1,203 research outputs found
Toric Geometry, Sasaki-Einstein Manifolds and a New Infinite Class of AdS/CFT Duals
Recently an infinite family of explicit Sasaki-Einstein metrics Y^{p,q} on
S^2 x S^3 has been discovered, where p and q are two coprime positive integers,
with q<p. These give rise to a corresponding family of Calabi-Yau cones, which
moreover are toric. Aided by several recent results in toric geometry, we show
that these are Kahler quotients C^4//U(1), namely the vacua of gauged linear
sigma models with charges (p,p,-p+q,-p-q), thereby generalising the conifold,
which is p=1,q=0. We present the corresponding toric diagrams and show that
these may be embedded in the toric diagram for the orbifold C^3/Z_{p+1}xZ_{p+1}
for all q<p with fixed p. We hence find that the Y^{p,q} manifolds are AdS/CFT
dual to an infinite class of N=1 superconformal field theories arising as IR
fixed points of toric quiver gauge theories with gauge group SU(N)^{2p}. As a
non-trivial example, we show that Y^{2,1} is an explicit irregular
Sasaki-Einstein metric on the horizon of the complex cone over the first del
Pezzo surface. The dual quiver gauge theory has already been constructed for
this case and hence we can predict the exact central charge of this theory at
its IR fixed point using the AdS/CFT correspondence. The value we obtain is a
quadratic irrational number and, remarkably, agrees with a recent purely field
theoretic calculation using a-maximisation.Comment: 54 pages, 5 figures; minor changes; further minor changes, ref [8]
added - published version; eqns 1.3, 1.4 remove
Mirror symmetry for log Calabi-Yau surfaces I
We give a canonical synthetic construction of the mirror family to a pair
(Y,D) of a smooth projective surface with an anti-canonical cycle of rational
curves, as the spectrum of an explicit algebra defined in terms of counts of
rational curves on Y meeting D in a single point. In the case D is
contractible, the family gives a smoothing of the dual cusp, and thus a proof
of Looijenga's 1981 cusp conjecture.Comment: 144 pages, 3 figures, Second version significantly shorter, 109
pages. The first version has a lot of material (particularly in the
introduction and material on cyclic quotient singularities) which does not
appear in the new version. Download version 1 if this material is desired.
Third and final version, small changes from Version 2, to appear in Publ.
IHE
Instantons, Topological Strings and Enumerative Geometry
We review and elaborate on certain aspects of the connections between
instanton counting in maximally supersymmetric gauge theories and the
computation of enumerative invariants of smooth varieties. We study in detail
three instances of gauge theories in six, four and two dimensions which
naturally arise in the context of topological string theory on certain
non-compact threefolds. We describe how the instanton counting in these gauge
theories are related to the computation of the entropy of supersymmetric black
holes, and how these results are related to wall-crossing properties of
enumerative invariants such as Donaldson-Thomas and Gromov-Witten invariants.
Some features of moduli spaces of torsion-free sheaves and the computation of
their Euler characteristics are also elucidated.Comment: 61 pages; v2: Typos corrected, reference added; v3: References added
and updated; Invited article for the special issue "Nonlinear and
Noncommutative Mathematics: New Developments and Applications in Quantum
Physics" of Advances in Mathematical Physic
Fixing All Moduli in a Simple F-Theory Compactification
We discuss a simple example of an F-theory compactification on a Calabi-Yau
fourfold where background fluxes, together with nonperturbative effects from
Euclidean D3 instantons and gauge dynamics on D7 branes, allow us to fix all
closed and open string moduli. We explicitly check that the known higher order
corrections to the potential, which we neglect in our leading approximation,
only shift the results by a small amount. In our exploration of the model, we
encounter interesting new phenomena, including examples of transitions where D7
branes absorb O3 planes, while changing topology to preserve the net D3 charge.Comment: 68 pages, 19 figures; v2: references adde
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