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19

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

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

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 ...


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 ...


10

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 ...


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 ...


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 ...


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 ...


7

Conformal field theories do not have a mass-gap, which is one of the assumptions [for the strong conclusions of non-mixing of Poincare spacetime symmetries vs internal symmetries] of the Coleman-Mandula no-go theorem. Similarly, for its superversion: the Haag-Lopuszanski-Sohnius no-go theorem. [In the supercase, the Poincare algebra is replaced with the ...


7

The actual paper by Haag, Łopuszański and Sohnius covers Conformal Supersymmetry, and it states explicitly that this extension is achieved by NOT assuming the mass gap.


7

This question is answered by Nima Arkani-Hamed in his Simons Center talk, at about 112 minutes in. His answer is that the structure of the amplituhedron itself does not directly use integrability of the theory in any way. It is only when you come to do the integrals themselves that integrability makes it possible. The amplituhedron itself is more linked to ...


7

The precise statement should be that massless fields in conformal field theories in 3+1 dimensions are necessarily free. This result was first proved by Buchholz and Fredenhagen. There are two modern proofs of this fact, one by Steven Weinberg (please see arXiv: hep-th/1210.3864v1) and the other by Yoh Tanimoto in the framework of algebraic quantum field ...


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 ...



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