https://cftzoo.net/ 2026-04-15T15:21:43+00:00 https://cftzoo.net/ https://cftzoo.net/lib/tpl/white/images/favicon.ico text/html 2021-04-07T02:01:30+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/6d_1_0_superconformal_field_theory?rev=1617760890&do=diff 6d (1,0) superconformal field theory holographic superconformal 6d External links * nLab text/html 2022-11-17T14:41:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/6d_2_0_superconformal_field_theory?rev=1668696062&do=diff 6d (2,0) superconformal field theory holographic superconformal 6d Methods Bootstrap <https://arxiv.org/pdf/1507.05637.pdf> External links * Wikipedia * nLab text/html 2021-04-07T02:03:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/abjm_superconformal_field_theory?rev=1617760980&do=diff ABJM superconformal field theory holographic superconformal 3d External links * The original paper by Aharony, Bergman, Jafferis, and Maldacena * Wikipedia text/html 2021-04-17T14:38:28+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/anyons?rev=1618670308&do=diff Anyons nonrelativistic 2d text/html 2021-05-10T05:34:42+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/banks_zaks_fixed_point?rev=1620624882&do=diff Banks-Zaks fixed point 4d A Banks-Zaks fixed point is an IR-stable CFT describing long-distance behavior of QCD for certain parameters. The Lagrangian of QCD is \[ \mathcal L= \sum_a \bar\psi_a (i \gamma\partial - m_a ) \psi_a - \frac 1 4 F^2 \] where color and spacetime indices have been suppressed. The flavor index $a$ runs over $N_f$ flavors of fermions. $N_f < \frac {33} 2$$N_f$$N_f \sim 12$ text/html 2021-04-08T00:01:24+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/bf_model?rev=1617840084&do=diff BF model topological 2d 3d 4d (Does this exist in 5d? 6d? 10d? 101d?) text/html 2021-04-07T23:52:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/chern-simons?rev=1617839530&do=diff Chern-Simons theory topological 3d text/html 2021-09-14T20:50:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/chiral_cfts?rev=1631652614&do=diff Chiral Conformal Field Theory Chiral conformal field theories are mathematical objects which are related to but distinct from two-dimensional conformal field theories. They transform under a single copy of the Virasoro algebra $\mathcal{V}$ and depend on $z$ but not $\bar z$. Notable examples include bosons compactified on any self-dual Euclidean lattice, and the text/html 2026-04-11T20:39:43+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/conformal-qed?rev=1775939983&do=diff Conformal QED 3d See <https://arxiv.org/abs/1508.06278> and <https://arxiv.org/abs/1508.06354>. text/html 2021-04-14T03:49:34+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/conformal_bootstrap?rev=1618372174&do=diff Conformal bootstrap External links * A review by Poland, Rychkov, Vichi * Slava Rychkov's list of open problems * Simons Collaboration text/html 2021-04-09T17:49:44+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/conformal_invariance?rev=1617990584&do=diff Conformal invariance Two dimensions See Virasoro symmetries. Euclidean space in 3d and beyond * Translations * Rotations * Scale transformations * Special conformal transformations text/html 2021-04-09T16:19:26+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/conformal_quantum_mechanics?rev=1617985166&do=diff Conformal quantum mechanics 1d Haven't gotten to this yet. Sources: * Qualls CFT lectures here * Model appears to have been introduced here * discussion on the non-invariance of the ground state here The most general Lagrangian consistent with time and scale invariance is given by % \begin{align} L = \frac{1}{2} \dot{Q}^2 - \frac{g^2}{2 Q^2} \end{align}$SL(2, \mathbb{R} \sim SO(2,1)$$SO(2,1)$ text/html 2021-04-09T16:18:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/defect_conformal_field_theory?rev=1617985138&do=diff Defect conformal field theory 1d text/html 2021-05-09T02:44:38+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/directed_percolation?rev=1620528278&do=diff Directed percolation 2d 3d 4d External links * Wikipedia text/html 2026-03-05T15:57:26+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/gaussian?rev=1772726246&do=diff Gaussian CFTs are theories with a quadratic action. They can be defined in arbitrary dimension. 