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AdS/CFT duality AdS/CFT duality

Quantization of coadjoint orbits

1. Motivation and overview

Section titled “1. Motivation and overview”
  • Symmetry →\to phase space. For a Lie group GG, each coadjoint orbit Ob⊂g\*\mathcal O_b\subset\mathfrak g^\* is a symplectic manifold with Kirillov–Kostant–Souriau (KKS) 2-form. Quantizing Ob\mathcal O_b often yields a unitary irrep—Kirillov’s orbit method [5].
  • AdS3_3 gravity and Virasoro. With Brown–Henneaux boundary conditions, 3D gravity’s asymptotic symmetry is Virasoro ×\times Virasoro with large central charge cc [1]. The boundary graviton phase space of a single chiral sector is a Virasoro coadjoint orbit; its phase-space path integral gives a concrete boundary QFT [3]. On the torus, the vacuum orbit produces the vacuum Virasoro character (one-loop exact), revealing why boundary gravitons alone do not form a modular-invariant CFT [3,7].
  • Analogy to AdS2_2. Compactifying the spatial circle (or truncating to the zero mode) reduces the Virasoro orbit action to the Schwarzian—precisely the boundary action of nearly-AdS2_2 gravity [3,8].

2. Warm-up: finite-dimensional picture (SU(2))

Section titled “2. Warm-up: finite-dimensional picture (SU(2))”

For G=SU(2)G=\mathrm{SU}(2), identify su(2)\*≃R3\mathfrak{su}(2)^\*\simeq\mathbb R^3; coadjoint orbits are 2-spheres of radius jj. The KKS form is (a multiple of) the area form; geometric quantization yields the spin-jj irrep. Keep this picture—orbit = phase space, quantization = representation—as intuition for Virasoro [5].

3. Virasoro group, coadjoint representation, and orbit classes

Section titled “3. Virasoro group, coadjoint representation, and orbit classes”

Virasoro algebra. Using shifted conventions, the modes obey

[Ln,Lm]=(n−m)Ln+m+c12 n3 δn+m,0 .[L_n,L_m]=(n-m)L_{n+m}+\frac{c}{12}\,n^3\,\delta_{n+m,0}\,.

A coadjoint element is (b(θ),t)(b(\theta),t) with pairing

⟨(b,t),(f,a)⟩=∫02πb(θ) f(θ) dθ+t a .\langle (b,t),(f,a)\rangle=\int_0^{2\pi} b(\theta)\,f(\theta)\,d\theta + t\,a\,.

Write the classical central parameter as CC (so that t=Ct=C). The infinitesimal coadjoint action is

δfb=f b′+2f′ b−C24πf′′′ ,δfC=0 ,\delta_f b = f\,b' + 2f'\,b - \frac{C}{24\pi}f''' \,,\qquad \delta_f C = 0\,,

and for a constant representative (b0,C)(b_0,C) the finite action by ϕ∈Diff(S1)\phi\in \mathrm{Diff}(S^1) reads

b(ϕ(θ))  =  b0 [ϕ′(θ)]2  −  C24π{ϕ(θ),θ} ,b(\phi(\theta)) \;=\; b_0\,[\phi'(\theta)]^2 \;-\; \frac{C}{24\pi}\{\phi(\theta),\theta\}\,,

with Schwarzian {ϕ,θ}=ϕ′′′ϕ′−32(ϕ′′ϕ′)2\{\phi,\theta\}=\dfrac{\phi'''}{\phi'}-\dfrac{3}{2}\Big(\dfrac{\phi''}{\phi'}\Big)^2 [2,3].

Orbit families by stabilizer S⊂Diff(S1)S\subset\mathrm{Diff}(S^1). For constant b0b_0 one finds [2,3]:

  • Normal (“ordinary”) orbits: b0≠−Cn248πb_0\neq -\dfrac{C n^2}{48\pi} for all n∈Zn\in\mathbb Z. Stabilizer S=U(1)S=U(1) (rotations), orbit Diff(S1)/U(1)\mathrm{Diff}(S^1)/U(1).
  • First exceptional (vacuum) orbit: b0=−C48πb_0=-\dfrac{C}{48\pi}. Stabilizer S=PSL(2,R)S=\mathrm{PSL}(2,\mathbb R) generated by L±1,L0L_{\pm1},L_0, orbit Diff(S1)/PSL(2,R)\mathrm{Diff}(S^1)/\mathrm{PSL}(2,\mathbb R).
  • Higher exceptional orbits: b0=−Cn248πb_0=-\dfrac{C n^2}{48\pi}, n>1n>1, stabilizer generated by L±n,L0L_{\pm n},L_0.

