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CFT Complexity and Penalty Factors

Stefano Baiguera, Nicolas Chagnet, Shira Chapman, Osher Shoval

arXiv:2507.22118v1hep-thquant-ph

TL;DR

Prior work largely treated conformal generators as equally costly, despite CFT complexity’s relevance to critical systems and holographic black-hole interiors. This paper derives a penalty-weighted state-space metric for generic Lie groups and applies it to one- and two-dimensional CFTs, finding that penalties can affect complexity and optimal trajectories.

  • Problem

    Prior work mostly assumed equal costs for conformal generators, while CFT complexity can inform the study of critical systems and holographic black-hole interiors.

    Source: §1 · p. 5

  • Method

    The authors construct a coset-space metric induced by a generic Lie-group circuit norm, relating the construction to submersions and extremization over stabilizer directions.

    Source: §3.1 · p. 12, §6 · p. 44

  • Results

    For 1D and 2D CFTs, turned-off penalties yield analytic results; L± penalties mildly redirect 1D trajectories, while small/large J0 respectively raises/lowers 2D complexity versus J0=0.

    Source: §1 · p. 6, §6 · p. 45

  • Takeaways & Limitations

    In 1D CFTs, larger L+ or L− penalties make trajectories using the cheaper generator with dilatations comparatively favored.

    Source: §6 · p. 45

    The analysis restricts penalties to choices satisfying (4.19), leaving the alternative with a nontrivial boundary between accessible states to appendix C.

    Source: §4.1.2 · p. 21

Abstract

from arXiv · show

Quantum complexity of conformal field theory (CFT) states has recently gained significant attention, both as a diagnostic tool in condensed matter systems and in connection with holographic observables probing black hole interiors. Previous studies have primarily focused on cases where all generators of the conformal group contribute equally to the cost of building a circuit. In this work, we present a general framework for studying the complexity of circuits in generic Lie groups, where penalty factors assign relative weights to different generators. Our approach constructs a metric on the coset space of quantum states, induced from a (pseudo-)Riemannian norm on the space of unitary circuits. The geodesics of this metric are interpreted as optimal circuits. The method builds on the formalism of (pseudo-)Riemannian submersions and connects naturally to other prescriptions in the literature, including cost function minimization along stabilizer directions and constructions based on coadjoint orbits. As a concrete application, we compute state complexity for states in one- and two-dimensional CFTs. For specific choices of penalty factors, our prescription yields a positive-definite metric with a viable interpretation as complexity; in other cases, the resulting metric is indefinite. In the viable regime, we derive analytic results when a specific penalty factor is turned off, develop perturbative expansions for small values of the penalty factors, and provide numerical results in the general case. We comment on the relation of our measure of complexity to holography.

1 Introduction

CFT complexity can probe critical systems and holographic black-hole interiors, but prior CFT studies considered equal costs for all generators. This work develops a penalty-weighted framework and applies it to one- and two-dimensional CFTs.

1 Introduction

  • Earlier CFT complexity studies considered isotropic costs, leaving the effects of unequal generator penalties unaddressed.CFTs are of interest both for studying systems near critical points and because their states are dual to black holes in holography.Source: §1 · p. 5
  • The paper constructs a penalty-weighted metric on state space for generic finite-dimensional Lie groups, including non-compact groups, using (pseudo-)Riemannian submersions.For non-compact groups, the stabilizer-direction prescription becomes an extremization; the paper also discusses a possible connection to coadjoint orbits.Source: §1 · p. 6
  • Only certain penalty ranges yield a positive-definite CFT complexity metric; analytic special cases, small-penalty expansions, and general numerical results cover 1D and 2D CFTs.Within the viable range, dilatation penalties have a relatively minor effect, while increasing penalties in other directions produce more significant changes.Source: §1 · p. 6
  • For isotropic costs, the paper relates its CFT complexity to AdS geodesics through the Fubini–Study metric and a massive particle’s bulk symplectic form.The authors describe this as a first step toward a complete holographic connection.Source: §1 · p. 6

2 Complexity geometry

Nielsen complexity treats circuit construction as finding a shortest path on the unitary manifold under a chosen cost function. For state complexity, the paper projects that geometry onto state space by accounting for directions that leave the reference state unchanged.

