We argue that the conjectural relation between the subleading term in the small-squashing expansion of the free energy of general three-dimensional CFTs on squashed spheres and the stress-tensor three-point charge $t_4$ proposed in arXiv:1808.02052: $F_{\mathbb{S}^3_{\varepsilon}}^{(3)}(0)=\frac{1}{630}π^4 C_{\scriptscriptstyle T} t_4$, holds for an infinite family of holographic higher-curvature theories. Using holographic calculations for quartic and quintic Generalized Quasi-topological gravities and general-order Quasi-topological gravities, we identify an analogous analytic relation between such term and the charges $t_2$ and $t_4$ valid for five-dimensional theories: $F_{\mathbb{S}^5_{\varepsilon}}^{(3)}(0)=\frac{2}{15}π^6 C_{ \scriptscriptstyle T} \left[1+\frac{3}{40} t_2+\frac{23}{630} t_4\right]$. We test both conjectures using new analytic and numerical results for conformally-coupled scalars and free fermions, finding perfect agreement.
We present several results concerning the free energy of odd-dimensional conformal field theories (CFTs) on squashed spheres. First, we propose a formula which computes this quantity for holographic CFTs dual to higher-curvature gravities with second-order linearized equations of motion. As opposed to standard on-shell action methods for Taub geometries, our formula is automatically UV-finite and only involves a simple evaluation of the corresponding bulk Lagrangian on an auxiliary pure-AdS space. The expression is closely related to the function determining the possible AdS vacua of the bulk …
We evaluate the partition function of the free and interacting O(N) vector model on a two-parameter family of squashed three spheres in the presence of a scalar deformation. We also find everywhere regular solutions of Einstein gravity coupled to a scalar field in AdS and in dS with the same double squashed boundary geometry. Remarkably, the thermodynamic properties of the AdS solutions qualitatively agree with the behavior predicted by the free O(N) model with a real mass deformation. The dS bulk solutions specify the semiclassical `no-boundary' measure over anisotropic deformations of inflat…
Factorisation of N = 2 theories on the squashed 3-sphere
最近已证明，N=2理论的配分函数在压缩3-球面上局部化为矩阵积分。通过显式地评估矩阵积分，我们表明，阿贝尔配分函数可以表示为两个块的产品之和。我们用涡旋理论的配分函数来确定第一块，其中等变参数hbar=2 Pi i b^2，定义在R^2 x S_1上，对应于椭球的b->0退化。第二块给出了在对偶R^2 x S_1上对应于1/b ->0退化的具有等变参数hbar^L=2 Pi i/b^2的涡旋理论的配分函数。椭球分区似乎提供了涡旋理论的hbar -> hbar^L模不变非微扰完备性。
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 $τ$, 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 …
We consider mass-deformed theories with ${\cal N}\geq2$ supersymmetry on round and squashed three-spheres. By embedding the supersymmetric backgrounds in extended supergravity we show that at special values of mass deformations the supersymmetry is enhanced on the squashed spheres. When the $3d$ partition function can be obtained by a limit of a $4d$ index we also show that for these special mass deformations only the states annihilated by extra supercharges contribute to the index. By using an equivalence between partition functions on squashed spheres and ellipsoids, we explain the recently …
We consider type IIB $SL(2,\mathbb{Z})$ symmetry to relate the partition functions of different 5d supersymmetric Abelian linear quiver Yang-Mills theories in the $Ω$-background and squashed $S^5$ background. By Higgsing S-dual theories, we extract new and old 3d mirror pairs. Generically, the Higgsing procedure yields 3d defects on intersecting spaces, and we derive new hyperbolic integral identities expressing the equivalence of the squashed $S^3$ partition functions with additional degrees of freedom on the $S^1$ intersection.
