Minimal Model Holography

• Minimal Model Holography is the AdS3/CFT2 correspondence that relates large‑N minimal-model CFTs to higher‑spin theories using solvable coset constructions.
• It employs exact realizations based on SU(N), SO(2N), and Kazama–Suzuki models to yield non-linear quantum W‑algebras and asymptotic symmetry matching.
• Partition function identities and detailed spectral matching provide evidence for both perturbative and non‑perturbative sectors across bosonic, supersymmetric, and even‑spin variants.

Searching arXiv for core papers on minimal model holography and related variants. arxiv_search(query="minimal model holography AdS3 W_N Kazama-Suzuki hs[lambda] review", max_results=10, sort_by="relevance") Minimal model holography is the AdS$_3$/CFT$_2$ correspondence relating large-$N$ ’t Hooft-like limits of two-dimensional minimal-model conformal field theories to three-dimensional higher-spin theories on AdS$_3$. In its canonical bosonic form, the boundary theory is the coset

$\frac{SU(N)_k \times SU(N)_1}{SU(N)_{k+1}}$

with central charge

$$c_{N,k}=(N-1)\Bigl[1-\frac{N(N+1)}{(N+k)(N+k+1)}\Bigr]\le N-1,$$

and the ’t Hooft coupling

$\lambda\equiv \frac{N}{N+k}\in[0,1].$

In the large-$N$ ’t Hooft limit, $N,k\to\infty$ with $\lambda$ fixed, one finds $_2$0, so the theory has $_2$1 degrees of freedom (Gaberdiel et al., 2012). This framework admits several closely related realizations: the original $_2$2 series, orthogonal even-spin variants, $_2$3 Kazama–Suzuki models, and special supersymmetric truncations. Across these cases, the common structure is that a solvable coset CFT is matched to a Chern–Simons higher-spin theory whose asymptotic symmetry algebra is a non-linear $_2$4-type algebra, with evidence supplied by symmetry matching, spectrum matching, and partition-function identities (Gaberdiel et al., 2012, Gaberdiel et al., 2011, Candu et al., 2012, Gaberdiel et al., 2014).

1. Bosonic $_2$5 minimal models and the higher-spin dual

The standard realization of minimal model holography uses the diagonal coset

$_2$6

whose central charge is

$_2$7

The large-$_2$8 ’t Hooft limit is defined by $_2$9 with $N$0 fixed, and in this limit $N$1 (Gaberdiel et al., 2012).

On the CFT side, each minimal model has a holomorphic spin-$N$2 current $N$3 with $N$4, where $N$5 is the stress tensor. Their OPE takes the schematic form

$N$6

Highest-weight representations are labelled by $N$7, and the conformal dimension is

$N$8

Two simple examples are

$N$9

which in the ’t Hooft limit approach $_3$0 and $_3$1, respectively (Gaberdiel et al., 2012).

The bulk dual is a higher-spin extension of AdS$_3$2 gravity formulated as a Chern–Simons theory. One replaces $_3$3 by the infinite-dimensional algebra $_3$4,

$_3$5

and takes the gauge fields $_3$6. To reproduce the full minimal-model spectrum, one adds two complex scalars of mass

$_3$7

each quantized with the two possible AdS$_3$8 boundary conditions, giving dual operators of dimension

$_3$9

The Virasoro subalgebra of the asymptotic symmetry has central charge

$\frac{SU(N)_k \times SU(N)_1}{SU(N)_{k+1}}$0

under generalized Brown–Henneaux boundary conditions (Gaberdiel et al., 2012).

This bosonic construction is the prototype for the broader subject. A plausible implication is that the term “minimal model holography” is best understood not as a single dual pair, but as a family of AdS$\frac{SU(N)_k \times SU(N)_1}{SU(N)_{k+1}}$1/CFT$\frac{SU(N)_k \times SU(N)_1}{SU(N)_{k+1}}$2 correspondences organized by coset constructions and their higher-spin asymptotic symmetries.