2.1 Nielsen complexity

  • Nielsen unitary complexity is the minimal cost-function length of a path from the identity to the target unitary.The path is generated by a time-dependent Hamiltonian expanded in Lie-algebra generators with real control functions.Source: §2.1 · pp. 7–8
  • State complexity minimizes unitary complexity over paths reaching the target state; stabilizer directions leave the reference state unchanged and can be removed locally to induce a state-space norm.For non-compact groups, the prescription becomes an extremization because the relevant distances need not be positive definite.Source: §2.1 · pp. 8–9

2.2 Choice of the cost function

  • The bi-invariant L2 cost treats generators equally, while penalty factors assign them different relative costs and can break bi-invariance.For non-compact groups, the penalty matrix has the same signature as the group’s inner product.Source: §2.2 · pp. 9–10
  • The Fubini–Study norm is a state-space cost that ignores stabilizer contributions; applying stabilizer minimization to a unitary metric induces the corresponding state-space metric.The bi-invariant unitary metric induces the Fubini–Study metric, while generic penalties can make the calculation cumbersome.Source: §2.2 · p. 11

3 State complexity and Riemannian submersions

The section extends state-complexity projection to generic Lie groups through (pseudo-)Riemannian submersions. It relates the resulting coset metric to stabilizer minimization, conserved charges, and coadjoint-orbit constructions.

3 State complexity and Riemannian submersions

  • State-complexity projection extends to generic, possibly non-compact Lie groups through (pseudo-)Riemannian submersions.The procedure is connected to stabilizer minimization and coadjoint-orbit constructions.Source: §3 · p. 11

3.1 General procedure

  • The construction separates stabilizer and coset directions, then projects the circuit cost to a coset metric via a smooth surjective submersion.The derivation uses a reductive symmetric coset structure; the procedure also applies to conformal groups SO(d, 2) for any spacetime dimension d.Source: §3.1 · pp. 12–14

3.2 Abelian example and relation to conserved charges

  • For an abelian stabilizer, setting the conserved charges K_a to zero yields the unique coset metric from the pseudo-Riemannian submersion.This condition is equivalent to f_a = 0.Source: §3.2 · p. 14

3.3 Relation to the minimization method

  • With bi-invariant cost and trivial penalties, the projected cost on the coset reduces to the Fubini–Study metric on CPN−1.The text identifies the trivial-penalty case as IIJ → δIJ.Source: §3.3 · p. 15
  • For non-compact groups with invertible penalty matrix, the stabilizer prescription is an extremization, not a minimization, because the circuit cost is indefinite.For compact groups, f_a = 0 instead amounts to minimization over stabilizer directions.Source: §3.3 · pp. 14–15

3.4 Comment on the relation with coadjoint orbits

  • The coadjoint-orbit construction reproduces the Fubini–Study metric for global conformal circuits, but a simple symplectic-manifold interpretation with penalties remains unresolved.The authors leave that interpretation as a question for future work.Source: §3.4 · pp. 15–16

4 One-dimensional CFT

The section develops complexity geometry for one-dimensional CFT states using circuits in the global conformal group and a primary reference state. It examines how penalty-weighted costs, including free dilatations, shape optimal circuits and their complexity through analytic, perturbative, and numerical study.

4.1.1 Homogeneous penalty factors

  • The homogeneous-penalty metric is the Poincaré unit disk, and its geodesic complexity is 2 arctanh(rT), diverging as the target approaches the disk boundary.The metric is obtained from the state-dependent Fubini–Study cost; the same metric can be recovered by projecting the group cost onto the coset.Source: §4.1.1 · pp. 18–19

4.1.2 General penalties - Hermitian basis

  • With general Hermitian-basis penalties, the dA′ and dB directions contribute with opposite signs, so the analysis restricts to penalties yielding a positive-definite metric throughout the unit disk.Along real λ, motion is generated by L+ alone; along imaginary λ, it is generated by L− alone.Source: §4.1.2 · pp. 20–21

4.1.3 General penalties - physical basis

  • In the physical {D, P, K} basis, diagonal penalties factorize stabilizer and coset directions, while nonzero J2 adds a tangent direction and makes the L± plane anisotropic.The resulting metric has reflection symmetries under Reλ → −Reλ and Imλ → −Imλ.Source: §4.1.3 · pp. 21–22

4.1.4 Hamiltonian generators along coset space trajectories

  • Near the disk origin, L0 produces angular rotations, while L+ and L− mainly change the real and imaginary parts of λ; raising their penalties favors motion away from their respective directions.Increasing I− favors real-axis motion, whereas increasing I+ favors imaginary-axis motion.Source: §4.1.4 · p. 23