'The free scalar' refers to the unique theory in flat space with a scalar field at the unitarity bound, $\Delta_\phi = \frac{d-2}{2}$. text/html 2021-09-17T16:09:17+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/generalized_minimal_models?rev=1631894957&do=diff Generalized Minimal Models 2d Generalized minimal models are 2d CFTs which have a discrete (but potentially countably infinite) number of degenerate representations in their spectra. Generalized minimal models exist for any central charge $c \in \mathbb{C}$. For $$ c = c_{p, q} = 1 - 6 \frac{(p - q)^2}{p q} \, ,$$ these theories become the $A$ text/html 2021-08-11T17:33:26+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/gross_neveu_model?rev=1628703206&do=diff Gross-Neveu model 3d The Gross-Neveu model is a theory of $N$ interacting fermions defined by the Lagrangian density \[ \mathcal L = \bar\psi i \gamma^\mu \partial_\mu \psi + \frac{g^2}{2N} (\bar\psi \psi)^2 \text. \] This model is usually studied in two dimensions, where it is asymptotically free for any $N$. However, for $2 < d < 4$, the theory is asymptotically safe: $N$ text/html 2021-08-11T17:36:24+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/gross_neveu_yukawa_model?rev=1628703384&do=diff Gross-Neveu-Yukawa model 3d Special cases N=4 Vojta Zhang Sachdev 2000, Herbut 2006, Classen, Herbut, Scherer 2017 N=8 Raghu, Qi, Honerkamp, Zhang 2017; Moon, Ku, Kim, Balents 2012; Herbut, Janssen 2014 Large-N limit See also * Gross-Neveu model text/html 2026-03-20T15:48:36+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/ising?rev=1774021716&do=diff Ising model 3d The Ising model is an interacting CFT in three dimensions. For related models in other dimensions, see . The Ising model is the $N=1$ case of the $O(N)$ model. The Ising CFT also goes by the name “Wilson-Fisher fixed point” (in any $d < 4$). Another terminological difficulty: $\sigma$$\epsilon$$$ \Delta_\sigma = 0.5181489(10)\text{ and } \Delta_\epsilon = 1.412625(10)\text. $$$$ \lambda_{\sigma\sigma\epsilon} = 1.0518537(41)\text{ and } \lambda_{\epsilon\epsilon\epsilon} = 1.53… text/html 2021-06-29T22:36:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/liouville_theory?rev=1625006169&do=diff Liouville theory 2d Liouville theory is an interacting 2d CFT with a continuous spectrum of scalar states. Bootstrap Picture Let us start by assuming that we have a unitary theory with $c>1$. We must also assume that all Virasoro primaries are scalar fields: with these assumptions, the theory is entirely fixed. \begin{align} Z(\tau, \bar \tau) = \sum_{h, \bar h} d_{h, \bar h} \chi_h(\tau) \bar{\chi}_{\bar h}(\bar \tau) \end{align}$c>1$\begin{align} \chi_h(\tau) = q^{h - c/24} \prod_{n = … text/html 2026-03-19T00:09:26+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/list_of_cfts?rev=1773878966&do=diff List of CFTs Here is a list of CFTs. They are organized by dimension. Note that some CFTs, like the gaussian theories and percolation, can be defined in any number of dimensions. This allows their conformal data to be analytically continued between dimensions, connecting otherwise apparently different CFTs.$T_\mu{}^\mu = 0$$T_{00} = H = 0$$c<1$$c>1$$d>6$$d>6$$d$$d$$d$$p$$(d+1,1)$$S= \int d^d x O(x)$$x$$\phi$$\Delta$$\phi^4$$d$$\Delta \to \Delta_{\phi, \text{Ising}}$$1/8$$d=2$$0.518$$d=3$$T^{\mu… text/html 2021-04-17T14:15:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/litim_sannino_model?rev=1618668916&do=diff Litim-Sannino model 4d <https://arxiv.org/pdf/1406.2337.pdf> text/html 2026-03-19T00:31:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/minimal_models?rev=1773880282&do=diff Minimal Models 2d Minimal models are 2d conformal field theories whose spectra consist of a finite number of representations of the Virasoro algebra. Construction The requirement for of a finite number of representations is quite constraining; generically the OPE $ \mathcal{O}_1 \mathcal{O}_2 = \sum_i C_{12i} \mathcal{O}_i$ will contain an infinite number of primaries. In some cases, the right-hand side contains a finite number of operators because the $C_{12i}$$i$$\mathcal{O}_1$$\mathcal{O… text/html 2021-04-28T23:18:51+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/monster_cft?rev=1619651931&do=diff Monster CFT 2d holographic External links * Witten's paper * Wikipedia on moonshine