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2
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1answer
309 views

Equivalent definitions of primary fields in CFT

I have come across two similar definitions of primary fields in conformal field theory. Depending on what I am doing each definition has its own usefulness. I expect both definitions to be compatible ...
1
vote
0answers
186 views

'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 ...
2
votes
1answer
145 views

Constraining two-point functions of boundary operators on the disk

I'm trying to understand the constraints on the disk CFT correlation function $\langle O_1(y_1)O_2(y_2)\rangle$, where the $O_i$'s are boundary operators that are not necessarily primary. It's a ...
3
votes
0answers
126 views

Derivation of the enhancement of U(1)$_L$ x U(1)$_R$ to SU(2)$_L$ x SU(2)$_R$ at the self-dual radius

Towards the end of the paragraph with the title String theory's added value 2: enhanced non-Abelian symmetries at self-dual radii and abstract C with current algebras of this article, it is explained ...
1
vote
1answer
168 views

Exercise QFT and CFT

Consider the action functional $S[z;t_1,t_2]=\int_{t_1}^{t^2}[g(z,\bar{z})\dot{z}\dot{\bar{z}}]^{\frac{1}{2}}dt$ with $z(t)$ a complex path with end points $z_i=z(t_i),\; i=1,2$. $g(z,\bar{z})$ is a ...
1
vote
1answer
173 views

1-Dimensional Sigma Models

I'm currently interested in 1-dimensional (linear) Sigma Models. In the theory of 2-Dimensional GLSM, the fields can be viewed as an embedding of the worldsheet in some target Manifold of higher ...
3
votes
0answers
85 views

Relating the deformation of Calabi-Yau metrics and the conformal quantum field theories

(v2) As I read e.g. in this question, the nice holonomy group features of Calabi-Yau manifolds are valuable regarding supersymmetry (I suspect because it's a symmetry involving the target manifold, ...
3
votes
1answer
153 views

The neutrality condition and the (non)-vanishing of the one-point correlator for the bosonic vertex operator

Consider the massless scalar field Hamiltonian, \begin{align} H = \frac{1}{2}\int \Pi^2- (\partial_x\phi)^2 dx \end{align} with $\Pi \sim \partial_t\phi$ the conjugate field of $\phi$. This ...
4
votes
2answers
238 views

Conformal fields on compactified manifolds? An apparent paradox!

I would appreciate it if someone tells me how a cft on a compactified manifold (e.g. by means of periodic boundary conditions) can be meaningful? The global conformal invariance is broken due to the ...
5
votes
0answers
82 views

Defects in 3+1 TFTs/2+1 CFTs

I would like to know of good pedagogic references to learn about the notion of "defects" in TFTs and CFTs. I am specially interested in 3+1 TFTs (.and probably about their relation to 2+1 CFTs..) In ...
3
votes
0answers
189 views

Central charge at the fixed point of the ${\cal N}=2$ Landau-Ginzburg theory in $1+1$ dimensions

Let me first believe that the ${\cal N}=2$ Landau-Ginzburg theory does in the IR flow to a non-trivial fixed point and that if the potential is of the form $\Phi ^k$ then the central charge of the CFT ...
3
votes
2answers
427 views

A certain regularization and renormalization scheme

In a certain lecture of Witten's about some QFT in $1+1$ dimensions, I came across these two statements of regularization and renormalization, which I could not prove, (1) $\int ^\Lambda \frac{d^2 ...
6
votes
0answers
173 views

Toda equations and surface operator

I would like to know the reason why the equation (14) in the paper by Yamada is called the Toda equation. \begin{equation} \left[\frac12\sum_{i=1}^N\left(y_i\frac{\partial}{\partial ...
2
votes
1answer
197 views

A certain $\cal{N}=2$ superconformal theory (or is it?)

I want to look at the following theory in $1+1$ dimensions with $\Phi$ being the chiral superfield, $L = \int d^2x d^4\theta \bar{\Phi}\Phi - \int d^2x d^2\theta \frac{\Phi^{k+2}}{k+2} - \int d^2x ...
5
votes
2answers
357 views

Branch-point twist fields and operator insertions on a Riemann manifold

I am having trouble understanding how Eq (2.6) in this paper (PDF) $$Z[\mathcal{L},\mathcal{M}_{n}]\propto\langle\Phi(u,0)\tilde{\Phi}(v,0)\rangle_{\mathcal{L}^{(n)},\mathbb{R}^{2}}$$ generalizes to ...
0
votes
0answers
70 views

