We identify a set of “energy” functionals on the space of metrics in a given Kähler class on a Calabi-Yau manifold, which are bounded below and minimized uniquely on the Ricci-flat metric in that class. Using these functionals, we recast the problem of numerically solving the Einstein equation as an optimization problem. We apply this strategy, using the “algebraic” metrics (metrics for which the Kähler potential is given in terms of a polynomial in the projective coordinates), to the Fermat quartic and to a one-parameter family of quintics that includes the Fermat and conifold quintics. We show that this method yields approximations to the Ricci-flat metric that are exponentially accurate in the degree of the polynomial (except at the conifold point, where the convergence is polynomial), and therefore orders of magnitude more accurate than the balanced metrics, previously studied as approximations to the Ricci-flat metric. The method is relatively fast and easy to implement. On the theoretical side, we also show that the functionals can be used to give a heuristic proof of Yau’s theorem.

Using the fact that every worldsheet is ruled by two (light-cone) copies of worldlines, the recent classification of off-shell supermultiplets of $N$-extended worldline supersymmetry is extended to construct standard off-shell and also unidextrous (on the half-shell) supermultiplets of worldsheet $(p, q)$-supersymmetry with no central extension. In the process, a new class of error-correcting (even-split doubly-even linear block) codes is introduced and classified for $p + q \leqslant 8$, providing a graphical method for classification of such codes and supermultiplets. This also classifies quotients by such codes, of which many are not tensor products of worldline factors. Also, supermultiplets that admit a complex structure are found to be depictable by graphs that have a hallmark twisted reflection symmetry.

We identify a large class $\mathcal{R}$ of three-dimensional $\mathcal{N} = 2$ superconformal field theories. This class includes the effective theories $T_M$ of M5-branes wrapped on 3-manifolds $\mathcal{M}$, 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 $\mathcal{R}$ might be viewed as an extreme quantum generalization of the Bloch group; in particular, the equivalence relation among theories in class $\mathcal{R}$ is a quantum-field-theoretic "2 to 3 move." We proceed to study the supersymmetric index of theories in class $\mathcal{R}$, 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 $SL(2,\mathbb{C})$ Chern-Simons theory on $\mathcal{M}$ with a previously unexplored "integration cycle."

By exploiting a "mixed" non-symmetric Freudenthal-Rozenfeld-Tits magic square, two types of coset decompositions are analyzed for the non-compact special Kähler symmetric rank-3 coset $E_{7(-25)}/ [(E_{6(-78)} \times U(1)) / \mathbb{Z}_3]$, occurring in supergravity as the vector multiplets'scalar manifold in $\mathcal{N} = 2, \mathcal{D} = 4$ exceptional Maxwell-Einstein theory. The first decomposition exhibits maximal manifest covariance, whereas the second (triality-symmetric) one is of Iwasawa type, with maximal $SO(8)$ covariance. Generalizations to conformal non-compact, real forms of nondegenerate, simple groups "of type E7" are presented for both classes of coset parametrizations, and relations to rank-3 simple Euclidean Jordan algebras and normed trialities over division algebras are also discussed.

We study analytic aspects of $\mathrm{U}(n)$ gauge theory over a toric noncommutative manifold $M_\theta$. We analyse moduli spaces of solutions to the self-dual Yang-Mills equations on $\mathrm{U}(2)$ vector bundles over four-manifolds $M_\theta$, showing that each such moduli space is either empty or a smooth Hausdorff manifold whose dimension we explicitly compute. In the special case of the four-sphere $S^4_\theta$ we find that the moduli space of $\mathrm{U}(2)$ instantons with fixed second Chern number $k$ is a smooth manifold of dimension $8k - 3$.