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25
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1answer
631 views

Sigma Models on Riemann Surfaces

I'm interested in knowing whether sigma models with an $n$-sheeted Riemann surface as the target space have been considered in the literature. To be explicit, these would have the action ...
20
votes
5answers
167 views

Connections and applications of SLE in physics

In probability theory, the Schramm–Loewner evolution, also known as stochastic Loewner evolution or SLE, is a conformally invariant stochastic process. It is a family of random planar curves that are ...
19
votes
1answer
74 views

What is known about the classification of N=4 SCFTs with central charge 6?

I was talking about K3 surfaces with some physicists, and one of them told me that the N=4 superconformal field theories with central charge 6 are expected to be relatively scarce. In particular, one ...
18
votes
2answers
432 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 ...
17
votes
3answers
1k views

Why does dilation invariance often imply proper conformal invariance?

Why does a quantum field theory invariant under dilations almost always also have to be invariant under proper conformal transformations? To show your favorite dilatation invariant theory is also ...
16
votes
4answers
3k views

A pedestrian explanation of conformal blocks

I would be very happy if someone could take a stab at conveying what conformal blocks are and how they are used in conformal field theory (CFT). I'm finally getting the glimmerings of understanding ...
16
votes
1answer
538 views

What does it mean for a QFT to not be well-defined?

It is usually said that QED, for instance, is not a well-defined QFT. It has to be embedded or completed in order to make it consistent. Most of these arguments amount to using the renormalization ...
16
votes
2answers
498 views

S-Matrix in $\mathcal{N}=4$ Super-Yang Mills

This is a general question, but what is meant when people refer to the S-Matrix of $\mathcal{N}=4$ Super Yang Mills? The way I understood it is the S-Matrix is only well defined for theories with a ...
15
votes
1answer
136 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 ...
14
votes
1answer
164 views

Do Gauge Theories (CFTs) Have Phase Transitions as the 't Hooft Coupling is Varied?

With an eye toward AdS/CFT, I'm wondering if large $N$ CFTs have a (quantum) phase transition as the 't Hooft coupling is varied. To be more specific -- if I look at correlation functions of ...
13
votes
2answers
342 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 ...
13
votes
2answers
111 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 ...
12
votes
5answers
2k views

What is the conserved quantity of a scale-invariant universe?

Consider that we have a system described by a wavefunction psi(x). We then make an exact copy of the system, and anything associated with it, (including the inner cogs and gears of the elementary ...
12
votes
1answer
588 views

Operator-state correspondence in QFT

The operator-state correspondence in CFT gives a 1-1 mapping between operators $\phi(z,\bar{z})$ and states $|\phi\rangle$, $$ |\phi\rangle=\lim_{z,\bar{z}\mapsto 0} \phi(z,\bar{z}) |0\rangle $$ where ...
12
votes
3answers
2k views

Beginners questions concerning Conformal Field Theory

I started reading about Conformal Field Theory a few weeks ago. I'm from a more mathematical background. I do know Quantum Mechanics/Classical Mechanics, but I'm not really an expert when it comes ...
12
votes
2answers
398 views

Why are there no particles in conformal theories?

In Matt Strassler's recent post (here) he makes the statement that scale invariant (I assume he means conformally invariant, more generally) theories have no particles in them. What's the reason for ...
12
votes
1answer
544 views

Is there a “covariant derivative” for conformal transformation?

A primary field is defined by its behavior under a conformal transformation $x\rightarrow x'(x)$: $$\phi(x)\rightarrow\phi'(x')=\left|\frac{\partial x'}{\partial x}\right|^{-h}\phi(x)$$ It's fairly ...
12
votes
2answers
116 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 ...
11
votes
2answers
1k 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 ...
11
votes
1answer
1k views

What is Euler Density?

Can someone please explain to me what Euler Density is? I have encountered it in Weyl anomaly related issues in various articles. Most of them assumes that its familiar, but I couldn't find any ...
11
votes
1answer
170 views

When are the OPE relevant?

I've seen OPEs commonly used in 2d CFT, it's quite apparent to me that, in that case, it dresses a bridge between the algebraic and the operator formalism especially when combined with radial ordering ...
11
votes
1answer
321 views

What is the CFT dual to pure gravity on AdS$_3$?

Pure $2+1$-dimensional gravity in $AdS_3$ (parametrized as $S= \int d^3 x \frac{1}{16 \pi G} \sqrt{-g} (R+\frac{2}{l^2})$) is a topological field theory closely related to Chern-Simons theory, and at ...
10
votes
1answer
282 views

Scale invariance plus unitarity implies conformal invariance?

