Naofumi Hama, Kazuo Hosomichi, Sungjay Lee
We extend the formula for partition functions of N=2 superconformal gauge theories on S^3 obtained recently by Kapustin, Willett and Yaakov, to incorporate matter fields with arbitrary R-charge assignments. We use the result to check that the self-mirror property of N=4 SQED with two electron hypermultiplets is preserved under a certain mass deformation which breaks the supersymmetry to N=2.
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We show in (1+2) dimensions that supersymmetric (SUSY) BF action for a (minimal and off-shell) spin-(1,3/2) supermultiplet is a unique SUSY invariant one required from the closure property of commutator algebra for SUSY transformations.
We comment on a recent puzzle regarding renormalization group invariance of exact results in SUSY theories, and argue that a purported renormalization of the dynamical scale Λdoes not in fact occur.
We study Euclidean 3D N=2 supersymmetric gauge theories on squashed three-spheres preserving isometries SU(2) x U(1) or U(1) x 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 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) x U(1) isometry, its measure and integrand are identified wi…
We compute the force between oppositely charged W bosons in the large N limit of Yang-Mills with 16 supercharges broken to SU(N) x U(1) by a finite Higgs vev. We clarify some issues regarding Wilson line computations and show that there is a regime in which the force between W bosons is independent of separation distance.
We describe a stabilization mechanism for fuzzy $S^4_N$ in the Euclidean IIB matrix model due to vacuum energy in the presence of a positive mass term. The one-loop effective potential for the radius contains an attractive contribution attributed to supergravity, while the mass term induces a repulsive contribution for small radius due to SUSY breaking. This leads to a stabilization of the radius. The mechanism should be pertinent to recent results on the genesis of 3+1-dimensional space-time in the Minkowskian IIB model.
We study aspects of Wess-Zumino models related to SUSY and R-symmetry breaking at tree-level. We present a recipe for constructing a wide class of tree-level SUSY and R-breaking models. We also deduce a general property shared by all tree-level SUSY breaking models that has broad application to model building. In particular, it explains why many models of direct gauge mediation have anomalously light gauginos (even if the R-symmetry is broken spontaneously by an order one amount). This suggests new approaches to dynamical SUSY breaking which can generate large enough gaugino masses.
This note gives a brief review of the integrable structures presented in the Seiberg-Witten approach to the N=2 SUSY gauge theories with emphasize on the case of the gauge theories with matter hypermultiplets included (described by spin chains). The web of different N=2 SUSY theories is discussed.
One can always write the mass matrix for Weyl spinors so that it is symmetric. However this is often not a good idea. It is usually incompatible with irreducibility of the fermion representations. As a result, a symmetrized mass term hides important symmetries and creates misleading difficulties that are not genuinely part of the theory. This is true for the Standard Model for example, and for its supersymmetric versions. There is a related subtlety, involving symmetrization of the interaction terms, that is central to the SUSY breaking mechanism of Cybersusy.
These are expanded notes of lectures given at the summer school "Gif 2000" in Paris. They constitute the first part of an "Introduction to supersymmetry and supergravity" with the second part on supergravity by J.-P. Derendinger to appear soon. The present introduction is elementary and pragmatic. I discuss: spinors and the Poincaré group, the susy algebra and susy multiplets, superfields and susy lagrangians, susy gauge theories, spontaneously broken susy, the non-linear sigma model, N=2 susy gauge theories, and finally Seiberg-Witten duality.
We consider an N=4 supersymmetric gauge theory on a curved space. We try to generalize Pestun's localization calculation on the four-sphere to a more general class of curved spaces. We calculated the Q-exact term to localize the path-integral, and when it becomes positive definite, we obtain a configuration where the path-integral localizes. We also evaluate the super Yang-Mills action in this configuration.
We describe surfaces in R^{N^2-1} generated by the holomorphic solutions of the supersymmetric CP^{N-1} model. We show that these surfaces are described by the fundamental projector constructed out of the solutions of this model and that in the CP^{N-1} case the corresponding surface is a sphere. Although the coordinates of the sphere are superfields the sphere's curvature is constant. We show that for N>2 the corresponding surfaces can also be constructed from the similar projector.
A simplified calculation of the structure constants of the diffeomorphism group of the two-sphere is presented
The low energy effective actions of the $N=2$ SUSY $SU(N_c)$ QCD are considered at the symmetric point on the moduli space. The classes of such theories have similar spectral curves. This fact allows us to show that all these models have the same structure of the coupling matrix and to show that the $N_f=2N_c$ spectral curve can not be presented as a double covering of the sphere. We calculate first instanton contributions to the coupling matrix and get nonperturbative $β$-functions in the $SU(2)$ gauge theory with non-zero bare masses of the matter hypermultiplets.
