We classify Freund-Rubin backgrounds of eleven-dimensional supergravity of the form AdS_4 x X^7 which are at least half BPS; equivalently, smooth quotients of the round 7-sphere by finite subgroups of SO(8) which admit an (N>3)-dimensional subspace of Killing spinors. The classification is given in terms of pairs consisting of an ADE subgroup of SU(2) and an automorphism defining its embedding in SO(8). In particular we find novel half-BPS quotients associated with the subgroups of type D_n (for n>5), E_7 and E_8 and their outer automorphisms.
We study the elliptic genera of two-dimensional orbifold CFTs, where the orbifolding procedure introduces du Val surface singularities on the target space. The N=4 character decompositions of the elliptic genus contributions from the twisted sectors at the singularities obey a consistent scaling property, and contain information about the arrangement of exceptional rational curves in the resolution. We also discuss how these twisted sector elliptic genera are related to twining genera and Hodge elliptic genera for sigma models with K3 target space.
We consider a double OPE limit of the planar four-point function of stress tensor multiplets in N = 4 SYM theory. Loop integrands for this correlator have been constructed to very high order, but the corresponding integrals are explicitly known only up to three loops. Fortunately, the double coincidence limit of the four-loop integrals can be found by the method of expansion by regions, which reduces the problem of computing the four-point integrals to the evaluation of a large set of massless propagator integrals. These can in turn be evaluated by IBP reduction. The OPE limit of the stress te…
We study vacua and BPS spectra of canonical surface defects of class $\mathcal{S}$ theories in different decoupling limits using ADE spectral networks. In some regions of the IR moduli spaces of these 2d-4d systems, the mixing between 2d and 4d BPS states is suppressed, and the spectrum of 2d-4d BPS states becomes that of a 2d $\mathcal{N}=(2,2)$ theory. For some decoupling limits, we identify the 2d theories describing the surface defects with nonlinear sigma models and coset models that have been previously studied. We also study certain cases where the decoupling limit of a surface defect e…
We study the BPS invariants of the preferred Calabi-Yau resolution of ADE polyhedral singularities C^3/G given by Nakamura's G-Hilbert schemes. Genus 0 BPS invariants are defined by means of the moduli space of torsion sheaves as proposed by Sheldon Katz. We show that these invariants are equal to half the number of certain positive roots of an ADE root system associated to G. This is in agreement with the prediction via Gromov-Witten theory.
We compute R-charges of the BPS-monopole operators in $\mathcal{N} = 3$ $\widehat{ADE}$ Chern-Simons quiver gauge theories, along the lines of the work of Benna, Klebanov and Klose in \cite{bkk}. These theories have a weakly coupled UV completion in terms of $\mathcal{N}=3$ supersymmetric Chern-Simons Yang-Mills theories. In the UV limit the monopole operators are well approximated by classical solutions. We construct classical BPS and anti-BPS monopole solutions to these theories which preserve $\frac{1}{3}$ supersymmetry all along the RG flow. We compute the $SU(2)_R$ charges in these backgr…
Non-BPS D-brane Solutions in Six Dimensional Orbifolds
从Eyras和Panda最近构造的IIB/ $(-1)^{F_L}I_4$的非BPS D 0-膜解出发，利用T-对偶构造了Sen预言的IIB/ $(-1)^{F_L}I_4$和IIA/ $(-1)^{F_L}I_4$的非BPS D 2-膜和D 1-膜解。而D1-膜耦合（电和磁）到NS 5A-膜的自对偶2-形式的潜力，这是目前在IIA型orbifold结构的扭曲部门。最后，我们讨论了这些膜的11维解释，起源于一个非BPS的M-理论定向折叠的$Ω_ρI_5$的M1-膜的解决方案。
We study the BPS spectrum of E-strings in a situation where the global E_8 symmetry is broken down to D_4 + D_4 by a certain twist. We find that the refined BPS index in this setup serves as a reduced BPS index of E-strings, which gives a novel trigonometric generalization of the Nekrasov partition function for four-dimensional N=2 supersymmetric SU(2) gauge theory with N_f=4 massless flavors. We determine the perturbative part of this index and also first few unrefined instanton corrections. By using these results together with the modular anomaly equation, the genus expansion of the free ene…
