nLab AdS-CFT correspondence



Duality in string theory

Quantum field theory



General idea

The AdS-CFT correspondence at its heart is the observation (Witten 98, Section 2.4) that the classical action functionals for various fields coupled to Einstein gravity on anti de Sitter spacetime are, when expressed as functions of the asymptotic boundary-values of the fields, of the form of generating functions for correlators/n-point functions of a conformal field theory on that asymptotic boundary, in a large N limit.

This is traditionally interpreted as a concrete realization of a vague “holographic principle” according to which quantum gravity in bulk spacetimes is controlled, in one way or other, by “boundary field theories” on effective spacetime boundaries, such as event horizons. The original and main motivation for the holographic principle itself was the fact that the apparent black hole entropy in Einstein gravity scales with the area of the event horizon instead of the black hole’s bulk volume (which is not even well-defined), suggesting that gravity encodes or is encoded by some boundary field theory associated with horizons; an idea that, in turn, seems to find a concrete realization in open/closed string duality in the vicinity of, more generally, black branes. The original intuition about holographic black hole entropy has meanwhile found remarkably detailed reflection in (mathematically fairly rigorous) analysis of holographic entanglement entropy, specifically via holographic tensor networks, which turn out to embody key principles of the AdS/CFT correspondence in the guise of quantum information theory, with concrete applications such as to quantum error correcting codes.

The AdS/CFT correspondence itself crucially involves the exceptional isomorphism between the isometry group of anti de Sitter spacetime AdS d+1AdS_{d+1} (the anti de Sitter group) and the conformal group of Minkowski spacetime of dimension dd: the connected component of both is the special orthogonal group SO(d,2)SO(d,2). But the AdS/CFT correspondence is deeper and more subtle than this group theory underlying it, in particular in how it puts fields and states on the gravity side in correspondence with sources and correlators on the field theory side, respectively.

In extrapolation of these elementary computations, the AdS/CFT correspondence conjecturally extends to a more general identification of states of gravity (quantum gravity) on asymptotically anti de Sitter spacetimes of dimension d+1d+1 with correlators/n-point functions of conformal field theories on the asymptotic boundary of dimension dd (Gubser-Klebanov-Polyakov 98 (12), Witten 98, (2.11)), such that perturbation theory on one side of the correspondence relates to non-perturbation on the other side.

While this works to some extent quite generally (see e.g. Natsuume 15 for review), allowing applications such as AdS/CFT in condensed matter physics and AdS/CFT in quantum chromodynamics, the tightest form of the correspondence relates the 1/N expansion of superconformal field theories (super Yang-Mills theories) on the asymptotic boundaries of near-horizon limits of NN coincident black M2-branes/D3-branes/M5-branes to corresponding sectors of the string theory/M-theory quantum gravity in the bulk spacetime away from the brane.

Before the proposal for the actual matching rule of ADS/CFT (Gubser-Klebanov-Polyakov 98 (12), Witten 98, (2.11)) it was by matching of BPS-states in these situations that the existence of an AdS/CFT correspondence was proposed in Maldacena 97a, Maldacena 97b; these articles are now widely regarded as the origin of the idea of the AdS/CFT correspondence.

A quick way to see that the supersymmetric-cases of AdS/CFT for near horizon geometries of M2-branes, D3-branes and M5-branes must be special is to observe that these are the only dimensions in which there are super anti de Sitter spacetime-enhancements of anti de Sitter spacetime, matching the classification of simple superconformal symmetries, see there:

ddNNsuperconformal super Lie algebraR-symmetryblack brane worldvolume
superconformal field theory
via AdS-CFT
A3A\phantom{A}3\phantom{A}A2k+1A\phantom{A}2k+1\phantom{A}AB(k,2)\phantom{A}B(k,2) \simeq osp(2k+1|4)A(2k+1 \vert 4)\phantom{A}ASO(2k+1)A\phantom{A}SO(2k+1)\phantom{A}
A3A\phantom{A}3\phantom{A}A2kA\phantom{A}2k\phantom{A}AD(k,2)\phantom{A}D(k,2)\simeq osp(2k|4)A(2k \vert 4)\phantom{A}ASO(2k)A\phantom{A}SO(2k)\phantom{A}M2-brane
BLG model
ABJM model
A4A\phantom{A}4\phantom{A}Ak+1A\phantom{A}k+1\phantom{A}AA(3,k)𝔰𝔩(4|k+1)A\phantom{A}A(3,k)\simeq \mathfrak{sl}(4 \vert k+1)\phantom{A}AU(k+1)A\phantom{A}U(k+1)\phantom{A}D3-brane
D=4 N=4 SYM
D=4 N=2 SYM
D=4 N=1 SYM
A6A\phantom{A}6\phantom{A}AkA\phantom{A}k\phantom{A}AD(4,k)\phantom{A}D(4,k) \simeq osp(8|2k)A(8 \vert 2k)\phantom{A}ASp(k)A\phantom{A}Sp(k)\phantom{A}M5-brane
D=6 N=(2,0) SCFT
D=6 N=(1,0) SCFT

(Shnider 88, also Nahm 78, see Minwalla 98, section 4.2)

It had already been observed in (Duff-Sutton 88, see Duff 98, Duff 99) that the field theory of small perturbation of a Green-Schwarz sigma-model for a fundamental brane stretched over the asymptotic boundary of the AdS near horizon geometry of its own black brane-incarnation is, after diffeomorphism gauge fixing, a conformal field theory. This was further developed in Claus-Kallosh-Proeyen 97, DGGGTT 98, Claus-Kallosh-Kumar-Townsend 98, Pasti-Sorokin-Tonin 99. See also at super p-brane – As part of the AdS-CFT correspondence

More recently, for the archetypical case of AdS/CFT relating N=4 D=4 super Yang-Mills theory to type IIB string theory on super anti de Sitter spacetime AdS 5×S 5AdS_5 \times S^5, fine detailed checks of the correspondence have been performed (Beisert et al. 10, Escobedo 12), see the section Checks below.

