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13
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2answers
2k views

What is the physical interpretation of the S-matrix in QFT?

A few closely related questions regarding the physical interpretation of the S-matrix in QFT: I am interested in both heuristic and mathematically precise answers. Given a quantum field theory when ...
3
votes
1answer
185 views

Question on Section 9.1.3 in “Conformal Field Theory” by Philippe Di Francesco et. al

Question on Section 9.1.3 in "Conformal Field Theory" by Philippe Di Francesco et. al. The basic idea of the Coulomb-gas formalism is to place a background charge in the system, making the $U(1)$ ...
2
votes
1answer
190 views

What is the operator for the edge current of a fracional quantum Hall state?

The edge of a fractional quantum Hall state is a chiral conformal field theory. In the Laughlin case it corresponds to the chiral boson, $$ S = \frac{1}{4\pi} \int dt dx ...
4
votes
0answers
160 views

Wilson lines, boundary conditions, surface defects of TQFTs

I asked the following question in mathematics stack exchange but I'd like to have answers from physicists too; I have been studying (extended) topological quantum field theories (in short TQFTs) from ...
2
votes
1answer
156 views

constraint on scaling dimension

How can we show that for any scalar operator $\Delta\geq1$ (where $\Delta$ is the scaling dimension)? Where can I find a reference for reading where it comes from?
1
vote
1answer
175 views

Definition of CFT

A standard QFT cannot be defined as a set of Poincare-invariant correlation functions because this does not take into account the possibility of non-perturbative effects (e.g. instantons) Can we ...
2
votes
2answers
384 views

Invariance of Maxwell's Equations under inverting variables - Reference and use

Some months ago, an ArXiv paper mentioned in passing that Maxwell's Equations were invariant under reciprocating the variables, or at least this results in a dual set of Maxwell Equations. (Actually I ...
8
votes
1answer
414 views

Clarification on “central charge equals number of degrees of freedom”

It's often stated that the central charge c of a CFT counts the degrees of freedom: it adds up when stacking different fields, decreases as you integrate out UV dof from one fixed point to another, ...
1
vote
0answers
279 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 ...
3
votes
1answer
297 views

2D Ghost CFT and two-point functions

For some reason I am suddenly confused over something which should be quit elementary. In two-dimensional CFT's the two-point functions of quasi-primary fields are fixed by global $SL(2,\mathbb ...
9
votes
4answers
2k views

Conformal transformation/ Weyl scaling are they two different things? Confused!

I see that the weyl transformation is $g_{ab} \to \Omega(x)g_{ab}$ under which Ricci scalar is not invariant. I am a bit puzzled when conformal transformation is defined as those coordinate ...
4
votes
1answer
77 views

How can we have massive states of strings and CFT on the string worldsheet at the same time?

Ok, so we can have conformal invariance on a string world sheet. However, it is well known that to preserve conformal symmetry we require states to be massless. So how is it that string theories ...
2
votes
1answer
184 views

Conformal Invariance and the OPE

I am reading Polchinski's String Theory now and am having trouble with a couple of things mentioned on Page 45 under "Conformal Invariance and the OPE". Here is the paragraph I am reading. The ...
2
votes
1answer
328 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
147 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
174 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
185 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
89 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
158 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
249 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
83 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
203 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
449 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
175 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
204 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
407 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
250 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
448 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
309 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
346 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 ...
7
votes
2answers
529 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
175 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
118 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
178 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
338 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
99 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
292 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
195 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
88 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
170 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
203 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
127 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
388 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
382 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
117 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
193 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 ...