We show that supersymmetric gauge theories with a product gauge group admit quasi-instantons, which, like instantons, sit at the absolute minimum of the action in the corresponding topological sector. However, unlike ordinary instantons which preserve part of the supersymmetry, these break all supersymmetries of the action. We also provide a generalization where quasi-instantons take value in the associated Lie algebra of the recently proposed 3-algebras. When the structure constants and the metric of the algebra are covariantly constant, the gauge bundle admits new topological sectors with quasi-instantons at the minimum of the action.
We clarify certain aspects of instanton operators in five-dimensional supersymmetric gauge theories. In particular, we show how, in the pointlike limit, they become supersymmetric and provide the natural bridge with the instantonic states contributing to the index as well as with the zero mode counting leading to broken current multiplets.
Supersymmetric instanton solutions in four dimensional Euclidean ungauged Einstein-Maxwell theory are analysed and classified according to the fraction of supersymmetry they preserve, using spinorial geometry techniques.
An introduction to the instanton formalism in supersymmetric gauge theories is given. We explain how the instanton calculations, in conjunction with analyticity in chiral parameters and other general properties following from supersymmetry, allow one to establish exact results in the weak and strong coupling regimes. Some key applications are reviewed, the main emphasis is put on the mechanisms of the dynamical breaking of supersymmetry.
We discuss the reality properties of the fermionic collective coordinates in Euclidean space in an instanton background and construct hermitean actions for N = 4 and N = 2 supersymmetric Euclidean Yang-Mills theories.
The role of instantons in describing non-perturbative aspects of globally supersymmetric gauge theories is reviewed. The cases of theories with N=1, N=2 and N=4 supersymmetry are discussed. Special attention is devoted to the intriguing relation between instanton solutions in field theory and branes in string theory.
In this letter we argue that instanton-dominated Green's functions in N=2 Super Yang-Mills theories can be equivalently computed either using the so-called constrained instanton method or making reference to the topological twisted version of the theory. Defining an appropriate BRST operator (as a supersymmetry plus a gauge variation), we also show that the expansion coefficients of the Seiberg-Witten effective action for the low-energy degrees of freedom can be written as integrals of total derivatives over the moduli space of self-dual gauge connections.
We present the (0,4) superspace version of Witten's sigma model construction for ADHM instantons. We use the harmonic superspace formalism, which exploits the three complex structures common to both (0,4) supersymmetry and self-dual Yang-Mills theory. A novel feature of the superspace formulation is the manifest interplay between the ADHM construction and its twistor counterpart.
Fermionic modes of D-instanton wormholes from broken local supersymmetry
Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka Metropolitan University（大阪公立大学）；College of Arts and Sciences, J. F. Oberlin University
We study the non-perturbative dynamics of an unoriented Z_5-quiver theory of GUT kind with gauge group U(5) and chiral matter. At strong coupling the non-perturbative dynamics is described in terms of set of baryon/meson variables satisfying a quantum deformed constraint. We compute the effective superpotential of the theory and show that it admits a line of supersymmetric vacua and a phase where supersymmetry is dynamically broken via gaugino condensation.
In supersymmetric quantum mechanics, the non-Abelian Berry phase is known to obey certain differential equations. Here we study N=(0,4) systems and show that the non-Abelian Berry connection over R^{4n} satisfies a generalization of the self-dual Yang-Mills equations. Upon dimensional reduction, these become the tt* equations. We further study the Berry connection in N=(4,4) theories and show that the curvature is covariantly constant.
We study deformed supersymmetry in N=2 supersymmetric U(N) gauge theory in non(anti)commutative N=1 superspace. Using the component formalism, we construct deformed N=(1,1/2) supersymmetry explicitly. Based on the deformed supersymmetry, we discuss the C-dependence of the correlators. We also study the C-deformation of the instanton equation for the gauge group U(2).
Extending recent N=1 and N=2 results, we propose an explicit formula for the integration measure on the moduli space of charge-n ADHM multi-instantons in N=4 supersymmetric SU(2) gauge theory. As a consistency check, we derive a renormalization group relation between the N=4, N=2, and N=1 measures. We then use this relation to construct the purely bosonic (``N=0'') measure as well, in the classical approximation in which the one-loop small-fluctuations determinants is not included.
