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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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Metric under conformal transformation

I have a question regarding the conformal factor $\Omega(x)$ when dealing with a conformal transformation. We know that under a change of coordinates $x\rightarrow x^{'}=x^{'}(x)$ our metric changes ...
Spoonszzz's user avatar
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Statement clarification: When do we have $\Delta = \frac{d-1}{2} q_R$?

This statement is for SCFT, where $\Delta$ is the conformal weight, $q_R$ is the R-charge. How many supersymmetries do we need? Do we need chiral primary scalar, or just chiral scalar? How to derive ...
Kangning Liu's user avatar
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Super Hilbert space of SYM

I hope this message finds you well. I am currently try to understand explicitly, at least in some sense, $d=4$, ${\cal N}=4$ super Yang-Mills theory. What is the explicit construction of the super ...
d'Alembert's user avatar
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Coulomb gas correlators on the sphere

I am trying to understand Appendix B.2 of Nakayama's notes (https://arxiv.org/abs/hep-th/0402009), wherein he derives the correlation functions of the following action (Coulomb gas) on the sphere: $$ ...
Jay Padayasi's user avatar
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Verifying a Conformal Ward Identity for the Free Boson

I've always been very uncomfortable with the following two conformal Ward identities [c.f. Di Francesco, pg 107, Eqs. (4.66) and (4.67)]: $$ \langle (T^{\rho \nu} - T^{\nu \rho})X \rangle = -i \sum_i \...
Zack's user avatar
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Splitting Scalar into Holomorphic and Anti-Holomorphic Parts

I am reading Tong’s string theory lecture notes. On page 78, he splits the 2d free scalar into left- and right-moving parts, seemingly using the classical equation of motion as justification. Why is ...
Sam's user avatar
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Weyl transformation of induced metric

Consider the Weyl/conformal transformation in four dimenions $$\tilde{g} \enspace = \enspace \Omega^2 g \quad \Longrightarrow \quad \sqrt{-|\tilde{g}|} \enspace = \enspace \Omega^4 \sqrt{-|g|}$$ The ...
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Ward identity for special conformal transformation in d dimensions

I am reading CFT from the yellow book ( "Conformal Field Theory" by Francesco, Mathieu, Sénéchal ). In section 4.3.2, they calculate three Ward identities corresponding to (i) translation ...
baba26's user avatar
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3 votes
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Cyclic Universe Problems

In Penroses's hypothesis, at the end of each iteration the universe undergoes a conformal transformation, meaning distances are rescaled. If I am right, it implies that a planet from the previous ...
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Why Is There No Oscillator Representation for Operators in Planar ${\cal N}=4$ SYM Theory?

I'm studying the planar ${\cal N}=4$ Super Yang-Mills (SYM) theory and I'm curious about the representations of its operators, specifically the Hamiltonian and the dilatation operator. In many quantum ...
iron's user avatar
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OPE limit of four-point function in de Sitter space

I have been trying to read the paper 'Cosmological Collider Physics'. This paper studies several things, of which the most interesting to me was studying the correlation function in de Sitter space by ...
Chandra Prakash's user avatar
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Conceptual Difference Between OPE and Propagator

I'm specifically working with a 2d free scalar CFT. In this case, the propagator is $$\langle X(\sigma) X(\sigma')\rangle=-\frac{\alpha'}{2}\ln(\sigma-\sigma')^2\tag{p.78}$$ while the OPE between $X(\...
Sam's user avatar
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Definition of the Conformal algebra generators

In the textbook "Introduction to Conformal Field Theory" by Blumenhagen and Plauschinn (2009), Section 2.1, the generators of the Lie algebra corresponding to the conformal group for the ...
Joseph Shtok's user avatar
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Conformal equivalent to Schwarzschild metric

