# Tag Info

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For a survey on what was known in 1999 on the subject, there is the review A Hiker's Guide to K3 - Aspects of N=(4,4) Superconformal Field Theory with central charge c=6 by Werner Nahm and Katrin Wendland. I have not been following this subject, so I am not sure whether the current picture is substantially different. Added Some of the papers citing that ...

19

The answer is not known, but many believe it is: "Yes, every CFT has an AdS dual." However, whether the AdS dual is weakly-coupled and has low curvature -- in other words whether it's easy to do calculations with it -- is a different question entirely. We expect, based on well-understood examples (like $\mathcal N=4$ SYM dual to Type IIB strings on ...

15

The symmetry you are asking about is usually called a scale transformation or dilation and it, along with Poincare transformations and conformal transformations is part of the group of conformal isometries of Minkowski space. In a large class of theories one can construct an "improved" energy-momentum tensor $\theta^{\mu \nu}$ such that the Noether current ...

12

Another use of the SLE approach seems to be (I haven't read the papers below much beyond their abstracts) as a tool to probe for the presence of conformal invariance in various systems, when a direct (numerical or experimental) verification is difficult. In this approach, (i) one extracts suitable non self-crossing paths, (ii) one determines (empirically) ...

12

There are already nice answers from both a physical and mathematical perspective, explaining the basic idea - given the algebra of holomorphic operators (or equivalently the symmetry algebra) of a CFT, we can write down a collection of equations (the Ward identities) that the partition function of the theory must satisfy on any Riemann surface. The space of ...

