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 ...

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Correlation function for ghosts in 2D CFT

In Di Fracenso, page 117, it is explained that the correlation function for two primary fields $\phi_1,\phi_2$ of weights $h_1,h_2$ is constrained to be of the form ...
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In a 2D CFT of a free boson $X$, is $X$ a primary field?

A primary field $\mathcal{O}(w,\bar{w})$ with weight $(h,\bar{h})$ is defined by having the following OPEs with the stress tensor: ...
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What is known about the physics of Planckbrane (another brane) in Randall–Sundrum model?

Randall–Sundrum model imagines that our universe is a five-dimensional anti-de Sitter space and the elementary particles except for the graviton are localized on a (3 + 1)-dimensional brane or branes. ...
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299 views

What uniquely defines a CFT?

So, I am quite new to CFT (and a as descriptive answer as possible would be appreciated). I want to know what uniquely defines a CFT in 2D and otherwise. Firstly in 2D, What defines a CFT? So I ...
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210 views

Virasoro operators commutation relations

For the commutation relation in quantising the bosonic string $$\left[L_n,L_{m}\right]=(n-m)L_{n+m}+\frac{D}{12}n(n^2-1)\delta_{n+m,0}$$ we can then calculate this for $m=-n$ in between the vacuum ...
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Cardy counting for time dependent theories

In certain limits of 2 dimensional CFT's, it is possible to compute the entropy of the theory in terms of the density of states which is given by the Cardy formula $$S = 2\pi \left(\sqrt{\frac{c_R ...
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46 views

CFT and temperature

I have tried to think about this for some time but could not really go anywhere. Sorry for the sloppy question and thanks for any pointer. My question is about CFT at finite temperature and ...
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Conformal field theories that are not conformally invariant

Is it possible to construct a field theory that is Weyl-invariant but not conformally invariant? Any references?
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Question about the superconformal index

According to arXiv:1507.08553v1, the superconformal index, defined by $$I(\beta_j) = \mbox{Tr}_{\mathcal{H}}(-1)^F e^{-\gamma\{Q,Q^\dagger\}}e^{-\sum_{j}\beta_j t_j}$$ is independent of the ...
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1answer
40 views

Fields in the action of the Non-linear Sigma Model (WZW)

I am trying to understand the action of the nonlinear sigma model in the context of understanding WZW-models. On Wikipedia, its action is given as ...
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1answer
239 views

Can a scalar field transform nontrivially under a local special conformal transformation?

Is there any way to have a scalar field that transforms non-trivially under local special conformal transformations? Just by the index structure, I can see that the possibilities are $$\begin{align} ...
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42 views

What is a “dynamically generated scale” physically?

A theory like QCD with massless quarks in four dimensions has no explicit mass parameters in its classical Lagrangian. At the quantum level however, instead a mass scale Λ is generated dynamically at ...
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270 views

Wick's theorem for calculating OPE

I am trying to understand a calculation using Wick's theorem. Let $T(z)$ be the analytic part of a stress-energy tensor, and $\phi(z)$ a free boson field. Now, ...
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46 views

Normal Ordering in String Theory: Polchinsky vs. all others

Polchinsky defines normal ordering in string theory as: $$:X^\mu(z,\bar z)X^\nu(w,\bar w): = X^\mu(z,\bar z) X^\nu(w, \bar w) + \frac{\alpha'}{2} \eta^{\mu\nu} \log |z-w|^2$$ and for more ...
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156 views

Why can we not choose the stress tensor in a CFT to be identically symmetric?

The stress tensor for a conformal field theory (or any quantum field theory) can be derived from the action $S$ by the functional derivative $$T^{\mu \nu} ~=~ -\frac{2}{\sqrt{|g|}}\frac{\delta ...
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Generally speaking, given two CFTs, how can we check that they are equivalent?

