We review cosmological inflation and its realization in quantum field theory and in string theory. This material is a portion of a book, also entitled "Inflation and String Theory", to be published by Cambridge University Press.

Inflation and String Theory

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

We compute the supersymmetric partition function of $$ \mathcal{N} $$
 = 1 supersymmetric gauge theories with an R-symmetry on $$ {\mathrm{\mathcal{M}}}_4\cong {\mathrm{\mathcal{M}}}_{g,p}\times {S}^1 $$
 , a principal elliptic fiber bundle of degree p over a genus-g Riemann surface, Σ
 g
 . Equivalently, we compute the generalized supersymmetric index $$ {I_{\mathrm{\mathcal{M}}}}_{{}_{g,p}} $$
 , with the supersymmetric three-manifold $$ {\mathrm{\mathcal{M}}}_{g,p} $$
 as the spatial slice. The ordinary $$ \mathcal{N} $$
 = 1 supersymmetric index on the round three-sphere is recovered as a special case. We approach this computation from the point of view of a topological A-model for the abelianized gauge fields on the base Σ
 g
 . This A-model — or A-twisted two-dimensional $$ \mathcal{N} $$
 = (2, 2) gauge theory — encodes all the information about the generalized indices, which are viewed as expectations values of some canonically-defined surface defects wrapped on T 2 inside Σ
 g
 × T 2. Being defined by compactification on the torus, the A-model also enjoys natural modular properties, governed by the four-dimensional ’t Hooft anomalies. As an application of our results, we provide new tests of Seiberg duality. We also present a new evaluation formula for the three-sphere index as a sum over two-dimensional vacua.

/pdf/arxiv-mathcal-n-1-supersymmetric-indices-and-the-four-2wuwb2disb.pdf

arXiv : $\mathcal{N}{=}1$ supersymmetric indices and the four-dimensional A-model

We review recent progress in studying mesons within gauge/gravity duality, in the context of adding flavour degrees of freedom to generalizations of the AdS/CFT correspondence. Our main focus is on the “top-down approach” of considering models constructed within string theory. We explain the string-theoretical constructions in detail, aiming at non-specialists. These give rise to a new way of describing strongly coupled confining large-N gauge theories similar to large-N QCD. In particular, we consider gravity dual descriptions of spontaneous chiral symmetry breaking, and compare with lattice results. A further topic covered is the behaviour of flavour bound states in finite-temperature field theories dual to a gravity background involving a black hole. We also describe the “bottom up” phenomenological approach to mesons within AdS/QCD. Some previously unpublished results are also included.

/pdf/mesons-in-gauge-gravity-duals-2b3nqyh1ca.pdf

Mesons in gauge/gravity duals

We compute the elliptic genera of general two-dimensional N=(2,2) and N=(0,2) gauge theories. We find that the elliptic genus is given by the sum of Jeffrey-Kirwan residues of a meromorphic form, representing the one-loop determinant of fields, on the moduli space of flat connections on T^2. We give several examples illustrating our formula, with both Abelian and non-Abelian gauge groups, and discuss some dualities for U(k) and SU(k) theories. This paper is a sequel to the authors' previous paper arXiv:1305.0533.

Elliptic genera of 2d N=2 gauge theories

We apply localization techniques to compute the partition function of a two-dimensional ${\mathcal{N}=(2,2)}$ R-symmetric theory of vector and chiral multiplets on S2. The path integral reduces to a sum over topological sectors of a matrix integral over the Cartan subalgebra of the gauge group. For gauge theories which would be completely Higgsed in the presence of a Fayet–Iliopoulos term in flat space, the path integral alternatively reduces to the product of a vortex times an antivortex partition functions, weighted by semiclassical factors and summed over isolated points on the Higgs branch. For applications, we evaluate the partition function for some U(N) gauge theories, showing equality of the path integrals for theories conjectured to be dual by Hori and Tong and deriving new expressions for vortex partition functions.