Geometry. For C>0C>0 and b0≥−C48πb_0\ge -\dfrac{C}{48\pi} (vacuum and normal orbits), the orbits admit a natural complex structure compatible with the KKS form; iωi\omega is positive—so these orbits are Kähler [2,3].

4. The KKS symplectic form and a convenient potential

Section titled “4. The KKS symplectic form and a convenient potential”

Start from ω(X1,X2)=−⟨(b,C),[F1,F2]⟩\omega(X_1,X_2)=-\langle (b,C),[F_1,F_2]\rangle with Fi=(fi,ai)F_i=(f_i,a_i) Virasoro vectors. Using the coadjoint action and integrating by parts gives (for two tangent diffeos f1,f2f_1,f_2)

ω  =  −∫02π ⁣dθ  [b (f1f2′−f2f1′)−C48π(f1f2′′′−f2f1′′′)] .\omega \;=\; -\int_0^{2\pi}\!d\theta\;\Big[b\,\big(f_1 f_2'-f_2 f_1'\big)-\frac{C}{48\pi}\big(f_1 f_2'''-f_2 f_1'''\big)\Big]\,.

Parametrize the orbit by ϕ(θ)\phi(\theta); on the orbit b(θ)=b(ϕ)b(\theta)=b(\phi) above and δfϕ=f ϕ′\delta_f\phi=f\,\phi'. Rewriting in field space yields the compact form

ω  =  −∫02π ⁣dθ  [C48π dϕ′∧dϕ′′(ϕ′)2+b0 dϕ∧dϕ′] ,(4.1)\omega \;=\; -\int_0^{2\pi}\!d\theta\;\Big[\frac{C}{48\pi}\,\frac{d\phi'\wedge d\phi''}{(\phi')^2} + b_0\,d\phi\wedge d\phi'\Big]\,,\tag{4.1}

with presymplectic potential

α  =  ∫02π ⁣dθ  [C48π ϕ′′ dϕ′(ϕ′)2+b0 ϕ′ dϕ] ,dα=ω .(4.2)\alpha \;=\; \int_0^{2\pi}\!d\theta\;\Big[\frac{C}{48\pi}\,\frac{\phi''\,d\phi'}{(\phi')^2} + b_0\,\phi'\,d\phi\Big]\,,\qquad d\alpha=\omega\,. \tag{4.2}

These expressions are invariant under SS up to exact forms and will seed the path integral [3].

5. Phase-space path integral (Alekseev–Shatashvili) and gauge structure

Section titled “5. Phase-space path integral (Alekseev–Shatashvili) and gauge structure”

For any orbit with Hamiltonian HH, the first-order action is

S[ϕ]  =  ∫dt  [α(ϕ,ϕ˙)−H(ϕ)] .S[\phi]\;=\;\int dt\;\Big[\alpha(\phi,\dot\phi) - H(\phi)\Big]\,.

Setting H=0H=0 gives the Alekseev–Shatashvili (AS) geometric action

SAS[ϕ]  =  −∫d2x  [C48π ∂tϕ′ ϕ′′(ϕ′)2  +  b0 ϕ ∂tϕ′] ,d2x=dt dθ ,(5.1)S_{\text{AS}}[\phi]\;=\;-\int d^2x\;\Big[\frac{C}{48\pi}\,\frac{\partial_t\phi'\,\phi''}{(\phi')^2} \;+\; b_0\,\phi\,\partial_t\phi'\Big]\,,\qquad d^2x=dt\,d\theta\,,\tag{5.1}

with a gauge redundancy along the stabilizer SS (time-dependent rotations for normal orbits; time-dependent PSL(2,R)\mathrm{PSL}(2,\mathbb R) for the vacuum orbit). Choosing H∝L0H\propto L_0 makes cylinder time evolution manifest (useful for torus thermodynamics), without changing ω\omega [4,3].

Interpretation. ϕ\phi is the boundary reparameterization; SASS_{\text{AS}} is its geometric action on Diff(S1)/S\mathrm{Diff}(S^1)/S. In the vacuum case, truncating to the zero mode reproduces the Schwarzian of nearly-AdS2_2 [3,8].

