Questions tagged [conformal-field-theory]
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In 2D, the infinite-dimensional algebra of local conformal transformations normally permits exact solution or classification of such theories. Further use for CFT applications to string theory, statistical mechanics, and condensed matter physics.
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Quantum corrections in Holography
AdS/CFT stablish that there is some kind of correspondence between the ${\cal N}=4$ SYM theory and strings in $AdS_5\times S^5$ space-time. I know, for instance that 1/2 BPS operators like Tr$(\phi_1^...
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Coupling three Ising chains via an energy-energy-energy interaction
I want to note that this question is related to another one I asked involving just two chains coupled by an energy-energy interaction. I'm choosing to ask them separately because I suspect they may ...
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Coupling two Ising chains via an energy-energy interaction
Consider the transverse-field Ising model on a chain with periodic boundary conditions:
$$ H = -\sum_{i=1}^{L} \sigma_{i}^z \sigma_{i+1}^z + h \sigma_{i}^x$$
There's a phase transition at $h=1$, which ...
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Momentum Operator in Radial Quantization
I have a Hilbert space of a CFT in radial quantization.
In particular, I have a Hamiltonian on a QFT/multibody system on a 2D sphere. The Hamiltonian acts as $D$ in the 3D CFT.
If I have a state $|O\...
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What does it mean by "minimally coupled action" and "Weyl invariant"?
The scalar field action, minimally coupled to the background field looks like
$$S= \int d^Dx \sqrt{-g} [\partial_{\mu} \phi \partial_{\nu} \phi g^{\mu\nu} - \lambda \phi^{\beta}]$$ Now in general the ...
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Derivation of the Conformal Ward Identity in Di Francesco et al
I am reading section 5.2.2. (titled The Conformal Ward Identity) from Conformal Field Theory by Di Francesco et al. The authors write
\begin{align}
\partial_\mu(\epsilon_\nu T^{\mu\nu}) &= \...
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How to derive WZW model’s energy-momentum tensor? The result is of course the Sugawara construction
I want to know how to derive WZW’s energy-momentum tensor. We know WZW action is
$$
S_{WZW}[g]=\frac{k}{16\pi}\int d^2x Tr(\partial g^{-1}\partial g) - \frac{ik}{24\pi}
\int_B d^3y \epsilon_{abc} Tr(h^...
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Wick rotation of CFT three-point function
Let $\langle O_1\cdots O_n\rangle$ be a Euclidean CFT$_d$ correlation function. I know that we can analytically continue to Lorentzian signature as follows. Let $x_i = (\tau_i,\mathbf{x}_i)\in\mathbb{...
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Fermionic generator from coset current algebra
I am recently studying some chiral algebra from physics perspective. I found many chiral algebras have nice coset realization of current algebra. However, most of the literatures are very algebraic. ...
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How do quantum Fields know about changes in normal ordering?
According to Transformation of the energy-momentum tensor under conformal transformations
The schwartzian term in the transformation properties arises due to the stress tensor being defined as the ...
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CFT algebra calculation [closed]
Hi I'm reading https://arxiv.org/abs/2006.13280 and following its calculation but I'm stuck in page16, eqn 4.4a.
I got different result than the one in the paper, but I can't find the point I missed.
...
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Noether charge for dilatations in terms of creation and anihilation operators
I am trying to compute the conserved charge for a continuous diatation symmetry for the massless real scalar field in four dimensions terms of creation and annihilation operators. Then I have,
$$\...
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How do Dedekind's eta function arise while computing the partition function of a compact scalar field over circle?
I am following the book String Theory in a nutshell (From Elias Kiritsis). In chapter 4.18, it takes a theory following the action:
$$S=\frac{1}{4\pi l_s^2}\int X\square X\ d\sigma,\tag{4.18.1}$$
$$ \...
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Central Charge Calculation of $SL_k(2,\mathbb{R})$ WZW Model
According to P. Francesco et al. conformal field theory book the central charge of the enveloping Virasoro algebra of the affine Lie algebra $\hat{g}_k$ corresponding with Lie algebra $g$ which ...
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About Noether currents for conformal symmetries
I´m reading "Conformal algebra on Fock space and conjugate pairs of operators" of
Klaus Sibold and Eden Burkhard, there the authors write all Noether currents in terms of the energy momentum ...
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Trace of stress tensor in 2D average null energy condition
I was looking through Zamolodchikov's derivation of the $c$-theorem and stumbled across an equation which says the following -
$$\Theta = T^\mu_\mu = 4T_{z\bar{z}}.$$
As far as I understand, for two ...
