Localization in supergravity
James Sparks
本文介绍了超引力中等变定位的应用，重点在于四维理论和超对称黑洞的研究。
我们介绍了超引力中的等变定位，重点在于其在四维理论和超对称黑洞中的应用。
We give an introduction to equivariant localization in supergravity, focusing on the application to four-dimensional theories and supersymmetric black holes.
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We extend previous investigations on the construction of extremal supersymmetric and non-supersymmetric solutions in the H-FGK formalism to "unconventional" solutions with anharmonic terms. We show how the use of "fake" charge vectors equivariant under duality transformations simplifies and clarifies the task of identification of the attractors of the theory.
Equivariant localization in supergravity
该研究证明含R-对称性Killing矢量的超对称超引力解存在等变闭形式，可借助Berline-Vergne-Atiyah-Bott不动点定理计算物理可观测量，仅靠拓扑数据与R-对称矢量就能得到结果，无需求解超引力方程。
Equivariant localization for higher derivative supergravity
研究高导数超引力中等变闭形式的计算方法，无需求解运动方程即可得到超对称可观测量的闭式表达，应用于共形超引力与反物质理论。
Equivariant localization for $D=5$ gauged supergravity
本文研究了 $D=5$ 欧几里得规范超引力的超对称解，利用额外 Killing 矢量降维至 $D=4$，通过等变局部化计算在壳作用，并借此求出对偶 SCFT 的超对称 Casimir 能量与指标。
Equivariant localization and gluing rules in 4d $\mathcal{N}=2$ higher derivative supergravity
本文证明了高阶导数超引力中引力构建模块形式的猜想并检验了粘合规则，结合BVAB等变局部化公式与Omega背景下的作用评估，利用超共形引力形式确保结果在牛顿常数任意微扰阶有效，并应用于M2-brane配分函数全息预测。
Equivariant localization for AdS/CFT
本研究将等变局部化方法应用于AdS/CFT，借助超对称AdS解的等变闭形式与不动点公式，无需求解超引力方程即可计算BPS可观测量，复现了已知超引力解结果并算出未知显式解的中心荷。
Odd-Dimensional Localization in Supergravity
研究奇数维超引力理论的局域化原理，通过结合规范不变部分的等变扩展与陈 - 西蒙斯反常形式的等变完备形成相对等变上同调类，得出通用不动点公式，并在$D = 11$和$D = 5$超引力中举例说明，有多种应用。
Equivariant localization for $D=4$ gauged supergravity
本文利用 Killing 旋量双线性建立 $D=4$ 规范超引力中的等变闭多重形式，从而通过局域化计算通量积分与在壳作用量，并联系体不动点和边界 SCFT 数据。
We develop the formalism of supersymmetric localization in supergravity using the deformed BRST algebra defined in the presence of a supersymmetric background as recently formulated in arxiv:1806.03690. The gravitational functional integral localizes onto the cohomology of a global supercharge $Q_\text{eq}$, obeying $Q_\text{eq}^2=H$, where $H$ is a global symmetry of the background. Our construction naturally produces a twisted version of supergravity whenever supersymmetry can be realized off-shell. We present the details of the twisted graviton multiplet and ghost fields for the superconfor…
It is shown that Parisi-Sourlas supersymmetry of stochastic quantization is a Cartan model of equivariant cohomology. Equivariant cohomological structure of stochastic quantization of linear and non-linear sigma models are discussed. Witten's nonabelian localization principle is applied to the stochastic quantization of matrix models. As a result the equivalence between the original matrix model and the corresponding quenched Eguchi-Kawai model is established.
Superform Approach to Equivariant Localization in Supergravity
本文提出了一种超形式方法，用于超引力中的协变局部化，通过构造超对称背景上的闭合超形式，生成协变闭合的多项式形式，并扩展了最近的结果。
Equivariant localization in supergravity in odd dimensions
本文提出了一种在奇数维超引力中利用背景减法正则化方法对超对称解的本征动作进行局部化的公式，并通过拓扑数据推导了AdS_5中黑洞的熵函数。
We apply equivariant localization to supersymmetric quantum mechanics and show that the partition function localizes on the instantons of the theory. Our construction of equivariant cohomology for SUSY quantum mechanics is different than the ones that already exist in the literature. A hidden bosonic symmetry is made explicit and the supersymmetry is extended. New bosonic symmetry is the square of the new fermionic symmetry. The D term is now the parameter of the bosonic symmetry. This construction provides us with an equivariant complex together with a Cartan differential and makes the use of…
We present a brief introduction to the Berline-Vergne localization formula which expresses the integral of an equivariant cohomology class as a sum over zeros of a vector field to which that class is related.