We analize instanton generated superpotentials for three dimensional N=2 supersymmetric gauge theories obtained by compactifying on S^1 N=1 four dimensional theories. In the N_f=1 case, we find that the vacua in the decompactification limit is given by the singular points of the Coulomb branch of the N=2 four dimensional theory (we also consider the massive case). The decompactification limit of the superpotential for pure gauge theories without chiral matter is interpreted in terms of `t Hooft's fractional instanton amplitudes.
Theories with axionic scalars admit three different Euclidean formulations, obtained by Wick rotation, Wick rotation combined with analytic continuation of the axionic scalars, and Wick rotation combined with Hodge dualization. We investigate the relation between these formulations for a class of theories which contains the sigma models of N=2 vector multiplets as a special case. It is shown that semi-classical amplitudes can be expressed equivalently using the two types of axionic actions, while the Hodge dualized version gives a different value for the instanton action unless the integration…
We study N=2 super Yang-Mills theory with gauge group SU(N) from the point of view of the Whitham hierarchy. We develop a new recursive method to compute the whole instanton expansion of the prepotential using the theta function associated to the root lattice of the group. Explicit results for the one and two-instanton corrections in SU(N) are presented. Interpreting the slow times of the hierarchy as additional couplings, we show how they can be promoted to spurion superfields that softly break N=2 supersymmetry down to N=0. This provides a family of nonsupersymmetric deformations of the theo…
We define and study projective special para-Kahler manifolds and show that they appear as target manifolds when reducing five-dimensional vector multiplets coupled to supergravity with respect to time. The dimensional reductions with respect to time and space are carried out in a uniform way using an epsilon-complex notation. We explain the relation of our formalism to other formalisms of special geometry used in the literature. In the second part of the paper we investigate instanton solutions and their dimensional lifting to black holes. We show that the instanton action, which can be define…
In this paper we construct the Lagrangian and Hamiltonian formulations of N=4 supersymmetric systems describing the motion of an isospin particle on a conformally flat four-manifold with SO(4) isometry carrying the non-Abelian field of a BPST instanton. The conformal factor can be specified to yield various particular systems, such as superconformally invariant mechanics as well as a particle on the four-sphere, the pseudosphere or on R x S^3. The isospin degrees of freedom arise as bosonic components of an additional fermionic N=4 supermultiplet, whose other components are rendered auxiliary …
We relate the non-perturbative exact results in supersymmetry to perturbation theory using several different methods: instanton calculations at weak or strong coupling, a method using gaugino condensation and another method relating strong and weak coupling. This allows many precise numerical checks of the consistency of these methods, especially the amplitude of instanton effects, and of the network of exact solutions in supersymmetry. However, there remain difficulties with the instanton computations at strong coupling.
The elucidation of the properties of the instantons in the topologically trivial sector has been a long-standing puzzle. Here we claim that the properties can be summarized in terms of the geometrical structure in the configuration space, the valley. The evidence for this claim is presented in various ways. The conventional perturbation theory and the non-perturbative calculation are unified, and the ambiguity of the Borel transform of the perturbation series is removed. A `proof' of Bogomolny's ``trick'' is presented, which enables us to go beyond the dilute-gas approximation. The prediction …
We propose explicit formulae for the integration measure on the moduli space of charge-n ADHM multi-instantons in N=1 and N=2 supersymmetric gauge theories. The form of this measure is fixed by its (super)symmetries as well as the physical requirement of clustering in the limit of large spacetime separation between instantons. We test our proposals against known expressions for n < 3. Knowledge of the measure for all n allows us to revisit, and strengthen, earlier N=2 results, chiefly: (1) For any number of flavors N_F, we provide a closed formula for F_n, the n-instanton contribution to the S…
Witten's linear sigma model for ADHM instantons possesses a natural $(0,4)$ supersymmetry. We study generalizations of the infrared limit of the model that are invariant under $(4,4)$ supersymmetry. In the case of four space-time dimensions a background with a conformally flat metric and torsion is required. The geometry is specified by a single real scalar function satisfying Laplace's equation. It gives rise to 't Hooft instantons for the gauge group $SU(2)$, instead of the general ADHM instantons for an $SO(n)$ gauge group in the case $(0,4)$.
We show that instantons violate a supersymmetric identity in a classically supersymmetric Higgs model with no massless fermions. This anomalous breaking arises because the correct perturbative expansion in the instanton sector is not supersymmetric. The attempt to construct a manifestly supersymmetric expansion generates infra-red divergences.
