6 research outputs found
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 D-brane probes for both and finite . 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
M2-Branes and Fano 3-folds
A class of supersymmetric gauge theories arising from M2-branes probing
Calabi-Yau 4-folds which are cones over smooth toric Fano 3-folds is
investigated. For each model, the toric data of the mesonic moduli space is
derived using the forward algorithm. The generators of the mesonic moduli space
are determined using Hilbert series. The spectrum of scaling dimensions for
chiral operators is computed.Comment: 128 pages, 39 figures, 42 table
Quiver gauge theories: beyond reflexivity
Reflexive polygons have been extensively studied in a variety of contexts in mathematics and physics. We generalize this programme by looking at the 45 different lattice polygons with two interior points up to SL(2,â€) equivalence. Each corresponds to some affine toric 3-fold as a cone over a Sasaki-Einstein 5-fold. We study the quiver gauge theories of D3-branes probing these cones, which coincide with the mesonic moduli space. The minimum of the volume function of the Sasaki-Einstein base manifold plays an important role in computing the R-charges. We analyze these minimized volumes with respect to the topological quantities of the compact surfaces constructed from the polygons. Unlike reflexive polytopes, one can have two fans from the two interior points, and hence give rise to two smooth varieties after complete resolutions, leading to an interesting pair of closely related geometries and gauge theories
Graph Zeta Function and Gauge Theories
Along the recently trodden path of studying certain number theoretic
properties of gauge theories, especially supersymmetric theories whose vacuum
manifolds are non-trivial, we investigate Ihara's Graph Zeta Function for large
classes of quiver theories and periodic tilings by bi-partite graphs. In
particular, we examine issues such as the spectra of the adjacency and whether
the gauge theory satisfies the strong and weak versions of the graph
theoretical analogue of the Riemann Hypothesis.Comment: 35 pages, 7 Figure
Mesonic Chiral Rings in Calabi-Yau Cones from Field Theory
We study the half-BPS mesonic chiral ring of the N=1 superconformal quiver
theories arising from N D3-branes stacked at Y^pq and L^abc Calabi-Yau conical
singularities. We map each gauge invariant operator represented on the quiver
as an irreducible loop adjoint at some node, to an invariant monomial, modulo
relations, in the gauged linear sigma model describing the corresponding bulk
geometry. This map enables us to write a partition function at finite N over
mesonic half-BPS states. It agrees with the bulk gravity interpretation of
chiral ring states as cohomologically trivial giant gravitons. The quiver
theories for L^aba, which have singular base geometries, contain extra
operators not counted by the naive bulk partition function. These extra
operators have a natural interpretation in terms of twisted states localized at
the orbifold-like singularities in the bulk.Comment: Latex, 25pgs, 12 figs, v2: minor clarification
Calabi-Yau Volumes and Reflexive Polytopes
We study various geometrical quantities for CalabiâYau varieties realized as cones over Gorenstein Fano varieties, obtained as toric varieties from reflexive polytopes in various dimensions. Focus is made on reflexive polytopes up to dimension 4 and the minimized volumes of the SasakiâEinstein base of the corresponding CalabiâYau cone are calculated. By doing so, we conjecture new bounds for the SasakiâEinstein volume with respect to various topological quantities of the corresponding toric varieties. We give interpretations about these volume bounds in the context of associated field theories via the AdS/CFT correspondence