What has the reaction been towards the recent paper claiming to have a proof that scale invariance plus unitarity implies conformal invariance in 4d?
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 ...
10
votes
1answer
165 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 ...
10
votes
0answers
209 views

Orbifold CFT of SU(2)/G and SO(3)/G

In this paper by Dijkgraaf, Vafa, Verlinde, Verlinde, orbifold CFT is discussed. In that paper, it outlined that orbifold CFT provides a way to generate the new theories from the old known ones. ...
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 ...
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 ...
9
votes
2answers
471 views

Algebraic/Axiomatic QFT vs Topological QFT

Can anybody please tell me a good source investigating the relation between Algebraic/Axiomatic Quantum Field Theory (AQFT) and Topological Quantum Field Theory (TQFT)? Or is there none?
9
votes
1answer
113 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 ...
9
votes
4answers
359 views

Symmetries of a Free Massless Scalar in Two Dimensions

On p. 49 of Polchinski's book, he says: "Incidentally, the free massless scalar in two dimensions has a remarkably large amount of symmetry -- much more than we will have occasion to mention." Does ...
9
votes
1answer
193 views

Why do we assume local conformal transformations are symmetries in 2D CFT

The global conformal group in 2D is $SL(2,\mathbb{C})$. It consists of the fractional linear transforms that map the Riemann sphere into itself bijectively and is finite dimensional. However, when ...
9
votes
1answer
243 views

Radial quantization and infrared divergences

I am reading Ginspard lectures "Applied CFT" http://arxiv.org/abs/hep-th/9108028 which is not my first material on the subject. He tries to motivates radial quantization on the reason that ...
9
votes
1answer
249 views

Anomalously broken conformal symmetry

I'm trying to understand an argument made by Bardeen in On Naturalness in the Standard Model. The argument is about quadratic divergences in Standard Model. My notation is that the SM Higgs potential ...
9
votes
0answers
137 views

Mathematical motivation of OPE?

In Peskin & Schroeder (and also Cheng which I have skimmed through) they motivate the Operator Product Expansion with a lot of words. Is there any way to motivate it mathematically, e.g. Taylor ...
8
votes
3answers
121 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 ...
8
votes
1answer
506 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 ...
8
votes
2answers
573 views

Superconformal theories

Can anyone tell me where can I read about the notion of "short" and "long" representations? Like what they are etc. From where can I learn the arguments which show that the bosonic subalgebra of ...
8
votes
1answer
392 views

Is Conformal Symmetry Local or Global?

I'm just brushing up on a bit of CFT, and I'm trying to understand whether conformal symmetry is local or global in the physics sense. Obviously when the metric is viewed as dynamical then the ...
8
votes
2answers
337 views

Central charge in a $d=2$ CFT

I've always been confused by this very VERY basic and important fact about two-dimensional CFTs. I hope I can get a satisfactory explanation here. In a classical CFT, the generators of the conformal ...
8
votes
1answer
169 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 ...
8
votes
1answer
234 views

Motivation for the Deformed Nekrasov Partition Function

I have recently been doing research on the AGT Correspondence between the Nekrasov Instanton Partition Function and Louiville Conformal Blocks (http://arxiv.org/abs/0906.3219). When looking at the ...
8
votes
1answer
393 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, ...
8
votes
1answer
283 views

About the general expression of trace anomaly and CFT partition functions

I have put up a question here, http://mathoverflow.net/questions/139685/proof-of-the-general-expression-for-anomaly-in-a-cft-and-its-partition-function Here I am putting up a slightly different ...
8
votes
1answer
451 views

Why does tachyon arise in bosonic string theory?

I am looking for precise mathematical and physical reasons which cause the presence of tachyon in bosonic string theory(specially closed bosonic string theory). Has it to do with the specific form of ...
8
votes
1answer
581 views

What exactly is meant by the conformal group of Minkowski space?

This is sort of a silly question because I'm a total beginner, and I debated whether it was better to ask here or on Math.SE. I decided on here because it's about how physicists use terminology, even ...
8
votes
0answers
185 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 ...
7
votes
3answers
168 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 ...
7
votes
2answers
120 views

Dilation operator in CFT viewed as 'hamiltonian'?

From the commutation relations for the conformal Lie algebra, we may infer that the dilation operator plays the same role as the Hamiltonian in CFTs. The appropriate commutation relations are ...
7
votes
1answer
255 views

String theory - OPE and primary operators

First, a disclaimer: I am new to Physics SE, and I am primarily a mathematician, not a physicist. I apologise in advance for the possibly poor quality of the question, any and thank you for your ...