Search

Non-relativistic conformal field theory describes many-body physics at unitarity. The correlation functions of the system are fixed by the requirement of conformal invariance. In this article, we discuss the correlation functions of scalar operators in non-relativistic conformal field theories in momentum space. We discuss the solution of conformal Ward identities and express 2,3, and 4-point functions as a function of energy and momentum. We also express the 3- and 4-point functions in the momentum space as the one-loop and three-loop Feynman diagram computations, respectively. Lastly, we generalize the discussion to the momentum space correlation functions in the presence of a boundary., Comment: 12 pages, 2 figures, A section on momentum space correlation function in the presence of boundary added, typos are removed, references are added

The partition functions of free bosons as well as fermions on $AdS_2$ are not smooth as a function of their masses. For free bosons, the partition function on $AdS_2$ is not smooth when the mass saturates the Breitenlohner-Freedman bound. We show that the expectation value of the scalar bilinear on $AdS_2$ exhibits a kink at the BF bound and the change in slope of the expectation value with respect to the mass is proportional to the inverse radius of $AdS_2$. For free fermions, when the mass vanishes the partition function exhibits a kink. We show that expectation value of the fermion bilinear is discontinuous and the jump in the expectation value is proportional to the inverse radius of $AdS_2$. We then show the supersymmetric actions of the chiral multiplet on $AdS_2\times S^1$ and the hypermultiplet on $AdS_2\times S^2$ demonstrate these features. The supersymmetric backgrounds are such that as the ratio of the radius of $AdS_2$ to $S^1$ or $S^2$ is dialled, the partition functions as well as expectation of bilinears are not smooth for each Kaluza-Klein mode on $S^1$ or $S^2$. Our observation is relevant for evaluating one-loop partition function in the near horizon geometry of extremal black holes., Comment: 34 pages, 6 figures, clarifying comments added

Non-relativistic conformal field theory is significant to understand various aspects of an ultra-cold system. In this paper, we study a non-relativistic system of two-component fermions interacting with a complex boson with Yukawa-like interactions near $d=4$-spatial dimensions in the presence of a quenched disorder. The homogeneous theory flows to an interacting fixed point describing a unitary fermion system. In the presence of the disorder, we find that the system has an interesting phase structure in the space of the coupling constants and exhibits an interacting disorder fixed point in $\epsilon$-expansion. The correlation function obeys Lifshitz scaling behaviour at the disorder fixed point with the anisotropic exponent being $z=2+\gamma_E$. We also study the disorder system at finite temperature and compute the leading contribution to the 1PI effective action., Comment: 25 pages, 6 figures

We study non-relativistic conformal field theory on a flat space in the presence of a planar boundary. We compute correlation functions of primary operators and obtain the expression for the boundary conformal block. We also discuss the non-relativistic conformal field theory on a general curved background in the presence of a boundary. As an example, we discuss the spectrum of boundary primary operator and compute scaling dimensions in a fermionic theory near one and three spatial dimensions., Comment: 20 pages

Disordered systems are interesting for many physical reasons. In this article, we study the renormalization group property of quenched disorder systems in the presence of a boundary. We construct examples of scalar field theories in various dimensions with both classical and quantum disorder localized at the boundary. We study these theories in $\e$-expansion and discuss properties of fixed points of the renormalization group flow., Comment: 23 pages

We present a formula for the quantum entropy of supersymmetric five-dimensional spinning black holes in M-theory compactified on $CY_3$, i.e., BMPV black holes. We use supersymmetric localization in the framework of off-shell five dimensional $N=2$ supergravity coupled to $I = 1,\dots,N_V + 1$ off-shell vector multiplets. The theory is governed at two-derivative level by the symmetric tensor $\mathcal{C}_{IJK}$ (the intersection numbers of the Calabi-Yau) and at four-derivative level by the gauge-gravitational Chern-Simons coupling $c_I$ (the second Chern class of the Calabi-Yau). The quantum entropy is an $N_V + 2$-dimensional integral parameterised by one real parameter $\varphi^I$ for each vector multiplet and an additional parameter $\varphi^0$ for the gravity multiplet. The integrand consists of an action governed completely by $\mathcal{C}_{IJK}$ and $c_{I}$, and a one-loop determinant. Consistency with the on-shell logarithmic corrections to the entropy, the symmetries of the very special geometry of the moduli space, and an assumption of analyticity constrains the one-loop determinant up to a scale-independent function $f(\varphi^0)$. For $f=1$ our result agrees completely with the topological M-theory conjecture of Dijkgraaf, Gukov, Nietzke, and Vafa for static black holes at two derivative level, and provides a natural extension to higher derivative corrections. For rotating BMPV black holes, our result differs from the DGNV conjecture at the level of the first quantum corrections.

