Some time ago I was reading articles about W-algebras and W-gravity by Chris Hull http://arxiv.org/abs/hep-th/9302110 and Christopher Pope http://arxiv.org/abs/hep-th/9112076 . This appeared to be very fascinating thing.
Every textbook on the string theory more or less starts with Virasoro algebra and Virasoro operators $L_n$. These operators are modes of the energy-momentum tensor of a string that is an object of spin 2. This could be found from it OPE
$$T(z)T(w)\sim \frac{\partial T(w)}{z-w}+\frac{2T(w)}{(z-w)^2}+\frac{c/2}{(z-w)^4}.$$
The Operator Product Expansion simly represents commutator in the radial quantization and can be understood as some fancy way of writing of Virasoro commutation relations. And 2 in the second term of OPE directly related to the fact, that $T(z)$ has spin 2.
But now we ask, what if we try to add in the theory operators of spin higher than 2? E.g. try to introduce some $W(z)$ that has spin 3. To check the consistency we need to close algebra of these operators, i.e. we need to ensure that OPE of $W(z)$ and $T(z)$ with each other and with themselves gives again the same set of operators. This works fine until we commute $W$ with itself:
$$W(z)W(w)\sim P[\Lambda] + \Pi [T].$$
This OPE on its RHS gives some expression with $T$ as it should be and some expression with new operator of spin $4$. Disaster, the algebra is not closed.
Actually, this follows from simple calculation. The maximal spin that appears in RHS of OPE of two operators with spins $s_1$ and $s_2$ is equal to $s_1+s_2-2$. Thus in the case of spin $2$ we do not face the problem of higher spins:
$$2+2-2=2.$$
But $3+3-2=4$ and this is going to last forewer with higher and higher spins appearing at each step.
There are two ways to deal with this problem. One is to consider nonlinear realisation of operator $\Lambda$ (schematically):
$$\Lambda=(TT)-\frac{3}{10}\partial^2T.$$
This closes the algebra and leaves us with only two operators $T$ and $W$ with highest spin equal to 3. This is what is called $W_3$ algebra. In the similar fashion you can cinstruct $W_N$ algebra for any $N$. But one should take into account that OPE's here are enormous are very complicated.
Another way is more obvious and in some sence more simple. We note, that the algebra is closed for $N=\infty$:
$$\infty+\infty-2=\infty.$$
This leads to $W_{\infty}$ algebra and to writing infinitely mane OPE's. Hopefully they can be arranged into some polinomials.
$W$ algebras arise naturaly in non-critical string-theories http://arxiv.org/abs/hep-th/9309115, in two-dimensional gravity, AdS/CFT correspondense http://arxiv.org/abs/arXiv:1009.6087.
Every textbook on the string theory more or less starts with Virasoro algebra and Virasoro operators $L_n$. These operators are modes of the energy-momentum tensor of a string that is an object of spin 2. This could be found from it OPE
$$T(z)T(w)\sim \frac{\partial T(w)}{z-w}+\frac{2T(w)}{(z-w)^2}+\frac{c/2}{(z-w)^4}.$$
The Operator Product Expansion simly represents commutator in the radial quantization and can be understood as some fancy way of writing of Virasoro commutation relations. And 2 in the second term of OPE directly related to the fact, that $T(z)$ has spin 2.
But now we ask, what if we try to add in the theory operators of spin higher than 2? E.g. try to introduce some $W(z)$ that has spin 3. To check the consistency we need to close algebra of these operators, i.e. we need to ensure that OPE of $W(z)$ and $T(z)$ with each other and with themselves gives again the same set of operators. This works fine until we commute $W$ with itself:
$$W(z)W(w)\sim P[\Lambda] + \Pi [T].$$
This OPE on its RHS gives some expression with $T$ as it should be and some expression with new operator of spin $4$. Disaster, the algebra is not closed.
Actually, this follows from simple calculation. The maximal spin that appears in RHS of OPE of two operators with spins $s_1$ and $s_2$ is equal to $s_1+s_2-2$. Thus in the case of spin $2$ we do not face the problem of higher spins:
$$2+2-2=2.$$
But $3+3-2=4$ and this is going to last forewer with higher and higher spins appearing at each step.
There are two ways to deal with this problem. One is to consider nonlinear realisation of operator $\Lambda$ (schematically):
$$\Lambda=(TT)-\frac{3}{10}\partial^2T.$$
This closes the algebra and leaves us with only two operators $T$ and $W$ with highest spin equal to 3. This is what is called $W_3$ algebra. In the similar fashion you can cinstruct $W_N$ algebra for any $N$. But one should take into account that OPE's here are enormous are very complicated.
Another way is more obvious and in some sence more simple. We note, that the algebra is closed for $N=\infty$:
$$\infty+\infty-2=\infty.$$
This leads to $W_{\infty}$ algebra and to writing infinitely mane OPE's. Hopefully they can be arranged into some polinomials.
$W$ algebras arise naturaly in non-critical string-theories http://arxiv.org/abs/hep-th/9309115, in two-dimensional gravity, AdS/CFT correspondense http://arxiv.org/abs/arXiv:1009.6087.