Localization principle is a powerful analytic tool in supersymmetric gauge theories which enables one to perform supersymmetric path integrals explicitly. Many important formulae have been obtained, and they led to a major breakthrough in the understanding of gauge theories at strong coupling as well as the dynamics of branes in M-theory. Some of those results are reviewed, focusing especially on Pestun's solution to four-dimensional N=2 supersymmetric gauge theories on S^4 and the subsequent developments on three or four-dimensional gauge theories on spheres.
We construct Euclidean 5d supersymmetric gauge theories on the five-sphere with vector and hypermultiplets. The SUSY transformation and the action are explicitly determined from the standard Noether procedure as well as from off-shell supergravity. Using localization techniques, the path-integral is shown to be restricted to the integration over a generalization of instantons on CP^2 and the Coulomb moduli.
We consider supersymmetric gauge theories on round 3-sphere with a certain background gauge field. The Lagrangians break the usual symmetry because the background gauge field which we have turned on violates the isometry. In order to maintain the supersymmetry, we choose unfamiliar charged Killing spinors as N = 2 SUSY parameters. We perform localization calculous within this setup and find the double sine function as we expected. We comment on more direct relationship between theories on round sphere and squashed sphere via Weyl transformation.
In these notes, we emphasize the rôle of spontaneous broken global discrete symmetries acting on the moduli space of $N=2$ susy Yang-Mills theories and show how they can be used, together with the BPS condition, as a spectrum generating symmetry. In particular, in the strong-coupling region, all BPS states come in multiplets of this broken symmetry. This played a key rôle in the determination of the strong-coupling spectra.
Many models of meta-stable supersymmetry (SUSY) breaking lead to a very light scalar pseudo-Nambu Goldstone boson (PNGB), P, associated with spontaneous breakdown of a baryon number like symmetry in the hidden sector. Current particle physics data provide no useful constraints on the existence of P. For example, the predicted decay rates for both K --> pi + P, b--> s + P and Upsilon --> photon + P are many orders of magnitude below the present experimental bounds. We also consider astrophysical implications of the PNGB and find a significant constraint from its effect on the evolution of red g…
This note covers various aspects of recent attempts to describe membranes ending on fivebranes using fuzzy geometry. In particular, we examine the Basu-Harvey equation and its relation to the Nahm equation as well as the consequences of using a non-associative algebra for the fuzzy three-sphere. This produces the tantalising result that the fuzzy funnel solution corresponding to Q coincident membranes ending on a five-brane has $Q^{3/2}$ degrees of freedom.
Duality in N=2 SUSY SU(2) Yang-Mills Theory: A pedagogical introduction to the work of Seiberg and Witten
这些是在巴黎高等师范学校和斯特拉斯堡纪念克劳德·伊奇克森会议上的介绍性演讲的笔记。我回顾了相当详细的工作，并希望在教学的方式塞伯格和威滕的$N=2$超对称$SU(2)$规范理论没有额外的问题。本演示文稿基本上遵循他们的原始工作，除了在最后一节中获得的低能量有效作用量，强调更多的monodromies和微分方程之间的关系，而不是使用椭圆曲线。
In this note we give an explicit formula for the preserved Killing spinors in deformed string theory backgrounds corresponding to integrable Yang--Baxter deformations realized via (sequences of) TsT transformations. The Killing spinors can be expressed only in terms of the bi-vector $Θ$ which encodes the deformation. This formula is applicable to deformed backgrounds related to $r$-matrices of various ranks, including those that do not satisfy the unimodularity condition and give rise to backgrounds in generalized supergravity. We conjecture that our formula also remains valid for integrable d…
Notes on Equivalences and Higgs Branches in N=2 Supersymmetric Yang-Mills Theory
本文研究了如何通过Higgs破缺，即通过给方夸克赋予期望值，来理解$N=2$ SUSY Yang-Mills理论的有效场论与物质之间的各种等价关系。我们给出了明确的表达式为广泛的一类例子的平坦方向。
In these notes we review Klimcik's construction of noncommutative gauge theory on the fuzzy supersphere. This theory has an exact SUSY gauge symmetry with a finite number of degrees of freedom and thus in principle it is amenable to the methods of matrix models and Monte Carlo numerical simulations. We also write down in this article a novel fuzzy supersymmetric scalar action on the fuzzy supersphere.
