43

First let us recall why string theory is attractive. As of now, we have two experimentally verified, but mutually incompatible theories describing fundamental physical phenomena. The standard model of particle physics is a quantum field theory describing all the elementary particle interactions except for gravitation. General relativity is a classical theory ...


37

A physicist would answer this question as follows. (Everything I'll say can be expressed in a way that the purest of mathematicians would understand, but that translation would take a lot of work, so I'll only do it on demand.) In physics we have units of mass ($M$), length ($L$) and time ($T$). In special relativity we have a fundamental constant $c$, ...


28

The problem with this question, for mathematicians, and actually for anyone, is that the term "string theory" is not well-defined, making the question of falsifiability much more complicated. The most well-defined interpretation of "string theory" would be the superstring in 10 flat space-time dimensions, which is defined by a series expansion. The details ...


24

In arXiv:1707.08388, I calculate that the cohomology class you described has order 24 and that it is not a characteristic class in the ordinary sense.


24

This identity was actually proven 24 years ago in S.O. Warnaar and P.A. Pearce, "Exceptional structure of the dilute A 3 model: E8 and E7 Rogers-Ramanujan identities" J.Phys. A27 (1994) L891-L898 The proof essentially establishes a finite polynomial identity whose limit under one of the parameters becomes the desired series. I want to remark that Nahm ...


18

There is a super analog constructed just as you describe with the Conway group $Co_0$ replacing the Monster and commuting with the superconformal algebra. The construction is described in detail in: John F. R. Duncan and Sander Mack-Crane, The Moonshine module for Conway’s group, arXiv:1409.3829. and in John Duncan's paper: John F. R. Duncan, Super-...


17

It's not clear from the post if you are talking about classical or quantum field theories. In QFT, conformal invariance implies scale invariance. If the theory has a mass $m$ then, as John explained, this defines a characteristic length, roughly $l=m^{-1}$ which has to be preserved by symmetries like scale transformations. So that means that the mass $m$ ...


16

The relationship between the Ising model (spins on a lattice) and conformal field theory holds only in the immediate vicinity of the critical point, when correlation lengths go to infinity and all details on the scale of the lattice constant become irrelevant. The relationship is explained, for example, in these lecture notes. Let me walk you through them. ...


16

Schellekens' enumeration is exhaustive in the following sense: the degree 1 subspace of the meromorphic CFT/vertex algebra is naturally a Lie algebra, and it is known that this Lie algebra must be one of the 71 that Schellekens wrote down. Each of these 71 Lie algebras is realised as the weight 1 piece of some holomorphic c=24 vertex algebra, but it is ...


15

To the best of my knowledge of the literature on this topic, the answer is: not really (a few exceptions appear below). Let me first give a rigorous statement of the problem: the physical equation $\beta = 0$ can be expressed as $\sum_{k=0}^{\infty} \epsilon^k \beta_k = 0$, where $\epsilon$ is a physical parameter one imagines is "small," and where $\...


15

(1) Look first at the references in Schramm and Smironv and Lowler. They refer to some important physics papers. Also look at the survey of Langlands, Pouliot, Saint-Aubin, Conformal invariance in two-dimensional percolation. Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 1, 1–61. This is almost a "physics paper", but Langlands is a mathematician, and it has ...


13

The nLab has a page devoted to this question: ncatlab.org/nlab/show/string+theory+FAQ I'll be glad to further expand this as need be.


13

Actually I think the idea of the holographic principle is that, as in a holograph, all the information in the 'bulk' is already present at the 'boundary'. So, it claims that any calculation involving bulk observables can be expressed in terms of boundary observables. It may not claim the reverse, though that could often be taken for granted! In ...


12

Given a $N=(2,2)$ two dimensional superconformal field theory (SCFT), one can construct two topological field theories called the $A$ and $B$ models. To each of these topological field theories, one should be able to associate a ($A_\infty$) triangulated category of boundary conditions, called category of branes. Thus corresponding to $A$ and $B$ models one ...


11

There has been quite a few developments as far as making this relation mathematically rigorous. See this paper and this one by Camia, Garban and Newman as well as this paper by Chelkak, Hongler and Izyurov for the scaling limit of the $\sigma$. As for the scaling limit of the $\varepsilon$ see the thesis by Hongler. It would be very nice if one could ...


11

This may be a less "clear-cut-no" than suggested by the mantra "there can be no massive particles in a CFT because that would introduce a scale": Anatol Odzijewicz has constructed a CFT for massive particles, in which the mass is allowed to vary as a result of interactions: The aim of this paper is the construction of a field theory for a massive ...