Thus regarded as a duality in string theory, the AdS/CFT correspondence is an incarnation of open/closed string duality, reflecting the fact that the physics on D-branes has two equivalent descriptions:

1) as a Yang-Mills-gauge theory coming from open strings attached to the brane

2) as a gravity theory coming from closed strings emitted/absorbed by the brane.

graphics grabbed from Schomerus 07, Figure 4, see also e.g. Peschanskia 09, Figure 1

This gives a vivid intuitive picture of the mechanism underlying the correspondence: An excitation of the gauge field on the brane goes along with an excitation of the field of gravity around the brane, and either is faithfully reflected in the other; at least in the suitable limits.

Small-NN corrections

The AdS/CFT correspondence has been widely discussed and is mostly understood by default only in the large N N limit and for large 't Hooft coupling, where the given gauge theory is dual to plain classical supergravity, which stands out as being particularly tractable and well-understood.

But it is expected (AGMOO99, p. 60) that the duality still applies in the opposite large 1/N limit, now involving on the gravity-side corrections

  1. from perturbative string theory (for small 't Hooft coupling, there are some checks of such stringy corrections) and

  2. from putative M-theory (for the full non-perturbative large 1/N limit, which remains largely unexplored):


(graphics adapted from SS22)

Notice that for real-world applications such as to the confinement/mass gap-problem of quantum chromodynamics, the value of NN typically is indeed small (the number of colors in quantum chromodynamics is N c=3N_c = 3) so that the string theory/M-theory-corrections to the AdS/QCD correspondence are going to be crucial for the full discussion of these applications:

In lack of a full formulation of M-theory (see M-theory – The open problem) approximate forms of the AdS/CFT correspondence away from the case of conformal invariance, supersymmetry, large N limit and/or exact anti de Sitter geometry are being argued to be of use for understanding quantum chromodynamics (for instance the quark-gluon plasma (Policastro-Son-Starinets 01, but most notably confined hadron-spectra – the AdS/QCD correspondence) and for various models in solid state physics (the AdS-CFT in condensed matter physics, see e.g. Hartnoll-Lucas-Sachdev 16).

More in detail, since the near horizon geometry of BPS black branes is conformal to the Cartesian product of anti de Sitter spaces with the unit nn-sphere around the brane, the cosmology of intersecting D-brane models realizes the observable universe on the asymptotic boundary of an approximately anti de Sitter spacetime (see for instance Kaloper 04, Flachi-Minamitsuji 09). The basic structure is hence that of Randall-Sundrum models, but details differ, such as notably in warped throat geometries, see Uranga 05, section 18.

These warped throat models go back to Klebanov-Strassler 00 which discusses aspects of confinement in Yang-Mills theory on conincident ordinary and fractional D3-branes at the singularity of a warped conifold. See also Klebanov-Witten 98

snippet grabbed from Uranga 05, section 18

here: “RS”=Randall-Sundrum model; “KS”=Klebanov-Strassler 00

In particular this means that AdS-CFT duality applies in some approximation to intersecting D-brane models (e.g. Soda 10, GHMO 16), thus allowing to compute, to some approximation, non-perturbative effects in the Yang-Mills theory on the intersecting branes in terms of gravity on the ambient warped throat \sim AdS (Klebanov-Strassler 00, section 6)

Matching single trace observables to string excitation

The single trace operators/observables in conformal field theories such as super Yang-Mills theories play a special role in the AdS-CFT correspondence: They correspond to single string excitations on the AdS-supergravity side of the correspondence, where, curiously, the “string of characters/letters” in the argument of the trace gets literally mapped to a superstring in spacetime (see the references below).

From Polyakov 02, referring to gauge fields and their single trace operators as letter and words, respectively:

The picture which slowly arises from the above considerations is that of the space-time gradually disappearing in the regions of large curvature. The natural description in this case is provided by a gauge theory in which the basic objects are the texts formed from the gauge-invariant words. The theory provides us with the expectation values assigned to the various texts, words and sentences.

These expectation values can be calculated either from the gauge theory or from the strongly coupled 2d sigma model. The coupling in this model is proportional to the target space curvature. This target space can be interpreted as a usual continuous space-time only when the curvature is small. As we increase the coupling, this interpretation becomes more and more fuzzy and finally completely meaningless.

From Berenstein-Maldacena-Nastase 02, who write ZZ for the elementary field observables (“letters”) Φ\mathbf{\Phi} above:

In summary, the “string of ZZs” becomes the physical string and that each ZZ carries one unit of JJ which is one unit of p +p_+. Locality along the worldsheet of the string comes from the fact that planar diagrams allow only contractions of neighboring operators. So the Yang Mills theory gives a string bit model where each bit is a ZZ operator.

On the CFT side these BMN operators of fixed length (of “letters”) are usefully identified as spin chains which, with the dilatation operator regarded as their Hamiltonian, are integrable systems (Minahan-Zarembo 02, Beisert-Staudacher 03).

This integrability allows a detailed matching between

under AdS/CFT duality (Beisert-Frolov-Staudacher-Tseytlin 03, …). For review see BBGK 04, Beisert et al. 10.


Polyakov gauge/string duality

Key ideas underlying what is now known as holographic duality in string theory and specifically as holographic QCD (see also at holographic light front QCD) were preconceived by Alexander Polyakov (cf. historical reminiscences in Polyakov (2008)) under the name gauge/string duality, in efforts to understand confined QCD (the mass gap problem) by regarding color-flux tubes (Wilson lines) between quarks as dynamical strings.

The logic here proceeds in the following steps (cf. Polyakov (2007), §1 and see the commentary below):

From Veneziano 2012, Fig 2.9
  1. flux tubes confine as dynamical strings

    The starting point is the hypothesis that the strong coupling of particles (such as quarks) by a (non-abelian) gauge field (such as the strong nuclear force) is embodied by the formation of “flux tubes” (“Wilson lines”) between pairs of such particles, which in themselves behave like strings with a given tension.

    [[Kogut & Susskind (1974), (1975); Wilson (1974); Polyakov (1979), (1980), (1987); Makeenko & Migdal (1981), following Nambu (1970), Gotō (1971)]]

    Under this assumption it would be:

    1. the flux tube/string‘s tension which keeps the particles at theirs endpoints confined,

    2. the excitation of these flux tubes/strings which follow Regge trajectories (such as of hadrons);

    3. the scattering of these flux tubes/strings which explains the observed Veneziano amplitudes,

    which are the main qualitative features to be explained.