M2-brane partition functions and HD supergravity from equivariant volumes
本文利用拓扑弦论等变体积相关结果建立M2膜配分函数的全息对应，推导普适艾里函数配分函数表示，验证多模型匹配结果并拓展至指标与其他膜系统。
It is shown, that the geometrical objects of Batalin-Vilkovisky formalism-- odd symplectic structure and nilpotent operator $Δ$ can be naturally uncorporated in Duistermaat--Heckman localization procedure. The presence of the supersymmetric bi-Hamiltonian dynamics with even and odd symplectic structure in this procedure is established. These constructions can be straightly generalized for the path-integral case.
Equivariant localization for wrapped M5-branes and D4-branes
该研究在AdS_{4,5}×M_2背景下，利用Killing旋量构造等变闭形式，通过不动点公式推导出重力块，验证了先前猜想。
The BRST algebra of supergravity is characterized by two different bilinears of the commuting supersymmetry ghosts: a vector $γ^μ$ and a scalar $ϕ$, the latter valued in the Yang-Mills Lie algebra. We observe that under BRST transformations $γ$ and $ϕ$ transform as the superghosts of, respectively, topological gravity and topological Yang-Mills coupled to topological gravity. This topological structure sitting inside any supergravity leads to universal equivariant cohomological equations for the curvatures 2-forms which hold on supersymmetric bosonic backgrounds. Additional equivariant cohomol…
Localization of the free energy in supergravity
本文推导了D=4、N=2规范超引力欧几里得超对称解自由能的通用局域化公式，无需求解方程即可计算，并给出了与全息对偶场论一致的新结果。
A supersymmetric Randall-Sundrum brane-world demands that not merely the graviton but the entire supergravity multiplet be trapped on the brane. To demonstrate this, we present a complete ansatz for the reduction of (D=5,N=4) gauged supergravity to (D=4,N=2) ungauged supergravity in the Randall-Sundrum geometry. We verify that it is consistent to lowest order in fermion terms. In particular, we show how the graviphotons avoid the `no photons on the brane' result because they do not originate from Maxwell's equations in D=5 but rather from odd-dimensional self-duality equations. In the case of …
Probing black holes with equivariant localization
通过协变局部化方法计算探测膜在超引力背景中的作用，研究了超对称探测D3膜在IIB超对称时空中的行为，揭示其对超对称指数的非微扰修正和缺陷算子插入。
We compute the quantum gravity partition function of M-theory on $AdS_4 \times X_7 $ by using localization techniques in four-dimensional gauged supergravity obtained by a consistent truncation on the Sasaki-Einstein manifold $X_{7}$. The supergravity path integral reduces to a finite dimensional integral over two collective coordinates that parametrize the localizing instanton solutions. The renormalized action of the off-shell instanton solutions depends linearly and holomorphically on the "square root" prepotential evaluated at the center of $AdS_{4}$. The partition function resembles the L…
The Faddeev-Popov gauge fixing in Yang-Mills theory is interpreted as equivariant localization. It is shown that the Faddeev-Popov procedure amounts to a construction of a symplectic manifold with a Hamiltonian group action. The BRST cohomology is shown to be equivalent to the equivariant cohomology based on this symplectic manifold with Hamiltonian group action. The ghost operator is interpreted as a (pre)symplectic form and the gauge condition as the moment map corresponding to the Hamiltonian group action. This results in the identification of the gauge fixing action as a closed equivariant…
We review equivariant localization techniques for the evaluation of Feynman path integrals. We develop systematic geometric methods for studying the semi-classical properties of phase space path integrals for dynamical systems, emphasizing the relations with integrable and topological quantum field theories. Beginning with a detailed review of the relevant mathematical background -- equivariant cohomology and the Duistermaat-Heckman theorem, we demonstrate how the localization ideas are related to classical integrability and how they can be formally extended to derive explicit localization for…
We study equivariant localization formulas for phase space path integrals when the phase space is a multiply connected compact Riemann surface. We consider the Hamiltonian systems to which the localization formulas are applicable and show that the localized partition function for such systems is a topological invariant which represents the non-trivial homology classes of the phase space. We explicitly construct the coherent states in the canonical quantum theory and show that the Hilbert space is finite dimensional with the wavefunctions carrying a projective representation of the discrete hom…