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Ricci Scalar Curvature under conformal transformation
Consider the Klein-Gordon equation in curved spacetime with metric $g$
$$\square_g \phi - \xi R \phi = 0$$
and consider a conformal transformation
$$g \longmapsto \tilde{g} = \Omega^{2} g \quad , \...
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Scale transformation of scalars in curved backgrounds
I am puzzled by the concept of scalar fields that arise in conformal field theory in curved backgrounds. In general relativity, so far as I understand it, a scalar field is basically a function ...
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WZW primary fields / correlations in terms of current algebra?
Cross-posted from a Mathoverflow thread! Answer there for a bounty ;)
Given the
$\mathfrak{u}_N$ algebra
with generators $L^a$ and commutation relations
$ [L^a,L^b] = \sum_c f^{a,b}_{c} L^c $ ,
the ...
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Reverse AdS/CFT correspondence?
$\text{AdS}_{n}/\text{CFT}_{n-1}$ correspondence provides a dictionary for one-to-one mapping observables in bulk gravity to boundary conformal field theories. However, does the reverse correspondence ...
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Phase diagram of the Ashkin-Teller model for unequal intraplane couplings
The 2d Ashkin-Teller model is a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetric model consisting of two planes of Ising models on the square lattice with an interplane four-spin coupling:
$$H = -\sum_{&...
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Is the celestial sphere we actually see the Riemann Sphere?
I've been watching a few lectures by R. Penrose where he seems to say that what we see around us is the Riemann sphere. He usually gives the example of an observer floating in deep space or if the ...
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Strange Wick rotation in the computation of string partition function
In order to compute the one-loop vacuum-to-vacuum amplitude for the bosonic string, one runs into
\begin{equation}
Z(\tau) = V_D (q \bar{q})^{-D/24} \int \frac{d^Dk}{(2 \pi)^D} \exp({- \pi \alpha^\...
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OPE Coefficients in Holography
I am having trouble reproducing a calculation from the paper "Holography from Conformal Field Theory".
In a 2d CFT, consider a generalized free field $\mathcal{O}$ with conformal dimension $\...
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How was the critical exponents related to the scaling dimensions of the local operators?
On "The Conformal Bootstrap: Theory, Numerical Techniques, and Applications"(arXiv:1805.04405
) by David Poland, Slava Rychkov, Alessandro Vichi page 5
Consider for example the critical ...
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Ground state vertex operator in superstring theory
In Tongs' "String Theory" lectures chapters 5.4.1-5.4.2 Tong referred to alternative way to find the mass of the different string states using suitable operators, and then integrating the ...
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Partial gauge fixing a point to infinity in conformal field theory [duplicate]
While deriving the structure of a 4 pt. function in CFT, we write the conformal block with respect to the cross ratios
$$ u = \frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}, \; v = \frac{x_{14}^2x_{23}^2}{...
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Faddeev-Popov Method for Gauge Fixing in CFT (Light-ray Operators)
I was attempting to go through the paper by Petr Kravchuk and David Simmons-Duffin: https://arxiv.org/abs/1805.00098 where I encountered the following
Just below E.4, it is mentioned that for the ...
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Derivation of Hofman-Maldacena Bounds
I am trying to study the section where the author of the article - https://gitlab.com/davidsd/lorentzian-cft-notes tries to outline the derivation of the Hofman-Maldacena bounds in Section 6.2. I have ...
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Gauge Fixing in Derivation of Lorentzian OPE Inversion Formula in 2D CFT
I have been looking through the following article: https://arxiv.org/abs/1711.03816 and wish to understand the derivation from scratch. The definition of conformal partial waves and the object of ...
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Finding the OPE coefficients when the correlator and block decomposition is given
I have the following expression which arises while studying minimal models in CFT. The 4-point amplitude of scalars is given by
$$G(z,\overline{z}) = ((1-z)(1-\overline{z}))^{\frac{m-1}{m+1}}\mathcal{...
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Spectral gap (barrier) under centain symmetry
I have recently been taught the problem of preparing the ground state with an adiabatic process, which requires the instantaneous Hamiltonian to have an energy gap between the ground and the first ...
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When is a continuous phase transition in a translationally-invariant 2$d$ classical spin system not described by a CFT?
I'm trying to better understand when I should expect conformal invariance to emerge at a continuous phase transition.
I'm imagining classical, $q$-state spins living on the vertices of a 2d square ...