  2. quantum flux tubes probe effective higher dimensions

    But if so, famous quantum effects make such flux tubes/strings behave like propagating in an effective/emergent higher-dimensional spacetime:

    with only the endpoints of the flux tube/string constrained to lie in the original lower dimensional spacetime

    [[Polyakov (1998), (1999)]]

    which now appears (in modern language that Polyakov did not originally use) as a “brane” inside a higher dimensional bulk spacetime.

    Notice that in this picture the observable physics that we set out to describe takes place on the brane (underlying which is typically flat Minkowski spacetime!) at the asymptotic boundary of a higher-dimensional bulk spacetime, while the (potentially large) extra dimensions of a possibly \sim AdS-bulk remain primarily unobservable. In fact, in Polyakov’s original picture the extra 5th dimension is not so much a spacetime dimension but a parameter for the thickness of the flux tube, which becomes non-vanishing due to quantum effects [[cf. Polyakov (2008), p. 3]].

  3. large/small NN confined gauge theory is holographic string theory/M-theory

    Thus the description of strongly coupled matter via flux tubes/strings now reveals a holographic situation where strongly-coupled quantum fields on intersecting branes are equivalently described by a theory of quantum gravity mediated by strings propagating in a higher dimensional bulk spacetime.

    In relation to gauge string duality this is due to Gubser, Klebanov & Polyakov (1998), which is now understood as part of AdS/CFT duality, but it is actually meant to be more general, cf. Polyakov & Rychkov (2000).

While this dual bulk string theory is itself strongly-coupled unless the “number of coincident branes” is humongous (the “large-N limit”) and thus unrealistic after all, the difference is that recognizing the branes as physical objects reveals a web of concrete hints as to the string’s strongly-coupled (non-perturbative) completion, going under the working title M-theory, cf. at AdS-CFT – Small NN corrections.

In summary, the plausible approach of understanding strongly-coupled quantum gauge theories by regarding their flux tubes as dynamical strings seems to recast the Millennium Problem of understanding strongly-coupled matter into the problem of formulating M-theory: Given M-theory, it ought to be possible to find intersecting brane models of single (or a small number of coincident) M-branes (such as the Witten-Sakai-Sugimoto model M5-brane system) on whose worldvolume the desired strongly-coupled field theory is realized (such as QCD).

Notice the decisive early insight of Alexander Polyakov here: While the idea that strings somehow describe hadronic bound states was the very origin of string theory in the early 1970s (“dual resonance models”), the mainstream abandoned this perspective in the later 1970s when the critical dimension and the full spectrum of the string became fully understood (cf. Goddard-Thorn no-ghost theorem) and declared that instead string should be understood as a grand unified theory of everything including quantum gravity (see e.g. the historical review of Veneziano (2012), esp. pp. 30-31 which still clings to this perspective). From here it was only through the long detour of first discovering, inside this grander theory: D-branes (and M5-branes) in the 1990s, then their near-horizon AdS-CFT duality just before the 2000s and then another decade of exploring intersecting D-brane models that the community in the 2010s came back full circle to Polyakov’s holographic perspective on QCD, now dubbed holographic QCD, in which strings are flux tubes that propagate not (alone) in the observable 4 spacetime dimensions but in a primarily unobservable (meanwhile known as Randall-Sundrum-like) higher-dimensional bulk spacetime – a holographic description of reality that Polyakov (1999) referred to as the wall of the cave, in allusion to Plato (cf. also Polyakov (2008), p. 6).

Our whereabouts in this remarkable picture are still often misunderstood today: If string theory is a theory of nature, then, it seems, we see the wall but not the cave: we live on a \simMinkowskian brane intersection at the (asympotic) boundary of a primarily unobserved \simanti de Sitter bulk – which may better be thought of not as physical space but as a configuration space of quantum flux.


At the heart of the duality is the observation that the classical action functionals for various fields coupled to Einstein gravity on anti de Sitter spacetime are, when expressed as functions of the asymptotic boundary values of the fields, equal to the generating functions for the correlators/n-point functions of a conformal field theory on that asymptotic boundary.

These computations were laid out in Witten 98, section 2.4 “Some sample computation”. These follow from elementary manipulation in differential geometry (involving neither supersymmetry nor string theory). A good exposition is in Hartnoll-Lucas-Sachdev 16, Section 1.6

For the more ambitious matching of the spectrum of the dilatation operator of N=4 D=4 super Yang-Mills theory to the corresponding spectrum of the Green-Schwarz superstring on the super anti de Sitter spacetime AdS 5×S 5AdS_5 \times S^5 detailed checks are summarized in Beisert et al. 10, Escobedo 12

graphics grabbed from Escobedo 12

Comparison to string scattering amplitudes beyond the planar SCFT limit: ABP 18.

Numerical checks using lattice gauge theory are reviewed in Joseph 15.

Exact duality checks pertaining to the full stringy regime for AdS 3 / CFT 2 AdS_3/CFT_2 : Eberhardt-Gaberdiel 19a, Eberhardt-Gaberdiel 19b, Eberhardt-Gaberdiel-Gopakumar 19. For more see the references there.


The solutions to supergravity that preserve the maximum of 32 supersymmetries are (e.g. HEGKS 08 (1.1))

as well as their Minkowski spacetime and plane wave limits. These are the main KK-compactifications for the following examples-

AdS 5/CFT 4AdS_5 / CFT_4 – Horizon limit of D3-branes

type II string theory on 5d anti de Sitter spacetime (times a 5-sphere) is dual to N=4 D=4 super Yang-Mills theory on the worldvolume of a D3-brane at the asymptotic boundary

(Maldacena 97, section 2)

(Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 3 and 4)

AdS 7/CFT 6AdS_7 / CFT_6 – Horizon limit of M5-branes

We list some of the conjectured statements and their evidence concerning the case of AdS 7/CFT 6AdS_7/CFT_6-duality.

The hypothesis (Maldacena 97, section 3.1) (see (Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.1) for a review) is that

is holographically related to


effectively this relation was already used to computed the 5-brane partition function in the abelian case from the states of abelian 7d Chern-Simons theory. (The quadratic refinement of the supergravity C-field necessary to make this come out right is what led to Hopkins-Singer 02 and hence to the further mathematical development of differential cohomology and its application in physics.)