We compute the partition function of $\mathcal N=2$ supersymmetric mixed dimensional QED on a squashed hemisphere using localization. Mixed dimensional QED is an abelian gauge theory coupled to charged matter fields at the boundary. The partition function is a function of the complex gauge coupling $\tau$, the choice of R-symmetry and the squashing deformation. The superconformal R-symmetry is determined using the 3-dimensional F-maximization. The free energy as a function of squashing deformation allows computing correlation functions that contain the insertion of the energy-momentum tensor. We compute the 2-point correlation function of the boundary energy-momentum tensor by differentiating the free energy with respect to the squashing parameter. We comment on the behaviour of the 2-point function as we change the complex coupling $\tau$., Comment: 19 pages, 2 figures

We use localization to compute the partition function of a four dimensional, supersymmetric, abelian gauge theory on a hemisphere coupled to charged matter on the boundary. Our theory has eight real supercharges in the bulk of which four are broken by the presence of the boundary. The main result is that the partition function is identical to that of ${\mathcal N}=2$ abelian Chern-Simons theory on a three-sphere coupled to chiral multiplets, but where the quantized Chern-Simons level is replaced by an arbitrary complexified gauge coupling $\tau$. The localization reduces the path integral to a single ordinary integral over a real variable. This integral in turn allows us to calculate the scaling dimensions of certain protected operators and two-point functions of abelian symmetry currents at arbitrary values of $\tau$. Because the underlying theory has conformal symmetry, the current two-point functions tell us the zero temperature conductivity of the Lorentzian versions of these theories at any value of the coupling. We comment on S-dualities which relate different theories of supersymmetric graphene. We identify a couple of self-dual theories for which the complexified conductivity associated to the U(1) gauge symmetry is $\tau/2$., Comment: 36 pages, 6 figures; v2 refs added

We develop the method of Green's function to evaluate the one loop determinants that arise in localization of supersymmetric field theories on $AdS$ spaces. The theories we study have at least ${\cal N}=2$ supersymmetry and normalisable boundary conditions are consistent with supersymmetry. We then show that under general assumptions the variation of the one loop determinant with respect to the localizing background reduces to a total derivative. Therefore it receives contributions only from the origin of $AdS$ and from asymptotic infinity. From expanding both the Greens function and the quadratic operators at the origin of $AdS$ and asymptotic infinity, we show that the variation of the one loop determinant is proportional to an integer. Furthermore, we show that this integer is an index of a first order differential operator. We demonstrate that these assumptions are valid for Chern-Simons theories coupled to chiral multiplets on $AdS_2\times S^1$. Finally we use our results to show that $U(N_c)$ Chern-Simons theory at level $k$ coupled to $N_f$ chiral multiplets and $N_f$ anti-chiral multiplets in the fundamental obeys level-rank duality on $AdS_2\times S^1$., Comment: 38 pages

We analyze the localization equations relevant to the quantum entropy of spinning supersymmetric black holes in five-dimensional asymptotically flat space. The precise problem is to classify all solutions to the off-shell supersymmetry equations in N=2 supergravity coupled to $n_\text{v}+1$ vector multiplets around the near-horizon black hole. We rewrite these equations in terms of the bosonic spinor bilinears that exist in the geometry for an arbitrary background. We then focus on the vector multiplet fluctuations around the near-horizon attractor region of the supersymmetric black hole, and classify all smooth solutions to the localization equations in this background for different choices of analytic continuation. For the choice of analytic continuation consistent with the 4d/5d lift, we find that the most general localization solution for the five-dimensional black hole problem is an~$(n_\text{v}+1)$-dimensional manifold, which is precisely the lift of the localization manifold for supersymmetric black holes in four-dimensional asymptotically flat space.