6. Gauge fixing, measure, and the role of zero modes

Section titled “6. Gauge fixing, measure, and the role of zero modes”

Quantization requires (i) quotienting by time-dependent SS and (ii) a measure compatible with ω\omega.

  • Vacuum orbit (S=PSL(2,R)S=\mathrm{PSL}(2,\mathbb R)). Expand ϕ(θ,t)=θ+ε(θ,t)\phi(\theta,t)=\theta+\varepsilon(\theta,t) and fix the three global modes ε−1,0,1\varepsilon_{-1,0,1} (the sl2\mathfrak{sl}_2 directions). The one-loop determinant then runs over Fourier modes n∈Z∖{−1,0,1}n\in\mathbb Z\setminus\{-1,0,1\} [3].
  • Normal orbit (S=U(1)S=U(1)). Fix the zero mode associated with rigid rotations. The one-loop determinant runs over n∈Z∖{0}n\in\mathbb Z\setminus\{0\} [3].

The natural measure is the Liouville measure induced by ω\omega (equivalently, by the Kähler metric). In practice this plus the Faddeev–Popov determinant yields exact torus partition functions for these quadratic theories [3].

7. Torus partition functions and central-charge shifts

Section titled “7. Torus partition functions and central-charge shifts”

Let q=e2πiτq=e^{2\pi i\tau}. For a single chiral sector:

  • Vacuum orbit Diff(S1)/PSL(2,R)\mathrm{Diff}(S^1)/\mathrm{PSL}(2,\mathbb R): the partition function equals the vacuum Virasoro character
Zvac(τ)  =  q−c24∏n=2∞11−qn ,c  =  C+13 ,(7.1)Z_{\text{vac}}(\tau)\;=\;q^{-\frac{c}{24}}\prod_{n=2}^{\infty}\frac{1}{1-q^n}\,, \qquad c \;=\; C + 13\,, \tag{7.1}

where the shift by +13+13 comes from the quotient by sl2\mathfrak{sl}_2 and the associated measure/ghost contributions [3,7].

  • Normal orbit Diff(S1)/U(1)\mathrm{Diff}(S^1)/U(1): one obtains a single Verma module
ZVerma(τ)  =  q h−c24∏n=1∞11−qn ,c  =  C+1 ,h−c−124  =  2πb0 .(7.2)Z_{\text{Verma}}(\tau)\;=\;q^{\,h-\frac{c}{24}}\prod_{n=1}^{\infty}\frac{1}{1-q^n}\,, \qquad c \;=\; C + 1\,,\qquad h-\frac{c-1}{24}\;=\;2\pi b_0\,. \tag{7.2}

Modular lesson. A single orbit yields a single module (vacuum or Verma) and is not modular invariant; a complete CFT requires adding other primaries/sectors. This mirrors the “pure gravity” discussion in AdS3_3 [3,7].

8. Stress tensor, correlators, and Virasoro blocks

Section titled “8. Stress tensor, correlators, and Virasoro blocks”

The reparameterization field reorganizes stress-tensor dynamics. After linearization on a normal orbit, one reads off the renormalized (c,h)(c,h) from the T(w)T(0)T(w)T(0) OPE or cylinder two-point function—reproducing c=C+1c=C+1 and hh in (7.2). Coupling bilocals to ϕ\phi produces a tidy 1/c1/c Feynman expansion for Virasoro conformal blocks (light–light and heavy–light), matching known one-loop results and providing a practical large-cc calculus [3].

9. Holographic interpretation and the Schwarzian limit

Section titled “9. Holographic interpretation and the Schwarzian limit”

In AdS3_3 gravity, the vacuum orbit quantizes boundary gravitons—the Virasoro descendants of the vacuum in the dual CFT. Euclidean BTZ and constant-stress backgrounds correspond to normal orbits with b0>−C48πb_0>-\dfrac{C}{48\pi}; their quantization gives single highest-weight sectors. The theory is weakly coupled at large cc (effective coupling ∼1/C\sim 1/C), paralleling loop expansions in the bulk [1,3,7]. Truncating to the zero mode recovers the Schwarzian and all its familiar properties (one-loop exactness, SL(2)SL(2) gauge redundancy) [3,8].