Using the ideas from the BPS/CFT correspondence, we give an explicit recursive formula for computing supersymmetric Wilson loop averages in 3d $\mathcal{N}=2$ Yang-Mills-Chern-Simons $U(N)$ theory on the squashed sphere $S^3_b$ with one adjoint chiral and two antichiral fundamental multiplets, for specific values of Chern-Simons level $κ_2$ and Fayet-Illiopoulos parameter $κ_1$. For these values of $κ_1$ and $κ_2$ the north and south pole turn out to be completely independent, and therefore Wilson loop averages factorize into answers for the two constituent $D^2 \times S^1$ theories. In partic…
We present evidence that N=1* SUSY Yang-Mills provides a deconstruction of a six-dimensional gauge theory compactified on a two-sphere. The six-dimensional theory is a twisted compactification of N=(1,1) SUSY Yang-Mills theory of the type considered by Maldacena and Nunez (MN). In particular, we calculate the full classical spectrum of the N=1* theory with gauge group U(N) in its Higgs vacuum. In the limit N goes to infinity, we find an exact agreement with the Kaluza-Klein spectrum of the MN compactification.
We study the patterns in the duality of a wide class of N=1 supersymmetric gauge theories in four dimensions. We present many new generalizations of the classic duality models of Kutasov and Schwimmer, which have themselves been generalized numerous times in works of Intriligator, Leigh and the present authors. All of these models contain one or two fields in a two-index tensor representation, along with fields in the defining representation. The superpotential for the two-index tensor(s) resembles A_k or D_k singularity forms, generalized from numbers to matrices. Looking at the ensemble of t…
We consider a free $p$-form gauge theory on a $d$-dimensional sphere of radius $R$ and calculate its free energy. We perform the calculation for generic values of $p$ and obtain the free energy as a function of $d, p$ and $R$. The result contains a $\log R$ term with coefficient proportional to $\left(2p+2-d \right)$, which is consistent with lack of conformal invariance for $p$ form theories in dimensions other than $2p+2$. We also compare the result for $p$-form and $(d-p-2)$-form theory which are classically Hodge dual to each other in $d$ dimensions and find that they agree for odd values …
Gauge theories in the presence of codimension two vortex defects are known to be related to the theories on orbifolds. By using this relation we study the localized path integrals of 2D N=(2,2) SUSY gauge theories with point-like vortex defects. We present a formula for the correlation functions of vortex defects inserted at the north and the south poles of squashed spheres. For Abelian gauge theories the correlators are locally constant as functions of the parameters of the defect, but exhibit discontinuity at some threshold values determined by the deficit angle at the poles and the R-charge…
In this note we compare even and odd fuzzy sphere constructions, their dimensional reductions and possible (M)atrix actions having them as solutions. We speculate on how the fuzzy 5-sphere might appear as a solution to the pp wave (M)atrix model.
We calculate the partition functions of supersymmetric gauge theories on S^5, which acquire non-perturbative contributions from instanton loops wrapping its Hopf fiber. The instantons on the CP^2 base equivariantly localize to 3 fixed points of SU(3) Cartans. Using our results, we study the index of the 6d (2,0) theory. We first study the partition functions of maximal SYM with AD(E) gauge groups, which agree with the gravity dual indices on AdS_7 in the large N limits. We also show that the most general partition function of the Abelian theory agrees with the index for a free 6d tensor superm…
The Power of Holomorphy -- Exact Results in 4D SUSY Field Theories
在超对称理论中，超势和规范动力学项系数的全纯性导致了强有力的结果。它们是重要的非重整化定理的潜在概念原因。它们还使我们能够研究这些理论的精确非微扰动力学。我们发现明确实现已知的现象，以及新的四维强耦合场论。这为禁闭和手征对称性破缺提供了新的线索。本文是根据在锡拉丘兹大学举行的PASCOS（94）会议上发表的讲话编写的。
Non-commutative corrections to the classical expression for the fuzzy sphere area are found out through the asymptotic expansion for its heat kernel trace. As an important consequence, some quantum gravity deviations in the luminosity of black holes must appear. We calculate these deviations for a static, spherically symmetric, black-hole with a horizon modeled by a fuzzy sphere. The results obtained could be verified through the radiation of black holes formed in the Large Hadron Collider (LHC).
