In the first part of this section, we review the construction of these singularities in a language adapted to the considerations that follow in the rest of this article. A class of BPS solutions of type IIB supergravity was constructed with SO(4)×SO(4) isometry, one timelike Killing vector and a non-trivial self-dual 5-form field strength F(5).

The solutions can be broken down as follows:

• ds2 = -h-2(dt+Vidxi) 2 +h2(dy2 + δijdxidxj) +yeGdΩ23+ye−GdeΩ23
• F(5) = Fμνdxμ∧dxν∧dΩ + eFμνdxμ∧dxν∧deΩ
• F = e3G∗4eF

With y≥0, we can define a function z=z(x1, x2, y) which determines the entire solution:

• z ≡ 1/2 tanhG
• h-2 = 2ycoshG= y√ (1/2 - z)(1/2 + z)
• dV = 1/y ∗3dz
• F = d(Bt(dt+V)) + dˆB
• ˜F = d(˜Bt(dt+V)) + dˆ˜B
• Bt = - 1/4 y2e2G ˜Bt = - 1/4 y2e-2G
• dˆB = - 1/4 y3 ∗3d θ 1/2 + z y2 ↔ dˆ˜B = 1/4 y3 ∗3d θ 1/2 - z y2

Boundary conditions

The solution is well defined for z restricted to the range -1/2 ≤ z ≤ 1/2. This equation implies that z takes its maximum and minimum on the boundary of its domain of definition Σ⊂R2×R+. A solution of the supergravity equations is determined by boundary conditions in the {x1, x2, y} space.

As we proceed to study the moduli of these geometries, these solutions fall into three classes - non-singular, null-singular, and time machines with a time-like naked singularity. To understand these classes further, we delve into the LLM construction, looking specifically at boundary conditions, asymptotic behavior as well as regular solutions and dual picture.

Additionally, we also study more general boundary conditions and singularities. Conclusions from these explorations are then used to discuss singular solutions and their generic properties.

Through this process, we uncover some interesting features such as closed time-like curves passing through any point of the spacetime and the possibility of unbounded from below negative mass excitations.

In conclusion, our observations indicate the possible existence of a chronology protection mechanism for this class of geometries. These observations allow us to show that there should exist a general proof of the chronology protection conjecture in this sector of supergravity in terms of a unitary CFT indicating.

The purpose of dealing with these abstract concepts is not merely the pursuit of academic knowledge but to deepen our understanding of the universe and its complex dimensions. This study provides a foundation for further work in the area of theoretical physics and string theories.