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,...

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Tadpole-free condition

Tadpole-free is a very important condition for perturbative string theory (which is equivalent to the theory to be expanded around the "right" vacuum). For simplicity, let's consider closed string ...
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In Rational Gaussian Model, why must $R^2$ be rational?

I am reading Fusion Rules and Modular Transformations in 2D Conformal Field Theory and Verlinde talks about the rational gaussian model. The simplest class of RCFT's are the rational gaussian ...
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Question about correlation functions of 2d CFTs

I have a question regarding equation (2.22) in Ginsparg's lecture notes on CFTs. Equation (2.22) is $$ \langle T(z) \phi_1(w_1, {\bar w}_1) \cdots \rangle = \sum_{i=1}^n \left( \frac{h_i}{(z-w_i)^2} +...
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Identitity operator-state mapping in CFT

In a general CFT, how can the identity operator be mapped into a state? Can you give examples to demonstrate the systematic way to do the mapping? Take 2D CFT (in String Theory) as examples. For X-CFT,...
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156 views

Beta function of the non-linear sigma model

In chapter 7.1.1. inTong's notes about String Theory could someone sketch how can I show the statements that he nmakes around eq. 7.5 That the addition of the counterterm can be absorbed by ...
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191 views

Central charges c and topological ground state degeneracy GSD

A 2+1D topological field theory (topologically ordered states), implies that the topological ground state degeneracy (GSD) on $T^2$ torus (2D manifold without boundary). For example a level k U(1) ...
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Action of conformal generators on fields

I am calculating the action of the conformal generators on fields, to be more precise on wavefunctions. For now, I'm classical. I will just paste the part of my report on this to show what I am ...
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163 views

How to prove Eq. (2.4.5) in Polchinski's string theory book?

I got one more stupid question in Polchinski's string theory book. In p. 44, it is said The currents $$j(z)=i v(z) T(z), \tilde{j}(\bar{z}) = i v(z)^* \tilde{T}(\bar{z}) \tag{2.4.5}$$ are ...
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References for Understanding Minahan's N=4 SCFT review

This is about the same paper as this thread: Some questions about chapter I.1 (by Minahan) of the "Review of AdS/CFT Integrability" but it was never answered. I have some different ...
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115 views

Explicit evaluation of a radially ordered product

I am trying to understand the application of the operator product expansion to calculate the radially ordered product in the complex plain of $T_{zz}(z)\partial_w X^{\rho}(w)$ which should result in $...
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A Mathematician Who Wants to Learn Particle Physics [duplicate]

Possible Duplicate: Book recommendations I'm a grad student in pure math, wrapping up a thesis in Lie theory. After years of talking to mathematicians and physicists, I've decided that it's ...
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468 views

Breaking of conformal symmetry

I am wondering something about the breaking of conformal symmetry: I know that it can be broken at the quantum level, anomalously, but I never encountered or heard about a model where it is broken "à ...
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't Hooft's landscape of conformally constrained QFTs

As described in "A class of elementary particle models without any adjustable real parameters", "The Conformal Constraint in Canonical Quantum Gravity", and "Probing the small distance structure of ...
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71 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 $S_k\left(\gamma\right)=-\frac{k}{8\pi}\int_{S^2}\...
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66 views

A question from CFT (possibly due to the English expressions)

I am currently reading the book ''Conformal Field Theory'' and encountered a description about which I am very confused. I am afraid to say, this may be due to the fact that I am not a native English ...
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92 views

Special OPE in $\beta\gamma$ system

I would like to find the OPE $$\beta(z)\gamma(w)^{-1}\tag{1}$$ given $$\beta(z)\gamma(w)~\sim~\frac{1}{z-w}\tag{2}$$ from the $\beta\gamma$-system in CFT. Is it possible?
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191 views

Polchinski Equation (7.2.4)

On page 209 of Polchinski's string theory book he writes down the expectation value of a product of vertex operators on the torus; equation $(7.2.4)$. The derivation is analogous to an earlier ...
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121 views

Connections between Density Matrix Renormalization Group and Conformal Field Theory

Can we use the density matrix renormalization group (DMRG) method to understand problems in conformal field theory? I have been trying to find some connections, but nothing is coming up when I search. ...
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Universality classes

I would like to ask about the universality classes. I know that these are the statistical models which describes different phase transitions with different critical exponents. But I would like to know ...
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127 views

Partition functions for a (3+1)-d TQFT

It is well known that for a Chern-Simons theory defined on an arbitrary (2+1)-d oriented manifold, its partition function can be evaluated based on Witten's surgery method. My question is: is there a ...
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148 views

Relation of conformal symmetry and traceless energy momentum tensor

In usual string theory, or conformal field theory textbook, they states traceless energy momentum tensor $T_{a}^{\phantom{a}a}=0$ implies (Here energy momentum tensor is usual one which is symmetric ...
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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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67 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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42 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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62 views

OPE of parity even theories in CFT.

