Vishnu Jejjala

Marginal and relevant deformations of N=4 field theories and non-commutative moduli spaces of vacua

Abstract We study marginal and relevant supersymmetric deformations of the N=4 super Yang–Mills theory in four dimensions. Our primary innovation is the interpretation of the moduli spaces of vacua of these theories as non-commutative spaces. The construction of these spaces relies on the representation theory of the related quantum algebras, which are obtained from F -term constraints. These field theories are dual to superstring theories propagating on deformations of the AdS 5 ×S 5 geometry. We study D -branes propagating in these vacua and introduce the appropriate notion of algebraic geometry for non-commutative spaces. The resulting moduli spaces of D -branes have several novel features. In particular, they may be interpreted as symmetric products of non-commutative spaces. We show how mirror symmetry between these deformed geometries and orbifold theories follows from T-duality. Many features of the dual closed string theory may be identified within the non-commutative algebra. In particular, we make progress towards understanding the K-theory necessary for backgrounds where the Neveu–Schwarz antisymmetric tensor of the string is turned on, and we shed light on some aspects of discrete anomalies based on the non-commutative geometry.

Nonsupersymmetric smooth geometries and D1-D5-P bound states

We construct smooth nonsupersymmetric soliton solutions with D1-brane, D5-brane, and momentum charges in type IIB supergravity compactified on

Standard model on a D-brane

{T}^{4}\ifmmode\times\else\texttimes\fi{}{S}^{1}

SQCD: a geometric aperçu

, with the charges along the compact directions. This generalizes previous studies of smooth supersymmetric solutions. The solutions are obtained by considering a known family of

Entropy of near-extremal black holes in AdS5

U(1)\ifmmode\times\else\texttimes\fi{}U(1)

Exploring the vacuum geometry of N=1 gauge theories

invariant metrics, and studying the conditions imposed by requiring smoothness. We discuss the relation of our solutions to states in the CFT describing the D1-D5 system and describe various interesting features of the geometry.

A Calabi-Yau database: threefolds constructed from the Kreuzer-Skarke list

We present a consistent string theory model which produces a simple extension of the standard model, consisting of a D3-brane at a simple orbifold singularity. We envision this as a local singularity within a warped compactification. The phenomenology of the model has some novel features. We note that, for the model to be viable, the scale of stringy physics must be in the multi-TeV range. There are natural hierarchies in the fermion spectrum and there are several possible experimental signatures of the model.

TIME AND M-THEORY

We take new algebraic and geometric perspectives on the old subject of SQCD. We count chiral gauge invariant operators using generating functions, or Hilbert series, derived from the plethystic programme and the Molien-Weyl formula. Using the character expansion technique, we also see how the global symmetries are encoded in the generating functions. Equipped with these methods and techniques of algorithmic algebraic geometry, we obtain the character expansions for theories with arbitrary numbers of colours and flavours. Moreover, computational algebraic geometry allows us to systematically study the classical vacuum moduli space of SQCD and investigate such structures as its irreducible components, degree and syzygies. We find the vacuum manifolds of SQCD to be affine Calabi-Yau cones over weighted projective varieties.

Toric CFTs, permutation triples, and Belyi pairs

We construct the microstates of near-extremal black holes in AdS_5 x S5 as gases of defects distributed in heavy BPS operators in the dual SU(N) Yang-Mills theory. These defects describe open strings on spherical D3-branes in the S5, and we show that they dominate the entropy by directly enumerating them and comparing the results with a partition sum calculation. We display new decoupling limits in which the field theory of the lightest open strings on the D-branes becomes dual to a near-horizon region of the black hole geometry. In the single-charge black hole we find evidence for an infrared duality between SU(N) Yang-Mills theories that exchanges the rank of the gauge group with an R-charge. In the two-charge case (where pairs of branes intersect on a line), the decoupled geometry includes an AdS_3 factor with a two-dimensional CFT dual. The degeneracy in this CFT accounts for the black hole entropy. In the three-charge case (where triples of branes intersect at a point), the decoupled geometry contains an AdS_2 factor. Below a certain critical mass, the two-charge system displays solutions with naked timelike singularities even though they do not violate a BPS bound. We suggest a string theoretic resolution of these singularities.

Non-commutative moduli spaces, dielectric tori and T-duality

Using techniques of algorithmic algebraic geometry, we present a new and efficient method for explicitly computing the vacuum space of N=1 gauge theories. We emphasize the importance of finding special geometric properties of these spaces in connecting phenomenology to guiding principles descending from high-energy physics. We exemplify the method by addressing various subsectors of the MSSM. In particular the geometry of the vacuum space of electroweak theory is described in detail, with and without right-handed neutrinos. We discuss the impact of our method on the search for evidence of underlying physics at a higher energy. Finally we describe how our results can be used to rule out certain top-down constructions of electroweak physics.