12

Euler density is simply the integrand in $2n$ dimensions of the integral that is equal to the Euler characteristic. The Euler characteristic may be written as the integral of the following Euler density in $2n$ dimensions: $$E_{2n} = \frac{1}{2^n} R_{i_1 j_1 k_1 l_1} \dots R_{i_n j_n k_n l_n} \epsilon^{i_1 j_1 \dots i_n j_n} \epsilon^{k_1 l_1 \dots k_n l_n} ... 12 The operator-state correspondence says that all states in the theory can be created by operators which act locally in a small neighborhood of the origin. That is to say that the entire Hilbert space of a CFT can be thought of as living at a single point. The key here is that for CFTs we have radial quantization, and states evolve radially outwards unitarily ... 11 concerning the first question, related to the German text by Matthias Gaberdiel (greetings to him): Closed algebra implies symmetry It is enough to construct the generators of an algebra - in this case, conformal algebra - and calculate their commutators [L_m,L_n] and so on. If the commutators are linear combinations of other generators, we say that the ... 11 The S-matrix (scattering matrix) is the unitary operator S that determines the evolution of the initial state at t=-\infty to the final state at t=+\infty.$$|\psi(t=+\infty)\rangle = S |\psi(t=-\infty)\rangle$$This matrix/operator is therefore a collection of complex numbers that are ready to calculate the probabilities of various scattering ... 11 SLEs can be used in order to define a certain kind of QFT. You can check M. Douglas' talk, from page 28 forward: Foundations of Quantum Field Theory (PDF). There's also another great article, Conformal invariance and 2D statistical physics. You may also like SLE and the free field: Partition functions and couplings. Finally, there's an approach to try and ... 10 Now that we have a physicist's perspective, I don't feel too bad outlining conformal blocks from a mathematician's point of view. Presumably there is a dictionary connecting the two worlds, but I don't understand physics well enough to say coherent sentences about it. I apologize in advance for any confusion - this is not a very pedestrian topic. I'll ... 10 I recommend you Chapter 5 (page 150+) of the AdS Bible, http://arxiv.org/abs/hep-th/9905111 Concerning your individual questions, which are mostly answered at the beginning of that Chapter, the additional Virasoro generators correspond to bulk coordinate reparametrizations that preserve the metric at infinity, but they do map the ground state to ... 9 I did a bit of reading about this, and it turns out that conformal blocks are actually quite relevant to my research! So I figured it was worth the time to investigate in some more detail. I've never studied conformal field theory formally, but I hope I'm not writing anything outright wrong here. (I lost my first draft and had to reconstruct it, which is why ... 9 The Virasoro algebra is a centrally extended algebra. This means that in every representation, its central element must be represented by the unit operator. Thus (for a non-vanishing central charge) it cannot be fully implemented at the quantum level as a symmetry of the vacuum, otherwise one can get a contradiction of the type 1 |0\rangle = 0 |0\rangle . ... 8 I) Here we discuss the problem of defining a connection on a conformal manifold M. We start with a conformal class [g_{\mu\nu}] of globally^{1} defined metrics$$\tag{1} g^{\prime}_{\mu\nu}~=~\Omega^2 g_{\mu\nu}given by Weyl transformations/rescalings. Under mild assumption about the manifold M (para-compactness), we may assume that there ... 8 The example i know does not use 't Hooft's coupling, but i think it may address your question in a more straightforward way (i'll be "loose" with constants and numerical pre-factors, but will keep all the relevant info and details). Think of a 0-dimensional scalar field (bosonic D0-brane) with a quartic potential à la V(\phi) = \mu\,\phi^2 + ... 8 Ben-Zvi & Frenkel denote vertex algebras V,W,... They're using the labels specifically for the spaces of states, but one could also use them to refer the whole package. Alternately, one sometimes sees all caps abbreviations: YM_2, SYM_{4,G},... There is not to my knowledge any conventional notation for morphisms of field theories. 8 There is, I think, no really standard symbol for the generic (chiral) CFT used universally, but there is within the different formalizations. When chiral CFTs are modeled by vertex operator algebras, the standard symbol is usually "V" (for obvious reasons) as user388027 notes in his reply.. When chiral CFTs are modeled as conformal nets, then (as you ... 8 The Weyl transformation and the conformal transformation are completely different things (although they are often discussed in similar contexts). A Weyl transformation isn't a coordinate transformation on the space or spacetime at all. It is a physical change of the metric, g_{\mu\nu}(x)\to g_{\mu\nu}(x)\cdot \Omega(x). It is a transformation that changes ... 7 The left-hand integration is to be interpreted as over a domain R in the set \{(z,\bar z)\,|\, \bar z = z^*\} which defines a copy of \mathbb R^2 in \mathbb C^2. Let \sigma^1 and \sigma^2 be real coordinates on this surface. Using the results on page 33, we find that \begin{align} \partial_zv^z &= \frac{1}{2}(\partial_1 ... 7 Dear Anirbit, great questions. Long and short representations are not difficult to be defined and most introductory texts to supersymmetry explain them. However, some of them don't use this particular terminology, so let me tell you: long multiplets are representations that transform nontrivially under all supercharges. So you can't find a supercharge that ... 7 The quantity \partial_\mu \epsilon_\nu +\partial_\nu\epsilon_\mu is just the variation of the metric g_{\mu\nu} under the (infinitesimal) diffeomorphism you wrote, x^\mu\to x^\mu+\epsilon^\mu. Your equation just says\partial_\mu \epsilon_\nu +\partial_\nu\epsilon_\mu = C \delta_{\mu\nu} for some $C$ which means the condition that the variation of ...

7

Conformal field theory is the theory of scale invariance (or large-order behavior) in two dimensions. Scaling means dependence on angles only. In 2d, group of angle-preserving (conformal) transformations is infinite-dimensional, and in fact there are only a finite number of degrees of freedom in a 2d metric after conformal transformations and ...

7

As Lubos Motl mentions in a comment, for all practical purposes, the sought-for equation (v1) is proved via Wick's Theorem. It is interesting to try to generalize Wick's Theorem and to try to minimize the number of assumptions that goes into it. Here we will outline one possible approach. I) Assume that a family $(\hat{A}_i)_{i\in I}$ of operators ...

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