We perform a heat kernel asymptotics analysis of the nonperturbative superpotential obtained from wrapping of an M2-brane around a supersymmetric noncompact three-fold embedded in a (noncompact) G_2-manifold as obtained in [1], the three-fold being the one relevant to domain walls in Witten's MQCD [2], in the limit of small "zeta", a complex constant that appears in the Riemann surfaces relevant to defining the boundary conditions for the domain wall in MQCD. The MQCD-like configuration is interpretable, for small but non-zero zeta as a noncompact/"large" open membrane instanton, and for vanis…
It is well known that D-brane instantons can generate contributions to the effective superpotential of gauge theories living on D-branes which are perturbatively forbidden by global U(1) symmetries. We extend this idea to theories with supersymmetry breaking, studying the effect of D-brane instantons stretched between the SUSY-breaking and visible sectors. Analogously to what happens in the SUSY case, this mechanism can give rise to perturbatively forbidden soft terms (among other effects). We introduce and discuss general properties of instanton mediation. We illustrate our ideas in simple Ty…
We find the D(-1) and D1-brane instanton contributions to the hypermultiplet moduli space of type IIB string compactifications on Calabi-Yau threefolds. These combine with known perturbative and worldsheet instanton corrections into a single modular invariant function that determines the hypermultiplet low-energy effective acction.
By constructing a nilpotent extended BRST operator $\bs$ that involves the N=2 global supersymmetry transformations of one chirality, we show that the standard N=2 off-shell Super Yang Mills Action can be represented as an exact BRST term $\bs Ψ$, if the gauge fermion $Ψ$ is allowed to depend on the inverse powers of supersymmetry ghosts. By using this nonanalytical structure of the gauge fermion (via inverse powers of supersymmetry ghosts), we give field redefinitions in terms of composite fields of supersymmetry ghosts and N=2 fields and we show that Witten's topological Yang Mills theory ca…
We discuss the spontaneous breakdown of chiral symmetry in Quantum Chromodynamics by considering gluonic instanton configurations in the partition function. It is shown that in order to obtain nontrivial fermionic correlators in a two dimensional gauge theory for the strong interactions among quarks, a regular instanton background has to be taken into account. We work over massless quarks in the -fundamental- representation of SU(N_c). For large N_c, massive quarks are also considered.
We propose a robust supergravity model of dynamical supersymmetry breaking and gauge mediation, and a natural embedding in non-perturbative string theory with D-branes. A chiral field (and its mirror) charged under "anomalous" U(1)'s acts as a Polonyi field whose hierarchical Polonyi-term can be generated by string instantons. Further quartic superpotential terms arise naturally as a tree-level decoupling effect of massive string states. A robust supersymmetry breaking minimum allows for gauge mediation with soft masses at the TeV scale, which we realise for a globally consistent SU(5) GUT mod…
Models of spontaneous supersymmetry breaking generically have an R-symmetry, which is problematic for obtaining gaugino masses and avoiding light R-axions. The situation is improved in models of metastable supersymmetry breaking, which generically have only an approximate R-symmetry. Based on this we argue, with mild assumptions, that metastable supersymmetry breaking is inevitable. We also illustrate various general issues regarding spontaneous and explicit R-symmetry breaking, using simple toy models of supersymmetry breaking.
M(atrix) theory in PP-wave background possesses a discrete set of classical vacua, all of which preserves 16 supersymmetry and interpretable as collection of giant gravitons. We find Euclidean instanton solutions that interpolate between them, and analyze their properties. Supersymmetry prevents direct mixing between different vacua but still allows effect of instanton to show up in higher order effective interactions, such as analog of v^4 interaction of flat space effective theory. An explicit construction of zero modes is performed, and Goldstone zero modes, bosonic and fermionic, are ident…
We study instanton contributions to the superpotential of local F-theory compactifications which could potentially be used to engineer models of dynamical supersymmetry breaking. These instantons correspond to Euclidean 3-branes which form a threshold bound state with spacetime filling 7-branes. In certain cases, their contributions to the effective 4d superpotential can be determined in both perturbative string theory as well as directly via the topologically twisted theory on the 3-brane worldvolume, and in all cases we observe an exact match between these results. We further present an inst…
The partial breaking of supersymmetry in flat space can be accomplished using any one of three dual representations for the massive N=1 spin-3/2 multiplet. Each of the representations can be ``unHiggsed'', which gives rise to a set of dual N=2 supergravities and supersymmetry algebras.