Consider Schwarzschild spacetime in Eddington-Finkelstein coordinates $(v,r,\theta,\phi)$ $$g \enspace = \enspace -f(r) \, dv^2 + 2 \, dv \, dr + r^2 \, d\Omega^2 \quad , \qquad f(r) = 1 - \frac{2m}{r}...
Octavius's user avatar
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Isomorphism of Virasoro Algebra with Different Highest Weights

Recently I was reading the big yellow book on "Conformal Field Theory" by P. Francesco et.al, and in appendix 8.A.1, it defined a covariant linear map for fusion process among irreducible ...
Mohammad. Reza. Moghtader's user avatar
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Ward Identities in Conformal Theory

For a 2D free boson model, $$ S=\frac{1}{2} g \int d^2 x \; \partial_\mu \varphi\;\partial^\mu \varphi $$ The energy-momentum tensor should be $$ T_{\mu \nu}=g\left(\partial_\mu \varphi \partial_\nu \...
Si-Yuan Chen's user avatar
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Conformal invariance and mass terms in QFT

We know that a physically sensible QFT must be renormalizable. If I understand correctly, when this happens, the theory has "asymptotic freedom" and is conformally invariant past some high ...
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Deriving OPE between vertex operator: Di Francesco Conformal Field Theory equation 6.65

How does one get Di Francesco Conformal Field Theory equation 6.65: $$ V_\alpha(z,\bar{z})V_\beta(w,\bar{w}) \sim |z-w|^{\frac{2\alpha\beta}{4\pi g}} V_{\alpha+\beta}(w,\bar{w})+\ldots~?\tag{6.65}$$ ...
Jens Wagemaker's user avatar
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Boundary-condition-changing Operators for Free Boson BCFT with Dirichlet Boundary Conditions (or more general BCFTs)?

Is there any literature about boundary-condition-changing (b.c.c.) operators for the Free Boson with Dirichlet Boundary Conditions? The b.c.c. operators I'm interested in would replace boundary ...
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Operator Product Expansion (OPE) coefficients of free massless theory

Consider the action of the free massless bosonic theory in $2+1D$ $$ S = \int d^3x \partial_{\mu}\phi(x) \partial^{\mu} \phi(x). $$ The single-particle spectrum (on the surface of a sphere) is given ...
eon97's user avatar
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2 answers
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Could a universe be expanding if its physics were scale invariant?

Imagine a universe where every field is massless and has scale-invariance. Would the expansion/contraction of the universe still be happening there? would it be detectable? Would it affect the ...
P. C. Spaniel's user avatar
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Which states contribute to the largest gap for WZW model with $so(16)_1$? [closed]

I was told that the WZW model with $so(16)_1$ occurred at $(c,\bar c )=(8,8)$, and it had a gap, i.e. the smallest state with conformal dimension $\Delta = h+\bar h\neq 0$, and it was said to be $2$. ...
ShoutOutAndCalculate's user avatar
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Step in derivation of the short distance behavior of the heat kernel in Francesco's Conformal field theory book equation 5.190

Can someone explain equation 5.190 of Francesco's Conformal Field Theory? It is a step in the derivation of the short distance behavior of the heat kernel in Francesco's Conformal field theory book. ...
Jens Wagemaker's user avatar
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Finiteness of the Kac table for minimal model

I am currently reading CFT from Di Francesco. I am stuck at not understanding why the kac table for minimal models $(p, q)$ where $p$ and $q$ are co-prime, only has fields with conformal dimension $h_{...
Suriyah R K's user avatar
2 votes
1 answer
76 views

Why does a normal ordered product of operators (in CFT) have 0 expectation value?

Why does a normal ordered product of operators (in CFT) have 0 expectation value? The definition (Francesco - Conformal field theory pg. 174) of the normal ordered product of two operators $A(z)$ $B(z)...
Jens Wagemaker's user avatar
3 votes
0 answers
60 views

Can the Wilson-Fisher fixed point be reached from the massless $\phi^4$ action?