In this paper we propose the identification in AdS4 N = 2 gauged supergravity of the coefficient $τ_{RR}$ of 3D N = 2 SCFTs. We constraint the structure of this function in supergravity by combining the results from unitarity, holography and localization. We show that our conjectured function is minimized by the exact R-charge, corresponding to a gravitational attractor for the scalars in the special geometry. We identify this mechanism with the supergravity dual of the $τ_{RR}$-minimization. We check this proposal in the ABJM model, comparing with the expectations from localization and the Ad…
We discuss the problem of localization of 4D massless states in Randall-Sundrum 2 (one-brane) models. A Randall-Sundrum 2 construction starting from N=8 gauged supergravity in 5D Anti de Sitter space gives rise to an N=4 supergravity-matter system. We explicitly show that only the modes of the N=4 graviton supermultiplet localize on the 4D brane, streamlining and generalizing previous works. We also point out that while charged 1/4 BPS black holes do exist in the 4D theory, they are always produced in sets of total charge zero. This zero-charge configuration uplifts to a 5D metric without nake…
By considering the most general metric which can occur on a contractable two dimensional symplectic manifold, we find the most general Hamiltonians on a two dimensional phase space to which equivariant localization formulas for the associated path integrals can be applied. We show that in the case of a maximally symmetric phase space the only applicable Hamiltonians are essentially harmonic oscillators, while for non-homogeneous phase spaces the possibilities are more numerous but ambiguities in the path integrals occur. In the latter case we give general formulas for the Darboux Hamiltonians,…
We develop BRST quantization of gauge theories with a soft gauge algebra on spaces with asymptotic boundaries. The asymptotic boundary conditions are imposed on background fields, while quantum fluctuations about these fields are described in terms of quantum fields that vanish at the boundary. This leads us to construct a suitable background field formalism that is generally applicable to soft gauge algebras, and therefore to supergravity. We define a nilpotent BRST charge that acts on both the background and the quantum fields, as well as on the background and quantum ghosts. When the backgr…
The superconformal index and localizing higher derivative supergravity
通过等变局部化计算五维AdS旋转带电黑洞的应激能，与对偶场论在Cardy极限下的超共形指数计算取得精确匹配。
We analyze in detail the equivariant supersymmetry of the $G/G$ model. In spite of the fact that this supersymmetry does not model the infinitesimal action of the group of gauge transformations, localization can be established by standard arguments. The theory localizes onto reducible connections and a careful evaluation of the fixed point contributions leads to an alternative derivation of the Verlinde formula for the $G_{k}$ WZW model. We show that the supersymmetry of the $G/G$ model can be regarded as an infinite dimensional realization of Bismut's theory of equivariant Bott-Chern currents…
The topologically twisted index for ABJM theory with gauge group $U(N)_k \times U(N)_{-k}$ has recently been shown, in the large-$N$ limit, to reproduce the Bekenstein-Hawking entropy of certain magnetically charged asymptotically AdS$_4$ black holes. We numerically study the index beyond the large-$N$ limit and provide evidence that it contains a subleading logarithmic term of the form $-1/2\log N$. On the holographic side, this term naturally arises from a one-loop computation. However, we find that the contribution coming from the near horizon states does not reproduce the field theory answ…
We prove a localization formula for group-valued equivariant de Rham cohomology of a compact G-manifold. This formula is a non-trivial generalization of the localization formula of Berline-Vergne and Atiyah-Bott for the usual equivariant de Rham cohomology. As an application, we obtain a version of the Duistermaat-Heckman formula for Hamiltonian spaces with group-valued moment maps.