In (Witten 98, section 4) this construction is argued for from within the framework of AdS/CFT, explicitly identifying the 7d Chern-Simons theory here with the compactification of the 11-dimensional Chern-Simons term of the supergravity C-field in 11-dimensional supergravity, which locally is

S 11dSUGRA,CS(C 3) = AdS 7 S 4C 3G 4G 4 =N AdS 7C 3G 4. \begin{aligned} S_{11d SUGRA, CS}(C_3) &= \int_{AdS_7} \int_{S^4} C_3 \wedge G_4 \wedge G_4 \\ & = N \, \int_{AdS_7} C_3 \wedge G_4 \end{aligned} \,.

But in fact the quantum anomaly cancellation (GS-type mechanism) for 11d sugra introduces a quantum correction to this Chern-Simons term (DLM, equation (3.14)), making it locally become

S(ω,C 3) = AdS 7 S 4C 3G 4(G 4+I 8(ω)) =N AdS 7(C 3G 4+148CS p 2(ω)112CS 12p 1(ω)tr(F ωω)), \begin{aligned} S(\omega,C_3) &= \int_{AdS_7} \int_{S^4} C_3 \wedge G_4 \wedge (G_4 + I_8(\omega)) \\ & = N \, \int_{AdS_7} \left( C_3 \wedge G_4 + \frac{1}{48} CS_{p_2}(\omega) - \frac{1}{12} CS_{\frac{1}{2}p_1}(\omega) \wedge tr(F_\omega \wedge \omega) \right) \end{aligned} \,,

where now ω\omega is the local 1-form representative of a spin connection and where CS p 2CS_{p_2} is a Chern-Simons form for the second Pontryagin class and CS 12p 1CS_{\frac{1}{2}p_1} for the first.

That therefore not an abelian, but this nonabelian higher dimensional Chern-Simons theory should be dual to the nonabelian 6d (2,0)-superconformal QFT was maybe first said explicitly in (LuWang 2010).

Its gauge field is hence locally and ignoring the flux quantization subtleties a pair consisting of the abelian 3-form field CC and a Spin group Spin(6,1)Spin(6,1)-valued connection (see supergravity C-field for global descriptions of such pairs). Or maybe rather Spin(6,2)Spin(6,2) to account for the constraint that the configurations are to be asymptotic anti de Sitter spacetimes (in analogy to the well-understood situation in 3d quantum gravity, see there for more details).

Indeed, in (SezginSundell 2002, section 7) more detailed arguments are given that the 7-dimensional dual to the free 6d theory is a higher spin gauge theory for a higher spin gauge group extending the (super) conformal group SO(6,2)SO(6,2).

A non-perturbative description of this nonabelian 7d Chern-Simons theory as a local prequantum field theory (hence defined non-perturbatively on the global moduli stack of fields (twisted differential string structures, in fact)) was discussed in (FSS 12a, FSS 12b).

General discussion of boundary local prequantum field theories relating higher Chern-Simons-type and higher WZW-type theories is in (dcct 13, section 3.9.14). Specifically, a characterization along these lines of the Green-Schwarz action functional of the M5-brane as a holographic higher WZW-type boundary theory of a 7d Chern-Simons theory is found in (FSS 13).

Analogous discussion of the 6d theory as a higher WZW analog of a 7d Chern-Simons theory phrased in terms of extended quantum field theory is (Freed 12).

AdS 4/CFT 3AdS_4 / CFT_3 –Horizon limit of M2-branes

11d supergravity/M-theory on the asymptotic AdS 4AdS_4

spacetime of an M2-brane.

(Maldacena 97, section 3.2, Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.2, Klebanov-Torri 10)

AdS 3/CFT 2AdS_3 / CFT_2 – Horizon limit of Dpp-D(p+kp+k) brane bound states

(for more see at AdS3-CFT2 and CS-WZW correspondence)

D1-D5 brane system in type IIB string theory

(Maldacena 97, section 4)

(Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 5)

D6-D8 brane bound state with D2-D4 brane bound state defects in massive type IIA string theory

(Dibitetto-Petri 17, …)

D4-D8 brane bound state with D2-D6 brane bound state defects in massive type IIA string theory

(Dibitetto-Petri 18, …)

AdS 2/CFT 1AdS_2 / CFT_1

see at nearly AdS2/CFT1

Non-conformal duals

Horizon limit of DpDp-branes for arbitrary pp

(Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.3)

Horizon limit of NS5-brane

(Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.4)

QCD models

While all of the above horizon limits product super Yang-Mills theory, one can consider certain limits of these in which they look like plain QCD, at least in certain sectors. This leads to a discussion of holographic description of QCD properties that are actually experimentally observed.

(Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.2)

See the References – Applications – In condensed matter physics.

Further gauge theories induced by compactification and twisting

gauge theory induced via AdS-CFT correspondence

M-theory perspective via AdS7-CFT6F-theory perspective
11d supergravity/M-theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 4S^4compactificationon elliptic fibration followed by T-duality
7-dimensional supergravity
\;\;\;\;\downarrow topological sector
7-dimensional Chern-Simons theory
\;\;\;\;\downarrow AdS7-CFT6 holographic duality
6d (2,0)-superconformal QFT on the M5-brane with conformal invarianceM5-brane worldvolume theory
\;\;\;\; \downarrow KK-compactification on Riemann surfacedouble dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondenceD3-brane worldvolume theory with type IIB S-duality
\;\;\;\;\; \downarrow topological twist
topologically twisted N=2 D=4 super Yang-Mills theory
\;\;\;\; \downarrow KK-compactification on Riemann surface
A-model on Bun GBun_G, Donaldson theory


gauge theory induced via AdS5-CFT4
type II string theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 5S^5
\;\;\;\; \downarrow topological sector
5-dimensional Chern-Simons theory
\;\;\;\;\downarrow AdS5-CFT4 holographic duality
N=4 D=4 super Yang-Mills theory
\;\;\;\;\; \downarrow topological twist
topologically twisted N=4 D=4 super Yang-Mills theory
\;\;\;\; \downarrow KK-compactification on Riemann surface
A-model on Bun GBun_G and B-model on Loc GLoc_G, geometric Langlands correspondence


The full formalization of AdS/CFT is still very much out of reach, but maybe mostly for lack of trying.

