The Architecture Machine Revisited: Experiments exploring Computational Design-and- Build Strategies based on Participation

This article summarises a series of experiments at the Architectural Association between 2011 and 2017, which explore the intellectual notion of ‘the architecture machine’ as introduced by Nicholas Negroponte and the Architecture Machine Group at MIT in 1967. The group explored automated computational processes that could assist the process of generating architectural solutions by incorporating much greater levels of complexity at both large and small scales. A central idea to the mission of the Architecture Machine Group was to enable the future inhabitants to participate in the decision-making process on the spatial configurations. The group aimed to define architecture as a spatial system that could directly correlate with human social activities through the application of new computer technologies. Our research presented here focuses on technologies and workflows that trace and translate human activities into architectural structures in order to continue the research agenda set out by Negroponte and others in the 1970s. The research work discusses new scenarios for the creation of architectural structures, using mobile and low-cost fabrication devices, and generative design algorithms driven by sensory technologies. The research question focuses on how architects may script individual and unique processes for generating structures using rule-sets that organise materiality and spatial relationships in order to achieve a user-driven outcome. Our explorations follow a renewed interest in the paradigm where the architect is a ‘process designer’, aiming to generate emergent outcomes where the inherent complexity of the project is generated towards specific performance criteria related to human activities and inhabitation

Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics

We develop a systematic and efficient method of counting single-trace and multi-trace BPS operators with two supercharges, for world-volume gauge theories of $N$ D-brane probes for both $N \to \infty$ and finite $N$. The techniques are applicable to generic singularities, orbifold, toric, non-toric, complete intersections, et cetera, even to geometries whose precise field theory duals are not yet known. The so-called ``Plethystic Exponential'' provides a simple bridge between (1) the defining equation of the Calabi-Yau, (2) the generating function of single-trace BPS operators and (3) the generating function of multi-trace operators. Mathematically, fascinating and intricate inter-relations between gauge theory, algebraic geometry, combinatorics and number theory exhibit themselves in the form of plethystics and syzygies.Comment: 59+1 pages, 7 Figure

Counting Gauge Invariants: the Plethystic Program

We propose a programme for systematically counting the single and multi-trace gauge invariant operators of a gauge theory. Key to this is the plethystic function. We expound in detail the power of this plethystic programme for world-volume quiver gauge theories of D-branes probing Calabi-Yau singularities, an illustrative case to which the programme is not limited, though in which a full intimate web of relations between the geometry and the gauge theory manifests herself. We can also use generalisations of Hardy-Ramanujan to compute the entropy of gauge theories from the plethystic exponential. In due course, we also touch upon fascinating connections to Young Tableaux, Hilbert schemes and the MacMahon Conjecture.Comment: 51 pages, 2 figures; refs updated, typos correcte

Counting BPS operators in N=4 SYM

The free field partition function for a generic U(N) gauge theory, where the fundamental fields transform in the adjoint representation, is analysed in terms of symmetric polynomial techniques. It is shown by these means how this is related to the cycle polynomial for the symmetric group and how the large N result may be easily recovered. Higher order corrections for finite N are also discussed in terms of symmetric group characters. For finite N, the partition function involving a single bosonic fundamental field is recovered and explicit counting of multi-trace quarter BPS operators in free \N=4 super Yang Mills discussed, including a general result for large N. The partition function for BPS operators in the chiral ring of \N=4 super Yang Mills is analysed in terms of plane partitions. Asymptotic counting of BPS primary operators with differing R-symmetry charges is discussed in both free \N=4 super Yang Mills and in the chiral ring. Also, general and explicit expressions are derived for SU(2) gauge theory partition functions, when the fundamental fields transform in the adjoint, for free field theory.Comment: 38 pages, uses harvmac, v.2. references added, typos corrected, discussion of asymptotic counting included for more general chiral ring, v.3. typos corrected, discussion for su(2) simplified, to be published in Nuclear Physics

Effective action in a higher-spin background

We consider a free massless scalar field coupled to an infinite tower of background higher-spin gauge fields via minimal coupling to the traceless conserved currents. The set of Abelian gauge transformations is deformed to the non-Abelian group of unitary operators acting on the scalar field. The gauge invariant effective action is computed perturbatively in the external fields. The structure of the various (divergent or finite) terms is determined. In particular, the quadratic part of the logarithmically divergent (or of the finite) term is expressed in terms of curvatures and related to conformal higher-spin gravity. The generalized higher-spin Weyl anomalies are also determined. The relation with the theory of interacting higher-spin gauge fields on anti de Sitter spacetime via the holographic correspondence is discussed.Comment: 40 pages, Some errors and typos corrected, Version published in JHE

Equivariant characteristic classes of external and symmetric products of varieties

We obtain refined generating series formulae for equivariant characteristic classes of external and symmetric products of singular complex quasi-projective varieties. More concretely, we study equivariant versions of Todd, Chern and Hirzebruch classes for singular spaces, with values in delocalized Borel-Moore homology of external and symmetric products. As a byproduct, we recover our previous characteristic class formulae for symmetric products, and obtain new equivariant generalizations of these results, in particular also in the context of twisting by representations of the symmetric group.Comment: final version, accepted in Geometry & Topolog