nLab non-Lorentzian type II string theory

Idea

What one usually refers to as type II string theory is defined on Lorentzian geometry. It is thought that analogues of Type II string theory (and, more generally, of the other theories appearing in string theory, such as M-theory) may be defined on spaces with signature other than Lorentz.

In Hull (1998) it is argued that there exists a web of Type II theories related by space- or time-like T-duality, as well as S-duality, summarized in:

where in particular “IIA/B” refers to type II string theory and “$IIA^*/B^*$” to type II* string theory. These theories are described in terms of their spacetime action functional. For the Type IIA’s:

$$S_{IIA} = \int d^{10}x\sqrt{\vert g\vert} (e^{-2\Phi}(R+4(\partial\Phi)^2 )-(\pm e^{-2\Phi}H^2\pm G^2_2 \pm G^2_4))+\cdots$$

while for the Type IIB’s:

$$S_{IIA} = \int d^{10}x\sqrt{\vert g\vert} (e^{-2\Phi}(R+4(\partial\Phi)^2 )-(\pm e^{-2\Phi}H^2\pm G^2_1 \pm G^2_3\pm G^2_5 ))+\cdots$$

References

General:

• Christopher Hull. Timelike T-duality, de Sitter space, large N gauge theories and topological field theory, JHEP 07 (1998) 021 [hep-th/9806146]

• Christopher Hull. Duality and the signature of space-time, JHEP 9811 (1998) 017 [hep-th/9807127]

• Christopher Hull and Ramzi Khuri, Branes, times and dualities, Nucl.Phys. B536 (1998) 219 [hep-th/9808069]

• Christopher Hull. Duality and Strings, Space and Time. (1999) [hep-th/9911080]

• Hitoshi Nishino, Jim Gates. The $*$-Report. (1999). [hep-th/9908136]

On the relation between the super Lie algebras of II and $II^\ast$ and the orthosymplectic super Lie algebra $\mathfrak{osp}(1\vert 32)$:

• Eric Bergshoeff, Antoine Van Proeyen. The many faces of $OSp(1\vert 32)$. (2000). [hep-th/0003261]

• Eric Bergshoeff, Antoine Van Proeyen. The unifying superalgebra $OSp(1\vert 32)$. (2000). [hep-th/0010194]

The supposed branes in $II^\ast$, E-branes?, are discussed in:

• Somdatta Bhattacharya, Sudipta Mukherji, Shibaji Roy. On the effective action of a space-like brane. (2003). [hep-th/0308069]

That the boundary state for the Dirichlet S-brane constructed is a solution of type $II^\ast$ rather than type II string theory is in:

• Bruno Durin, Boris Pioline. Aspects of Dirichlet S-branes. (2005) [hep-th/0507059]

The suggestion that $II^\ast$ and its would-be M-theory lift may play a role in interpreting M-theory as defined on the base space of an octonionic projective space $\mathbb{O}P^2$ bundle with structure group the exceptional Lie group $F_4$ is in Remark 5, Section 2.3 of:

• Hisham Sati. On the geometry of the supermultiplet in M-theory. (2009). [arXiv:0909.4737]

On the relation of twisted $\mathcal{N}=2$ algebra, Para-quaternionic Kähler manifolds, and compactifications of $II^\ast$ on Calabi-Yau's:

• Maxime Médevielle, Thomas Mohaupt, and G. Pope. Type-II Calabi-Yau compactifications, T-duality and special geometry in general spacetime signature. J. High Energ. Phys. 2022, 48 (2022). [doi:10.1007/JHEP02(2022)048048)]