But see Anderson 04.

One proposal for a formalization of a toy version in the context of AQFT is Rehren duality. However, it does not seem that this actually formalizes AdS-CFT, but something else.

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
(D=2n)(D = 2n)type IIA\,\,
D0-brane\,\,BFSS matrix model
D4-brane\,\,D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane\,\,D=7 super Yang-Mills theory
(D=2n+1)(D = 2n+1)type IIB\,\,
D1-brane\,\,2d CFT with BH entropy
D3-brane\,\,N=4 D=4 super Yang-Mills theory
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection\,
string\,B2-field2d SCFT
NS5-brane\,B6-fieldlittle string theory
D-brane for topological string\,
M-brane11D SuGra/M-theorycircle n-connection\,
M2-brane\,C3-fieldABJM theory, BLG model
M5-brane\,C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane\,C6-field on G2-manifold
membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d


Polyakov gauge/string duality

Key ideas underlying what is now known as holographic duality in string theory and specifically as holographic QCD (see notably also at holographic light front QCD) were preconceived by Alexander Polyakov (cf. historical remarks in Polyakov (2008)) under the name gauge/string duality (cf. historical review in Polyakov (2008)), in efforts to understand confined QCD (the mass gap problem) by regarding color-flux tubes (Wilson lines) between quarks as dynamical strings:

Early suggestion that confined QCD is described by regarding the color-flux tubes as string-like dynamical degrees of freedoms:

  • John Kogut, Leonard Susskind, Vacuum polarization and the absence of free quarks in four dimensions, Phys. Rev. D 9 (1974) 3501-3512 [[doi:10.1103/PhysRevD.9.3501]]

  • Kenneth G. Wilson, Confinement of quarks, Phys. Rev. D 10 (1974) 2445 [[doi:10.1103/PhysRevD.10.2445]]

    (argument in lattice gauge theory)

  • John Kogut, Leonard Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D 11 (1975) 395 [[doi:10.1103/PhysRevD.11.395]]

    “The gauge-invariant configuration space consists of a collection of strings with quarks at their ends. The strings are lines of non-Abelian electric flux. In the strong coupling limit the dynamics is best described in terms of these strings. Quark confinement is a result of the inability to break a string without producing a pair. [[]]

    “The confining mechanism is the appearance of one dimensional electric flux tubes which must link separated quarks. The appropriate description of the strongly coupled limit consists of a theory of interacting, propagating strings. [[]]

    “This picture of the strongly coupled Yang-Mills theory in terms of a collection of stringlike flux lines is the central result of our analysis. It should be compared with the phenomenological use of stringlike degrees of freedom which has been widely used in describing hadrons.”

  • Alexander Polyakov, String representations and hidden symmetries for gauge fields, Physics Letters B 82 2 (1979) 247-250 [[doi:10.1016/0370-2693(79)90747-0]]

  • Alexander Polyakov, Gauge fields as rings of glue, Nuclear Physics B 164 (1980) 171-188 [[doi:10.1016/0550-3213(80)90507-6]]

    “The basic idea is that gauge fields can be considered as chiral fields, defined on the space of all possible contours (the loop space). The origin of the idea lies in the expectation that, in the confining phase of a gauge theory, closed strings should play the role of elementary excitations.”

  • Yuri Makeenko, Alexander A. Migdal, Quantum chromodynamics as dynamics of loops, Nuclear Physics B 188 2 (1981) 269-316 [[doi:10.1016/0550-3213(81)90258-3]]

    “So the world sheet of string should be interpreted as the color magnetic dipole sheet. The string itself should be interpreted as the electric flux tube in the monopole plasma.”

  • Alexander Polyakov, Gauge Fields and Strings, Routledge, Taylor and Francis (1987, 2021) [[doi:10.1201/9780203755082, oapen:20.500.12657/50871]]

[[old personal page]]: “My main interests this year [[1993?]] were directed towards string theory of quark confinement. The problem is to find the string Lagrangian for the Faraday’s ”lines of force“,which would reproduce perturbative corrections from the Yang-Mills theory to the Coulomb law at small distances and would give permanent confinement of quarks at large distances.”

Early suggestion, due to the Liouville field seen in the quantization of the bosonic string via the Polyakov action,

that such flux tubes regarded as confining strings are to be thought of a probing higher dimensional spacetime, exhibiting a holographic principle in which actual spacetime appears as a brane:

eventually culminating in the formulation of the dictionary for the AdS-CFT correspondence:

“Relations between gauge fields and strings present an old, fascinating and unanswered question. The full answer to this question is of great importance for theoretical physics. It will provide us with a theory of quark confinement by explaining the dynamics of color-electric fluxes.”

and the suggestion of finding the string-QCD correspondence:

“in the strong coupling limit of a lattice gauge theory the elementary excitations are represented by closed strings formed by the color-electric fluxes. In the presence of quarks these strings open up and end on the quarks, thus guaranteeing quark confinement. Moreover, in the SU(N)SU(N) gauge theory the strings interaction is weak at large NN. This fact makes it reasonable to expect that also in the physically interesting continuous limit (not accessible by the strong coupling approximation) the best description of the theory should involve the flux lines (strings) and not fields, thus returning us from Maxwell to Faraday. In other words it is natural to expect an exact duality between gauge fields and strings. The challenge is to build a precise theory on the string side of this duality.”

Historical reminiscences:

“Already in 1974, in his famous large NN paper, ‘t Hooft already tried to find the string-gauge connections. His idea was that the lines of Feynman’s diagrams become dense in a certain sense and could be described as a 2d surface. This is, however, very different from the picture of strings as flux lines. Interestingly, even now people often don’t distinguish between these approaches. In fact, for the usual amplitudes Feynman’s diagrams don’t become dense and the flux lines picture is an appropriate one. However there are cases in which t’Hooft’s mechanism is really working.”

