Some beautiful equations of mathematical physics

The basic ideas and the important role of gauge principles in modern elementary particle physics are outlined. There are three theoretically consistent gauge principles in quantum field theory: the spin-1 gauge principle of electromagnetism and the standard model, the spin-2 gauge principle of general relativity, and the spin-3/2 gauge principle of supergravity. (Dirac Prize lecture, November 1993. Two figures are unavailable, but should not be essential.)Comment: 18 page

Quantifying Homology Classes

We develop a method for measuring homology classes. This involves three problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements' size have the minimal sum. We provide a greedy algorithm to compute the optimal basis and measure classes in it. The algorithm runs in $O(\beta^4 n^3 \log^2 n)$ time, where $n$ is the size of the simplicial complex and $\beta$ is the Betti number of the homology group. Third, we discuss different ways of localizing homology classes and prove some hardness results

The Holography of F-maximization

We find new supersymmetric backgrounds of ${\cal N} = 8$ gauged supergravity in four Euclidean dimensions that are dual to deformations of ABJM theory on $S^3$. The deformations encode the most general choice of $U(1)_R$ symmetry used to define the theory on $S^3$. We work within an ${\cal N} = 2$ truncation of the ${\cal N} = 8$ supergravity theory obtained via a group theory argument. We find perfect agreement between the $S^3$ free energy computed from our supergravity backgrounds and the previous field theory computations of the same quantity based on supersymmetric localization and matrix model techniques.Comment: 48 pages; v2 minor improvement

Spatial Geometry and the Wu-Yang Ambiguity

We display continuous families of SU(2) vector potentials $A_i^a(x)$ in 3 space dimensions which generate the same magnetic field $B^{ai}(x)$ (with det $B\neq 0$). These Wu-Yang families are obtained from the Einstein equation $R_{ij}=-2G_{ij}$ derived recently via a local map of the gauge field system into a spatial geometry with $2$-tensor $G_{ij}=B^a{}_i B^a{}_j\det B$ and connection $\Gamma_{jk}^i$ with torsion defined from gauge covariant derivatives of $B$.Comment: Based on talks given by R. Khuri at PASCOS-94, Syracuse University, May 1994 and at Gursey Memorial Conference I, Istanbul, June 1994, 7 pages, TeX (typo in first Author's name is corrected.