We can calculate their central charges. And beyond this, are there some other general approaches? Or we must find an isomorphism of operators between the two theories?
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577 views

Zamolodchikov's c-theorem paper

I am reading the 1986 paper [1] where Zamolodchikov proves the c-theorem and I would like to understand how equations (7a), (7b) and (8) are derived from the Callan-Symanzik equation. For ...
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24 views

Reference request: 2D conformal field theory and functions on the triangular lattice

I don't have much of a physics background and was wondering if anyone knows what is meant by "conformally invariant" functions defined on the plaquettes of the honeycomb lattice (ie functions defined ...
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64 views

“Irreversibility” of the RG flow

In his remarkable work, Zamolodchikov proved a theorem regarding two dimensional QFT Renormalization Group (RG) flow, describing a monotonically decreasing function in the flow parameter which is ...
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49 views

Existence of lagrangians at strong coupling

It is well known that some QFT do not admit a lagrangian formulation (like the $(2,0)$ SCFT in $d=6$). Up to my understanding, all the examples that I know of non lagrangian theories are always ...
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25 views

Operator product expansion in 2D cft

My question is about an orbifold CFT theory in 2 dimension. Forgetting about the details, the problem is If I want to find the OPE of two operators, let's say, $O^{++}(z)$ and $O^{--}(z)$, i can ...
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79 views

Orbifold actions and twist operators

A twist operator $\sigma$ is the operator that acts on the untwisted vacuum $|0\rangle$ to create a twisted vacuum $\sigma|0\rangle$. States belonging to the twisted sector of an orbifold are built on ...
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Can the terms in the microscopic model with nonzero conformal spin generate some new term(s) under RG (renormalization group) flow?

As in the book Bosonization and Strongly Correlated Systems at page 66, it says that "We see that the original perturbation with nonzero conformal spin generates the perturbation with zero conformal ...
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64 views

Simple conceptual question conformal field theory

I come up with this conclusion after reading some books and review articles on conformal field theory (CFT). CFT is a subset of FT such that the action is invariant under conformal transformation ...
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594 views

Time-ordering vs normal-ordering and the two-point function/propagator

I don't understand how to calculate this generalized two-point function or propagator, used in some advanced topics in quantum field theory, a normal ordered product (denoted between $::$) is ...
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57 views

Central charge in energy-momentum tensor OPE

I think that general point of view about central charge in books is considering OPE $T(z) T(w)$ for different field theories and finding that general expression for the most singular term is about to ...
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64 views

Operator Product Expansion

I wonder why in OPE in CFT terms like $$ \frac{:O(z) O(w):}{(z-w)^2} $$ occur, for example in the OPE of Energy-momentum tensor with itself: $$T(z) T(w) = \frac{c/2}{(z-w)^4} + \frac{T(z)}{(z-w)^2} ...
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78 views

Is $\phi^4$ theory in 4d conformally invariant at the classial level?

I used to believe the three following statements to be true (at the classical level only): From scale invariance full conformal invariance follows. Scale invariance is present if there are no ...
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1answer
49 views

Globally defined solutions in bc CFT system

Consider $bc$-system which is 2-dimensional CFT of fermions: $S = \int_\Sigma d^2 z \ b \bar{\partial} c + h.c. $ where $\Sigma$ - 2-dimensional manifold of genus $p$, fields $b, c$ have ...
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185 views

Charged CFT observables and AdS/CFT

I have a simple question regarding the holographic dictionary when mapping operators on the CFT side to those in AdS. One piece of the dictionary is that a global symmetry maps onto a gauge symmetry ...
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1answer
59 views

Different OPE channels in bootstrap

Can someone quickly explain what exactly are those different channels (namely s,t,u) in OPE expansions frequently used in conformal bootstrap. Explanation with a simple example will be really helpful. ...
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114 views

1/m Laughlin state and $U(1)_M$ chiral CFT

I am a little confused that people claim that the edge theory of a 1/m Laughlin state corresponds to a $U(1)_m$ chiral CFT. This indicates there should be $m$ primary field operators in $U(1)_m$ ...
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113 views

Cones with deficit angle 2$\pi$ and euler characteristics

I've managed to confuse myself with cones and deficit angles. Let's consider a conical defect in 2 dimensions. So the metric is the usual one in polar coordinates, $$ ds^2 = dr^2 + r^2 d\phi^2,$$ ...
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Why is it that the conformal anomaly has to be scale invariant?

When reading about conformal anomalies, such as in this paper it is often stated that the anomaly (ie. $ \delta W[g]/ \delta \sigma$ where $ W[g]$ is the quantum effective action for gravity) must be ...
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1answer
56 views

Typo in CFT primary field transformation?

