A surprising relation between 4d $\mathcal{N}=2$ class $\mathcal{S}$ superconformal field theories of Type-$A$ and 6d $\mathcal{N}=(1,0)$ orbi-instanton theories is investigated. We find that all of the theories in the former class can be obtained by a series of deformations of the 4d theories arising from compactifying the latter on a torus. This is demonstrated by examining Fayet--Iliopoulos (FI) deformations of the $E_8$-shaped magnetic quivers of the orbi-instanton theories whose body fits into the affine $E_8$ Dynkin diagram with a tail attached. Turning on FI parameters at the appropriate gauge groups leads, in stages, to $E_7$-shaped, $E_6$-shaped, and general star-shaped quivers, where the latter are magnetic quivers for the class $\mathcal{S}$ theory of Type-$A$ on a sphere with punctures. Deforming a suitable star-shaped quiver, one obtains a magnetic quiver of the Type-$A$ class $\mathcal{S}$ theory with general genus and an arbitrary number of punctures. Given such a theory, we also propose the inverse algorithm, thereby determining a parent orbi-instanton theory. This is achieved by uplifting the corresponding magnetic quiver step by step to the star-shaped, $E_6$-shaped, $E_7$-shaped, and $E_8$-shaped quivers, where at each step all of the underbalanced nodes, possessing non-zero FI parameters, are dualized. The latter $E_8$-shaped quiver then characterizes the 6d orbi-instanton theory from which the class $\mathcal{S}$ theory in question originates.
D-instantons are used to probe the near-horizon geometry of D3-branes systems on orbifold spaces. For fractional D3-branes, D-instanton calculus correctly reproduces the gauge beta-function and U(1)_R anomaly of the corresponding N=2 non-conformal Super Yang-Mills theories. For D3-branes wrapping the orbifold singularity, D-instantons can be identified with gauge instantons on ALE space, providing evidence of AdS/CFT duality for gauge theories on curved spaces.
$N$ M5-branes probing the intersection between the orbifold $\mathbb{C}^2/Γ_\text{ADE}$ and an $E_8$ wall give rise to 6d $(1,0)$ SCFTs known as ADE-type orbi-instantons. At fixed $N$ and order of the orbifold, each element of $\text{Hom}(Γ_\text{ADE},E_8)$ defines a different SCFT. The SCFTs are connected by Higgs branch RG flows, which generically reduce the flavor symmetry of the UV fixed point. We determine the full hierarchy of these RG flows for type A, i.e. $\mathbb{C}^2/\mathbb{Z}_k$, for any value of $N$ and $k$. The hierarchy takes the form of an intricate Hasse diagram: each node re…
We compute the instanton partition functions of $\mathcal{N}=1$ SCFTs in class $\mathcal{S}_k$. We obtain this result via orbifolding Dp/D(p-4) brane systems and calculating the partition function of the supersymmetric gauge theory on the worldvolume of $K$ D(p-4) branes. Starting with D5/D1 setups probing a $\mathbb{Z}_\ell\times \mathbb{Z}_k$ orbifold singularity we obtain the $K$ instanton partition functions of 6d $(1,0)$ theories on $\mathbb{R}^4 \times T^2$ in the presence of orbifold defects on $T^2$ via computing the 2d superconformal index of the worldvolume theory on $K$ D1 branes wr…
We show how to construct the general action coupling (multi)instantons to gauge theories arising from branes probing arbitrary toric singularities. We give a general set of rules for how to construct such an action given the knowledge of the superpotential for the gauge theory. The main idea is to obtain the action by higgsing a theory whose instanton dynamics is known, namely an orbifold of N=4 super Yang-Mills. We find that the couplings of the fermionic zero-modes with the holomorphic fields are dictated by the structure of the superpotential describing the toric singularity. We present exp…
Noncommutative Donaldson-Thomas invariants for abelian orbifold singularities can be studied via the enumeration of instanton solutions in a six-dimensional noncommutative N=2 gauge theory; this construction is based on the generalized McKay correspondence and identifies the instanton counting with the counting of framed representations of a quiver which is naturally associated to the geometry of the singularity. We extend these constructions to compute BPS partition functions for higher-rank refined and motivic noncommutative Donaldson-Thomas invariants in the Coulomb branch in terms of gauge…
