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59 votes
Accepted

How do we give mathematical meaning to 'physical dimensions'?

Mathematically, the concept of a physical dimension is expressed using one-dimensional vector spaces and their tensor products. For example, consider mass. You can add masses together and you know how ...
Dmitri Pavlov's user avatar
40 votes
Accepted

Why is conformal invariance only possible for massless theories?

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 ...
John Baez's user avatar
  • 21.8k
26 votes

$H^4$ of the Monster

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.
Theo Johnson-Freyd's user avatar
24 votes
Accepted

Has the $E_8$-based generating function for squares numbers been proven?

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 ...
Gjergji Zaimi's user avatar
18 votes

What's the supersymmetric analogue of the Monster group?

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: ...
pho's user avatar
  • 281
17 votes

Why is conformal invariance only possible for massless theories?

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, ...
Abdelmalek Abdesselam's user avatar
16 votes
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71, the Monster, and c = 24 CFTs

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 ...
JethroVE's user avatar
  • 176
15 votes

How do we give mathematical meaning to 'physical dimensions'?

The action appears in an exponent, so it must be dimensionless. That then fixes the dimension of each term which appears in the action and "forbids you from proclaiming that $\phi$ is ...
Carlo Beenakker's user avatar
14 votes
Accepted

Mathematical predictions of AdS/CFT

Although it might seem futile, given how far most of the activity on AdS/CFT is from rigorous mathematics, I think this a good question, provided one is happy, for now, with (very) baby versions of ...
Abdelmalek Abdesselam's user avatar
14 votes
Accepted

From a physicist: How do I show certain superelliptic curves are also hyperelliptic?

This curve is not hyperelliptic unless $n=2$ or $N=2$. First, note that it is more convenient to write the curve as $$ w^n = \frac{ \prod_{\alpha=1}^N (z- u_\alpha )} {\prod_{\alpha=1}^N (z- v_\alpha)}...
Will Sawin's user avatar
  • 139k
11 votes

Why is conformal invariance only possible for massless theories?

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 ...
user25309's user avatar
  • 6,840
11 votes

Why is conformal invariance only possible for massless theories?

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 ...
Carlo Beenakker's user avatar
11 votes
Accepted

What are some mathematical consequences of the study of 6D $\mathcal N = (2,0)$ SCFT?

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 ...
Aaron Bergman's user avatar
11 votes

What are some mathematical consequences of the study of 6D $\mathcal N = (2,0)$ SCFT?

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 $\Sigma\times M_4$...
Surya Raghavendran's user avatar
11 votes
Accepted

Holomorphic maps from upper half plane to itself (or equivalently Poincare disc to itself)

The class of holomorphic maps satisfying 1 and 2 is well known and is frequently used. Let us begin with rational functions satisfying 1,2. They are all of the form: $$az+b-\sum_{n}\frac{c_n}{z-z_n}$$ ...
Alexandre Eremenko's user avatar
11 votes

Mathematical predictions of AdS/CFT

Since the AdS/CFT correspondence links quantum field theory to something as exotic as quantum gravity, I don't think there is any hope for precise mathematical statements coming out of that ...
Carlo Beenakker's user avatar
10 votes

Analogues of the Monster for central charges different from 24

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 ...
Theo Johnson-Freyd's user avatar
10 votes

Why is conformal invariance only possible for massless theories?

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 ...
user1504's user avatar
  • 5,879
9 votes

Why is conformal invariance only possible for massless theories?

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 ...
AccidentalFourierTransform's user avatar
9 votes
Accepted

Vector bundle structure of conformal block bundle

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 ...
Bin Gui's user avatar
  • 555
9 votes

Mathematical predictions of AdS/CFT

I am not an expert in this area but I know that Costello, Gaiotto, Paquette, and others have been studying topological and holomorphic twists of ADS/CFT. Unlike the full correspondence this one seems ...
Justin Hilburn's user avatar
9 votes
Accepted

Elegant proofs of $\bar{\partial}z^{-1} = 2\pi \delta_0$

The identification of $\partial_{\bar{z}}z^{-1}$ with a delta function follows directly from the Cauchy–Pompeiu formula $$f(\zeta) = \frac{1}{2\pi i}\int_{\partial D} \frac{f(z) \,dz}{z-\zeta} - \frac{...
Carlo Beenakker's user avatar
8 votes
Accepted

Is there a discrete lattice analogue of conformal transformations?

There are various ways of putting additional data on a fixed 2-complex and then defining a notion of "discrete conformal equivalence" between different assignments of data, see e.g. the answers to ...
j.c.'s user avatar
  • 13.5k
8 votes

Why is conformal invariance only possible for massless theories?

Why is "conformal invariance only possible for massless theories?" Actually, the answer is NO. Not only the massless, but also the infinite large massive theory can have "conformal invariance." ...
wonderich's user avatar
  • 10.4k
7 votes

$\text{Rep}(D(G))$ as representation category of a vertex operator algebra

A lot has happened in the last four years, and we now have lots of positive results. The current state of knowledge is given in Evans-Gannon, "Reconstruction and Local Extensions for Twisted Group ...
S. Carnahan's user avatar
  • 45.3k
7 votes
Accepted

Do we have a braided tensor category for vertex algebra modules by using conformal blocks on an arbitary compact Riemann Surface?

In general, you won't get a vertex tensor category, because you don't get well-defined unit behavior when you use conformal blocks on higher genus surfaces. Huang-Lepowsky assume the vertex operator ...
S. Carnahan's user avatar
  • 45.3k
7 votes
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Analogues of the Monster for central charges different from 24

As others have mentioned, there are many CFTs, but we can narrow down our list by looking at conditions that select for interesting automorphism groups. Perhaps the easiest is to consider holomorphic ...
S. Carnahan's user avatar
  • 45.3k
7 votes

Conformal blocks in genus zero

Well, the claim is bogus, so you can't expect the proof to hold much water. On the other hand, it may be instructive to try filling in details to see why it fails. First of all, we can't define ...
S. Carnahan's user avatar
  • 45.3k
7 votes
Accepted

Compactification of 6d (2, 0) SCFT on 4-manifolds

You could have a look at https://arxiv.org/abs/1806.02470 and references therein. EDIT (taking into account the comment): compactification of the $\mathcal{N}=(2,0)$ 6d superconformal field theory on ...
user25309's user avatar
  • 6,840
7 votes
Accepted

What is the strongest known RSW result in planar percolation?

This paper of Grimmett and Manolescu prove RSW for bond percolation on isoradial graphs with critical weights (see also this one). The critical weights are those for which the model satisfies the star-...
Kostya_I's user avatar
  • 8,937

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