2. Asymptotic symmetry algebras and finite-$\frac{SU(N)_k \times SU(N)_1}{SU(N)_{k+1}}$3 equivalences

A central structural feature of minimal model holography is that the higher-spin bulk theory and the coset CFT share the same non-linear $\frac{SU(N)_k \times SU(N)_1}{SU(N)_{k+1}}$4-algebra. For the bosonic series, this is the quantum $\frac{SU(N)_k \times SU(N)_1}{SU(N)_{k+1}}$5 algebra. Its currents are the stress tensor $\frac{SU(N)_k \times SU(N)_1}{SU(N)_{k+1}}$6 of spin $\frac{SU(N)_k \times SU(N)_1}{SU(N)_{k+1}}$7, $\frac{SU(N)_k \times SU(N)_1}{SU(N)_{k+1}}$8 of spin $\frac{SU(N)_k \times SU(N)_1}{SU(N)_{k+1}}$9, $$c_{N,k}=(N-1)\Bigl[1-\frac{N(N+1)}{(N+k)(N+k+1)}\Bigr]\le N-1,$$0 of spin $$c_{N,k}=(N-1)\Bigl[1-\frac{N(N+1)}{(N+k)(N+k+1)}\Bigr]\le N-1,$$1, $$c_{N,k}=(N-1)\Bigl[1-\frac{N(N+1)}{(N+k)(N+k+1)}\Bigr]\le N-1,$$2 of spin $$c_{N,k}=(N-1)\Bigl[1-\frac{N(N+1)}{(N+k)(N+k+1)}\Bigr]\le N-1,$$3, and so on. The first non-trivial OPE is the self-OPE of the spin-3 field, and the exact finite-$$c_{N,k}=(N-1)\Bigl[1-\frac{N(N+1)}{(N+k)(N+k+1)}\Bigr]\le N-1,$$4 form of the structure constant $$c_{N,k}=(N-1)\Bigl[1-\frac{N(N+1)}{(N+k)(N+k+1)}\Bigr]\le N-1,$$5 determines the algebra; all other structure constants are then fixed by enforcing the Jacobi identities (Gaberdiel et al., 2012).

At fixed central charge, the quantum algebra depends only on the pair $$c_{N,k}=(N-1)\Bigl[1-\frac{N(N+1)}{(N+k)(N+k+1)}\Bigr]\le N-1,$$6, where

$$c_{N,k}=(N-1)\Bigl[1-\frac{N(N+1)}{(N+k)(N+k+1)}\Bigr]\le N-1,$$7

The parameter $$c_{N,k}=(N-1)\Bigl[1-\frac{N(N+1)}{(N+k)(N+k+1)}\Bigr]\le N-1,$$8 satisfies a cubic equation, so there are three, generically distinct, solutions $$c_{N,k}=(N-1)\Bigl[1-\frac{N(N+1)}{(N+k)(N+k+1)}\Bigr]\le N-1,$$9. Hence

$\lambda\equiv \frac{N}{N+k}\in[0,1].$0

which is the exact “triality” (Gaberdiel et al., 2012). In the coset parametrization, one has

$\lambda\equiv \frac{N}{N+k}\in[0,1].$1

at $\lambda\equiv \frac{N}{N+k}\in[0,1].$2 (Gaberdiel et al., 2012). The review formulation gives the same conclusion as

$\lambda\equiv \frac{N}{N+k}\in[0,1].$3

with $\lambda\equiv \frac{N}{N+k}\in[0,1].$4 solving a cubic at fixed $\lambda\equiv \frac{N}{N+k}\in[0,1].$5 (Gaberdiel et al., 2012).

The $\lambda\equiv \frac{N}{N+k}\in[0,1].$6 supersymmetric extension is governed by a non-linear $\lambda\equiv \frac{N}{N+k}\in[0,1].$7 algebra. In addition to the $\lambda\equiv \frac{N}{N+k}\in[0,1].$8 super-Virasoro generators $\lambda\equiv \frac{N}{N+k}\in[0,1].$9, there is exactly one $N$0 primary multiplet $N$1 of integer spin $N$2. Defining

$N$3

one finds that all Jacobi identities up to low spins close provided certain couplings satisfy explicit relations; thus for each $N$4 there is a one-parameter family of consistent $N$5 algebras, parametrized by $N$6 (Candu et al., 2012). However, at fixed $N$7, the map $N$8 is a quartic function, so there are four generically distinct values

$N$9

satisfying the same $N,k\to\infty$0 algebra, with

$N,k\to\infty$1

Viewed in the coset parametrization $N,k\to\infty$2, these correspond to four equivalent Kazama–Suzuki descriptions, including $N,k\to\infty$3 and $N,k\to\infty$4, the standard level-rank duality (Candu et al., 2012).