4.2 State complexity with free dilatations

  • When D is free, the metric is positive definite without the general penalty constraint, phase shifts cost nothing, and the radial geodesic matches the homogeneous Fubini–Study path.The radial path is r(t) = tanh(arctanh(rT)t).Source: §4.2 · p. 25
  • With free dilatations, the optimal path is a straight radial segment followed by an arc, with the radial direction chosen along the less-penalized axis; the complexity matches the Fubini–Study cost up to radial rescaling.The straight segment lies along the real axis when I− > I+ and along the imaginary axis when I+ > I−.Source: §4.2 · p. 25

4.3 State complexity with general penalties

  • For general penalties, the optimal circuits follow affinely parametrized geodesics governed by equations that can be viewed as a classical two-dimensional trajectory in an effective potential.The equations extremize the squared cost Lagrangian rather than the cost itself.Source: §4.3 · p. 26

4.3.1 Analytical solution

  • When J2 = 0, exact optimal trajectories match the Fubini–Study stationary paths; complexity is independent of J1 and receives an overall rescaling from J3.This case makes the cost isotropic along the coset directions.Source: §4.3.1 · pp. 26–27

4.3.2 Perturbation theory

  • Through order ε², complexity is independent of the dilation penalty β1, while the P and K penalties matter; for small rT, the second perturbative term is subleading.The J2 = 0 limit recovers the expansion of the analytic result.Source: §4.3.2 · pp. 27–28

4.3.3 Numerical results

  • At finite J2, J1 changes complexity by only about 10^-3 through higher-order terms, a much smaller effect than the other penalties.The numerical study finds this dependence beyond the second-order perturbative result.Source: §4.3.3 · pp. 28–29, Figure 4 · p. 29
  • Increasing J3 raises the cost along both axes, while increasing J2 favors the real direction; at high anisotropy, indirect paths can reach states with a large imaginary component.The numerical study restricts to positive-definite metrics and target radii |rT| ≤ 0.5.Source: §4.3.3 · pp. 28, 30, 31, Figure 5 · p. 31

5 Two-dimensional CFT

This section studies complexity in two-dimensional CFTs, including factorized left- and right-moving trajectories and relative-complexity behavior under couplings between one-dimensional CFT copies. It examines how penalty choices and alternative couplings affect complexity and the positivity of the induced metric.

5 Two-dimensional CFT

  • Separate penalties in the two SO(1,2) copies preserve factorized trajectories; mixed penalties are needed to obtain nontrivial new geodesics.The section studies coupling the dilatation operations and reports that its complexity effect depends on the penalties, as detailed below.Source: §5 · pp. 32–33

5.1 Coupling (D, ¯D): coset space metric and analytic results

  • The coset metric is obtained by eliminating stabilizer directions; when J0 = 0, the two copies factorize and their state complexities combine as in equation (5.3).The projection can be performed using conserved charges or by minimizing the cost with respect to stabilizer velocities.Source: §5.1 · pp. 33–34

5.1.1 Positivity of the cost function

  • A positive-definite metric can arise either with J0 > 2 and J1 = ¯J1 = J3 = ¯J3 = 1, or with a gap between J1 and J3 that permits small J0.The paper identifies these as two regimes and provides example penalty sets for each.Source: §5.1.1 · pp. 35–36

5.1.2 Analytic solution: isotropic cost along the coset directions

  • For isotropic coset costs (J2 = ¯J2 = 0), the complexity is independent of J0 and J1, while J3 and ¯J3 rescale the respective Fubini–Study costs.The angular coordinates remain constant along the circuit, and the induced line element is positive definite.Source: §5.1.2 · pp. 36–37

5.2 Coupling (D, ¯D): perturbative expansion

  • With nonzero J2 and ¯J2, perturbation theory finds no J0 contribution through second order; the coupling dependence likely enters at third order.For small J0, positivity requires a gap between J1 and J3, and between their anti-holomorphic counterparts.Source: §5.2 · p. 38

5.3 Coupling (D, ¯D): numerical results

  • Numerical geodesics were solved by a shooting method; computations focus on |r|, |¯r| ≤ 0.8 in case 1 and ≤ 0.6 in case 2.The authors report qualitatively similar behavior for sampled target states closer to the boundary.Source: §5.3 · pp. 38–39