/pdf/partition-functions-of-n-2-2-gauge-theories-on-s-2-and-5b9wx6p47k.pdf

Partition functions of N = (2,2) gauge theories on S 2 and vortices

We study Seiberg-like dualities in three dimensional $ \mathcal{N} = 2 $
 supersymmetric theories, emphasizing Chern-Simons terms for the global symmetry group, which affect contact terms in two-point functions of global currents and are essential to the duality map. We introduce new Seiberg-like dualities for Yang-Mills-Chern-Simons theories with unitary gauge groups with arbitrary numbers of matter fields in the fundamental and antifundamental representations. These dualities are derived from Aharony duality by real mass deformations. They allow to initiate the systematic study of Seiberg-like dualities in Chern-Simons quivers. We also comment on known Seiberg-like dualities for symplectic and orthogonal gauge groups and extend the latter to the Yang-Mills case. We check our proposals by showing that the localized partition functions on the squashed S
 3 match between dual descriptions.

Comments on 3d Seiberg-like dualities

We apply localization techniques to compute the partition function of a two-dimensional N=(2,2) R-symmetric theory of vector and chiral multiplets on S^2. The path integral reduces to a sum over topological sectors of a matrix integral over the Cartan subalgebra of the gauge group. For gauge theories which would be completely Higgsed in the presence of a Fayet-Iliopoulos term in flat space, the path integral alternatively reduces to the product of a vortex times an antivortex partition functions, weighted by semiclassical factors and summed over isolated points on the Higgs branch. As applications we evaluate the partition function for some U(N) gauge theories, showing equality of the path integrals for theories conjectured to be dual by Hori and Tong and deriving new expressions for vortex partition functions.

/pdf/partition-functions-of-n-2-2-gauge-theories-on-s-2-and-4rizu31bzo.pdf

Partition functions of N=(2,2) gauge theories on S^2 and vortices

This paper addresses a long standing problem - to identify the chiral ring and moduli space (i.e. as an algebraic variety) on the Coulomb branch of an $ \mathcal{N} $
 = 4 superconformal field theory in 2+1 dimensions. Previous techniques involved a computation of the metric on the moduli space and/or mirror symmetry. These methods are limited to sufficiently small moduli spaces, with enough symmetry, or to Higgs branches of sufficiently small gauge theories. We introduce a simple formula for the Hilbert series of the Coulomb branch, which applies to any good or ugly three-dimensional $ \mathcal{N} $
 = 4 gauge theory. The formula counts monopole operators which are dressed by classical operators, the Casimir invariants of the residual gauge group that is left unbroken by the magnetic flux. We apply our formula to several classes of gauge theories. Along the way we make various tests of mirror symmetry, successfully comparing the Hilbert series of the Coulomb branch with the Hilbert series of the Higgs branch of the mirror theory.

/pdf/monopole-operators-and-hilbert-series-of-coulomb-branches-of-39rsjyf195.pdf

Monopole operators and Hilbert series of Coulomb branches of 3d \mathcal{N} = 4 gauge theories

This paper addresses a long standing problem - to identify the chiral ring and moduli space (i.e. as an algebraic variety) on the Coulomb branch of an N = 4 superconformal field theory in 2+1 dimensions. Previous techniques involved a computation of the metric on the moduli space and/or mirror symmetry. These methods are limited to sufficiently small moduli spaces, with enough symmetry, or to Higgs branches of sufficiently small gauge theories. We introduce a simple formula for the Hilbert series of the Coulomb branch, which applies to any good or ugly three-dimensional N = 4 gauge theory. The formula counts monopole operators which are dressed by classical operators, the Casimir invariants of the residual gauge group that is left unbroken by the magnetic flux. We apply our formula to several classes of gauge theories. Along the way we make various tests of mirror symmetry, successfully comparing the Hilbert series of the Coulomb branch with the Hilbert series of the Higgs branch of the mirror theory.

/pdf/monopole-operators-and-hilbert-series-of-coulomb-branches-of-yhoz3wjbh2.pdf

Stefano Cremonesi

Papers

Partition functions of N = (2,2) gauge theories on S 2 and vortices

Comments on 3d Seiberg-like dualities

Partition functions of N=(2,2) gauge theories on S^2 and vortices

Monopole operators and Hilbert series of Coulomb branches of 3d \mathcal{N} = 4 gauge theories

Monopole operators and Hilbert series of Coulomb branches of 3d N = 4 gauge theories