Bloch Group Research Articles

Invariants from triangulations of hyperbolic 3-manifolds

For any finite volume hyperbolic 3-manifold M we use ideal triangulation to define an invariant β(M) in the Bloch group B(C ). It actually lies in the subgroup of B(C ) determined by the invariant trace field of M . The Chern-Simons invariant of M is determined modulo rationals by β(M). This implies rationality and — assuming the Ramakrishnan conjecture — irrationality results for Chern Simons invariants.

Quantum states and precession: The two discoveries of NMR

Nuclear magnetic resonance in bulk matter was discovered independently by Purcell, Torrey, and Pound at Harvard and by Bloch, Hansen, and Packard at Stanford towards the end of 1945. Their experiments were so different that members of neither group were quick to recognize their own experiment in the other. The magnetic resonance phenomenon was conceptualized differently by the two groups, and the design of their experiments differed accordingly. The Purcell group thought of magnetic resonance in terms of transitions between quantum states while the Bloch group visualized magnetic moments being reoriented with respect to a magnetic field. The conceptual approach adopted by each group can be seen as a natural consequence of earlier influences.

New malonyl-CoA-dependent fatty acid elongation system in Mycobacterium smegmatis.

Beside de novo fatty acid synthetase, two kinds of fatty acid elongating systems have been found in Mycobacterium smegmatis; they are the malonyl-CoA-dependent, acyl carrier protein (ACP)-requiring system (fatty acid synthetase II, named by K. Bloch's group, 1969) and the acetyl-CoA-dependent ACP-non-requiring one (acetyl-CoA-dependent elongation of fatty acids reported by us, 1977). Now, a third fatty acid elongating system, malonyl-CoA-dependent and ACP-non-requiring, has been isolated from an extract of this microorganisms, separately from each of the previous two elongating systems. Primer specificity and cofactor requirements, especially of pyridine-nucleotide-coenzyme, of the last elongation system also distinguish it from the two previously known systems. All three systems, however, were found in a soluble fraction of M. smegmatis, therefore the regulatory mechanism for these elongation systems should be investigated hereafter.

Nahm’s conjecture and $Y$-systems

Nahm's conjecture relates $q$-hypergeometric modular functions to torsion elements in the Bloch group. An interesting class of such functions can be (conjecturally) obtained from a pair $(X,X')$ of diagrams, each of which is either a Dynkin diagram of type $ADE$ or a diagram of type $T$. Using properties of Y-systems, we prove that for a matrix of the form $A=\mathcal{C}(X)\otimes \mathcal{C}(X')^{-1}$ where $\mathcal{C}(X)$ and $\mathcal{C}(X')$ are the corresponding Cartan matrices, every solution of the equation $\mathbf{x}=(1-\mathbf{x})^A$ gives rise to a torsion element of the Bloch group.

A combinatorial invariant for spherical CR structures

We study a cross-ratio of four generic points of $S^3$ which comes from spherical CR geometry. We construct a homomorphism from a certain group generated by generic configurations of four points in $S^3$ to the pre-Bloch group $\mathcal{P}(\mathbb{C})$. If $M$ is a 3-dimensional spherical CR manifold with a CR triangulation, by our homomorphism, we get a $\mathcal{P}(\mathbb{C})$-valued invariant for $M$. We show that when applying to it the Bloch-Wigner function, it is zero. Under some conditions on $M$, we show the invariant lies in the Bloch group $\mathcal{B}(k)$, where $k$ is the field generated by the cross-ratio. For a CR triangulation of the Whitehead link complement, we show its invariant is a torsion in $\mathcal{B}(k)$ and for a triangulation of the complement of the 52-knot we show that the invariant is not trivial and not a torsion element.

3-Manifolds and 3d indices

We identify a large class R of three-dimensional N = 2 superconformal field theories. This class includes the effective theories T_M of M5-branes wrapped on 3-manifolds M, discussed in previous work by the authors, and more generally comprises theories that admit a UV description as abelian Chern–Simons-matter theories with (possibly non-perturbative) superpotential. Mathematically, class R might be viewed as an extreme quantum generalization of the Bloch group; in particular, the equivalence relation among theories in class R is a quantum-field-theoretic “2 to 3 move.” We proceed to study the supersymmetric index of theories in class R, uncovering its physical and mathematical properties, including relations to algebras of line operators and to 4d indices. For 3-manifold theories T_M, the index is a new topological invariant, which turns out to be equivalent to non-holomorphic SL(2,ℂ) Chern–Simons theory on M with a previously unexplored “integration cycle.”