We consider two simple examples of multi-instanton configurations in type II 4d N=1 superstring compactifications. The first one involves O(1) and U(1) D2-instantons embedded in T^6/(Z_2 x Z_2') geometry with SU(4) gauge symmetry coming from D6-branes and the second is related to quiver gauge theory of orientifolded orbifold of conifold containing fractional D(-1)-instantons and D3-branes. The additional zero modes of instantons which appear at the intersection points are lifted through interactions stemming from dimensionally reduced F- and D-terms. It is shown that multi-instanton configurat…
We consider N=4 theories on ALE spaces of $A_{k-1}$ type. As is well known, their partition functions coincide with $A_{k-1}$ affine characters. We show that these partition functions are equal to the generating functions of some peculiar classes of partitions which we introduce under the name 'orbifold partitions'. These orbifold partitions turn out to be related to the generalized Frobenius partitions introduced by G. E. Andrews some years ago. We relate the orbifold partitions to the blended partitions and interpret explicitly in terms of a free fermion system.
We consider systems of magnetized D9 branes on orbifolds supporting N=1 gauge theories. In such realizations, the matter multiplets arise from "twisted" strings connecting different stacks of branes. The introduction of Euclidean 5-branes wrapped on the six-dimensional compact space leads to instanton effects. We examine the interplay between the annuli diagrams with an E5 boundary and the holomorphicity properties of the effective low-energy supergravity action which describes the system, including its instanton corrections. Mostly based on arXiv:0709.0245.
The moduli space of instantons on an ALE space is studied using the moduli space of $\mathcal{N}=4$ field theories in three dimensions. For instantons in a simple gauge group $G$ on $\mathbb{C}^2/\mathbb{Z}_n$, the Hilbert series of such an instanton moduli space is computed from the Coulomb branch of the quiver given by the affine Dynkin diagram of $G$ with flavour nodes of unitary groups attached to various nodes of the Dynkin diagram. We provide a simple prescription to determine the ranks and the positions of these flavour nodes from the order of the orbifold $n$ and from the residual subg…
We analyze a class of dual pairs of heterotic and type I models based on freely-acting $\mathbb{Z}_2 \times \mathbb{Z}_2$ orbifolds in four dimensions. Using the adiabatic argument, it is possible to calculate non-perturbative contributions to the gauge coupling threshold corrections on the type I side by exploiting perturbative calculations on the heterotic side, without the drawbacks due to twisted moduli. The instanton effects can then be combined with closed-string fluxes to stabilize most of the moduli fields of the internal manifold, and also the dilaton, in a racetrack realization of th…
D-instanton calculus has proved to be able to probe the AdS near horizon geometry for $N$ D-branes systems which, when decoupled from gravity, yield four dimensional superconformal gauge theories with various matter content. In this work we extend previous analysis to encompass fractional brane models which give rise to non conformal N=2 Super Yang-Mills theories. Via D-instanton calculus we study the geometry of such models for finite $N$ and recover the $β$ function of the gauge coupling constants which is expected in non conformal gauge theories. We also give a topological matrix theory for…
In this note we explore the stringy interpretation of non-perturbative effects in N=1^* deformations of the A_{k-1} quiver models. For certain types of deformations we argue that the massive vacua are described by Nk fractional D3-branes at the orbifold polarizing into k concentric 5-brane spheres each carrying fractional brane charge. The polarization of the D3-branes induces a polarization of D-instantons into string world-sheets wrapped on the Myers spheres. We show that the superpotentials in these models are indeed generated by these world-sheet instantons. We point out that for certain p…
We study the $\mathcal N=2$ field theory realized by D3-branes on the ${\mathbb C}^2/{\mathbb Z}_2$ orbifold. The dual supergravity solution exhibits a repulson singularity cured by the enhancon mechanism. By comparing the open and closed string descriptions of a probe D-instanton, we can compute the exact non-perturbative profile of the supergravity twisted field, which determines the supergravity background. We then show how the non-trivial IR physics of the field theory translates into the stringy effects that give rise to the enhancon mechanism and the associated excision procedure.
