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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Anomalous dimension of double-trace operators

Is it true that if a single-trace operator, say, $O$ acquires an anomalous dimension $\gamma_o$, then the anomalous dimension of the double-trace operator $O^2$ is $2\gamma_o$? If no, can anyone ...
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Why the full conformal symmetry is $Vir\otimes \overline{Vir}$ instead of $Vir\oplus \overline{Vir}$

In 2D CFT, we have the Virasoro generators $L_m$ and the generators $\bar L_m$, which are such that $[L_m,\bar L_n]=0$. Hence I thought that the full conformal algebra was $Vir\oplus \overline{Vir}$. ...
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Normal ordering by contour integral in CFT

In chapter 6 of Di Francesco, they introduce the normal ordering $$ (AB)(w) = \oint_w \frac{ dz }{ 2\pi i (z-w) }A(z) B(w)\ .\tag{6.130}$$ So far so good. But then starting eq (6.139) $$ \oint_w \...
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Tachyon mode in Brane-Antibrane system (Type II)

Lets First start with coincident $D3$ brane- $D3$ Brane system. Superstring stretch between these two contain a Tachyon mode But GSO projection removes this. For open string, $D$ branes serve just as ...
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Can conformal transformations in $\mathbb{R}^{1,1}$ be analytically continued to $\mathbb{R}^{2,0}$?

In 1+1 dimensions, 2D Minkowski space, a conformal transformation is given by two real functions (of one variable). After Wick rotating the time dimension, giving us 2 dimensional Euclidean space, ...
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Which AdS/CFT correspondences have been found so far?

When I read about AdS/CFT correspondence, there always comes the most famous example of conjectured correspondence, which is the one between type IIB string theory (AdS side) and $\mathcal{N}=4$ ...
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Finite conformal transformations of fields from infinitesimal

I know that in conformal field theories conformal group acts not by pushforwards but (e.g. for scalar field $\phi$ with conformal dimension $\Delta$) $$ \phi(x) \mapsto \phi'(x') = \left| \frac{\...
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Why is $\rm{Conf}(\mathbb{R}^{1,1}) = \rm{Diff}(S^1) \times \rm{Diff}(S^1)$ and not $ \rm{Diff}(\mathbb{R}) \times \rm{Diff}(\mathbb{R})$?

The Minkowski metric for $\mathbb{R}^{1,1}$ is $$ ds^2 = dt^2 - dx^2 = du dv $$ for coordinates $$ u = t + x \hspace{1cm} v = t - x $$ If you do any coordinate transformation that acts independently ...
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What's the partition function and action of thermofield double state (TFD)?

We know that TFD state is consisted of two CFT at each side, and their partition functions are $$ Z_{L/R}= {\rm tr}\,e^{-\beta H_{L/R}}=\int e^{-S_E}\,\,\,, $$ where $S_E$ is just Euclidean action of ...
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What are the excitations in the near critical 2D-Ising model in a magnetic field?

Apparently it is well known that the 2D Ising model with $T=T_C$ in a small magnetic field has a mass gap and correlation length $\xi \sim h^{- \frac{8}{15}} $. Further, in a paper in 1989 ...
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67 views

Why do we regard $z$ and $\bar z$ as independent in CFT? [duplicate]

I have been studying String Theory and CFT for a while, and I am sad to say I do not know why we treat $z$ and $\bar z$ as independent variables, and why we go on to consider the algebra $Vir\oplus\...
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1answer
32 views

Wicks contractions of stress-energy tensor and plane partitions

I am working out the number of wick contraction of a number $n$ of stress-energy tensor in 4D CFT. The strategy is as follows: For 1 stress energy tensor $T_{\alpha\beta}$, you have only one ...
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3answers
69 views

Change of variable in 4-dimensional integral

If I have a measure $d^4 x$ and I want to perform a conformal transformation $x^\mu \rightarrow \frac{x^\mu}{x^2}$, how do I get that the transformed measure is $\frac{d^4 x}{x^8}$? I started by ...
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Finite size energy levels

I am reading this paper. In section 5 they consider a 2d QFT in finite size geometry (cylinder of radius $R$). They say that the energy levels of stationary states ($|n \rangle$) therein will behave ...
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1answer
56 views

If a RG fixed point (FP) is CFT, do all theories flowing into FP CFTs?