5.3.2 Case 2

  • For case 2, Crel ∼10^-2 and is always negative, so coupling permits shorter trajectories.This regime uses a large J0 and J1 = J3 in each sector, outside the perturbative regime.Source: §5.3.2 · pp. 40–41, Figure 7 · p. 41
  • Relative complexity can remain nonzero when one copy is near the origin, because the cost still depends on J0 when that copy is fixed.The submersion requires using the fixed copy’s stabilizer direction to set the relevant conserved charge to zero; relative complexity also increases in absolute value as target angles approach θ = θ̄ = π/4.Source: §5.3.2 · p. 42

5.4 Other couplings between one-dimensional CFTs

  • Two alternative left-right couplings are projected onto state space, yielding the two-copy Fubini–Study metric plus an interaction-dependent contribution.The considered couplings are between D and P̄ + K̄, and between the symmetric combinations (P, K̄) and (K, P̄).Source: §5.4 · p. 42

5.4.1 Coupling (D, ¯P + ¯K)

  • For the (D, P̄ + K̄) coupling, the metric is positive-definite at the origin for any penalties, but its signature changes across state space.The metric therefore does not define a valid complexity geometry everywhere; the positive-definite region around the origin ends at a non-trivial boundary in the Poincaré disk.Source: §5.4.1 · p. 43

5.4.2 Coupling (P, ¯K) + (K, ¯P)

  • For the (P, K̄) + (K, P̄) coupling, the metric is positive-definite at the origin when J0 < 1, but no J0 makes it positive throughout state space.The authors numerically investigate Nielsen complexity only in the positive-definite region, bounded by where the metric changes signature.Source: §5.4.2 · p. 44

6 Discussion

The discussion connects the coset-space construction to cost minimization and conserved charges, then reviews CFT applications and open directions. In the studied CFTs, penalty effects vary by setting, and a positive-definite state-space metric is not automatic.

Projection to the coset space

  • The authors map a Lie-group cost function to a coset-space complexity metric, including for the conformal group SO(d, 2).The map fa = 0 defines a (pseudo-)Riemannian submersion and is equivalent to extremizing the cost over the group manifold.Source: §6 · p. 44
  • For an abelian stabilizer, fa is related to a conserved charge for F2, and fa = 0 is equivalent to that charge vanishing.The discussion also relates this construction to coadjoint-orbit methods without penalties.Source: §6 · p. 45

Applications to CFTs

  • In 1D CFTs, penalizing L+ (L−) increases the cost of real (imaginary) motion and can favor alternative trajectories using the cheaper generator with dilatations.Dilatation penalties have only a mild effect, and the L± effect appears only for anisotropic penalties.Source: §6 · p. 45
  • In 2D CFTs, small J0 increases complexity and large J0 decreases it relative to J0 = 0, with the mild dependence visible only numerically for anisotropic L± penalties.The two copies are coupled through their dilatation control functions.Source: §6 · p. 45

Positivity of the metric

  • The induced metric need not be positive-definite; the paper constrains penalties or restricts target states to the region before the metric loses positive-definiteness.The authors leave the physical intuition behind these alternatives open.Source: §6 · p. 45

Generalization to higher dimensions

  • For d ≥ 3 CFTs, the general procedure applies, but solving the higher-dimensional Euler–Lagrange equations is technically more difficult.Special penalty choices may offer analytic or perturbative starting points.Source: §6 · p. 46

Relation to holography

  • For isotropic costs, CFT coherent-state phase space matches that of a massive AdS particle; extending this relation to arbitrary penalties remains future work.The projection condition fa = 0 corresponds to setting the associated symplectic potential to zero.Source: §6 · p. 46

Open directions

  • Further directions include penalized Virasoro complexity, curvature-based chaos questions, binding-complexity limits for coupled CFTs, and adapting golden gates to conformal-group circuits.The proposed questions include moving between conformal families, interpreting negative curvature, and finding gate sets that cover a Verma module.Source: §6 · pp. 46–47

A (Pseudo-)Riemannian submersions

The appendix defines submersions and explains how they induce a metric on a coset space. In this paper, the group manifold is projected to the space obtained by quotienting by the reference state's stabilizer.

