- Towards Loop Quantum Supergravity (LQSG)
Norbert Bodendorfer, Thomas Thiemann, Andreas Thurn
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But what is the paper actually about? It is an attempt to make contact between Loop Quantum Gravity (LQG) and Superstring Theory (ST). Both are approaches to a quantization of gravity, one of the big open problems in theoretical physics. LQG directly attacks the problem by a careful selection of variables and quantization procedure. String theory does not only aim at quantizing gravity, but at the same time at unifying also the other 3 interactions of the standard model by taking as fundamental the strings that give it its name. If quantizing gravity and unifying the standard model interactions are actually related problems, then string theorists are wise to attack them together. Yet, we don't know if they are related. In any case, it has turned out that gravity is necessarily contained in ST.
Both theories still struggle to reproduce general relativity and/or the standard model, and to make contact to phenomenology, though for very different reasons. This begs the question how the theories compare to each other, whether they give the same results for selected problems. Unfortunately, so far this has not been possible to answer because LQG has been developed for a 3+1 dimensional space-time, while ST famously or infamously, depending on your perspective, necessitates 6 additional dimensions that then have to be compactified. ST is also, as the name says, supersymmetric. It should be noted that these both features, supersymmetry and extra dimensions, are not optional but mandatory for ST to make sense.
I've always wondered why one hasn't extended LQG to higher dimensions since the idea of extra dimensions is appealing and somebody in the field who should have known better once told me it would be straight forward to do. It is however not so because one of the variables (a certain SU(2) Yang-Mills connection) used in the quantization procedure relies on a property (the equivalence of the fundamental and adjoint representations of SU(2)) that is fulfilled only in 3 spatial dimensions. So it took many years and two brilliant young students, Norbert Bodendorfer and Andreas Thurn, to come up with a variable that could be used in an arbitrary number of dimensions and to work through the maths which, as you can imagine, didn't get easier. It required to work around the difficulty that SO(1,D) is not compact and digging out a technique for gauge unfixing, a procedure that I had never heard of before.
Compared to the difficulty of adding dimensions, going supersymmetric can be done by simply generalizing the appropriate matter content which is contained in the supergravity actions, and constructing a supersymmetry constraint operator.
Taken together, this in principle allows one to compare the super extra loop quantized gravity to string theory, to which supergravity is a low energy approximation, though concrete calculations have yet to follow. One of the tasks on the to-do list the entropy of extremal supersymmetric black holes to see if LQG reproduces the ST results. (Or if not, which might be even more interesting.) Since LQG is a manifestly non perturbative approach, this relation to string theory might also help filling in some blanks in the AdS/CFT correspondence in areas where neither side of the duality is weakly coupled.