If I consider an OPE of some operators, which belong to a theory where parity is not violated, will I have a constraint on the kind of operators appearing in the right hand side ? For example, I ...
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127 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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123 views

What is the conformal mode of a metric?

I have a problem in terminology. This article talks about the conformal mode of a physical metric. I know what a conformal transformation is. But what is the conformal mode of a metric?
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61 views

Notion of distance in a Conformal Field Theory

I'm confused about the how the notion of distance is used in Conformal Field Theory. Let's take for example the Operator Product Expansion (OPE). In a conformal field theory, due to the scale ...
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How is the two-point function of an operator dual to a scalar ADS field obtained in ADS/CFT?

The two point function of an operator dual to a scalar field in ADS/CFT can obtained directly from computation of the on-shell action in momentum space and then taking it back to position space. The ...
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Where to look up calculations in CFT? [duplicate]

I am a mathematician, but occasionally I need the answers to questions that might already have been calculated by physicists. Recently I've found myself asking questions like "what is the space of 0-...
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27 views

Transformation law for the eigenfunction representation of dilations (scale transformations) in CFT

On page 32 section 2.1.2. of http://arxiv.org/abs/hep-th/9905111 a representation $\phi(x)$ for the conformal group is chosen such that $D\phi(x) = -i \Delta\phi(x) $. After that it is stated that ...
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CFT conformal weight vs. scaling dimension

I was wondering if anybody could clarify what the difference between the conformal scaling dimension $\Delta$ and the conformal weight $h$ is? Is it correctly understood that $\Delta$ is related to ...
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Entanglement in Quantum field theory [duplicate]

How is entanglement represented in a field theory? For instance how can I represent a maximally entangled state such as a Bell state? Would such an approach also apply in a Conformal field theory ...
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How to calculate anti analytic conformal weights

In books like Mussardo's "Statistical Field Theory" and "Conformal Field Theory" of Di Francesco et al, there is no clear explanation on how to calculate the anti analytic conformal weights $\bar \...
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What is a zero temperature horizon?

While reading the paper "Disorder horizons: Holography of randomly disordered fixed points" by Hartnoll and Santos, I came across this: We are interested in solutions with a zero temperature ...
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Does Operator Product Expansion form an algebra?

The operator product algebra in CFT is defined as $$\mathcal{O}_i(z,\bar{z})\mathcal{O}_j(\omega,\bar{\omega}) = \sum_{k} C^k_{ij}(z-\omega,\bar{z}-\bar{\omega})\mathcal{O}_k(\omega,\bar{\omega}).$$ ...
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How to show that $L_n^\dagger=L_{-n}$ for the Virasoro generators in CFT?

It seems to be a common knowledge that for the Virasoro generators in CFT the rule of hermitian conjugation reads $$L_n^\dagger=L_{-n}$$ There is probably more then one way to show this. I ask for a ...
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$i\epsilon$ in CFT correlation functions

M. Luescher in his talk on p.6 writes that the 2-point correlation function of a Hermitian local field $O_k$ of scaling dimension $d=3-k$ looks like $$ \langle 0| O_k(x) O_k(y) |0\rangle = A_k (x-y-i ...
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Applicability of Cardy's “doubling trick” to the 2D Ising Model

In Section 11.2.2 of the book on Conformal Field Theory by di Francesco, Mathieu, and Senechal (page 417), the two point function on the Upper Half Plane is written as being equal to the four point ...
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AdS3/CFT2 duality for free bulk scalar

I am looking at lecture notes by Kaplan. Through chapters 4, 5, and 6 he takes the free field in $\mathrm{AdS}_3$ to the boundary to create a CFT2 primary field. The result is equation (6.5): $$\...
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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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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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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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22 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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74 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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AdS/CFT: why is the fifth coordinate in AdS space inversely proportional to an energy scale?

In several different articles about the AdS/CFT correspondence, it is stated that one can show that the fifth coordinate $z$ on the AdS side, in coordinates such that the AdS metric becomes: $$ds^2 = ...
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What is the meaning of conformal mass on a branched manifold?

Consider the following QFT on the "branched" manifold, $\mathbb{H}^2 \times S^1_q$ where $S^1_q$ is an unit radius circle whose angular coordinate goes from $0$ to $2\pi q$, and hence the "branching". ...
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Dilations in momentum space

I don't quite understand what's going on here. Let's suppose I have a dilation in real space. The generator is $D=x^j \partial_j$, so an infinitesimal dilation is $\delta x^i = Dx^i = x^j \partial_j ...
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111 views

Deriving the energy-momentum tensor conservation equation in complex coordinates, Polchinski 2.4.2

I am trying to derive equation (2.4.2) in Polchinski's string theory textbook, $$\overline \partial T_{zz}=\partial T_{\overline z \overline z} = 0 \tag{2.4.2}.$$ Using the conservation equation, ...