Scale invariance Vs Conformal invariance [duplicate]

Possible Duplicate: Why does dilation invariance often imply proper conformal invariance? What exactly is the difference between the two? Can someone give an example of a theory which is ...
2
votes
1answer
213 views

Electromagnetic current-current correlators

Let the free electromagnetic current $J_\mu(x)$ be = $:\bar{\psi}(x)\gamma_\mu Q \psi(x):$ where $::$ is the normal ordering. In this expression why is $Q$ thought of as a "charge operator" instead ...
2
votes
1answer
423 views

What is the meaning of the concepts of “operator mixing” (and anomalous dimensions) [closed]

I am looking for an explanation about the idea of "operator mixing" and its associated concept about when anomalous dimension has to be thought of as a matrix. For example this idea is slightly ...
3
votes
1answer
1k views

Why/How is this Wick's theorem?

Let $\phi$ be a scalar field and then I see the following expression for the square of the normal ordered version of $\phi^2(x)$. $$T(:\phi^2(x)::\phi^2(0):) ~=~ 2<0|T(\phi(x)\phi(0))|0>^2 $$ ...
4
votes
1answer
282 views

Wick Order and Radial Ordering in CFT

I am not so much familiar with the computations tools of conformal field theory, and I just run into an exercise asking to demonstrate the following formula (related to the bosonic field case): ...
3
votes
2answers
323 views

How are functional determinants of Laplace-type operators used in physics?

Many mathematical papers concerning the $\zeta$-regularized Determinant of Laplace-type operators refer for motivation to the broad use of such determinants in mathematical physics, especially in ...
6
votes
2answers
490 views

Why are conformal transformations so prevalent in physics?

What is it about conformal transformations that make them so widely applicable in physics? These preserve angles, in other words directions (locally), and I can understand that might be useful. Also, ...
8
votes
1answer
164 views

Do thermodynamic quantities in CFT correspond to something different in AdS/CFT?

From what I've (hopefully) understood from the AdS/CFT correspondence, physical quantities have a dual version. For example, the position in the bulk is the scale size (in renormalization), and waves ...
6
votes
1answer
112 views

Poisson structure on moduli space of CFTs

The moduli space of CFTs with central charge 26 forms the classical phase space of bosonic string theory, in some sense. Similarily the moduli space of SCFTs with central charge 10 forms the classical ...
7
votes
3answers
164 views

String-theoretic significance of extended CFT

Extended TQFT and CFT have been puzzling me for while. While I understand the mathematical motivation behind them, I don't quite understand the physical meaning. In particular, it's not clear to me to ...
6
votes
1answer
306 views

Conformal transformation equation

I am currently reading Kiritsis's string theory book, and something bugs in the CFT (fourth) chapter. He derives the equation that should satisfy an infinitesimal conformal transformation $x^{\mu} ...
5
votes
2answers
94 views

Significance of massive states in string theory

A free superstring has an infinite tower of states with increasing mass. The massless states correspond to the fields of the corresponding SUGRA. In "Quantum Fields and Strings: A Course for ...
6
votes
3answers
262 views

Modular invariance for higher genus

As far as I understand, there are roughly 2 "common" kinds of 2D conformal field theories: Theories that are defined only on the plane, more precisely, on any surface of vanishing genus. Such a ...
8
votes
0answers
182 views

Holomorphic Factorization in CFT$_2$

Is a CFT$_2$ always holomorphically factorizable? I had this idea because that's what we usually see is taken in string theory e.g (taking $z$ and $\bar{z}$ as independent variables). E.g. Ginsparg ...
3
votes
0answers
87 views

How to construct a 2D conformal field on a mirrored annulus with a looping pattern?

I would like to construct a 2D conformal field on an annulus in which the inner and outer boundaries are like mirrors, and can be approximated by regular polygons (with the same number n of mirror ...
5
votes
1answer
162 views

Do a 1-dimensional conformal theory exist?

can we have in physic or can we speak about 1-d conformal theory in physics ?? for example in this one dimensional theory what would be the generators $ x \partial _{x} $ or $ \partial _{x} $ ??
9
votes
1answer
193 views

AdS/CFT at D = 3

AdS/CFT at D = 3 (on the AdS side) seems to have some special issues which I bundled into a single question The CFT is 2D hence it has an infinite-dimensional group of symmetries (locally). The ...
8
votes
3answers
120 views

“tmf$(n)$ is the space of supersymmetric conformal field theories of central charge $-n$”

I read this intriguing statement in John Baez' week 197 the other day, and I've been giving it some thought. The post in question is from 2003, so I was wondering if there has been any progress in ...
2
votes
2answers
373 views

Why is a critical quantum system described by a conformal theory in one higher dimension of space?