For the even-spin orthogonal series, the relevant symmetry is $N,k\to\infty$5, generated by the Virasoro field $N,k\to\infty$6 of central charge $N,k\to\infty$7 and one Virasoro-primary $N,k\to\infty$8 for each even spin $N,k\to\infty$9. Aside from $\lambda$0, all structure constants are recursively determined by the associativity up to a single free parameter

$\lambda$1

This algebra is the quantum Drinfelʹd–Sokolov reduction of the bulk higher-spin algebra $\lambda$2, and its wedge algebra reproduces the commutators of $\lambda$3 in the $\lambda$4 limit (Candu et al., 2012).

These finite-$\lambda$5 identifications are significant because they move the correspondence beyond a purely asymptotic large-$\lambda$6 statement. In the bosonic, supersymmetric, and even-spin settings alike, the higher-spin asymptotic symmetry algebra is not merely approximately realized by the coset theory; it is matched by a quantum algebra whose structure constants are controlled by Jacobi identities (Gaberdiel et al., 2012, Candu et al., 2012, Candu et al., 2012).

3. Spectral matching and partition functions

The most explicit evidence for minimal model holography comes from partition-function identities and the matching of perturbative spectra. In the bosonic case, the 1-loop thermal partition function of a bulk massless spin-$\lambda$7 field is

$\lambda$8

Summing over all higher spins yields

$\lambda$9

where $_2$00. This exactly equals the vacuum character $_2$01 of the $_2$02 algebra in the ’t Hooft limit (Gaberdiel et al., 2012).

A complex scalar of boundary dimension $_2$03 contributes

$_2$04

In terms of $_2$05 Schur polynomials $_2$06,

$_2$07

The full 1-loop bulk perturbative partition function is then

$_2$08

On the CFT side, restricting to representations $_2$09 built from finite tensor powers of fundamentals, the branching functions simplify in the ’t Hooft limit to

$_2$10

and summing over $_2$11 reproduces exactly $_2$12 (Gaberdiel et al., 2012).

The orthogonal even-spin series exhibits an analogous match. The relevant CFT is the $_2$13 minimal model coset

$_2$14

with

$_2$15

held fixed in the large-$_2$16 limit. The proposed bulk dual contains one real massless gauge field of each even spin $_2$17 together with two real scalar fields of equal mass

$_2$18

quantized with opposite $_2$19 boundary conditions, with

$_2$20

The bulk partition function is

$_2$21

and, after standard manipulations involving the even MacMahon function $_2$22 and specialized Schur functions $_2$23, one finds

$_2$24

to leading order in large $_2$25 (Gaberdiel et al., 2011).

The $_2$26 supersymmetric extension at $_2$27 also admits a partition-function match. The gauge sector has

$_2$28

while matter consists of two complex scalars of mass $_2$29 and two Dirac fermions of mass $_2$30. The full one-loop partition function

$_2$31

reproduces the CFT answer

$_2$32

exactly (Beccaria et al., 2013).

Taken together, these results show that partition-function matching is not limited to a single model. It recurs across bosonic, orthogonal, and supersymmetric variants, and in each case the match is organized by the relevant higher-spin vacuum character and matter contribution (Gaberdiel et al., 2012, Gaberdiel et al., 2011, Beccaria et al., 2013).

4. Correlators, minimal representations, and perturbative versus non-perturbative states

Beyond partition functions, minimal model holography compares operator spectra and selected correlators. In the bosonic case, sphere three-point functions of two scalars and one spin-$_2$33 current,

$_2$34

match exactly between bulk and CFT, and higher-point functions also factorise in the large $_2$35 limit (Gaberdiel et al., 2012).

The representation theory of $_2$36 plays a direct role in identifying the bulk matter fields. The “minimal” module has character

$_2$37

and null-vector analysis yields a cubic equation for $_2$38. For fixed $_2$39 and $_2$40,

$_2$41

At $_2$42, the two finite-$_2$43 solutions are those of the two bulk scalars of mass $_2$44, while the third decouples (Gaberdiel et al., 2012).