Curves of Marginal Stability in Two-Dimensional CP(N-1) Models with Z_N-Symmetric Twisted Masses
本文研究了N =（2，2）超对称的二维CP（N-1）模型的边际稳定性曲线.在这些理论中，可以引入扭曲质量项。BPS谱在弱耦合区和强耦合区具有不同的态数。这种光谱重构可以解释为两个区域被CMS分开，一些BPS态在CMS上衰变。我们研究了Z_N对称扭转质量的一种特殊情况。在这种情况下，由于Dorey的一般解决方案大大简化，CMS可以显式地找到。对于小- $N$值的数值结果。在大氮的限制，我们发现CMS分析高达ln N /N校正。
Smooth Freund-Rubin backgrounds of eleven-dimensional supergravity of the form AdS_4 x X^7 and preserving at least half of the supersymmetry have been recently classified. Requiring that amount of supersymmetry forces X to be a spherical space form, whence isometric to the quotient of the round 7-sphere by a freely-acting finite subgroup of SO(8). The classification is given in terms of ADE subgroups of the quaternions embedded in SO(8) as the graph of an automorphism. In this paper we extend this classification by dropping the requirement that the background be smooth, so that X is now allowe…
We study the BPS spectrum of four-dimensional $\mathcal{N}=2$ superconformal field theory of Argyres-Douglas type, obtained via twisted compactification of six-dimensional $A_{N-1}$ $(2,0)$ theory on a sphere with an irregular puncture, by using spectral networks. We give strong evidence of the equivalence of $\mathcal{N}=2$ superconformal field theories from six-dimensional theories of different ranks by systematically comparing the chamber structure and wall-crossing phenomena.
We review the connection of $Y$- and $Q$-systems with the BPS spectra of $4D$ $\mathcal{N}=2$ supersymmetric QFTs. For each finite BPS chamber of a $\mathcal{N}=2$ model which is UV superconformal, one gets a periodic $Y$-system, while for each finite BPS chamber of an asymptotically-free $\mathcal{N}=2$ QFT one gets a $Q$-system i.e. a rational recursion all whose solutions satisfy a linear recursion with constant coefficients (depending on the initial conditions). For instance, the classical $ADE$ $Y$-systems of Zamolodchikov correspond to the $ADE$ Argyres-Douglas $\mathcal{N}=2$ SCFTs, whi…
BPS bound states, supermembranes, and T-duality in M-theory
这是对II型超弦理论BPS束缚态的11维描述以及超膜在M理论中的作用的介绍性评论。第一部分描述了11维超引力的经典解，这些解在降维和T对偶下给出了IIA和IIB型弦论的NS-NS和R-R p-膜的束缚态。在某些情况下（例如IIB型弦论的（q_1，q_2）弦束缚态），这些非微扰的对象允许用基本2-膜进行简单的11维描述。计算了这类2-膜的BPS激发，并表明与BPS（q_1，q_2）弦束缚态的质谱完全匹配。相同束缚态的不同11 d表示可以用来提供振荡BPS态的不等价（T-对偶）描述。这允许测试T-对偶超越微扰理论，并在某些情况下，以逃避膜不稳定性去一个稳定的T-对偶表示。最后，我们总结了一些结果，这些结果表明在模参数空间的哪些区域，R ^9 x T ^2上的M理论的超膜描述似乎是足够的。
N=2 four dimensional gauge theories admit interesting half BPS surface operators preserving a (2,2) two dimensional SUSY algebra. Typical examples are (2,2) 2d sigma models with a flavor symmetry which is coupled to the 4d gauge fields. Interesting features of such 2d sigma models, such as (twisted) chiral rings, and the tt* geometry, can be carried over to the surface operators, and are affected in surprising ways by the coupling to 4d degrees of freedom. We will describe in detail a relation between the parameter space of twisted couplings of the surface operator and the Seiberg-Witten geome…
We analyze the supersymmetry transformations of gauged $SO(4)$ supergravity coupled to extra hypermultiplets in three dimensions, and find large families of smooth BPS solutions that preserve four supersymmetries. These BPS solutions are part of the consistent truncation of some families of six-dimensional superstrata. From the three-dimensional perspective, these solutions give rise to "smoothly-capped BTZ" geometries. We show how the twisting of the spin connection, the holomorphy of the fields, and the Chern-Simons connections all play an essential role in the existence of these supersymmet…
In this paper we would like to demonstrate how the known rules of anyon condensation motivated physically proposed by Bais \textit{et al} can be recovered by the mathematics of twist-free commutative separable Frobenius algebra (CSFA). In some simple cases, those physical rules are also sufficient conditions defining a twist-free CSFA. This allows us to make use of the generalized $ADE$ classification of CSFA's and modular invariants to classify anyon condensation, and thus characterizing all gapped domain walls and gapped boundaries of a large class of topological orders. In fact, this classi…