  • Alexander M. Polyakov, §1 in: Beyond Space-Time, in The Quantum Structure of Space and Time, Proceedings of the 23rd Solvay Conference on Physics, World Scientific (2007) [[arXiv:hep-th/0602011, pdf]]

  • Alexander M. Polyakov, From Quarks to Strings [[arXiv:0812.0183]]

    published as Quarks, strings and beyond, section 44 in: Paolo Di Vecchia et al. (ed.), The Birth of String Theory, Cambridge University Press (2012) 544-551 [[doi:10.1017/CBO9780511977725.048]]

    “By the end of ’77 it was clear to me that I needed a new strategy [[for understanding confinement]] and I became convinced that the way to go was the gauge/string duality. [[]]

    “Classically the string is infinitely thin and has only transverse oscillations. But when I quantized it there was a surprise – an extra, longitudinal mode, which appears due to the quantum ”thickening“ of the string. This new field is called the Liouville mode. [[]]

    “I kept thinking about gauge/strings dualities. Soon after the Liouville mode was discovered it became clear to many people including myself that its natural interpretation is that random surfaces in 4d are described by the strings flying in 5d with the Liouville field playing the role of the fifth dimension. The precise meaning of this statement is that the wave function of the general string state depends on the four center of mass coordinates and also on the fifth, the Liouville one. In the case of minimal models this extra dimension is related to the matrix eigenvalues and the resulting space is flat.”

    “Since this 5d space must contain the flat 4d subspace in which the gauge theory resides, the natural ansatz for the metric is just the Friedman universe with a certain warp factor. This factor must be determined from the conditions of conformal symmetry on the world sheet. Its dependence on the Liouville mode must be related to the renormalization group flow. As a result we arrive at a fascinating picture – our 4d world is a projection of a more fundamental 5d string theory. [[]]

    “At this point I was certain that I have found the right language for the gauge/string duality. I attended various conferences, telling people that it is possible to describe gauge theories by solving Einstein-like equations (coming from the conformal symmetry on the world sheet) in five dimensions. The impact of my talks was close to zero. That was not unusual and didn’t bother me much. What really caused me to delay the publication (Polyakov 1998) for a couple of years was my inability to derive the asymptotic freedom from my equations. At this point I should have noticed the paper of Klebanov 1997 in which he related D3 branes described by the supersymmetric Yang Mills theory to the same object described by supergravity. Unfortunately I wrongly thought that the paper is related to matrix theory and I was skeptical about this subject. As a result I have missed this paper which would provide me with a nice special case of my program. This special case was presented little later in full generality by Juan Maldacena (Maldacena 1997) and his work opened the flood gates.”

Original articles on AdS/CFT duality

The rough conjecture originates in:

The actual rule for matching bulk states to generating functions for boundary correlators/n-point functions is due to

See also:

Discussion of how Green-Schwarz action functionals for super pp-branes in AdS target spaces induce, after diffeomorphism gauge fixing, superconformal field theory on the worldvolumes (see singleton representation) goes back to

  • Mike Duff, C. Sutton, The Membrane at the End of the Universe, New Sci. 118 (1988) 67-71 (spire:268230)

and was further developed in

Review is in

The resulting super-conformal brane scan:

See also at super p-brane – As part of the AdS-CFT correspondence.

Sketch of a derivation of AdS/CFT:

Further references include:

Via exceptional field theory:

  • Oscar Varela, Super-Chern-Simons spectra from Exceptional Field Theory (arXiv:2010.09743)

Introductions and surveys

Surveys and introductions:

See also

Review of Yangian symmetry:

Lattice gauge theory computations

Review of lattice gauge theory-numerics for the AdS-CFT correspondence:

  • Anosh Joseph, Review of Lattice Supersymmetry and Gauge-Gravity Duality (arXiv:1509.01440)

Using the KK-compactification of D=4 N=4 super Yang-Mills theory to the BMN matrix model for lattice gauge theory-computations in D=4 N=4 SYM and for numerical checks of the AdS-CFT correspondence:

  • Masazumi Honda, Goro Ishiki, Sang-Woo Kim, Jun Nishimura, Asato Tsuchiya, Direct test of the AdS/CFT correspondence by Monte Carlo studies of N=4 super Yang-Mills theory, JHEP 1311 (2013) 200 (arXiv:1308.3525)

On single trace operators

The correspondence of single trace operators to superstring excitations under the AdS-CFT correspondence originates with these articles:

The identification of the relevant single trace operators with integrable spin chains is due to

which led to more detailed matching of single trace operators to rotating string excitations in

Review includes

SYK-model in AdS 2/CFT 1AdS_2/CFT_1

Discussion of the SYK-model as the AdS/CFT dual of JT-gravity in nearly AdS2/CFT1 and AdS-CFT in condensed matter physics:

Original articles:


Relation to black holes in terms of Majorana dimer states:

Relation to black holes in string theory and random matrix theory:

On non-perturbative effects and resurgence:

See also

Discussion of small N corrections via a lattice QFT-Ansatz on the AdS side:

  • Richard C. Brower, Cameron V. Cogburn, A. Liam Fitzpatrick, Dean Howarth, Chung-I Tan, Lattice Setup for Quantum Field Theory in AdS 2AdS_2 (arXiv:1912.07606)

See also:

  • Gregory J. Galloway, Melanie Graf, Eric Ling, A conformal infinity approach to asymptotically AdS 2×S n1AdS_2 \times S^{n-1} spacetimes (arXiv:2003.00093)

Random matrix theory in AdS 2/CFT 1AdS_2/CFT_1

On Jackiw-Teitelboim gravity dual to random matrix theory (via AdS2/CFT1 and topological recursion):

BFSS matrix model in AdS 2/CFT 1AdS_2/CFT_1

On AdS2/CFT1 with the BFSS matrix model on the CFT side and black hole-like solutions in type IIA supergravity on the AdS side:

and on its analog of holographic entanglement entropy:

See also

  • Takeshi Morita, Hiroki Yoshida, A Critical Dimension in One-dimensional Large-N Reduced Models (arXiv:2001.02109)

Flat space limit

The SYK model in flat space holography:

D1-D3 brane intersections in AdS 2/CFT 1AdS_2/CFT_1

On D1-D3 brane intersections in AdS2/CFT1:

Via T-duality from D6-D8 brane intersections:

Chord diagrams and weight systems in Physics

The following is a list of references that involve (weight systems on) chord diagrams/Jacobi diagrams in physics:

  1. In Chern-Simons theory

  2. In Dp-D(p+2) brane intersections

  3. In quantum many body models for for holographic brane/bulk correspondence:

    1. In AdS2/CFT1, JT-gravity/SYK-model

    2. As dimer/bit thread codes for holographic entanglement entropy

For a unifying perspective (via Hypothesis H) and further pointers, see:


In Chern-Simons theory

Since weight systems are the associated graded of Vassiliev invariants, and since Vassiliev invariants are knot invariants arising as certain correlators/Feynman amplitudes of Chern-Simons theory in the presence of Wilson lines, there is a close relation between weight systems and quantum Chern-Simons theory.