One of the simplest $(1,0)$ supersymmetric theories in six dimensions lives on the world volume of one M5 brane at a $D$ type singularity $\mathbb{C}^2/D_k$. The low energy theory is given by an SQCD theory with $Sp(k-4)$ gauge group, a precise number of $2k$ flavors which is anomaly free, and a scale which is set by the inverse gauge coupling. The Higgs branch at finite coupling $\mathcal{H}_f$ is a closure of a nilpotent orbit of $D_{2k}$ and develops many more flat directions as the inverse gauge coupling is set to zero (violating a standard lore that wrongly claims the Higgs branch remains…
We provide a direct world-sheet derivation of the couplings of NS-NS and R-R fluxes to various types of D-branes (including instantonic ones) by evaluating disk amplitudes among two open string vertex operators at a generic brane intersection and one closed string vertex representing the background fluxes. This world-sheet approach is in full agreement with the derivation of the flux couplings in the brane effective actions based on supergravity methods, but it is applicable also to more general brane configurations involving fields with twisted boundary conditions. As an application, we consi…
Exact Thresholds and Instanton Effects in String Theory
在这堂课中，我们总结了最近在理解具有16种超对称性的弦论中的瞬子效应方面的一些工作。特别地，我们利用T^4上的杂化弦和K_3上的IIA型弦在轨道折叠点上的对偶性来考虑F^4耦合，以及这种弦-弦对偶性的高维和低维形式。在微扰水平的一个非平凡的测试的二元性，需要几个神奇的身份，提出了一个纯粹的单循环杂种优势的振幅匹配到一个纯粹的树级II型的结果。在这种情况下，我们发现了各种各样的非微扰效应，包括K_3 x S^1上IIA型的D膜瞬子和K_3 x T^2上IIA型的NS 5膜瞬子。此外，分析的三维情况下，它拥有一个非微扰SO（8，24，Z）的U-对偶，揭示了卡鲁扎-克莱因5膜瞬子效应的存在下，在杂种优势和II型侧。
We construct noncommutative Donaldson-Thomas invariants associated with abelian orbifold singularities by analysing the instanton contributions to a six-dimensional topological gauge theory. The noncommutative deformation of this gauge theory localizes on noncommutative instantons which can be classified in terms of three-dimensional Young diagrams with a colouring of boxes according to the orbifold group. We construct a moduli space for these gauge field configurations which allows us to compute its virtual numbers via the counting of representations of a quiver with relations. The quiver enc…
The worldvolume theory of D3-branes probing four D7-branes and an O7-plane on $\mathbb{C}^2/\mathbb{Z}_2$ is given by a supersymmetric USp x USp gauge theory. We demonstrate that, at least for a particular choice of the holonomy at infinity, we can go to a dual description of the gauge theory, where we can add a Fayet-Iliopoulos term describing the blowing-up of the orbifold to the smooth ALE space. This allows us to express the moduli space of SO(8) instantons on the smooth ALE space as a hyperkähler quotient of a flat space times the Higgs branch of a class S theory. We also discuss a genera…
We study fractional D-branes in the Type-IIA theory on a non-compact orientifold of the orbifold C^3/Z_3 in the boundary state formalism. We find that the fractional D0-branes of the orbifold theory become unstable due to the presence of a tachyon, while there is a stable D-instanton whose tachyon gets projected out. We propose that the D-instanton is obtained after tachyon condensation. We evidence this by calculating the Whitehead group of the Abelian category of objects corresponding to the boundary states as being isomorphic to Z_2.