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Can we compute tree-level amplitudes in string theory using the fundamental domain of $SL(2; \mathbb{C})$?
I am not a specialist in string theory. I understand the computation of tree-level string amplitudes (Veneziano or Virasoro-Shapiro), where three variables are fixed using the symmetry $SL(2;\mathbb{C}...
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OPE of Energy-Momentum Tensor in Polchinski
I am learning Polchinski's textbook on string theory. In section 2.6 I get confused when Polchinski introduce two natural choices of complex coordinates of closed strings
$$w=\sigma^1+i\sigma^2,\tag{2....
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Form of the trace of the energy-momentum tensor in 2D spacetime
I'm going over this article by Davies, in which he derives the form of the energy-momentum tensor (emt) in 2D spacetime assuming a non vanishing trace anomaly.
He considers a metric of the form
$$
ds^...
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Conformal manifold of a supersymmetric field theory
I'm trying to understand what exactly is the conformal manifold of a theory. If I understand it right, the conformal manifold is the space of couplings.
From that point of view, it is just a subset ...
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Dimensional analysis for a theory with a single mass scale
I am reading this paper on $T\overline{T}$ deformation and have a question about eq.(2.7),
\begin{align}
\mu \frac{dS}{d\mu} =\frac{1}{2\pi}\int d^{2}x\ \sqrt{g}T^{i}_{i}.\tag{2.7}
\end{align}
The ...
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What does the WZ term in a WZW action means for string theory on group manifolds?
Let $G$ be a semi-simple Lie group. By Cartan's criterion its Killing form $B(X,Y)$ on $\frak g$ is non-degenerate. We can use it to define an inner product on the whole group by left translation
$${\...
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Polchinski's doubling trick for extending open string theory to the whole complex plane
Open string theory can be described on the upper-half complex plane. To simplify the description of open string theory, Polchinski asserts (eq. 2.6.28 in his Vol. I String Theory book) that it is ...
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How to understand non-invertible symmetries from stacking TQFTs?
I'm reading section 3.3.3 of https://arxiv.org/abs/2305.18296. The idea is to stack a 1d TQFT with G symmetry on a quantum field theory T with symmetry G, then gauge the diagonal G symmetry, the 1d ...
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Scaling dimension of stress tensor
In the paper I am reading, there is a statement that seems to use the fact that the scaling dimension of stress tensor is $\Delta=d$, and I would like to show that this is correct. I found a similar ...
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Massless limit of Dirac fermion correlation functions
In the 2D massless Dirac fermion CFT we have correlation functions like
$$\langle J(z,\bar{z})J(0)\rangle \sim \frac{1}{z^2},$$
where in terms of real Euclidean coordinates $x^0,x^1$, we have $z=x^0+...
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What are the conformal Ward identities associated with the correlation functions of the stress-energy tensor?
I found two papers on this matter, but am having trouble parsing the answer from either of them.
https://arxiv.org/abs/1911.05359
https://arxiv.org/abs/2108.06767
For that matter, what even are the ...
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Topological band crossing and central charge
Conventionally for a 1D topological insulator, SSH model for example, the low energy assumes a linear dispersion i.e. Dirac cone at a topological phase transition point, thus obeys the massless Dirac ...
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About 3D Ising model
In the 2D Ising model, Onsager provided an exact solution for the lattice model in 1944. However, despite numerous efforts, exact solutions for higher-dimensional Ising models have yet to be derived.
...
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Selecting Kac determinant solutions for Yang-Lee Minimal Model
I'm looking into the non-unitary minimal model $\mathcal M_{5,2}$ associated with the Yang-Lee edge singularity. I'm trying to justify which conformal dimensions we expect to appear (easy enough) but ...
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Beta function of CFT perturbed by several operators
As the title says, I am trying to derive the $\beta$ function of a CFT whose action is perturbed by several operators. The main refference I am following is https://arxiv.org/abs/1507.01960 starting ...
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Homothety exercise
Given the metric
$ds^2=-\rho dt^2-tdtd\rho+t^2d\Omega^2$
How to show that $T=T^{\alpha}\partial_{\alpha}=t\frac{d}{dt}$ is an homothety, so that $(L_T g )_{\mu\nu}=2g_{\mu\nu}$?
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Different version of conformal Ward identity
In the book by Di Franceso, Mathieu, Senechal, equation (5.46) shows that (assuming $\bar\epsilon = 0$)
$$
(*) \qquad \langle \delta_{\epsilon, 0} \mathcal{O}\rangle
= - \oint_\infty \frac{dz}{2\pi i ...
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