Historically this is the original application of chord diagrams/Jacobi diagrams and their weight systems, see also at graph complex and Kontsevich integral.

Reviewed in:

Applied to Gopakumar-Vafa duality:

  • Dave Auckly, Sergiy Koshkin, Introduction to the Gopakumar-Vafa Large NN Duality, Geom. Topol. Monogr. 8 (2006) 195-456 (arXiv:0701568)

See also

For single trace operators in AdS/CFT duality

Interpretation of Lie algebra weight systems on chord diagrams as certain single trace operators, in particular in application to black hole thermodynamics

In AdS 2/CFT 1AdS_2/CFT_1, JT-gravity/SYK-model

Discussion of (Lie algebra-)weight systems on chord diagrams as SYK model single trace operators:

  • Antonio M. García-García, Yiyang Jia, Jacobus J. M. Verbaarschot, Exact moments of the Sachdev-Ye-Kitaev model up to order 1/N 21/N^2, JHEP 04 (2018) 146 (arXiv:1801.02696)

  • Yiyang Jia, Jacobus J. M. Verbaarschot, Section 4 of: Large NN expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs, JHEP 11 (2018) 031 (arXiv:1806.03271)

  • Micha Berkooz, Prithvi Narayan, Joan Simón, Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction, JHEP 08 (2018) 192 (arxiv:1806.04380)


  • László Erdős, Dominik Schröder, Phase Transition in the Density of States of Quantum Spin Glasses, D. Math Phys Anal Geom (2014) 17: 9164 (arXiv:1407.1552)

which in turn follows

  • Philippe Flajolet, Marc Noy, Analytic Combinatorics of Chord Diagrams, pages 191–201 in Daniel Krob, Alexander A. Mikhalev,and Alexander V. Mikhalev, (eds.), Formal Power Series and Algebraic Combinatorics, Springer 2000 (doi:10.1007/978-3-662-04166-6_17)

With emphasis on the holographic content:

  • Micha Berkooz, Mikhail Isachenkov, Vladimir Narovlansky, Genis Torrents, Section 5 of: Towards a full solution of the large NN double-scaled SYK model, JHEP 03 (2019) 079 (arxiv:1811.02584)

  • Vladimir Narovlansky, Slide 23 (of 28) of: Towards a Solution of Large NN Double-Scaled SYK, 2019 (pdf)

  • Micha Berkooz, Mikhail Isachenkov, Prithvi Narayan, Vladimir Narovlansky, Quantum groups, non-commutative AdS 2AdS_2, and chords in the double-scaled SYK model [arXiv:2212.13668]

and specifically in relation, under AdS2/CFT1, to Jackiw-Teitelboim gravity:

In Dpp/D(p+2)(p+2)-brane intersections

Discussion of weight systems on chord diagrams as single trace observables for the non-abelian DBI action on the fuzzy funnel/fuzzy sphere non-commutative geometry of Dp-D(p+2)-brane intersections (hence Yang-Mills monopoles):

As codes for holographic entanglement entropy

From Yan 20

Chord diagrams encoding Majorana dimer codes and other quantum error correcting codes via tensor networks exhibiting holographic entanglement entropy:

For Dyson-Schwinger equations

Discussion of round chord diagrams organizing Dyson-Schwinger equations:

  • Nicolas Marie, Karen Yeats, A chord diagram expansion coming from some Dyson-Schwinger equations, Communications in Number Theory and Physics, 7(2):251291, 2013 (arXiv:1210.5457)

  • Markus Hihn, Karen Yeats, Generalized chord diagram expansions of Dyson-Schwinger equations, Ann. Inst. Henri Poincar Comb. Phys. Interact. 6 no 4:573-605 (arXiv:1602.02550)

Review in:

  • Ali Assem Mahmoud, Section 3 of: On the Enumerative Structures in Quantum Field Theory (arXiv:2008.11661)

On AdS 3/CFT 2AdS_3 / CFT_2

(For more see the references at AdS3/CFT2.)

An exact correspondence of the symmetric orbifold CFT of Liouville theory with a string theory on AdS 3AdS_3 is claimed in:

based on

See also

On black\;D6-D8-brane bound states in massive type IIA string theory, with defect D2-D4-brane bound states inside them realizing AdS3-CFT2 “inside” AdS7-CFT6:

On black\;D4-D8-brane bound states in massive type IIA string theory, with defect D2-D6-brane bound states inside them realizing AdS3-CFT2 “inside” AdS7-CFT6:

On AdS 4/CFT 3AdS_4 / CFT_3

On AdS 5/CFT 4AdS_5 / CFT_4

Computing dual string scattering amplitudes by AdS/CFT beyond the planar limit:

On AdS 7/CFT 6AdS_7 / CFT_6

We list references specific to AdS 7/CFT 6AdS_7/CFT_6.


it is argued that the conformal blocks of the 6d (2,0)-superconformal QFT are entirely controled just by the effective 7d Chern-Simons theory inside 11-dimensional supergravity, but only the abelian piece is discussed explicitly.