10. How to compute in practice (a checklist)

Section titled “10. How to compute in practice (a checklist)”
  1. Choose an orbit (vacuum vs normal) with parameters (C,b0)(C,b_0) and stabilizer SS [2].
  2. Write ω\omega and α\alpha via (4.1)–(4.2) in terms of ϕ(θ)\phi(\theta) [3].
  3. Pick HH: H=0H=0 for representation-theory questions; H∝L0H\propto L_0 for cylinder time/thermodynamics [4,3].
  4. Gauge-fix the SS redundancy (remove sl2\mathfrak{sl}_2 or the U(1)U(1) zero mode) and include the FP determinant [3].
  5. Linearize where appropriate (normal orbits become quadratic/free after a linear field redefinition) [3].
  6. Evaluate determinants to get characters (7.1)–(7.2); then compute TT-correlators or Virasoro blocks using the ϕ\phi propagator [3].

11. Common pitfalls

Section titled “11. Common pitfalls”
  • Modular invariance: One orbit ≠\neq one CFT. Add sectors/primaries to build a modular-invariant partition function [3,7].
  • Zero modes: Always remove the stabilizer modes (sl2\mathfrak{sl}_2 for vacuum; U(1)U(1) for normal) from determinants and the measure [3].
  • Positivity window: For C>0C>0 and b0≥−C48πb_0\ge -\dfrac{C}{48\pi} the orbits are Kähler and the quadratic theory is well-behaved; outside, expect non-unitary features [2,3].
  • cc-shifts: Do not confuse the classical parameter CC with the quantum central charge cc; shifts arise from the measure and quotient [3].

12. Minimal worked derivations (sketches)

Section titled “12. Minimal worked derivations (sketches)”

(a) From algebra to ω\omega in field variables. Insert the Virasoro bracket into ω(X1,X2)\omega(X_1,X_2), integrate by parts to move derivatives off fif_i, then use δfϕ=f ϕ′\delta_f\phi=f\,\phi' to rewrite everything in dϕd\phi; a suitable potential reproduces (4.1)–(4.2) [3].

(b) Torus characters. Expand ϕ(θ,t)=θ+ε(θ,t)\phi(\theta,t)=\theta+\varepsilon(\theta,t), gauge-fix the stabilizer modes, and keep quadratic terms. The Gaussian functional determinant over remaining Fourier modes gives the products in (7.1)–(7.2). The missing n=−1,0,1n=-1,0,1 (vacuum) vs n=0n=0 (normal) factors explain the different products and cc-shifts [3,7].

(c) Schwarzian limit. Restrict to ϕ(θ,t)=ϕ0(t)\phi(\theta,t)=\phi_0(t) (zero spatial mode). Then ω∼(C/48π) d(ϕ0′)∧d(ϕ0′′)/(ϕ0′)2\omega\sim (C/48\pi)\,d(\phi_0')\wedge d(\phi_0'')/(\phi_0')^2 reduces to the 1D structure whose action is ∝∫dt {ϕ0,t}\propto \int dt\,\{\phi_0,t\} after choosing HH appropriately—recovering the Schwarzian [3,8].

References

Section titled “References”

[1] J. D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207. (No arXiv)

[2] E. Witten, Coadjoint Orbits of the Virasoro Group, Commun. Math. Phys. 114 (1988) 1. (No arXiv)

[3] J. Cotler and K. Jensen, A Theory of Reparameterizations for AdS3_3 Gravity, JHEP 2019 (2), 079. arXiv:1808.03263.

[4] A. Alekseev and S. Shatashvili, Path Integral Quantization of the Coadjoint Orbits of the Virasoro Group and 2D Gravity, Nucl. Phys. B 323 (1989) 719. (No arXiv)

[5] A. A. Kirillov, Lectures on the Orbit Method, Grad. Studies in Math. 64, AMS (2004). (Book)

[6] A. Pressley and G. Segal, Loop Groups, Oxford University Press (1986). (Book)

[7] A. Maloney and E. Witten, Quantum Gravity in AdS3_3, JHEP 02 (2010) 029. arXiv:0712.0155.

[8] J. Maldacena, D. Stanford, and Z. Yang, Conformal symmetry and its breaking in two-dimensional Nearly AdS2_2 space, PTEP 2016 (12), 12C104. arXiv:1606.01857.