We analyze several issues concerning the singular vectors of the Topological N=2 Superconformal algebra. First we investigate which types of singular vectors exist, regarding the relative U(1) charge and the BRST-invariance properties, finding four different types in chiral Verma modules and twentynine different types in complete Verma modules. Then we study the family structure of the singular vectors, every member of a family being mapped to any other member by a chain of simple transformations involving the spectral flows. The families of singular vectors in chiral Verma modules follow a unique pattern (four vectors) and contain subsingular vectors. We write down these families until level 3, identifying the subsingular vectors. The families of singular vectors in complete Verma modules follow infinitely many different patterns, grouped roughly in five main kinds. We present a particularly interesting thirty-eight-member family at levels 3, 4, 5, and 6, as well as the complete set of singular vectors at level 1 (twenty-eight different types). Finally we analyze the Dörrzapf conditions leading to two linearly independent singular vectors of the same type, at the same level in the same Verma module, and we write down four examples of those pairs of singular vectors, which belong to the same thirty-eight-member family. NotationHighest weight (h.w.) states denote states annihilated by all the positive modes of the generators of the algebra, i.e. L n≥1 |χ = H n≥1 |χ = G n≥1 |χ = Q n≥1 |χ = 0 .Primary states denote non-singular h.w. states.Secondary or descendant states denote states obtained by acting on the h.w. states with the negative modes of the generators of the algebra and with the fermionic zero modes Q 0 and G 0 . The fermionic zero modes can also interpolate between two h.w. states at the same footing (two primary states or two singular vectors).Chiral topological states |χ G,Q are states annihilated by both G 0 and Q 0 , i.e. G 0 |χ G,Q = Q 0 |χ G,Q = 0.G 0 -closed topological states |χ G denote non-chiral states annihilated by G 0 , i.e. G 0 |χ G = 0.Q 0 -closed topological states |χ Q denote non-chiral states annihilated by Q 0 , i.e. Q 0 |χ Q = 0 (they are BRST-invariant since Q 0 is the BRST charge).G 0 -exact topological states are G 0 -closed or chiral states that can be expressed as the action of G 0 on another state: |γ = G 0 |χ . Q 0 -exact topological states are Q 0 -closed or chiral states that can be expressed as the action of Q 0 on another state: |γ = Q 0 |χ .No-label topological states |χ denote states that cannot be expressed as linear combinations of G 0 -closed and Q 0 -closed states.

Accurate investigations of quantum-level energies in molecular systems are shown to provide a testing ground to constrain the size of compactified extra dimensions. This is made possible by recent progress in precision metrology with ultrastable lasers on energy levels in neutral molecular hydrogen (H 2 , HD, and D 2 ) and molecular hydrogen ions (H 2 + , HD + , and D 2 + ). Comparisons between experiment and quantum electrodynamics calculations for these molecular systems can be interpreted in terms of probing large extra dimensions, under which conditions gravity will become much stronger. Molecules are a probe of spacetime geometry at typical distances where chemical bonds are effective (i.e., at length scales of an Å). Constraints on compactification radii for extra dimensions are derived within the Arkani-Hamed-Dimopoulos-Dvali framework, while constraints for curvature or brane separation are derived within the Randall-Sundrum framework. Based on the molecular spectroscopy of D 2 molecules and HD + ions, the compactification size for seven extra dimensions (in connection to M-theory defined in 11 dimensions) of equal size is shown to be limited to μ < R 0.6 m 7 . While limits on compactification sizes of extra dimensions based on other branches of physics are compared, the prospect of further tightening constraints from the molecular method is discussed.

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Verma modules of superconfomal algebras can have singular vector spaces with dimensions greater than 1. Following a method developed for the Virasoro algebra by Kent, we introduce the concept of adapted orderings on superconformal algebras. We prove several general results on the ordering kernels associated to the adapted orderings and show that the size of an ordering kernel implies an upper limit for the dimension of a singular vector space. We apply this method to the topological N = 2 algebra and obtain the maximal dimensions of the singular vector spaces in the topological Verma modules: 0, 1, 2 or 3 depending on the type of Verma module and the type of singular vector. As a consequence we prove the conjecture of Gato-Rivera and Rosado on the possible existing types of topological singular vectors (4 in chiral Verma modules and 29 in complete Verma modules). Interestingly, we have found two-dimensional spaces of singular vectors at level 1. Finally, by using the topological twists and the spectral flows, we also obtain the maximal dimensions of the singular vector spaces for the Neveu-Schwarz N = 2 algebra (0, 1 or 2) and for the Ramond N = 2 algebra (0, 1, 2 or 3).

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