We present distance measurements to 71 high redshift type Ia supernovae discovered during the first year of the 5-year Supernova Legacy Survey (SNLS). These events were detected and their multi-color light-curves measured using the MegaPrime/MegaCam instrument at the Canada-France-Hawaii Telescope (CFHT), by repeatedly imaging four one-square degree fields in four bands. Follow-up spectroscopy was performed at the VLT, Gemini and Keck telescopes to confirm the nature of the supernovae and to measure their redshift. With this data set, we have built a Hubble diagram extending to z = 1, with all distance measurements involving at least two bands. Systematic uncertainties are evaluated making use of the multiband photometry obtained at CFHT. Cosmological fits to this first year SNLS Hubble diagram give the following results: {Omega}{sub M} = 0.263 {+-} 0.042 (stat) {+-} 0.032 (sys) for a flat {Lambda}CDM model; and w = -1.023 {+-} 0.090 (stat) {+-} 0.054 (sys) for a flat cosmology with constant equation of state w when combined with the constraint from the recent Sloan Digital Sky Survey measurement of baryon acoustic oscillations.

The Supernova Legacy Survey: Measurement of Omega_m, Omega_l and w from the First Year Data Set

For 3-dimensional field theories with {\cal N}=2 supersymmetry the Euclidean path integrals on the three-sphere can be calculated using the method of localization; they reduce to certain matrix integrals that depend on the R-charges of the matter fields. We solve a number of such large N matrix models and calculate the free energy F as a function of the trial R-charges consistent with the marginality of the superpotential. In all our {\cal N}=2 superconformal examples, the local maximization of F yields answers that scale as N^{3/2} and agree with the dual M-theory backgrounds AdS_4 x Y, where Y are 7-dimensional Sasaki-Einstein spaces. We also find in toric examples that local F-maximization is equivalent to the minimization of the volume of Y over the space of Sasakian metrics, a procedure also referred to as Z-minimization. Moreover, we find that the functions F and Z are related for any trial R-charges. In the models we study F is positive and decreases along RG flows. We therefore propose the "F-theorem" that we hope applies to all 3-d field theories: the finite part of the free energy on the three-sphere decreases along RG trajectories and is stationary at RG fixed points. We also show that in an infinite class of Chern-Simons-matter gauge theories where the Chern-Simons levels do not sum to zero, the free energy grows as N^{5/3} at large N. This non-trivial scaling matches that of the free energy of the gravity duals in type IIA string theory with Romans mass.

Towards the F-Theorem: N=2 Field Theories on the Three-Sphere

We study Euclidean 3D $ \mathcal{N} = 2 $ supersymmetric gauge theories on squashed three-spheres preserving isometries SU(2) × U(1) or U(1) × U(1) We show that, when a suitable background U(1) gauge field is turned on, these squashed spheres support charged Killing spinors and therefore $ \mathcal{N} = 2 $ supersymmetric gauge theories We present the Lagrangian and supersymmetry rules for general gauge theories The partition functions are computed using localization principle, and are expressed as integrals over Coulomb branch For the squashed sphere with U(1) × U(1) isometry, its measure and integrand are identified with the building blocks of structure constants in Liouville or Toda conformal field theories with b ≠ 1

SUSY gauge theories on squashed three-spheres

For 3-dimensional field theories with $ \mathcal{N} = 2 $
 supersymmetry the Euclidean path integrals on the three-sphere can be calculated using the method of localization; they reduce to certain matrix integrals that depend on the R-charges of the matter fields. We solve a number ofsuch large N matrix models and calculate the free energy F as a function of the trial R-charges consistent with the marginality of the super potential. In all our $ \mathcal{N} = 2 $
 superconformal examples, the local maximization of F yields answers that scale as N
 3/2 and agree with the dual M-theory backgrounds AdS
 4 × Y, where Y are 7-dimensional Sasaki-Einstein spaces. We also find in toric examples that local F-maximization is equivalent to the minimization of the volume of Y over the space of Sasakian metrics, a procedure also referred to as Z-minimization. Moreover, we find that the functions F and Z are related for any trial R-charges. In the models we study F is positive and decreases along RG flows. We therefore propose the “F-theorem” that we hope applies to all 3-d field theories: the finite part of the free energy on the three-sphere decreases along RG trajectories and is stationary at RG fixed points. We also show that in an infinite class of Chern-Simons-matter gauge theories where the Chern-Simons levels do not sum to zero, the free energy grows as N
 5/3 at large N. This non-trivial scaling matches that of the free energy of the gravity duals in type IIA string theory with Romans mass.

Towards the F-theorem: $ \mathcal{N} = 2 $ field theories on the three-sphere

These are notes based on a series of lectures given at the KITP workshop Quantum Criticality and the AdS/CFT Correspondence in July, 2009. The goal of the lectures was to introduce condensed matter physicists to the AdS/CFT correspondence. Discussion of string theory and of supersymmetry is avoided to the extent possible.