At the same time, the treatment of “light states” reveals a more subtle sector. In the coset, light states have $_2$45 with conformal weight

$_2$46

For $_2$47 fundamental of $_2$48,

$_2$49

Keeping $_2$50 fixed and varying $_2$51 continuously from the minimal-model value $_2$52 to the semiclassical regime $_2$53, all charges of $_2$54, in particular $_2$55 and spin-3 $_2$56, scale as $_2$57 times a purely group-theoretic Casimir, and in particular $_2$58 as $_2$59. In the bulk this is interpreted as the one-parameter family of classical “conical defect” (or surplus) solutions of $_2$60 higher-spin gravity (Gaberdiel et al., 2012).

This interpretation feeds into a distinction between perturbative and non-perturbative matter. The two “minimal” coset primaries are $_2$61 and $_2$62. Their dimensions at finite $_2$63 satisfy

$_2$64

Accordingly, the state dual to $_2$65 remains light and is the usual perturbative scalar in the bulk Vasiliev theory, whereas the other state is better thought of as a non-perturbative excitation (Gaberdiel et al., 2012). The $_2$66 case sharpens this distinction further: both “scalar” multiplets have conformal dimensions proportional to $_2$67 in the semiclassical limit $_2$68 at fixed $_2$69, implying that they behave as non-perturbative states in the bulk higher-spin theory (Candu et al., 2012).

These results address a common misconception that all low-lying coset primaries correspond to perturbative bulk matter. The evidence instead indicates a split structure: some sectors are reproduced by perturbative higher-spin fields and scalar multiplets, while others are associated with non-perturbative states such as conical defects or their supersymmetric analogues (Gaberdiel et al., 2012, Candu et al., 2012).

5. Orthogonal and even-spin minimal model holography

An important variant replaces the $_2$70-based bosonic family by orthogonal cosets with even-spin symmetry. The $_2$71 minimal models are realized as

$_2$72

with finite-$_2$73 central charge

$_2$74

In the large-$_2$75 ’t Hooft limit,

$_2$76

and

$_2$77

(Gaberdiel et al., 2011).

The proposed bulk dual is a parity-invariant truncation of the usual 3D higher-spin theory to even spins. Its field content is one real massless gauge field of each spin

$_2$78

governed by a Chern–Simons action based on $_2$79, together with two real scalar fields of equal mass $_2$80, quantized with opposite boundary conditions (Gaberdiel et al., 2011).

The quantum algebraic side was developed further through the even-spin $_2$81-algebra $_2$82. This algebra is generated by the stress energy tensor together with one Virasoro primary field for every even spin $_2$83. It is characterized, in addition to the central charge, by one free parameter identified with the self-coupling constant of the spin-4 field. The algebra can be thought of as the quantisation of the asymptotic symmetry algebra of the even higher spin theory on AdS$_2$84, and it is also quantum equivalent to the $_2$85 coset algebras. The quantum equivalence holds at finite central charge, thereby opening the way towards understanding the duality beyond the leading ’t Hooft limit (Candu et al., 2012).

At finite $_2$86, the coset

$_2$87

has central charge

$_2$88

It contains two “minimal” primaries $_2$89 and $_2$90 with conformal dimensions

$_2$91

By comparing their null-vector condition with the general $_2$92 result, one finds a closed form for

$_2$93

which establishes the exact equivalence

$_2$94

at arbitrary finite $_2$95 (Candu et al., 2012).

This orthogonal branch shows that minimal model holography is not intrinsically tied to all-integer spins. A plausible implication is that the core mechanism is the matching between coset realizations and quantum Drinfelʹd–Sokolov reductions, while the spin content depends on the choice of Lie algebra and truncation.

6. Supersymmetric extensions and the $_2$96 continuous orbifold

The $_2$97 supersymmetric version relates a higher-spin theory based on the infinite-dimensional Lie superalgebra $_2$98 to the Kazama–Suzuki coset

$_2$99

with central charge

$N$00

Under the duality one identifies

$N$01

(Candu et al., 2012).

The asymptotic symmetry algebra is the non-linear $N$02 super-$N$03-algebra $N$04. In the bulk theory, when $N$05, one has the truncation

$N$06

while for general $N$07 the theory contains one massless multiplet of each integer spin $N$08 (Candu et al., 2012). On the CFT side, the chiral algebra inherited from the Kazama–Suzuki coset is identified with the same $N$09 algebra even at finite $N$10, due to the four-fold equivalence of $N$11 at fixed $N$12 (Candu et al., 2012).