Squashed toric sigma models are a class of sigma models whose target space is a toric manifold in which the torus fibration is squashed away from the fixed points so as to produce a neck-like region. The elliptic genera of squashed toric-Calabi-Yau manifolds are known to obey the modular transformation property of holomorphic Jacobi forms, but have an explicit non-holomorphic dependence on the modular parameter. The elliptic genus of the simplest one-dimensional example is known to be a mixed mock Jacobi form, but the precise automorphic nature for the general case remained to be understood. We show that these elliptic genera fall precisely into a class of functions called higher-depth mock modular forms that have been formulated recently in terms of indefinite theta series. We also compute a generalization of the elliptic genera of these models corresponding to an additional set of charges corresponding to the toric symmetries. Finally we speculate on some relations of the elliptic genera of squashed toric models with the Vafa-Witten partition functions of $\mathcal{N}=4$ SYM theory on $\mathbb{CP}^2$., Comment: 43 pages; v2: Typos corrected, brief comments and clarifications added, accepted for publication in JHEP

We study the role of boundary conditions on the one loop partition function of ${\cal N}=2$ chiral multiplet of R-charge $\Delta$ on $AdS_2\times S^1$. The chiral multiplet is coupled to a background vector multiplet which preserves supersymmetry. We implement normalizable boundary conditions in $AdS_2$ and develop the Green's function method to obtain the one loop determinant. We evaluate the one loop determinant for two different actions: the standard action and the $Q$-exact deformed positive definite action used for localization. We show that if there exists an integer $n$ in the interval $D: ( \frac{\Delta-1}{2L}, \frac{\Delta}{2L} )$, where $L$ being the ratio of radius of $AdS_2$ to that of $S^1$, then the one loop determinants obtained for the two actions differ. It is in this situation that fields which obey normalizable boundary conditions do not obey supersymmetric boundary conditions. However if there are no integers in $D$, then fields which obey normalizable boundary conditions also obey supersymmetric boundary conditions and the one loop determinants of the two actions precisely agree. We also show that it is only in the latter situation that the one loop determinant obtained by evaluating the index of the $D_{10}$ operator associated with the localizing action agrees with the one loop determinant obtained using Green's function method., Comment: 34 pages

We study a class of two-dimensional N=(2,2) sigma models called squashed toric sigma models, using their Gauged Linear Sigma Models (GLSM) description. These models are obtained by gauging the global U(1) symmetries of toric GLSMs and introducing a set of corresponding compensator superfields. The geometry of the resulting vacuum manifold is a deformation of the corresponding toric manifold in which the torus fibration maintains a constant size in the interior of the manifold, thus producing a neck-like region. We compute the elliptic genus of these models, using localization, in the case when the unsquashed vacuum manifolds obey the Calabi-Yau condition. The elliptic genera have a non-holomorphic dependence on the modular parameter $\tau$ coming from the continuum produced by the neck. In the simplest case corresponding to squashed $\mathbb{C}/\mathbb{Z}_{2}$ the elliptic genus is a mixed mock Jacobi form which coincides with the elliptic genus of the N=(2,2) SL(2,R)/U(1) cigar coset., Comment: 35 pages, 4 figures

Conformal symmetry relates the metric on $AdS_2 \times S^{1}$ to that of $S^3$. This implies that under a suitable choice of boundary conditions for fields on $AdS_2$ the partition function of conformal field theories on these spaces must agree which makes $AdS_2 \times S^{1}$ a good testing ground to study localization on non-compact spaces. We study supersymmetry on $AdS_2\times S^1$ and determine the localizing Lagrangian for ${\cal N}=2$ supersymmetric Chern-Simons theory on $AdS_2\times S^1$. We evaluate the partition function of ${\cal N}=2$ supersymmetric Chern-Simons theory on $AdS_2 \times S^1$ using localization, where the radius of $S^1$ is $q$ times that of $AdS_2$. With boundary conditions on $AdS_2\times S^1$ which ensure that all the physical fields are normalizable and lie in the space of square integrable wave functions in $AdS_2$, the result for the partition function precisely agrees with that of the theory on the $q$-fold covering of $S^3$., Comment: 28 pages

We compute the anomalous dimensions of a class of operators of the form $(\bar\psi\psi)^p$ and $(\bar\psi\psi)^p\psi$ to leading order in $\epsilon$ in the Gross-Neveu model in $2+\epsilon$ dimensions. We use the techniques developed in arXiv: 1505.00963., Comment: 16 pages, some explanations in section 2 improved, references added and typos corrected