Most textbooks and papers work out the derivation of the Wilson-Fisher fixed point for $\phi^4$ starting from the massive action (in Euclidean space) $$S = \int d^d x \biggl( \frac{1}{2} \partial_\mu \...
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What is the definition of a conformal symmetry? [duplicate]

I have been very confused by this after some recent reading. So as far as I know, a conformal transformation (according to the definition in di Francesco et. al.'s book on CFT) is an active coordinate ...
QFTheorist's user avatar
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Is the causal structure completely determined by the Weyl tensor alone?

By causal/conformal structure I mean the context of Malament's 1977 theorem. If I understand correctly this means that any two spacetimes which agree about all of the future-directed continuous ...
Daniel Grimmer's user avatar
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Calculation of conformal dimension for Ising model in two dimensional space

Recently I was reading Ph. Di Francesco's book, "Conformal Field Theory", and in section 7.4.2 where it discussed about Ising model, conformal dimensions $(h,\bar{h})$ are deduced from ...
Mohammad. Reza. Moghtader's user avatar
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Which Lorentzian metrics are comformally equivalent to some Einstein metric or some Ricci flat metric?

I know that given a conformally flat Lorentzian metric, I can implement a Weyl transformation, $$g_\text{ab}\mapsto\bar{g}_\text{ab}=\Omega^2 g_\text{ab}$$ to flatten it as $\bar{g}_\text{ab}=\eta_\...
Daniel Grimmer's user avatar
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2 answers
123 views

How to see that the Ising CFT has $c = 1/2$ while the quantum XY CFT has $c = 1$ via Jordan-Wigner?

It is well known that the CFT at the critical point of the 1+1d transverse field Ising model has central charge 1/2. This can be attributed to the fact that, after a Jordan-Wigner transformation, the ...
Midnight Conqueror's user avatar
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References for Bulk-Boundary correspondence

I have been reading the Quantum Hall Effect from Prof. David Tong's lecture notes. In the Edge Mode chapter, he talked about Bulk-Boundary correspondence where he reproduces the Laughlin wavefunctions....
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When are 2d RCFTs group-theoretical?

An RCFT is defined to have finite conformal primaries. Verlinde lines lebeled by these primaries form a fusion category. These topological defects are called category symmetry in modern language. ...
Yankun Ma's user avatar
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Distance conjecture being false in $\phi^4$ theory

One part of Distance conjecture states that free theory (Higher spin) are at infinite distance away from any arbitrary point on conformal manifold where the distance is measured with respect to ...
aitfel's user avatar
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1 vote
1 answer
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What implements finite conformal transformations in two dimensions?

In a two dimensional conformal field theory I have two sets of generators giving a representation of the Virasoro algebra $$L_n, \bar{L}_n, n \in \mathbb{Z}$$ $$[L_n,L_n] = (m-n) L_{m+n} + c\frac{m(m^...
DerHutmacher's user avatar
3 votes
1 answer
129 views

What are single-, double- and multi-trace operators in AdS/CFT?

Can someone explain what are single-, double- and multi-trace operators are in AdS/CFT? I am a senior undergrad and only recently started studying AdS/CFT from TASI lectures and could not make much ...
QFTheorist's user avatar
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Picture Number in String Vertex Operator

How can I know what is the Picture of a particular vertex operator? For example in 8.3.15 in Polchinski's book Vol.1, the Vertex Operators for the Enhanced Gauge symmetry are given by \begin{equation}...
Roddy 's user avatar
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3 votes
3 answers
189 views

Closed form expression of 2D CFT integral

I am currently working on a 2d CFT and am wanting to compute a complex plane integral, making sure I take into consideration potential contact terms as well. The integral in question is $$ \int_{\...
NoName's user avatar
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3 votes
2 answers
181 views

Why does no one add Einstein-Hilbert term to CFT in AdS/CFT?