Holographic Duality with a View Toward Many-Body Physics

We derive a general formula of an index for three dimensional $ \mathcal{N} = 2 $ super-conformal field theories with general R-charge assignments to chiral multiplets by using the localization method in S2 × S1 background. As examples we compute the index for theories in a few mirror pairs, and confirm the agreement of the indices in each mirror pair.

Index for three dimensional superconformal field theories with general R-charge assignments

We compute the thermal free energy for all renormalizable Chern Simon the- ories coupled to a single fundamental bosonic and fermionic field in the 't Hooft large N limit. We use our results to conjecture a strong weak coupling duality invariance for this class of theories. Our conjectured duality reduces to Giveon Kutasov duality when restricted to N = 2 supersymmetric theories and to an earlier conjectured bosonization duality in an appropriate decoupling limit. Consequently the bosonization duality may be regarded as a deformation of Giveon Kutasov duality, suggesting that it is true even at large but finite N.

Chern Simons duality with a fundamental boson and fermion

We study the thermal partition function of level k U(N) Chern-Simons theories on S
 2 interacting with matter in the fundamental representation. We work in the ’t Hooft limit, $ N,k\to \infty $
 , with $ \lambda ={N \left/ {k} \right.} $
 and $ \frac{{{T^2}{V_2}}}{N} $
 held fixed where T is the temperature and V
 2 the volume of the sphere. An effective action proposed in arXiv:1211.4843 relates the partition function to the expectation value of a ‘potential’ function of the S1 holonomy in pure Chern-Simons theory; in several examples we compute the holonomy potential as a function of λ. We use level-rank duality of pure Chern-Simons theory to demonstrate the equality of thermal partition functions of previously conjectured dual pairs of theories as a function of the temperature. We reduce the partition function to a matrix integral over holonomies. The summation over flux sectors quantizes the eigenvalues of this matrix in units of $ \frac{{2\pi }}{k} $
 and the eigenvalue density of the holonomy matrix is bounded from above by $ \frac{1}{{2\pi \lambda }} $
 . The corresponding matrix integrals generically undergo two phase transitions as a function of temperature. For several Chern-Simons matter theories we are able to exactly solve the relevant matrix models in the low temperature phase, and determine the phase transition temperature as a function of λ. At low temperatures our partition function smoothly matches onto the N and λ independent free energy of a gas of non renormalized multi trace operators. We also find an exact solution to a simple toy matrix model; the large N Gross-Witten-Wadia matrix integral subject to an upper bound on eigenvalue density.

Phases of large N vector Chern-Simons theories on S 2 × S 1

We study the thermal partition function of level $k$ U(N) Chern-Simons theories on $S^2$ interacting with matter in the fundamental representation. We work in the 't Hooft limit, $N,k\to\infty$, with $\lambda = N/k$ and $\frac{T^2 V_{2}}{N}$ held fixed where $T$ is the temperature and $V_{2}$ the volume of the sphere. An effective action proposed in arXiv:1211.4843 relates the partition function to the expectation value of a `potential' function of the $S^1$ holonomy in pure Chern-Simons theory; in several examples we compute the holonomy potential as a function of $\lambda$. We use level rank duality of pure Chern-Simons theory to demonstrate the equality of thermal partition functions of previously conjectured dual pairs of theories as a function of the temperature. We reduce the partition function to a matrix integral over holonomies. The summation over flux sectors quantizes the eigenvalues of this matrix in units of ${2\pi \over k}$ and the eigenvalue density of the holonomy matrix is bounded from above by $\frac{1}{2 \pi \lambda}$. The corresponding matrix integrals generically undergo two phase transitions as a function of temperature. For several Chern-Simons matter theories we are able to exactly solve the relevant matrix models in the low temperature phase, and determine the phase transition temperature as a function of $\lambda$. At low temperatures our partition function smoothly matches onto the $N$ and $\lambda$ independent free energy of a gas of non renormalized multi trace operators. We also find an exact solution to a simple toy matrix model; the large $N$ Gross-Witten-Wadia matrix integral subject to an upper bound on eigenvalue density.

Phases of large $N$ vector Chern-Simons theories on $S^2 \times S^1$

In this paper we discuss SU(N) Chern-Simons theories at level k with both fermionic and bosonic vector matter. In particular we present an exact calculation of the free energy of the N = 2 supersymmetric model (with one chiral field) for all values of the `t Hooft coupling in the large N limit. This is done by using a generalization of the standard Hubbard-Stratanovich method because the SUSY model contains higher order polynomial interactions.

Shuichi Yokoyama

Papers

Index for three dimensional superconformal field theories with general R-charge assignments

Chern Simons duality with a fundamental boson and fermion

Phases of large N vector Chern-Simons theories on S 2 × S 1

Phases of large $N$ vector Chern-Simons theories on $S^2 \times S^1$

Supersymmetric Chern-Simons theories with vector matter