11

The usual mantra is usually formulated in the context of quantum field theory on Minkowski space. In a classical field theory, there is no particles and no mass for particles. In a quantum relativistic field theory, there are particles and it makes sense to talk about their mass (technically, a particle in quantum field theory is one of the irreducible ...


10

I apologize since my answer will involve shameless self-promotion. You can find one example of this kind on page 36 of my slides. In this example one Frobenius algebra is commutative and another is not Morita equivalent to a commutative algebra.


10

It's claimed on Page 54 of Bakalov and Kirillov's Lectures on Tensor Categories and Modular Functors that the formula in Scott's post above is first proven in Moore and Seiberg's Classical and Quantum Conformal Field Theory). I've only just grabbed that paper, but equation (A.7) gives a generalization of the formula above to "n-point function characters at ...


10

This is rather an addition to my comment, as Scott Morrision already give the same answer. Usually, what's meant by the Verlinde formula is that the fusion coefficients $N_ {ij}^k$ can be determined by the S-matrix by the formula: $$ N_{ij}^k = \sum_l \frac{S_{jl}S_{il} (S^{-1})_{lk}}{S_{0l}}. $$ While this formula looks mysterious, it basically says that ...


10

One possible answer to your question is discussed in Witten's paper Three-dimensional gravity revisited. Since the beginning of the subject, it has been conjectured that the Moonshine VOA is the unique self-dual VOA with $c=24$ and no spin=1 fields. As Marcel mentions in the comments, the spin=1 fields comprise the Lie algebra of the automorphism group, so ...


10

So, you must be talking about quantum field theories, as mass is something we associated with particles, which are a quantum phenomenon in field theories. In QFT, there are operators which create particles. This means (neglecting various details, so please forgive the imprecision), that: There is a distinguished element of the Hilbert space of states, $|...


9

I have no idea what a ``symmetric haploid Frobenius algebra" is, so my answer may be terribly naive, but given an even-self dual lattice of rank $d$ there is a well known lattice construction of a rational (actually holomorphic) conformal field theory with central charge $c=d$. There are two even self-dual lattices of rank $16$ ($E_8 \times E_8$ and $Spin(32)...


9

I'm not sure exactly what kind of information you want, and CFT is an enormous subject, but here is some information on the physical interpretation of the complex coordinates and correlation functions along with an example of their mathematical interpretation in a special CFT. An ordinary refrigerator magnet contains a ferromagnetic material. The atoms in ...


9

It seems that the previous answers describe Verlinde's formula for a modular tensor category, or a slight weakening of that condition. Moore and Seiberg essentially proved the formula under the assumption that the sectors of a rational CFT form a modular tensor category (although I. Frenkel hadn't invented the name yet). However, chiral CFTs are much ...


9

Note that the Schrödinger action, say in flat space-time, $$ S=\int\mathrm dx\ \psi^*\left(i\frac{\mathrm d}{\mathrm dt}+\frac{1}{2m}\nabla^2\right)\psi $$ has a length scale (mass) $m$, and yet it is scale invariant. In fact, it is invariant under the so-called Schrödinger group, which includes the centrally extended Galilei Group (the Bargmann Group) ...


9

If you take the (2,0) theory and put it on a manifold which is $T^2 \times M_4$, it is known to reduce to $\mathcal{N} = 4$ super-Yang Mills theory on $M_4$. That theory exhibits S-duality, which has been shown to be related to geometric Langlands. In particular, S-duality is part of an $SL(2,\mathbb{Z})$ symmetry, and the $G \leftrightarrow \widehat{G}$ ...


9

In general, mathematical outputs of SUSY field theories often become more accessible after performing some twist, and the same is true of the 6d (2,0) SCFT. Considering the theory on $\Sigma\times M_4$, it admits a twist (first studied by Beem-Rastelli I believe) that's holomorphic along $\Sigma$ and topological along $M_4$. Some discussion of the ...


9

I also had the same question when I was reading Kohno's book. My opinion is that that book is only an introduction to the topic of conformal blocks and is not rigorous. For rigorous treatment you need to read the classical paper Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries by Tsuchiya, Ueno, Yamada. There are also many ...


8

This morning I just discovered these corrigenda of K. Iohara and Y. Koga (in which my name is cited in acknowledgement). In fact, three years ago, I have contacted K. Iohara (author, with Y. Koga, of the book Representation Theory of the Virasoro Algebra) about this error (but I didn't know they fixed it). I do not yet read these corrigenda into details, ...


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