Submersion

  • A submersion is a smooth map whose differential is surjective at every point.Its differential's kernel defines the vertical directions, while the orthogonal complement gives the horizontal directions.Source: A (Pseudo-)Riemannian submersions · p. 48

Pseudo-Riemannian submersion

  • A pseudo-Riemannian submersion maps horizontal directions isometrically onto the tangent space of the base.The horizontal space is defined as the orthogonal complement of the vertical space, which is the differential's kernel.Source: A (Pseudo-)Riemannian submersions · p. 48

Projection over the coset space

  • Here, the manifold is the Lie-group manifold and the base is the coset obtained by quotienting by the reference state's stabilizer.Source: A (Pseudo-)Riemannian submersions · p. 48
  • Under the stated projection assumptions, the coset has a unique induced pseudo-Riemannian metric for which the projection is a pseudo-Riemannian submersion.The appendix theorem supports the uniqueness of the metric obtained by the paper's procedure.Source: A (Pseudo-)Riemannian submersions · p. 49

B Fundamental representation of the conformal group

The section describes how the paper represents conformal circuits using Euclidean conformal generators while imposing conditions to obtain Lorentzian circuits.

Conformal generators

B.1 Inner product on a Lie group

This section sets up a Lie-group inner product using the Killing form and relates it to a trace expression in the conformal group's fundamental representation.

Inner product on a Lie group

  • A Lie algebra has a non-degenerate, symmetric Killing form, introduced here as a starting point for defining an inner product.Source: §B.1 · p. 50

Fundamental representation

  • The paper focuses on the fundamental matrix representation of so(d, 2), with generators represented in Md+2,d+2(C).Source: §B.1 · p. 50

Inner product in the fundamental representation

  • In the fundamental representation, the bilinear form can be expressed as a trace over matrices; the paper defines the corresponding inner product there.The inner product and bilinear form are related up to a normalization that depends on the Lie group's adjoint-representation quadratic Casimir.Source: §B.1 · p. 51

Adjoint-action property

  • The inner product and Killing form are invariant under the adjoint action, with the Killing-form relation expressed using the inverse adjoint action.Source: §B.1 · p. 51

B.2 Explicit example – SO(1,2)

The appendix gives an explicit SO(1,2) example, representing its generators as matrices and checking the associated conjugation and inner-product identities. The resulting quadratic form is indefinite, as expected for the non-compact group.

B.2 Explicit example – SO(1,2)

  • The generators D, P, and K span so(1,2), which is locally isomorphic to sl(2, R), and obey the listed commutation relations.The appendix verifies that the inner product agrees with the bilinear form in this example and extends the identity to any two Lie-algebra elements for N = 1.Source: §B.2 · pp. 51–52
  • The fundamental representation maps the generators to 3 × 3 matrices and uses η = diag(−1, −1, 1) to define Hermitian conjugation.The text checks that this conjugation is consistent with the stated generator identities.Source: §B.2 · p. 52
  • The SO(1,2) quadratic form is indefinite, with ⟨P, K⟩ = −2, ⟨D, D⟩ = 1, and ⟨L±, L±⟩ = −1.These are the non-vanishing inner products listed for the normalized Hermitian basis.Source: §B.2 · p. 52

C Cost functions with non-trivial boundary over the space of states

Rather than restrict penalty factors to keep the metric positive-definite everywhere, this section allows arbitrary penalties and identifies where the state-space metric remains positive-definite. The metric’s singular, signature-changing boundary limits the region interpreted as a complexity geometry.

C Cost functions with non-trivial boundary over the space of states

  • With arbitrary penalty factors, the state-space metric is treated as a complexity geometry only inside the region where it is positive-definite.The boundary is determined by the locus where the metric becomes singular and changes signature.Source: C Cost functions with non-trivial… · pp. 52–53
  • If distances on the coset space quantify the cost of constructing states, the boundary obstructs reaching states outside the allowed region.This is the stated interpretation of the boundary, not a claim that all such states are physically inaccessible.Source: C Cost functions with non-trivial… · p. 53
  • The appendices examine the boundary in one-dimensional CFTs and numerically compute two-dimensional state complexity for two left-right coupling cases.Source: C Cost functions with non-trivial… · p. 53

C.1 Boundary in one-dimensional CFTs

For one-dimensional CFTs, positivity of the state-space metric is tested using its leading principal minors. With positive penalties, the resulting signature-change curve bounds the region in which the complexity geometry is defined.

C.1 Boundary in one-dimensional CFTs

  • The cost function is positive-definite when α(θ) > 0 and I0(r^2 + 1)^2 − 2r^2α(θ) > 0, assuming I0 > 0.These inequalities are obtained from the two leading principal minors of the metric.Source: §C.1 · p. 53
  • For I± ≥ 0, the boundary B is the locus I0(r^2 + 1)^2 − 2r^2[I+ + I− + (I− − I+)cos(2θ)] = 0.The penalties are left arbitrary to locate where the metric changes signature.Source: §C.1 · p. 53
  • The complexity geometry is defined only for target states inside B; if that region is convex, geodesics from the origin to allowed targets should remain within the positive-definite region.Source: §C.1 · p. 53

C.2 Coupling (D, ¯P + ¯K)

For the specified two-dimensional CFT coupling, relative complexity rises radially in both copies, while its angular dependence differs between them. A small-coupling expansion accounts for the left-copy angular pattern and radial growth.