The fact that this Chern-Simons term is in fact a nonabelian higher dimensional Chern-Simons theory in d=7d = 7, due the quantum anomaly cancellation, is clear from the original source, equation (3.14) of

but seems not to be noted explicitly in the context of AdS 7/CFT 6AdS_7/CFT_6 before the references

  • H. Lü, Yi Pang, Seven-Dimensional Gravity with Topological Terms Phys.Rev.D81:085016 (2010) (arXiv:1001.0042)

  • H. Lu, Zhao-Long Wang, On M-Theory Embedding of Topologically Massive Gravity Int.J.Mod.Phys.D19:1197 (2010) (arXiv:1001.2349)

More on the relation between the M5-brane and supergravity on AdS 7×S 4AdS_7 \times S^4 and arguments for the SO(5)SO(5) R-symmetry group on the 6d theory from the 7d theory are given in

See also

  • M. Nishimura, Y. Tanii, Local Symmetries in the AdS 7/CFT 6AdS_7/CFT_6 Correspondence, Mod. Phys. Lett. A14 (1999) 2709-2720 (arXiv:hep-th/9910192)

Discussion of the CFT 6CFT_6 in AdS 7/CFT 6AdS_7/CFT_6 via conformal bootstrap:


arguments are given that the 7d theory is a higher spin gauge theory extension of SO(6,2)SO(6,2).

Generalization beyond exact AdS / exact CFT

Discussion for cosmology of intersecting D-brane models (ambient \sim anti de Sitter spacetimes with the \sim conformal intersecting branes at the asymptotic boundary) includes the following (see also at Randall-Sundrum model):

  • Igor Klebanov, Matthew Strassler, Supergravity and a Confining Gauge Theory: Duality Cascades and χ SB\chi^{SB}-Resolution of Naked Singularities, JHEP 0008:052, 2000 (arXiv:hep-th/0007191)

  • Igor Klebanov, Edward Witten, Superconformal Field Theory on Threebranes at a Calabi-Yau Singularity, Nucl.Phys.B536:199-218, 1998 (arXiv:hep-th/9807080)

  • Nemanja Kaloper, Origami World, JHEP 0405 (2004) 061 (arXiv:hep-th/0403208)

  • Angel Uranga, section 18 of TASI lectures on String Compactification, Model Building, and Fluxes, 2005 (pdf)

  • Antonino Flachi, Masato Minamitsuji, Field localization on a brane intersection in anti-de Sitter spacetime, Phys.Rev.D79:104021, 2009 (arXiv:0903.0133)

  • Jiro Soda, AdS/CFT on the brane, Lect.Notes Phys.828:235-270, 2011 (arXiv:1001.1011)

  • Shunsuke Teraguchi, around slide 21 String theory and its relation to particle physics, 2007 (pdf)

  • Gianluca Grignani, Troels Harmark, Andrea Marini, Marta Orselli, The Born-Infeld/Gravity Correspondence, Phys. Rev. D 94, 066009 (2016) (arXiv:1602.01640)

pp-Waves as Penrose limits of AdS p×S qAdS_p \times S^q spacetimes

Discussion of pp-wave spacetimes as Penrose limits (Inönü-Wigner contractions) of AdSp x S^q spacetimes and of the corresponding limit of AdS-CFT duality:


See also:

  • Michael Gutperle, Nicholas Klein, A Penrose limit for type IIB AdS 6AdS_6 solutions (arXiv:2105.10824)

Applications to physics

To gravity

Discussion of event horizons of black holes in terms of AdS/CFT (the “firewall problem”) is in

  • Kyriakos Papadodimas, Suvrat Raju, An Infalling Observer in AdS/CFT (arXiv:1211.6767)

To black hole interiors:

The SYK model gives us a glimpse into the interior of an extremal black hole…That’s the feature of SYK that I find most interesting…It is a feature this model has, that I think no other model has

To symmetries in gravity:

To the quark-gluon plasma

Applications of AdS-CFT to the quark-gluon plasma of QCD:

Expositions and reviews include

Holographic discussion of the shear viscosity of the quark-gluon plasema goes back to

Other original articles include:

  • Hovhannes R. Grigoryan, Paul M. Hohler, Mikhail A. Stephanov, Towards the Gravity Dual of Quarkonium in the Strongly Coupled QCD Plasma (arXiv:1003.1138)

  • Brett McInnes, Holography of the Quark Matter Triple Point (arXiv:0910.4456)

To particle physics

For more see at AdS/QCD correspondence.

To fluid dynamics

Application to fluid dynamics – see also at fluid/gravity correspondence:

To condensed matter physics

On AdS-CFT in condensed matter physics:

Textbook account

Further reviews include the following:

Applications in mathematics

To the volume conjecture

Suggestion that the statement of the volume conjecture is really AdS-CFT duality combined with the 3d-3d correspondence for M5-branes wrapped on hyperbolic 3-manifolds:

  • Dongmin Gang, Nakwoo Kim, Sangmin Lee, Section 3.2_Holography of 3d-3d correspondence at Large NN, JHEP04(2015) 091 (arXiv:1409.6206)

  • Dongmin Gang, Nakwoo Kim, around (21) of: Large NN twisted partition functions in 3d-3d correspondence and Holography, Phys. Rev. D 99, 021901 (2019) (arXiv:1808.02797)

To deep learning in neural networks

On the deep learning algorithm on neural networks as analogous to the AdS/CFT correspondence:

  • Yi-Zhuang You, Zhao Yang, Xiao-Liang Qi, Machine Learning Spatial Geometry from Entanglement Features, Phys. Rev. B 97, 045153 (2018) (arxiv:1709.01223)

  • W. C. Gan and F. W. Shu, Holography as deep learning, Int. J. Mod. Phys. D 26, no. 12, 1743020 (2017) (arXiv:1705.05750)

  • J. W. Lee, Quantum fields as deep learning (arXiv:1708.07408)

  • Koji Hashimoto, Sotaro Sugishita, Akinori Tanaka, Akio Tomiya, Deep Learning and AdS/CFT, Phys. Rev. D 98, 046019 (2018) (arxiv:1802.08313)

Philosophy of AdS/CFT

  • Radin Dardashti, Richard Dawid, Sean Gryb, Karim Thébault (2020). On the Empirical Consequences of the AdS/CFT Duality. In Nick Huggett, Keizo Matsubara, and Christian Wüthrich (Eds.), Beyond Spacetime: The Foundations of Quantum Gravity (pp. 284-303). Cambridge: Cambridge University Press. (doi:10.1017/9781108655705.016, arXiv:1810.00949)

Last revised on September 19, 2023 at 10:51:53. See the history of this page for a list of all contributions to it.