Quantum toroidal algebras are obtained from quantum affine algebras by a further affinization, and, like the latter, can be used to construct integrable systems. These algebras also describe the symmetries of instanton partition functions for 5D $\mathcal{N}=1$ supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold $S^1\times\mathbb{C}^2/\mathbb{Z}_p$ where the action of $\mathbb{Z}_p$ is determined by two integer parameters $(ν_1,ν_2)$. The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of $\mathf…
We consider SU(3)-equivariant dimensional reduction of Yang-Mills theory over certain cyclic orbifolds of the 5-sphere which are Sasaki-Einstein manifolds. We obtain new quiver gauge theories extending those induced via reduction over the leaf spaces of the characteristic foliation of the Sasaki-Einstein structure, which are projective planes. We describe the Higgs branches of these quiver gauge theories as moduli spaces of spherically symmetric instantons which are SU(3)-equivariant solutions to the Hermitian Yang-Mills equations on the associated Calabi-Yau cones, and further compare them to…
We analyze instantons in the very strongly coupled large-$N$ limit ($N\to\infty$ with $g^2$ fixed) of large-$N$ gauge theories, where the effect of the instantons remains finite. By using the exact partition function of four-dimensional ${\cal N}=2^*$ gauge theories as a concrete example, we demonstrate that each instanton sector in the very strongly coupled large-$N$ limit is related to the one in the 't Hooft limit ($N\to\infty$ with $g^2N$ fixed) through a simple analytic continuation. Furthermore we show the equivalence between the instanton partition functions of a pair of large-$N$ gauge…
In this note, we establish several interesting connections between the supergroup gauge theories and the super integrable systems, i.e. gauge theories with supergroups as their gauge groups and integrable systems defined on superalgebras. In particular, we construct the super-characteristic polynomials of super-Toda lattice and elliptic double Calogero-Moser system by considering certain orbifolded instanton partition functions of their corresponding supergroup gauge theories. We also derive an exotic generalization of sl(2) XXX spin chain arising from the instanton partition function of SQCD …
We study quiver gauge theories on the round and squashed seven-spheres, and orbifolds thereof. They arise by imposing $G$-equivariance on the homogeneous space $G/H=\mathrm{SU}(4)/\mathrm{SU}(3)$ endowed with its Sasaki-Einstein structure, and $G/H=\mathrm{Sp}(2)/\mathrm{Sp}(1)$ as a 3-Sasakian manifold. In both cases we describe the equivariance conditions and the resulting quivers. We further study the moduli spaces of instantons on the metric cones over these spaces by using the known description for Hermitian Yang-Mills instantons on Calabi-Yau cones. It is shown that the moduli space of i…
We consider SU(2)-equivariant dimensional reduction of Yang-Mills theory on manifolds of the form $M\times S^3/Γ$, where $M$ is a smooth manifold and $S^3/Γ$ is a three-dimensional Sasaki-Einstein orbifold. We obtain new quiver gauge theories on $M$ whose quiver bundles are based on the affine ADE Dynkin diagram associated to $Γ$. We relate them to those arising through translationally-invariant dimensional reduction over the associated Calabi-Yau cones $C(S^3/Γ)$ which are based on McKay quivers and ADHM matrix models, and to those arising through SU(2)-equivariant dimensional reduction over …
Gieseker-Nakajima moduli spaces $M_{k}(n)$ parametrize the charge $k$ noncommutative $U(n)$ instantons on ${\bf R}^{4}$ and framed rank $n$ torsion free sheaves $\mathcal{E}$ on ${\bf C\bf P}^{2}$ with ${\rm ch}_{2}({\mathcal{E}}) = k$. They also serve as local models of the moduli spaces of instantons on general four-manifolds. We study the generalization of gauge theory in which the four dimensional spacetime is a stratified space $X$ immersed into a Calabi-Yau fourfold $Z$. The local model ${\bf M}_{k}({\vec n})$ of the corresponding instanton moduli space is the moduli space of charge $k$ …