Localization on $AdS_3 \times S^2$ I: the 4d/5d connection in off-shell Euclidean supergravity
本文研究 $AdS_3 \times S^2$ 超引力泛函积分的局域化，通过 S$^2$ 扭转构造对偶于 $(0,4)$ SCFT$_2$ 椭圆亏格的五维欧几里得背景，并利用欧几里得 4d/5d 提升获得离壳 BPS 解。
Localization of the 5D supergravity action and Euclidean saddles for the black hole index
本文研究五维无规范超引力在壳作用量的等变局域化，构造具有两个独立转动和任意电荷的欧几里得超对称非极端黑洞鞍点，揭示其连接极端黑洞与双中心无地平线微态几何的插值行为，并证明作用量不依赖温度但受解析延拓影响。
IIB an equivariantly localized puncture
本研究通过均等局部化方法计算IIB超引力中AdS_3解的可观测量，并应用于研究穿孔和缺陷，获得2d SCFT的中心电荷新结果。
In this thesis, we examine in detail the notion of black hole entropy in Quantum Field Theories, with a specific focus on supersymmetric black holes and the perturbative and non-perturbative quantum corrections to the classical area-law of Bekenstein-Hawking. To examine such corrections, we employ the formalism of Sen's Quantum Entropy Function where the complete quantum entropy of a supersymmetric black hole in theories of supergravity is defined as a path-integral in the near-horizon region of the black hole. Evaluation of this path-integral can then be conducted exactly using localization c…
We use the Atiyah-Singer index theorem to derive the general form of the one-loop corrections to observables in asymptotically anti-de Sitter (AdS$_4$) supersymmetric backgrounds of abelian gauged supergravity. Using the method of supergravity localization combined with the factorization of the supergravity action on fixed points (NUTs) and fixed two-manifolds (Bolts) we show that an analogous factorization takes place for the one-loop determinants of supergravity fields. This allows us to propose a general fixed-point formula for the logarithmic corrections to a large class of supersymmetric …
Patch-wise localization with Chern-Simons forms in five dimensional supergravity
利用叶状结构的等变局域化，计算含矢量多重态的五维规范超引力中一般超对称解的壳作用，得到仅依赖拓扑数据的通用公式，适用于多中心黑洞、黑环等物理实例，并建立拓扑数据与热力学量的联系。
We find the most general bosonic solution to the localization equations describing the contributions to the quantum entropy of supersymmetric black holes in four-dimensional N=2 supergravity coupled to n_v vector multiplets. This requires the analysis of the BPS equations of the corresponding off-shell supergravity (including fluctuations of the auxiliary fields) with AdS2 \times S2 attractor boundary conditions. Our work completes and extends the results of arXiv:1012.0265 that were obtained for the vector multiplet sector, to include the fluctuations of all the fields of the off-shell superg…
Localizing Romans supergravity
本文利用等变局域化推导六维欧几里得 Romans 超引力在壳作用量的一般公式，证明其完全由 R 对称 Killing 矢量和拓扑数据决定，并复现已知结果、预测新解且与全息对偶匹配。
We study two-dimensional $\mathcal{N}=(2,2)$ supersymmetric gauged linear sigma models (GLSM) on the $Ω$-deformed sphere, $S^2_Ω$, which is a one-parameter deformation of the $A$-twisted sphere. We provide an exact formula for the $S^2_Ω$ supersymmetric correlation functions using supersymmetric localization. The contribution of each instanton sector is given in terms of a Jeffrey-Kirwan residue on the Coulomb branch. In the limit of vanishing $Ω$-deformation, the localization formula greatly simplifies the computation of $A$-twisted correlation functions, and leads to new results for non-abel…
Localization and Attraction
该研究利用等变局域化方法，为4维和5维超引力中的各类超对称黑洞构造离壳熵函数，推广了吸引子机制，还得到了对偶二维超对称共形场论中心荷的离壳表达式。
We give a brief introduction to the Berline-Vergne localization formula for the finite-dimensional setting and indicate how the Duistermaat-Heckman formula is derived from it. We consider applications of the localization formula when it is specialized to a maximal dimensional co-adjoint orbit. In particular, the case when the co-adjoint orbit is a quotient $G/T$ of a connected Lie group $G$ modulo a maximal torus $T$ is analyzed in detail. We describe also a generalization of the localization formula to non-compact group actions.
