I spent sometime in the early autumn months thinking about the cosmological constant problem (CC). This was actually secondary, because my primary note taking focused more on S-duality and manifestly duality invariant actions, non-perturbative corrections, and the dilaton. But my supervisor, Tony, has spent a lot of time thinking about this problem, with one of his big ideas being vacuum energy sequestering, so naturally there is motivation whenever we get the chance. There has also been some renewed interest in the CC problem in the context of generalised double sigma models and double field theory. In general, there is a lot of interesting cosmology to be investigated here.

I’m currently drafting a post on the CC problem from the view of string theory. This will hopefully provide the reader with a thorough introduction. But as a passing comment in this short note, it suffices to say that the role of the CC in string theory is generally mysterious. In standard textbook analysis, one sees that the mystery starts with the massless sector contribution, with the dilaton central to the discussion; but the mysteriousness comes further into focus once the role of dual geometry is investigated and the peculiar change of the CC under duality transformation. Intuitively, I am inclined to think that a piece of the picture is missing.

One idea I find interesting to play with involves adding extra fields. Another idea people play with is redefining the dilaton. An example comes from a breakthrough paper by Tseytlin and Wulff [1].

Admittedly, I wasn’t aware of this paper until my early autumn investigations. Within it, a 30 year old problem is solved using the Green-Schwarz (GS) formulation of supergravity theory. The short version is that, in the standard GS formulation of Type IIB string theory there is a problem with the number of degrees of freedom. The space-time fermions have 32 components. An on-shell condition reduces the degrees of freedom to 16, but it needs to be 8. It was later discovered that kappa-symmetry is present in the theory, which is a non-trivial gauge symmetry, and this symmetry may be used to reduce the remaining 8 degrees of freedom. However, issues remained in proving a number of associated conjectures – that is, until Tseytlin and Wulff formulated generalised type IIB SUGRA on an arbitrary background.

The key observation is that generalised SUGRA is equivalent to standard SUGRA plus an extra vector field. Furthermore, one of the characteristics is that, under generalised T-duality, there is a modification of the dilaton such that a non-linear term is added [2]. I think this is quite interesting, and it is something I want to look at more deeply in the future.

Although the context of the calculation is completely different to my own investigations, it is worth noting that this generalised Type IIB theory can be obtained from double field theory. Perhaps not surprisingly, I have seen some pin their hopes that generalised SUGRA could contribute to solving the cosmological constant problem (and potentially also give de Sitter vacua). That seems premature, from my vantange; but in any case it is an interesting bit of work by Tseytlin, Wulff, and others.

References

[1] Tseytlin, A.A., Wulff, L., \textit{Kappa-symmetry of superstring sigma model and generalized 10d supergravity equations}. (2016). [arXiv:1605.04884 [hep-th]].

[2] Tseytlin, A.A., et al, Scale invariance of the $\eta$-deformed $AdS5 \times S5$ superstring, T-duality, and modified type II equations. (2016). [arXiv:1511.05795 [hep-th]].

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