Suppose that a RG (renormalization group) fixed point of some RG trajectory (or flow) is a CFT. Then do theories in this trajectory have to be CFTs as well?
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Virasoro algebra commutation (part 2)

This was a sub-question in my previous post that I ask separately now. In Introduction to Conformal Field Theory by Blumenhagen and Plauschinn (springer link) the Virasoro algebra is introduced the ...
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Idea behind boundary states in BCFT

In Blumenhagen's book on CFT, in the BCFT chapter he introduces the concept of a boundary state. TO do this, he first explains how there is a duality between the one-loop open string worldsheet and ...
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1answer
60 views

Water boiling and 3D Ising model

I've been told for a long time that water boiling near critical temperature and the 3D Ising model near critical temperature are described by the same laws, and give a CFT. This is usually mentioned ...
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158 views

Transformation of the energy-momentum tensor under conformal transformations

I am reading the yellow book of Di Francesco about conformal field theory, and there is a step that he takes that I cannot follow while deriving the transformation law of the energy-momentum tensor ...
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Integrability condition of perturbations of Wess-Zumino-Witten (WZW) models

When one tries to analyze the renormalization group of marginal perturbations of Wess-Zumino-Witten (WZW) model in 1+1d, only those "integrable perturbations" can be computed analytically. I wonder ...
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Difference between $\tilde{\textrm{Diff}}_+(S^1)$ and ${\textrm{Diff}}_+(S^1)$

In this paper, where Liouville theory is being studied on a strip, after equation 2.3 it is mentioned that the conformal transformations of the strip are given by the same chiral and anti-chiral ...
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1answer
92 views

Virasoro Algebra commutation

In Introduction to Conformal Field Theory by Blumenhagen and Plauschinn (springer link) the Virasoro algebra is introduced the central extension of the Witt algebra. They give the central extension $$\...
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Veneziano amplitude from 3-point constants

Consider an open bosonic string in the critical dimension at $g_s = 0$ (only the sphere contributes to the string amplitude). The scattering of 4 tachyons is given by the Veneziano amplitude. I'm ...
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51 views

2-sheeted Riemann surface with 2 branch cuts and Torus

A 2-Sheeted Riemann surface, with 2 branch cuts has a genus 1. A ring torus also has a genus 1 (In fact, section 13.4 of John Terning's book, modern supersymmetry and dynamics and duality claims that ...
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50 views

Systematic way of calculating 3-point worldsheet amplitudes

I'm looking for a systematic way of deriving the 3-point functions $\left< V_1(z_1, \bar{z}_1) V_2(z_2, \bar{z}_2)V_3(z_3, \bar{z}_3)\right>$ of the worldsheet CFT of a closed bosonic string. ...
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1answer
45 views

Mobius invariance of the worldsheet 3-point function

Consider the CFT that corresponds to a gauge-fixed closed bosonic string. Ground level string states are described by vertex operators such as $$V(p) = :\exp(i p_{\mu} X^{\mu}(z, \bar{z})):$$ which ...
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1answer
37 views

References regarding Green's function on a square domain in 2D

Premise: I know this question would be better suited to MathSE, but since I endeavour to solve a free CQFT on a bounded domain, I'm confident I'll find a more exhaustive answer here. I'm trying to ...
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1answer
56 views

Generators of conformal transformations change of basis

I recently started going through Introduction to Conformal Field Theory by Blumenhagen and Plauschinn ( springer link ). On page 11, they glue together the generators of conformal transformations as ...
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82 views

OPE of three operators

In the process of thinking about this question, I realized that I don't understand something very fundamental about operator product expansions. Consider a product of 3 local operators in a 2d CFT: $...
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55 views

Factor of $1/2$ in the Sugawara construction

I'm trying to reproduce the Sugawara construction calculation using this reference (page 14). The normal-ordering of two local operators is defined as $$ N(XY)(w)=\frac{1}{2\pi i} \oint_w \frac{dx}{...
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Book with explanation of Kazama-Suzuki models

The book "Introduction to Conformal Field Theory" by Blumenhagen and Plauschinn (BP) covers coset construction and minimal models, but it stops there. The original papers by Kazama and Suzuki seem to ...
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In 2d CFT, why the $T_{zz}$ component of energy-momentum tensor is holomorphic even at quantum level?

In 2d Conformal Field Theory, the $T_{zz}$ component of energy-momentum tensor is treated as a holomorphic function $T(z)=T_{zz}$ at quantum level such as in OPE involved energy-momentum tensor. I ...
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Why is it that the equation of a massless scalar field *must* be conformal invariant?