In this paper we briefly review the main idea of the localization technique and its extension suitable in supersymmetric gauge field theory. We analyze the partition function of the vector multiplets with supercharges and its blocks on the even- and odd-dimensional spheres and squashed spheres. We exploit so-called Faà di Bruno's formula and show that multipartite partition functions can be written in the form of expansion series of the Bell polynomials. Applying the restricted specialization argument we show that $q$-infinite-product representation of partition functions admits presentation i…
我们证明了在压缩的S^3和S^2 x S^1上的三维N=2 R对称陈-西蒙斯物质理论的超对称配分函数可以用所谓的希格斯分支定域方法来计算，而不是更标准的库仑分支定域方法。对于可以用Fayet-Iliopoulos项完全Higgsed的理论，路径积分主要由位于几何中两个圆上的BPS涡弦决定。这样，配分函数就直接采用涡弦乘以反涡弦的配分函数在经典库仑分支上的有限个点上的和的形式。
We discuss the Cardy limit of 3d supersymmetric partition functions which allow the factorization into the hemisphere indices: the generalized superconformal index, the refined topologically twisted index and the squashed sphere partition function. In the Cardy limit, the hemisphere index can be evaluated by the saddle point approximation where there exists a dominant saddle point contribution, which we call the Cardy block. The Cardy block turns out to be a simple but powerful object as it is a building block of other partition functions in the Cardy limit. The factorization to the Cardy bloc…
We consider the partition function for Euclidean $SU(N)$ super Yang-Mills on a squashed seven-sphere. We show that the localization locus of the partition function has instanton membrane solutions wrapping the six "fixed" three-spheres on the $\mathbb{S}^7$. The ADHM variables of these instantons are fields living on the membrane world volume. We compute their contribution by localizing the resulting three-dimensional supersymmetric field theory. In the round-sphere limit the individual instanton contributions are singular, but the singularities cancel when adding the contributions of all six …
We discuss the large $N$ factorization properties of five-dimensional supersymmetric partition functions for CFT with a holographic dual. We consider partition functions on manifolds of the form $\mathcal{M}= \mathcal{M}_3 \times S^2_ε$, where $ε$ is an equivariant parameter for rotation. We show that, when $\mathcal{M}_3$ is a squashed three-sphere, the large $N$ partition functions can be obtained by gluing elementary blocks associated with simple physical quantities. The same is true for various observables of the theories on $\mathcal{M}_3=Σ_\mathfrak{g} \times S^1$, where $Σ_\mathfrak{g}$…
The Geometry of Supersymmetric Partition Functions
考虑紧致流形M上的超对称场论，得到了其配分函数Z_M对参数依赖性的约束。我们的主要焦点是Z_M对M几何的依赖性，以及耦合到连续味对称的背景规范场。对于N=1的四维U（1）R对称理论，M必是一个具有Hermite度量的复流形.我们发现Z_M与度量无关，全纯地依赖于复结构模。背景规范场定义了M上的全纯向量丛，Z_M是相应丛模的全纯函数。我们还对具有U（1）R对称性的三维N=2理论进行了平行分析，其中M上的必要几何结构是具有横向Hermite度量的横向全纯叶理（THF）.我们再次发现Z_M与度规无关，全纯地依赖于THF的模。我们讨论了几个应用，包括与超对称指数相关的半纯于S^3 x S^1或S^2 x S^1的流形，以及半纯于S^3（压扁球）的流形。在Z_M已明确计算的例子中，我们的结果解释了它的许多观察到的性质。
We reinterpret the OSV formula for the on-shell action/entropy function of asymptotically flat BPS black holes as a fixed point formula that is formally equivalent to a recent gluing proposal for asymptotically AdS$_4$ black holes. This prompts a conjecture that the complete perturbative answer for the most general gravitational building block of 4d $\mathcal{N}=2$ supergravity at a single fixed point takes the form of a Nekrasov-like partition function with equivariant parameters related to the higher-derivative expansion of the prepotential. In turn this leads to a simple localization-like p…
We study a perturbative expansion of the squashed 3-sphere ($S^3_b$) partition function of 3d $\mathcal{N}=2$ gauge theories around the squashing parameter $b= 1$. Our proposal gives the coefficients of the perturbative expansion as a finite sum over the saddle points of the supersymmetric-localization integral in the limit $b \rightarrow 0$ (the so-called Bethe vacua), and the contribution from each Bethe vacua can be systematically computed using saddle-point methods. Our expansion provides an efficient and practical method for computing basic CFT data ($F,C_T,C_{JJ}$ and higher-point correl…
We initiate the study of intersecting surface operators/defects in four-dimensional quantum field theories (QFTs). We characterize these defects by coupled 4d/2d/0d theories constructed by coupling the degrees of freedom localized at a point and on intersecting surfaces in spacetime to each other and to the four-dimensional QFT. We construct supersymmetric intersecting surface defects preserving just two supercharges in N = 2 gauge theories. These defects are amenable to exact analysis by localization of the partition function of the underlying 4d/2d/0d QFT. We identify the 4d/2d/0d QFTs that …
We study the supersymmetric partition function of 4d supersymmetric gauge theories with a U(1) R-symmetry on Euclidean $S^3\times S_β^1$, with $S^3$ the unit-radius squashed three-sphere, and $β$ the circumference of the circle. For superconformal theories, this partition function coincides (up to a Casimir energy factor) with the 4d superconformal index. The partition function can be computed exactly using supersymmetric localization of the gauge theory path-integral. It takes the form of an elliptic hypergeometric integral, which may be viewed as a matrix-integral over the moduli space of th…
We study the large $N$ limit of the superconformal index of a large class of 5d $\mathcal{N}=1$ superconformal field theories and show it is given by the square of the partition function on the squashed five-sphere. We show this simple relation implies a Cardy formula in 5d, which is valid in an "extended" regime in which fugacities are finite and $N$ is large. For theories with weakly coupled gravity duals we conjecture this large $N$ Cardy formula universally accounts for the microscopic entropy of spinning black holes in AdS$_{6}$. We check this explicitly for known black hole solutions in …
We present a statistical mechanical model whose random variables are solid partitions, i.e. Young diagrams built by stacking up four dimensional hypercubes. Equivalently, it can be viewed as the model of random tessellations of ${\bf R}^{3}$ by squashed cubes of four fixed orientations. The model computes the refined index of a system of D0-branes in the presence of $D8$-$\overline{D8}$ system, with a $B$-field strong enough to support the bound states. Mathematically, it is the equivariant K-theoretic version of integration over the Hilbert scheme of points on ${\bf C}^{4}$ and its higher ran…