It is a common belief among field theorists that path integrals can be computed exactly only in a limited number of special cases, and that most of these cases are already known. However recent developments, which generalize the WKBJ method using equivariant cohomology, appear to contradict this folk wisdom. At the formal level, equivariant localization would seem to allow exact computation of phase space path integrals for an arbitrary partition function! To see how, and if, these methods really work in practice, we have applied them in explicit quantum mechanics examples. We show that the pa…
Equivariant Localization for N=2 Theories with Hypermultiplets
Ω背景下的等变局域化技术被应用于CP²上的N=2超对称场论，计算等变关联函数并与低能场论结果对比，确定了不同规范群配分函数的差异因子，还结合S对偶讨论了相关结果。
Ten-dimensional localization
该研究在十维直接构造等变闭多形式用于有质量IIA和IIB型超引力，将形式主义应用于IIA型两类解，重现并推广壳上作用已知结果，无需截断，还匹配场论配分函数，结果可扩展到其他II型超引力解。
Equivariant Cohomology, BRST Quantization, and Analytic Localization: A Unified Framework
本文提出一个统一框架，将等变上同调、BRST量子化与解析局部化联系起来，通过Kalkman变换和Witten变形推导ABBV局部化公式。
Equivariant localization and holography
本文探讨等变局部化理论在全息中的应用，通过Berline-Vergne和Molien-Weyl公式计算等变体积，并将其应用于超对称场论反常多项式积分、轨形指标及超引力自由能的引力块因子化证明。
The Komar charges of type II supergravity and supersymmetric localization
本文计算了IIA和IIB型超引力中任意Killing矢量的广义Komar荷，并研究了其用Page荷和特殊微分形式的重写，用于黑p-膜热力学；进一步证明了超对称Killing矢量下动量映射的性质。
Supersymmetric field theories can be studied exactly on suitable off-shell supergravity backgrounds. We show that in two dimensions such backgrounds are identifiable with BRST invariant backgrounds of topological gravity coupled to an abelian topological gauge multiplet. This latter background is required for the consistent coupling of the topological `matter' YM theory to topological gravity. We make use of this topological point of view to obtain, in a simple and straightforward way, a complete classification of localizing supersymmetric backgrounds in two dimensions. The BRST invariant topo…
Equivariant localization theory is a powerful tool that has been extensively used in the past thirty years to elegantly obtain exact integration formulas, in both mathematics and physics. These integration formulas are proved within the mathematical formalism of equivariant cohomology, a variant of standard cohomology theory that incorporates the presence of a symmetry group acting on the space at hand. A suitable infinite-dimensional generalization of this formalism is applicable to a certain class of Quantum Field Theories (QFT) endowed with supersymmetry. In this thesis we review the formal…
We develop a theory of equivariant factorization algebras on varieties with an action of a connected algebraic group $G$, extending the definitions of Francis-Gaitsgory [FG] and Beilinson-Drinfeld [BD1] to the equivariant setting. We define an equivariant analogue of factorization homology, valued in modules over $\text{H}^\bullet_G(\text{pt})$, and in the case $G=(\mathbb{C}^\times)^n$ we prove an equivariant localization theorem for factorization homology, analogous to the classical localization theorem [AtB]. We establish a relationship between $\mathbb{C}^\times$ equivariant factorization …
We study twists of supergravity theories and supersymmetric field theories, using a version of the pure spinor superfield formalism. Our results show that, just as the component fields of supersymmetric multiplets are the vector bundles associated to the equivariant Koszul homology of the variety of square-zero elements in the supersymmetry algebra, the component fields of the holomorphic twists of the corresponding multiplets are the holomorphic vector bundles associated to the equivariant Koszul homology of square-zero elements in the twisted supersymmetry algebra. The BRST or BV differentia…
We find a one-dimensional protected subsector of $\mathcal{N}=4$ matter theories on a general class of three-dimensional manifolds. By means of equivariant localization, we identify a dual quantum mechanics computing BPS correlators of the original model in three dimensions. Specifically, applying the Atiyah-Bott-Berline-Vergne formula to the original action demonstrates that this localizes on a one-dimensional action with support on the fixed-point submanifold of suitable isometries. We first show that our approach reproduces previous results obtained on $S^3$. Then, we apply it to the novel …
Indices of M5 and M2 branes at finite $N$ from equivariant volumes, and a new duality
研究6d(2,0)理论在有限N M5膜上的超对称指标，通过拓扑弦理论和超引力推导M2膜的超对称指标，并提出M2/M5对偶性的新扩展。
Review of localization in geometry: equivariant cohomology, characteristic classes, Atiyah-Bott formula, Atiyah-Singer equivariant index formula, Mathai-Quillen formalism
Equivariant volume extremization and holography
本文以内部几何或锥的等变体积表述全息引力极值化，通过缠绕四维轨形的膜超引力解验证其可恢复并预测中心荷与自由能，且证实 M5、D4 膜引力块公式。
We derive a fixed-point formula for integrals on moduli spaces of stable maps to projective spaces of even dimension. This gives a formula for the equivariant open Gromov-Witten invariants of (RP^{2m},CP^{2m}) and the structure constants of the equivariant Fukaya A-infinity algebra of RP^2m in CP^{2m} with bulk deformations. The formula involves contributions from Givental's correlators for the closed theory and the descendent integrals of discs, and specializes to give a new expression for the Welschinger count of real rational curves in the plane passing through some real and conjugation inv…
We consider the supergravity dual descriptions of non-conformal super Yang-Mills theories realized on the world-volume of Dp-branes. We use the dual description to compute stress-energy tensor and current correlators. We apply the results to the study of dilatonic brane-worlds described by non-conformal field theories coupled to gravity. We find that brane-worlds based on D4 and D5 branes exhibit a localization of gauge and gravitational fields. We calculate the corrections to the Newton and Coulomb laws in these theories.
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