Note on a numerical equality regarding the eta invariant on Berger spheres
本文通过数值计算发现Berger球面上的Dirac APS eta不变量与圆球面上的Dirac共形反常一致，并用广义Bernoulli多项式验证了已知猜想，同时得到了Weingart生成函数。
We study a recently discovered family of 1/8-BPS supersymmetric Wilson loops in N=4 super Yang-Mills theory and their string theory duals. The operators are defined for arbitrary contours on a two-sphere in space-time, and they were conjectured to be captured perturbatively by 2d bosonic Yang-Mills theory. In the AdS dual, they are described by pseudo-holomorphic string surfaces living on a certain submanifold of AdS_5 x S^5. We show that the regularized area of these string surfaces is invariant under area preserving diffeomorphisms of the boundary loop, in agreement with the conjecture. Furt…
Gauge theories in 2+1 dimensions can admit monopole operators in the potential. Starting with the theory without monopole potential, if the monopole potential is relevant there is an RG flow to the monopole-deformed theory. Here, focusing on U(Nc) SQCD with Nf flavors and N=2 supersymmetry, we show that even when the monopole potential is irrelevant, the monopole-modified theory Tm can exist and enjoy Seiberg-like dualities. We provide a renormalizable UV completion of Tm and an electric-magnetic dual description Tm'. We subject our proposal to various consistency checks such as mass deformati…
The UV fixed point theory of SU(2) gauge theory with N_f = 0,1,...,7 flavors is believed to have the enlarged E_{N_f +1} flavor symmetry. Actually it is not easy to check this conjecture because the UV theory is strongly-coupled, however, computation of certain SUSY protected quantities provides strong evidence for the enhancement of flavor symmetry. We study the superconformal index for SU(2) gauge theory with N_f = 0, 1 flavors in details, and we give a support for the enhancement by studying combinatorial structure of the superconformal indexes of these theories. We also give a nontrivial e…
Note on explicit construction of conformal generators on the fuzzy sphere
本研究针对三维共形场论的模糊球正则化方案，提出借助微观哈密顿量显式构造平移与特殊共形变换生成元的方法，可精准识别自旋和标度维数满足特定条件的共形初级态。
Pseudo-goldstinos appear in the scenario of multi-sector SUSY breaking. Unlike the true goldstino which is massless and absorbed by the gravitino, pseudo-goldstinos could obtain mass from radiative effects. In this note, working in the scenario of two-sector SUSY breaking with gauge mediation, we explicitly calculate the pseudo-goldstino mass at the leading three-loop level and provide the analytical results. In our calculation we consider the general case of messenger masses (not necessarily equal) and include the higher order terms of SUSY breaking scales. Our results can reproduce the numer…
A note on the functional determinant of higher-derivative scalar fields on sphere products
该注记研究S^q×S^p上标量场的GJMS算子泛函行列式，发现奇数维时其仅由总维数和p的奇偶性决定，偶p时为S^d标准值两倍，奇p时为零。
In a recent article, we were motivated by the question of whether any of the remarkable condensed matter phenomena, such as the quantum Hall effect (QHE), the Integer quantum Hall effect (IQHE) etc., could potentially be observed in the extreme astrophysical environments of neutron stars. As a prequel to that work, and with the aim of understanding better the role of the geometry of the conducting surface on the structure of Landau levels, in this article we study the quantum dynamics of a quantum particle on a squashed sphere. More specifically, we study the dynamics of a single particle on a…
We study timelike and null geodesics in a non-singular black hole metric proposed by Hayward. The metric contains an additional length-scale parameter $\ell$ and approaches the Schwarzschild metric at large radii while approaches a constant at small radii so that the singularity is resolved. We tabulate the various critical values of $\ell$ for timelike and null geodesics: the critical values for the existence of horizon, marginally stable circular orbit and photon sphere. We find the photon sphere exists even if the horizon is absent and two marginally stable circular orbits appear if the pho…
In supersymmetric (SUSY) field theory, there exist configurations which formally satisfy SUSY conditions but are not on original path integral contour. We refer to such configurations as complexified supersymmetric solutions (CSS). In this paper we discuss that CSS provide important information on large order behavior of weak coupling perturbative series in SUSY field theories. We conjecture that CSS with a bosonic (fermionic) free parameter give poles (zeroes) of Borel transformation of perturbative series whose locations are uniquely determined by actions of the solutions. We demonstrate thi…
I present a set of theories which display non-trivial `t Hooft anomaly matching for QCD with $F$ flavors. The matching theories are non-Abelian gauge theories with "dual" quarks and baryons, rather than the purely confining theories of baryons that `t Hooft originally searched for. The matching gauge groups are required to have an $F\pm 6$ dimensional representation. Such a correspondence is reminiscent of Seiberg's duality for supersymmetric (SUSY) QCD, and these theories are candidates for non-SUSY duality. However anomaly matching by itself is not sufficiently restrictive, and duality for Q…
We note that hadronic susy (empirical quark-diquark) symmetry can be expanded into the lepton sector, and that for three generations the counting of degrees of freedom is the one we need to build charged supermultiplets. For this to cure hierarchy, Higgs modeling becomes restricted.
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