents

For a free minimally coupled scalar $\psi$ of mass $m$, use $(\nabla_E^2-m^2)\psi=0$ in this Euclidean metric. Since $\sqrt G=(R/z)^{d+1}$ and $G^{zz}=z^2/R^2$, the equation becomes

$$\left[z^2\partial_z^2-(d-1)z\partial_z +z^2\partial_i\partial_i-m^2R^2\right]\psi=0.$$

At fixed finite boundary momentum, the boundary-derivative term is subleading as $z\to0$. Substituting $\psi\sim z^\Delta$ gives

$$\Delta(\Delta-d)=m^2R^2, \qquad \Delta_\pm=\frac d2\pm\nu, \qquad \nu=\sqrt{\frac{d^2}{4}+m^2R^2}.$$

Real roots require the Breitenlohner–Freedman bound $m^2R^2\ge-d^2/4$. These are possible local powers; a quantum theory still needs a boundary condition. In the standard AdS scalar dictionary, fixing the leading source coefficient gives operator dimension $\Delta_+$. Under suitable boundary conditions a different choice is available in a restricted mass range. See Klebanov and Witten 1999, §§2.1–2.2, pp. 3–6, Eqs. (2.1)–(2.9), PDF; these are printed preprint page labels.

The restriction is visible without assuming the full holographic correspondence. Continue to Lorentzian AdS and use the ordinary Klein–Gordon norm. For a mode with radial power $z^\Delta$, the near-boundary radial integral is proportional to

$$\int_0 dz\,z^{1-d+2\Delta}.$$

For the slower branch $\Delta_-$, its power is $1-2\nu$, so it converges for $\nu<1$. Distinct real roots above the BF endpoint require $\nu>0$. Thus for $d\ge2$ the strict window for the two usual quantizations is

$$0<\nu<1 \quad\Longleftrightarrow\quad -\frac{d^2}{4}<m^2R^2<-\frac{d^2}{4}+1.$$

The choice also involves the boundary terms and admissible boundary conditions; the norm here is not the Euclidean bulk $L^2$ norm. The repeated-root logarithms at $\nu=0$ and the marginal slow-branch norm at $\nu=1$ are outside this discussion. Boundary conditions, quantization, and deformations develops the boundary-value problem.

What these calculations establish

The auxiliary metric reproduces a classical area action under positive-metric assumptions. Quantum Weyl consistency constrains target backgrounds in a controlled derivative expansion. Physical vertex conditions restrict spacetime masses in a specified string theory. AdS geometry supplies a dilation isometry and scalar asymptotic powers. Establishing a particular gauge/gravity correspondence requires additional dynamical evidence beyond all four calculations.

These distinctions also organize the course’s earlier themes. Duality can make useful variables local; RG identifies controlled long-distance descriptions; conformal symmetry restricts spectra and correlators; compactness permits defect sectors whose weights determine their importance. A fluctuating-surface description becomes useful when those tools establish its domain and predictions.

Exercises

Exercise 1: Eliminating the worldsheet metric

Start from the pure kinetic Euclidean Polyakov action

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

Let $T_s>0$, let the auxiliary and target metrics be positive, and take a smooth immersion so that $h_{ab}$ is positive and nondegenerate. Show that varying with respect to $g^{ab}$ makes $g_{ab}$ proportional to $h_{ab}$, and that substituting this result gives

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

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 and area suppression

In the regulated counting model above, suppose the number of surfaces of physical area $A$ with a fixed boundary behaves as $N(A)\sim e^{s_0A}$. If each has weight $e^{-T_0A}$, find the effective exponential suppression rate. What happens as it approaches zero, and what has this calculation left undetermined?

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. As $T_{\rm eff}\to0^+$, this exponential suppression disappears. That cancellation alone does not establish a continuum limit; subleading weights, the measure, and other interactions still matter. 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 breakdown scale

Let

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

be the one-loop equation for the two-dimensional $O(N)$ model with $N>2$ and $0<\alpha_0=\alpha(\ell_0)\ll1$. Solve for $\alpha(\ell)$ and estimate the scale $\xi$ where the weak-coupling approximation fails.

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 formal one-loop scale estimates the onset of strong coupling, not an exact correlation length or its prefactor. Perturbation theory fails before its denominator actually reaches zero. In energy variables $\mu=1/\ell$, the coupling decreases logarithmically in the ultraviolet.

Exercise 4: Marginality of the closed-string tachyon vertex

Use the free critical bosonic string and analytic continuation specified above. Normal ordering of every exponential is implicit in this exercise. For the covariance

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

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

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

Find the linearized marginality condition for the integrated matter vertex

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

and translate it into the Lorentzian mass. Here $z$ is a worldsheet coordinate, not the AdS radius.

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

For $R>0$, $z>0$ and $\lambda>0$, 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

Use the free minimally coupled AdS scalar with $d\ge2$, the ordinary Lorentzian Klein–Gordon norm, and boundary conditions with the appropriate boundary terms. Starting from

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

solve for $\Delta$, identify the standard branch, and state the strict mass window permitting the usual alternate quantization. Logarithmic or marginal endpoint quantizations are outside the exercise.

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 $\Delta_+$.

References

• Callan, C. G., Jr., D. Friedan, E. J. Martinec, and M. J. Perry. “Strings in Background Fields.” Nuclear Physics B 262 (1985): 593–609. DOI: 10.1016/0550-3213(85)90506-1. Author-hosted Open PDF.
• Cardy, John. Boundary Conformal Field Theory. Consulted arXiv revision 2, 20 February 2008; original preprint 2004. arXiv:hep-th/0411189v2. Open PDF.
• Klebanov, Igor R., and Edward Witten. “AdS/CFT Correspondence and Symmetry Breaking.” Nuclear Physics B 556 (1999): 89–114. DOI: 10.1016/S0550-3213(99)00387-9. Open PDF: arXiv:hep-th/9905104v2, revised 3 June 1999.
• Polyakov, A. M. Gauge Fields and Strings. Contemporary Concepts in Physics, Vol. 3. Harwood Academic Publishers, 1987. DOI: 10.1201/9780203755082.
• Tong, David. String Theory. University of Cambridge Part III lecture notes, January 2009. Official notes and chapter PDFs.