Is this a typo? Shouldn't the first formula be $$ \phi'(z', \bar{z}') = \lambda^h \bar{\lambda}^\bar{h} \phi(z, \bar{z}) $$ ? For example, with $$ \lambda = 2 $$ the pair of points $(1,1)$ gets ...
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1answer
19 views

State operator map, and scalar fields in $R\times S^{D-1}$ to $R^D$

This question is generalized version of my previous questions State operator map in $R \times S^{D-1}$ to $R^D$ , State-operator map, and scalar fields and State operator correponding $i.e$ $S^1\times ...
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61 views

How should the path integral change under a dilation?

Let's say I have a two-point function of a scalar field in flat space: $$ \langle \phi(x)\phi(y)\rangle = \int \mathcal D \phi \, \phi(x)\phi(y)\,e^{iS[\phi]} $$ Then I dilate things: $$ \langle ...
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1answer
33 views

State operator map in $R \times S^{D-1}$ to $R^D$

This question is relevant in my former question State-operator map, and scalar fields and State operator corrponding $i.e$ $S\times S$ to $R^2$. (which was wrong, corrected one was states in $R \times ...
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1answer
30 views

State-operator map, and scalar fields

Up so far, i have been studied state-operator correspondence, $i.e$, i have been questioned State operator corrponding $i.e$ $S\times S$ to $R^2$ which was wrong question. By studing Ginsparg's ...
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OPE of two quasi-primaries involves only other quasi-primaries and their derivatives

From Blumenhagen and Plauschinn Introduction to Conformal Field Theory, 2009: "The proof that the OPE of two quasi-primary fields involves indeed just other quasi-primary fields and their ...
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Target space Lorentz symmetry of superstrings in noncritical dimensions

Is the target space Lorentz symmetry of noncritical strings ($D \neq 26$) or superstrings ($D \neq 10$) broken or is it not? Naive arguments suggest that the mentioned anomaly does not exist, since ...
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64 views

Orbifold with discrete torsion

I'm trying to understand some of the early works of Vafa and Witten [1-3]. The way I look at orbifolds is they are the quotient space $M/G$. This is simply a quotient manifold when the action of $G$ ...
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1answer
64 views

Error in Kac's “Vertex algebra for beginners” proof that a Wightman QFT gives rise to a vertex algebra?

Given $$ i[Q_k,\Phi_a(x)]=((x_0^2-x_1^2)\partial_{x_k}-2\eta_k x_k E - 2\Delta_a \eta_k x_k) \Phi_a(x), \quad (1.1.8) $$ applying a coordinate change $t= x_0-x_1$, $\bar{t}= x_0+x_1$ and defining $$ ...
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Definition of primary fields actually leads to a Witt algebra with a minus sign?

Let's take as an example Di Francesco et al. but every source I am aware of is doing the same. First of all, the Virasoro algebra is usually defined as $$[L_m,L_n] = (m - n)L_{m+n} + \frac{c}{12} m ...
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CFTs in the phase space of QFTs [closed]

In the cases I have encountered a CFT is often realised as a RG fixed POINT of the RG flow. Is it also possible to have a whole family/mine/manifold of CFTs instead?
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94 views

Relationship between modular transformations and anyon braiding

In the context of anyon braiding, we have $S$ and $T$ matrices which describe the mutual and self statistics of anyons. In the context of conformal field theory on a torus, we have modular ...
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Linear Dilaton CFT

I´m doing the exercises on the Tong lectures of String Theory, in particular Problem Sheet 2: Consider the tensor: $T(z) = \frac{-1}{\alpha '} :\partial X(z) \partial X(z): - Q \partial^2 X$. By ...
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38 views

DBI action with Weyl Invariance

The DBI action, given by $S_{Dp}=-T_p\int d^{p+1}\xi e^{-\phi(\xi)}\sqrt{-\textrm{det}\left(G_{ab}(\xi)+B_{ab}(\xi)+2\pi{\alpha}'F_{ab}(\xi)\right)}$ is diff and Poincaré invariant. I want to ...
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In what situation will a conformal transformation leave the vacuum invariant

Recently I am reading Francesco's CFT. In section 5.4.2, it considers a generic CFT living on the entire complex plane and maps this theory on a cylinder of circumference L by the transformation: $$ ...