C.2 Coupling (D, ¯P + ¯K)

  • Relative complexity increases radially for both copies; it appears independent of the left-copy target angle but increases as the right-copy angle approaches zero.Figure 9 plots the relative complexity for J0 = 1/2, with the black dot marking the right-copy target and quadrant points marking left-copy targets.Source: §C.2 · p. 53, Figure 9 · p. 54
  • At J0 = 1/2, the right-hand trajectory remains within roughly 2% of its free shape, motivating a small-J0 expansion with the right copy replaced by its free trajectory.The expansion of the interacting line element is taken up to O(J0^2).Source: §C.2 · p. 54
  • At leading order around J0 = 0, the relative-complexity term vanishes at right-copy angle π/2 and is maximal at angle 0.The terms in the line element increase with radial distance; at fixed right-copy angle and radius, relative complexity decreases as rT increases.Source: §C.2 · p. 55

C.3 Coupling (P, ¯K) + (K, ¯P)

For the specified coupling, relative complexity peaks around the right-copy target and rises with its radial coordinate. The trajectories remain close to Fubini–Study geodesics, with equal target angles making those geodesics extremal.

C.3 Coupling (P, ¯K) + (K, ¯P)

  • Relative complexity peaks around the right-copy target, and increasing its radial coordinate raises the cost; the figure varies the left-copy target across the quadrant.The figure uses J0 = 1/2.Source: §C.3 · p. 55, Figure 10 · p. 56
  • Although complexity varies with the endpoints, trajectories remain close to Fubini–Study geodesics; equal target angles make those geodesics extremal for (5.31).Evaluating complexity on these trajectories gives a quantity depending only on θT−¯θT.Source: §C.3 · p. 55

D Holographic interpretation

  • The appendix’s holographic interpretation treats the coset-space metric projection for trivial penalties, while leaving the general-penalty extension to section 6.Source: D Holographic interpretation · p. 56

D.1 Geodesics of massive particles in embedding space

The appendix relates coherent states in CFTd to timelike massive-particle geodesics in AdSd+1, then uses conserved charges and circuit coordinates to describe those trajectories and their symplectic structure.

D.1 Geodesics of massive particles in embedding space

  • Coherent states in CFTd are related to timelike geodesics of massive particles in AdSd+1.The appendix summarizes this relation and its prior study.Source: §D.1 · p. 57
  • The particle moves on the embedding-space hyperboloid X² = −R², with momentum satisfying P² = −m².Its action uses a Lagrange multiplier to impose the hyperboloid constraint along the trajectory.Source: §D.1 · p. 57
  • The SO(d, 2) symmetry yields conserved charges, and complex vectors z and z∗ uniquely describe the reduced phase space of timelike geodesics.The stated identities constrain the conserved charges in terms of these vectors.Source: §D.1 · p. 57
  • A coordinate change maps the complex vectors (z, z∗) to the circuit parameters λ, connecting the particle-geodesic description to unitary circuits.Source: §D.1 · p. 57
  • At t∗ = R, the particle symplectic potential matches the field-theory Maurer–Cartan expectation and the bulk presymplectic potential.The discussion sets this up as a connection to the Fubini–Study metric on the CFT coset space.Source: §D.1 · p. 58

D.2 Novel interpretation

The holographic orthogonality condition is recast as the Q = 0 condition associated with projecting a metric to the coset. The discussion then relates bulk line elements and the bulk symplectic form to the Fubini–Study geometry.

D.2 Novel interpretation

  • The bulk constraint ˙X·δX = 0 is equivalent to Q = 0, the condition associated with pseudo-Riemannian submersion to the coset; for the Fubini–Study line element it reads KD = 0.Source: §D.2 · p. 58
  • The final bridge uses the Kähler relation between a metric and its symplectic form to connect the D.12 line element with the bulk symplectic form.Source: §D.2 · p. 59
  • The derived line element δs²₂ is time-independent, so its relation to the other metric holds at all times.The derivation uses P·δP = X·δX = 0.Source: §D.2 · p. 59
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