These questions are linked, so I've asked them in a single post: Why is a critical one-dimensional many-body system a two-dimensional conformal field theory?- Why the switch from 1D to 2D? What does ...
5
votes
1answer
350 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 ...
9
votes
1answer
112 views

Conformal QFTs for D > 2

Which conformal QFTs do we know for spacetime dimension d > 2? I know that for D = 4 we have N = 4 SYM and some N = 2 supersymmetric Yang-Mills + matter models. What is the complete list of such ...
4
votes
1answer
184 views

Massive excitations in Conformal Quantum Field Theory

Single particle states in quantum field theory appear as discrete components in the spectrum of the Poincare group's action on the state space (i.e. in the decomposition of the Hilbert space of ...
12
votes
2answers
115 views

How much of the Capelli-Itzykson-Zuber ADE-classification of su(2)-conformal field theories can one see perturbatively?

In their celebrated work, Capelli Itzykson and Zuber established an ADE-classification of modular invariant CFTs with chiral algebra $\mathfrak{su}(2)_k$. How much of that classification can one ...
4
votes
3answers
530 views

Special conformal transformations and locality

In the conformal symmetry, used in some QFT theories, the infinitesimal generators, applying to space-time, are all linear (translations, rotations, boosts, dilatation), except the special conformal ...
10
votes
2answers
62 views

Examples of heterotic CFTs

I'm trying to get a global idea of the world of conformal field theories. Many authors restrict attention to CFTs where the algebras of left and right movers agree. I'd like to increase my intuition ...
5
votes
1answer
48 views

Derivation of Eq. 7.12 in the review paper of Kraus

I'm reading "Lectures on black holes and the $AdS_3/CFT_2$ correspondence" by Kraus. http://arxiv.org/abs/hep-th/0609074 I don't know how one can obtain Eq.7.12. My stupid question is how to obtain ...
5
votes
1answer
579 views

Explanation of Cardy's “a theorem”

There seems to have been some discussion of Cardy's "a-theorem" recently: “It is shown that, for d even, the one-point function of the trace of the stress tensor on the sphere, Sd, when suitably ...
13
votes
2answers
338 views

Which CFTs have AdS/CFT duals?

The AdS/CFT correspondence states that string theory in an asymptotically anti-De Sitter spacetime can be exactly described as a CFT on the boundary of this spacetime. Is the converse true? Does any ...
8
votes
1answer
498 views

AGT conjecture and WZW model

In 2009 Alday, Gaiotto and Tachikawa conjectured an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov ...
10
votes
1answer
162 views

Characters of $\widehat{\mathfrak{su}}(2)_k$ and WZW coset construction

I am currently studying affine Lie algebras and the WZW coset construction. I have a minor technical problem in calculating the (specialized) character of $\widehat{\mathfrak{su}}(2)_k$ for an affine ...
15
votes
1answer
135 views

Miura transform for W-algebras of exceptional type

Miura transform for W-algebras of classical types can be found in e.g. Sec. 6.3.3 of Bouwknegt-Schoutens. Is there a similar explicit Miura transform for W-algebras of exceptional types, say, E6? It's ...
17
votes
2answers
398 views

Edge theory of FQHE - Unable to produce Green's function from anticommutation relations and equation of motion?

I'm studying the edge theory of the fractional quantum Hall effect (FQHE) and I've stumbled on a peculiar contradiction concerning the bosonization procedure which I am unable to resolve. Help! In ...
5
votes
3answers
87 views

which letter to use for a CFT?

In math, one says "let $G$ be a group", "let $A$ be an algebra", ... For groups, the typical letters are $G$, $H$, $K$, ... For algebras, the typical letters are $A$, $B$, ... I want to say ...
5
votes
1answer
93 views

Choice and identification of vacuums in AdS/CFT

I know how we define a vacuum in flat space QFT and also in a curved space QFT. But, can somebody tell me how do the choice of vacuum state in (say) the CFT side of AdS/CFT changes the choice of ...
13
votes
2answers
109 views

Uniqueness of supersymmetric heterotic string theory

Usually we say there are two types of heterotic strings, namely $E_8\times E_8$ and $Spin(32)/\mathbb{Z}_2$. (Let's forget about non-supersymmetric heterotic strings for now.) The standard argument ...