As I work through AdS/CFT exercises, it struck me that there seemed no one doing the following. Suppose we have a holographic CFT. By some reeconstruction method, we can write CFT operators in terms ...
Bulldozer's user avatar
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1 answer
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Choice of spacetime foliation while quantising a conformal field theory

I was reading Rychkov's EPFL lectures on $D\geq 3$ CFT (along with these set of TASI lectures) and in chapter 3, he starts discussing radial quantisation and OPE (operator product expansion). I ...
QFTheorist's user avatar
5 votes
1 answer
174 views

Book recommendation for CFT in condensed matter theory

I've been looking for sources about conformal field theory (CFT) applications in condensed matter theory (CMT) like bosonization, critical phenomena, and QFT anomalies. I have studied CFT from ...
1 vote
1 answer
72 views

Inverstion matrix as parallel transport in CFT

In Gauge/Gravity Duality by Ammon and Erdmenger (pg. 105), and CFT Lecture Notes by Osborn (pg. 4), it is stated that the interval between two points $x$ and $y$ transforms under general conformal ...
Darkseid's user avatar
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-1 votes
2 answers
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Extracting the dimension of an operator from algebra

I may misinterpret the question. In the lecture note of conformal field theory, arXiv:2207.09474, it says the following where for $P^\mu=i\partial_\mu$ and $D=ix^\mu \partial_\mu$. I am confused ...
Tanmoy Pati's user avatar
1 vote
0 answers
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Stress-energy tensor of a $2\text{d}$ conformal field theory [closed]

Does stress-energy tensor of a $2\text{d}$ conformal field theory split into holomorphic and anti-holomorphic parts as follows? In a conformal field theory, stress-energy tensor $$T_{\mu\nu} = \frac{1}...
user avatar
1 vote
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Why is $\epsilon$ at most quadratic in CFT with $d\geq 3$? [duplicate]

I am trying to read through these notes on CFT, and author reaches a point in chapter $2$ saying: $$\Big(\eta_{\mu\nu}\square + (d-2)\partial_{\mu}\partial_{\nu}\Big)(\partial\cdot\epsilon) = 0\tag{2....
Mahammad Yusifov's user avatar
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1 answer
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Large central charge limit for Virasoro blocks

On page 3 of this paper (https://hal.science/hal-00627906v3/document), the authors say that in the $c\to\infty$ limit only the global generators will survive when computing the conformal block. In ...
furious.neutrino's user avatar
4 votes
2 answers
71 views

Is there any meaning or statistical distribution associated with the Jacobi's $\theta$ functions?

The Jacobi's theta functions $$\theta_1(0,\tau )=0$$ $$\theta_2(\tau) =\sum_{n\in \mathbb{Z}} q^{(n+\frac{1}{2})^2 /2 }$$ $$\theta_3(\tau) =\sum_{n\in\mathbb{Z}} q^{n^2/2}$$ $$\theta_4(\tau) =\sum_{n\...
ShoutOutAndCalculate's user avatar
1 vote
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Conjugate momenta in Radial Quantization

When we radially quantize a conformal field theory, is there at least formally a notion of a conjugate momentum $\Pi$ to the primary fields $O$ which would satisfy an equal radius commutation relation ...
pseudo-goldstone's user avatar
1 vote
1 answer
39 views

Reading operators from Kac tables and performing operator product expansions with them

I don't know much CFT, but I'm hoping to learn some of its main results. I'm particularly interested in minimal models in the context of quantum spin chains. I tried reading BPZ, but I'm realizing ...
user196574's user avatar
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1 vote
2 answers
108 views

Inconsistency in Virasoro expansion of stress energy

As explained in Axiom 2.3 on page 7 of https://arxiv.org/abs/1609.09523, the independence of the stress tensor $$T(y)=\sum_{n}\frac{L_n}{(y-z)^{n+2}}$$ on the choice of expansion point $z$ leads to ...
phonon's user avatar
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