text/html 2021-06-10T06:37:43+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/n_1_sym?rev=1623307063&do=diff N=1 supersymmetric Yang-Mills $\mathcal N=1$ super Yang-Mills is supersymmetric, but not conformal. External links * Tachikawa's lectures text/html 2021-04-06T20:54:04+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/n_2_sym?rev=1617742444&do=diff N=2 supersymmetric Yang-Mills $\mathcal N=2$ super Yang-Mills is supersymmetric, but not conformal. text/html 2021-04-08T00:04:07+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/n_4_sym?rev=1617840247&do=diff N=4 supersymmetric Yang-Mills superconformal holographic 4d Related theories See also $\mathcal N=1$ super Yang-Mills and $\mathcal N=2$ super Yang-Mills, neither of which is conformal. text/html 2021-06-29T21:30:41+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/narain_theories?rev=1625002241&do=diff Narain theories Narain theories are free 2d CFTs where the target space is compactified on a lattice. 2d text/html 2026-03-19T00:24:06+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/null_results?rev=1773879846&do=diff Null Results This page is meant to be a resource for informally keeping track of null results, which often are not published but which still may be quite useful to other researchers trying to do the same thing. Minimal models One might want to find the unitary $\mathcal{M}_{m+1,m}$$\frac{1}{2}(\partial \phi)^2 + g \phi^{2(m-1)}$$d=d_c -\epsilon$$d_c = 2(m-1)/(m-2)$$m$$d=2$$d_c$$\epsilon$ text/html 2021-11-13T12:27:39+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/null_results_for_cfts?rev=1636806459&do=diff Null Results text/html 2022-12-21T04:46:05+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/o2_model?rev=1671597965&do=diff $O(2)$ model 3d The $O(2)$ model is a well studied special case of the $O(N)$ model. It also goes by the name “XY model”. Operator content Three of the leading operators in the spectrum: $\phi$ (an $O(2)$ vector), $s$ (a singlet), and $t$ (a symmetric rank-2 tensor i.e. a charge 2 operator) have been determined precisely and rigorously. The scaling dimensions are \[ \Delta_\phi = 0.519088(\mathbf{22}),~\Delta_s = 1.51136(\mathbf{22}),~\Delta_t = 1.23629(\mathbf{11}). \]\[ \lambda_{\phi\phi s… text/html 2023-01-30T03:02:45+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/o3_model?rev=1675047765&do=diff $O(3)$ model 3d The $N=3$ special case of the $O(N)$ model also goes by the name “Heisenberg model”. Operator content Three of the leading operators in the spectrum have had their dimensions precisely determined: $\phi$ (an $O(3)$ vector), $s$ (a singlet), and $t$ (a symmetric rank-2 tensor). The scaling dimensions are \[ \Delta_\phi = 0.518936(\mathbf{67}),~\Delta_s = 1.59488(\mathbf{81}),~\Delta_t = 1.20954(\mathbf{32}). \]$t_4$\[ \Delta_{t_4} < 2.99056. \]\[ \lambda_{\phi\phi s} =0.524261… text/html 2021-05-07T19:49:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/o4_model?rev=1620416954&do=diff $O(4)$ model 3d This is the $N=4$ special case of the $O(N)$ model. Physical realizations The chiral transition in two-flavor QCD is conjectured to be an $O(4)$ transition: <https://arxiv.org/pdf/hep-ph/9504310.pdf> text/html 2026-03-20T15:48:03+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/on_model?rev=1774021683&do=diff $O(N)$ model 3d holographic The special cases of the $O(2)$ model and $O(3)$ model are particularly well-studied. Higher values of $N$ do come up occasionally, see for instance the $O(4)$ model. The Ising model is the case $N=1$, and the case $N=0$ is covered by self-avoiding walks. This also goes by the name of the “n-vector model”. Its Ginzburg-Landau description is the critical point of an O$(N)$$\phi_i$$\phi^4 =(\phi_i \phi_i)^2$$\sigma$$N-1$$(N)$$N$$N+1$ text/html 2021-05-05T19:06:49+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/percolation?rev=1620241609&do=diff Percolation 2d 3d 4d 5d logarithmic non-unitary Percolation is a logarithmic CFT. Among other things, this means it is not unitary. This page focuses on “standard percolation” --- see below for alternative definitions that yield different CFTs. Lattice theories Percolation can be defined as either bond or site percolation, and on any geometry of lattice. In both cases, by $p=1$$p$$p_c = 0.5$$d\rightarrow\infty$$z$$z=1$$z=2$$\beta = 1$$p_c = (z-1)^{-1}$ text/html 2021-06-29T21:31:21+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/runkel-watts?rev=1625002281&do=diff Runkel-Watts theory 2d text/html 2021-09-16T17:24:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/runkel-watts_theory?rev=1631813095&do=diff Runkel-Watts Theory Runkel-Watts theory is a 2d CFT with central charge $c = 1$. It can be constructed from the $A$-series of minimal models, which are parametrized by integers $p$ and which have central charge $$ c = 1 - 6 \frac{1}{p(p+1)} \, . $$ The Runkel-Watts theory is defined as the limiting theory that arises from taking $p \to \infty$$c < 1$$c < 1$ text/html 2021-04-30T16:59:42+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/schrodinger_group?rev=1619801982&do=diff Schrödinger group The Schrödinger group is the group of symmetries that define a non-relativistic CFT. External links * Wikipedia text/html 2021-11-13T13:29:40+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/self_avoiding_walk?rev=1636810180&do=diff Self-avoiding walk logarithmic non-unitary 2d 3d A self-avoiding random walk (or self-avoiding walk, or SAW) is a path on a $d$-dimensional lattice in $\mathbb{Z}^d$ which never visits the same point more than once. As a statistical model, they are defined as the set of all such paths with length $n$$n \to \infty$$n\rightarrow 0$$O(0)$ text/html 2026-03-05T15:41:42+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/start?rev=1772725302&do=diff Welcome to the CFT Zoo You might want to look at the list of CFTs. Or the list of null results for CFTs. About the zoo This is a collaborative wiki, styled after the famed complexity zoo. Editing this wiki is open to all, but you must create an account first. Click on the pencil in the upper-right to get started. If you have trouble creating an account, check your spam folder. Failing that, don't hesitate to email me. text/html 2021-06-27T22:26:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/supersymmetric_localization?rev=1624832817&do=diff Supersymmetric localization External links * Pestin, Zabzine * An Introduction to Localisation and Supersymmetry in Curved Space (Lecture notes) * Localization techniques in quantum field theories * Szabo * Wikipedia text/html 2021-04-07T23:40:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/syk_model?rev=1617838855&do=diff SYK model holographic 1d This page has not yet been written External links * Maldacena, Stanford text/html 2021-04-29T20:03:08+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/transfer_matrix?rev=1619726588&do=diff Transfer matrix The transfer matrix is the object that connects the Hamiltonian formulation of a theory to a path integral. Consider as an example a two-level quantum mechanical system with Hamiltonian: \begin{equation} H = -\mu \sigma_x \text. \end{equation} We can derive a (Euclidean) path integral for this Hamiltonian by starting from the thermal partition function and inserting many copies of the identity operator $\left|0\right>\left<0\right| + \left|1\right>\left<1\right|$\begin{equation… text/html 2021-04-30T17:06:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/unitary_fermi_gas?rev=1619802369&do=diff Unitary Fermi gas nonrelativistic Physical realizations External links text/html 2021-09-15T17:52:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/virasoro_symmetries?rev=1631728375&do=diff Virasoro Algebra The Virasoro algebra $\mathcal{V}$ is defined by the commutation relation $$ [ L_n, L_m] = (n - m) L_{n + m} + \frac{c}{12} (n - 1)n(n + 1) \delta_{n, -m}$$ the number $c$ is called the central charge. 2d conformal field theories transform under two copies of the Virasoro algebra, $\mathcal{V} \, \times \, \bar{ \mathcal{V}} $. Chiral Conformal Field Theory transform under a single copy of the Virasoro algebra.$f(z)$$$ z \to z + \epsilon z^{n + 1}$$$n$$z$$$ \ell_n = - z^{n … text/html 2026-03-05T14:58:52+00:00 Anonymous (anonymous@undisclosed.example.com) https://cftzoo.net/yanglee?rev=1772722732&do=diff The Yang-Lee edge singularity is a non-unitary CFT found by tuning the magnetic field in the Ising model to a critical imaginary value. It has Ginzburg-Landau description as a scalar field with interaction $i g \phi^3$, and so is weakly coupled in $d=6-\epsilon$ 1. External links