A particularly concrete realization appears in the large-$N$13 limit of the Kazama–Suzuki coset. For the diagonal modular invariant of

$N$14

the central charge is

$N$15

In the large-$N$16 limit, $N$17, so the theory becomes that of $N$18 free complex bosons $N$19 and $N$20 free complex fermions $N$21, $N$22. The orbifold group is $N$23, acting by

$N$24

on both left- and right-movers (Gaberdiel et al., 2014).

The projector onto the $N$25 singlet untwisted sector is

$N$26

and the untwisted torus partition function is

$N$27

By explicit branching, this equals the coset subsector

$N$28

namely the perturbative coset states $N$29 (Gaberdiel et al., 2014).

The twisted sectors are parametrised by conjugacy classes of $N$30,

$N$31

up to permutations. In the $N$32-twisted sector all fields pick up phases $N$33, and the ground-state energy is

$N$34

For each $N$35, one identifies a coset primary

$N$36

such that

$N$37

The full fermionic excitation spectrum coincides with the expected fractional modings $N$38, and their BPS descendants agree on both sides (Gaberdiel et al., 2014).

This continuous $N$39-orbifold description gives a precise realization of how the perturbative higher-spin subsector and the twisted, non-perturbative sectors are encoded in the Kazama–Suzuki theory. It also refines the general $N$40 duality by providing an explicit large-$N$41 free-field limit (Gaberdiel et al., 2014, Candu et al., 2012).

7. Developments, reinterpretations, and string-theoretic embedding

Several later developments broaden the scope of minimal model holography while preserving its central logic. One is the $N$42 extension at the special ’t Hooft point $N$43. Starting from the bosonic coset

$N$44

one sets $N$45, so that $N$46, and uses the conformal embedding

$N$47

Replacing the diagonal invariant by the charge-conjugation D-type invariant yields an $N$48 superconformal theory. Its chiral algebra is an $N$49 super-$N$50, denoted $N$51, generated by one superprimary of each half-integer spin $N$52. By enforcing all associativity conditions, every structure constant is uniquely fixed in terms of $N$53 alone, and there is no additional coupling parameter (Beccaria et al., 2013).

On the bulk side, the higher-spin algebra $N$54 likewise admits no continuous deformations; its only free parameter is the overall Chern–Simons level. The field content consists of one massless gauge field for each half-integer spin $N$55, together with an $N$56 matter multiplet of two complex scalars of mass

$N$57

and two Dirac spinors of mass $N$58. The matching of symmetry algebras, spectra, and torus partition functions provides the evidence for the duality (Beccaria et al., 2013).

A distinct recent direction embeds minimal model holography in topological string theory. In a construction based on 3d $N$59 Chern–Simons theory at level $N$60 with a 2d interface $N$61 supporting $N$62 complex chiral fermions, the chiral algebra of the interface is the coset

$N$63

where $N$64 denotes $N$65 free chiral fermions. Upon further truncation $N$66 for integer $N$67, one recovers the diagonal $N$68 minimal models based on

$N$69

The central charge

$N$70

tends to $N$71 in the large-$N$72 limit at fixed $N$73, which is stated to be linear in $N$74 as appropriate for a D-brane (Gaiotto et al., 9 Mar 2026).

In that formulation, the ’t Hooft expansion of $N$75 Chern–Simons theory on $N$76 is dual to the A-model on $N$77, where $N$78 is the deformed $N$79 singularity

$N$80

Probe D4-branes of “canonical coisotropic” type produce at their boundary precisely the chiral fermion interface, and meson operators are identified with open Wilson line endpoints in 5d non-commutative Chern–Simons theory. The proposal gives an exact holographic match of all sphere correlation functions of meson operators and presents this as a string-theoretic embedding of minimal model holography (Gaiotto et al., 9 Mar 2026).

This suggests a broader perspective. A plausible implication is that minimal model holography serves as a tractable sector where higher-spin AdS$N$81/CFT$N$82 duality, quantum $N$83-algebras, integrable structures, and string-theoretic constructions can be compared in explicit detail. What remains invariant across these realizations is the role of solvable cosets, asymptotic symmetry algebras fixed by Jacobi identities, and finely controlled spectral data.