We propose a systematic approach to computing the BPS spectra of any 5d/6d supersymmetric quantum field theory in Coulomb phases, which admits either gauge theory descriptions or geometric descriptions, based on the Nakajima-Yoshioka's blowup equations. We provide a significant generalization of the blowup equation approach in terms of both properly quantized magnetic fluxes on the blowup $\hat{\mathbb{C}}^2$ and the effective prepotential for 5d/6d field theories on the Omega background which is uniquely determined by the Chern-Simons couplings on their Coulomb branches. We employ our method …
We compute one-loop threshold corrections to non-abelian gauge couplings in four-dimensional heterotic vacua with spontaneously broken $\cal N = 2 \to \cal N = 0$ supersymmetry, obtained as Scherk-Schwarz reductions of six-dimensional K3 compactifications. As expected, the gauge thresholds are no-longer BPS protected, and receive contributions also from the excitations of the RNS sector. Remarkably, the difference of thresholds for non-abelian gauge couplings is BPS saturated and exhibits a universal behaviour independently of the orbifold realisation of K3. Moreover, the thresholds and their …
In theories with extended supersymmetry the protected observables of UV superconformal fixed points are found in a number of contexts to be encoded in the BPS solitons along an IR Coulomb-like phase. For six-dimensional SCFTs such a role is played by the BPS strings on the tensorial Coulomb branch. In this paper we develop a uniform description of the worldsheet theories of a BPS string for rank-one 6d SCFTs. These strings are the basic constituents of the BPS string spectrum of arbitrary rank six-dimensional models, which they generate by forming bound states. Motivated by geometric engineeri…
在非对易空间R^{2n}_theta x G/H上构造了Yang-Mills方程的显式BPS解和非BPS解.给定一个G-表示，通过对G/H上的一个特殊丛进行扭曲，得到了R^{2n}_theta x G/H上的一个G-等变联络的G-等变U（k）丛.这些空间上的U（k）Donaldson-Uhlenbeck-Yau方程在R^{2n}_θ上的特殊规范规范理论中化为涡旋型方程。Seiberg-Witten方程是一个特殊的例子。非对易BPS组态的部分等距，这是从一个等变的Atiyah-Bott-Shapiro建设制定。它们可以被解释为充满空间的膜-反膜系统中的D 0-膜。
We study the BPS spectrum and vacuum moduli spaces in dimensional reductions of Chern-Simons-matter theories with N>=2 supersymmetry to zero dimensions. Our main example is a matrix model version of the ABJM theory which we relate explicitly to certain reduced 3-algebra models. We find the explicit maps from Chern-Simons quiver matrix models to dual IKKT matrix models. We address the problem of topologically twisting the ABJM matrix model, and along the way construct a new twist of the IKKT model. We construct a cohomological matrix model whose partition function localizes onto a moduli space …
We study half-BPS surface operators in N=2 supersymmetric QCD in four dimensions with gauge group SU(2) and four fundamental flavours. We compute the twisted chiral superpotential that describes the effective theory on the surface operator using equivariant localization as well as the Seiberg-Witten data. We then use the constraints imposed by S-duality to resum the instanton contributions to the twisted superpotential into elliptic functions and (quasi-) modular forms. The resummed results match what one would obtain from the description of surface operators as the insertion of a degenerate o…
The 6d $(2,0)$ theory has codimension-one symmetry defects associated to the outer-automorphism group of the underlying ADE Lie algebra. These symmetry defects give rise to twisted sectors of codimension-two defects that are either regular or irregular corresponding to simple or higher order poles of the Higgs field. In this paper, we perform a systematic study of twisted irregular codimension-two defects generalizing our earlier work in the untwisted case. In a class S setup, such twisted defects engineer 4d ${\mathcal N}=2$ superconformal field theories of the Argyres-Douglas type whose flav…