I'm reading a paper [1], p.111 where it is said that: However, the equation of scalar field with zero mass must be conformal invariant while equation $\square\varphi=0$ does not satisfy this ...
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55 views

Invariance of Liouville action under rescaling

I was studying the Liouville action $$S=\frac{1}{8\pi} \int d^2 x\ \left[ \partial_\mu \phi \partial^\mu \phi + e^{\beta\phi} \right] \tag{1}$$ under the following general form of transformation: $$...
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Use of classical equations of motion inside correlation functions

I am reading this paper by Zamolodchikov about the expectation value of $T \bar{T}$ in $2d$ QFT and I don't understand how he uses the classical equations of motion. For instance, classically, in any ...
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Scaling dimension and system size

I am reading a paper (Sliding Luttinger liquid phases ) which is trying to obtain the scaling dimension of several operators in (condensed matter) field theory. In this paper, the authors mentioned ...
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1answer
113 views

Unitarity representations of CFT in arbitrary dimensions

There is a well defined notion of unitarity of representations in Euclidean Conformal field theories that follows from the requiring unitarity in the Lorentzian space. Under this notion, all states ...
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38 views

Asymptotically free/flat

What does the expression: "...the theory becomes asymptotically free/conformal" mean? If it means that the spacetime $M$ on which the fields are defined is e invariant under conformal ...
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2answers
121 views

Expectation value of descendant fields

I'm trying to calculate the following quantity: $ \left<(L_{-1}\phi)(w_1)(L_{-1}\phi)(w_2) \ldots (L_{-1}\phi)(w_N) \right>$ where $\phi(w_i)$ is a primary operator and $L_{-1}$ is the ...
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Fermionic contribution to central charge in $\mathcal{N}=2$ Super Yang-Mills?

I am trying to replicate the calculation of the central charge for $\mathcal{N}=2$ Super Yang-Mills, by following Weinberg's textbook in section 27.9. He calculates it by finding how one supercurrent ...
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2answers
140 views

Rigorously why there should be an operator product expansion in conformal field theory?

This is probably something quite trivial I'm not getting. I'm studying CFT (conformal field theory) through David Tong's lecture notes and on page 9 he says: We now define the operator product ...
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2answers
136 views

$T \bar{T}$ OPE

In page 157 of Di Francesco (Conformal Field Theory) it is said that the holomorphic and antiholomorphic components of the energy-momentum tensor have the trivial OPE $T(z) \bar{T}(\bar{w}) \sim 0$....
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What is going on in “nonlinear gravity from entanglement in conformal field theories”?

EDIT: I am now convinced that the sign of the logarithmic terms in the equations after 3.29 and 3.30 are wrong (unless I have missed something else). These identites come from looking at ...
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51 views

CFT in momentum space

Is there a way to see the conformal symmetry in momentum space in a CFT? I mean if I can recover the conformal group in some way in momentum space.
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Is string theory self-consistent? (Conformal anomaly)

Recently I attended a very short course on string theory. We went through the standard presentation in light-cone gauge for brevity. We ‘derived’ the Einstein field equation in the following manner. ...
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Does the dictionary always map the bulk operator to the CFT operator?

Using the (extrapolate) dictionary, one can map a bulk field to a boundary CFT operator. The mapped operator is always a CFT operator? How is it guaranteed?
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How are shadows and projector related?

While computing the conformal partial waves, it seems to me that $$\int d^dx |O\rangle\langle\tilde O| = \mathcal{N}^{-1}\sum_{n}|P^n O\rangle\langle P^n O|$$ where $\tilde O$ is the shadow dual of $...
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38 views

Second book on QFT before understanding Conformal Field Theory [duplicate]

I'm a newcomer to Quatum Field Theory, recently I started to work with the very basic book of Lancaster. My final destination is to learn Conformal Field Theory applied to Quantum Phase Transitions ...
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Non-trivial content of AdS/CFT for a generic EFT on AdS

I have a very generic and naive question on the actual content (and usefulness) of the AdS/CFT conjecture in the low energy approximation where one considers a low energy QFT on AdS, comprising ...
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47 views

How does AdS/CFT enact and not just be static geometry?

I understand the duality between the two regions of phase space (as Maldacena described it) that are Anti-de Sitter geometry and conformal field theory as an asymptotic grafting on of scale-invariant ...