In this talk we will study the partial breaking of supersymmetry in flat and anti de Sitter space. We will see that partial breaking in flat space can be accomplished using either of two representations for the massive N=1 spin-3/2 multiplet. We will "unHiggs" each representation and find a new N=2 supergravity and a new N=2 supersymmetry algebra. We will also see that partial supersymmetry breaking in AdS space can give rise to a new N=2 supersymmetry algebra, one that is necessarily nonlinearly realized.
These lectures give an introduction to the problem of finding a realistic and natural extension of the standard model based on spontaneously broken supersymmetry. Topics discussed at some length include the effective field theory paradigm, coupling constants as superfield spurions, gauge mediated supersymmetry breaking, and anomaly mediated supersymmetry breaking, including an extensive introduction to supergravity relevant for phenomenology.
A family of superpotentials is constructed which may be relevant to supersymmetry breaking in 4 dimensional (0,1) heterotic string models. The scale of supersymmetry breaking, as well as the coupling constant, would be stable.
This paper is devoted to the construction of a family of linear sigma models with $(0,4)$ supersymmetry which should flow in the infrared to the stringy version of Yang-Mills instantons on ${\bf R}^4$. The family depends on the full set of expected parameters and is obtained by using the data that appear in the ADHM construction of instantons.
We investigate mechanisms that can trigger supersymmetry breaking in open string vacua. The focus is on backgrounds with D-branes and orientifold planes that have an exact string description, and allow to study some of the quantum effects induced by supersymmetry breaking.
We study partial supersymmetry breaking from ${\cal N}=2$ to ${\cal N}=1$ by adding non-linear terms to the ${\cal N}=2$ supersymmetry transformations. By exploiting the necessary existence of a deformed supersymmetry algebra for partial breaking to occur, we systematically use ${\cal N}=2$ projective superspace with central charges to provide a streamlined setup. For deformed ${\cal O}(2)$ and ${\cal O}(4)$ hypermultiplets, besides reproducing known results, we describe new models exhibiting partial supersymmetry breaking with and without higher-derivative interactions.
We use string duality to describe instanton induced spontaneous supersymmetry breaking in string compactifications with additional background fields. Dynamical supersymmetry breaking by space-time instantons in the heterotic string theory is mapped to a tree level breaking in the type II string which can be explicitly calculated by geometric methods. It is argued that the instanton corrections resurrect the no-go theorem on partial supersymmetry breaking. The point particle limit describes the non-perturbative scalar potential of a SYM theory localized on a hypersurface of space-time. The N=0 …
We give a new twist to an old-fashioned topic in quantum field theory describing violations of the chiral charge conservation of massless fermions through Adler-Bell-Jackiw anomalies in the background of instanton fields in the context of non-relativistic Lifshitz theories. The results we report here summarize in a nut-shell our earlier work on the subject found in arXiv:1103.5693 and arXiv:1110.1332. We present simple examples where index computations can be carried out explicitly focusing, in particular, to gravitational models of Lifshitz type and highlight their differences from ordinary g…
We propose "conformal supersymmetry breaking" models, which tightly relate the conformal breaking scale (i.e. R-symmetry breaking scale) and the supersymmetry breaking scale. The both scales are originated from the constant term in the superpotential through the common source of the R-symmetry breaking. We show that dynamical tuning between those mass scales significantly reduces the degree of fine-tuning necessary for generating the almost vanishing cosmological constant.
We construct a model with D5 branes wrapped on a deformed and resolved $A_6$ singularity which realizes metastable supersymmetry breaking and minimal gauge mediation. Supersymmetry is broken at tree level by the F--term of singlet which also obtains a VEV as required in gauge mediation. Three nodes of the singularity are used to break supersymmetry whereas the other three realize gauge mediation. The supersymmetry breaking scale is suppressed due to brane instanton effects which are computed using a geometric transition.
We describe how a soft supersymmetry breaking Lagrangian arises naturally in the context of almost-commutative geometries that fall within the classification of those having a supersymmetric particle content as well as a supersymmetric spectral action. All contributions to such a Lagrangian are seen to either be generated automatically after introducing gaugino masses to the theory or coming from the second Seeley-DeWitt coefficient that is already part of the spectral action. In noncommutative geometry, a supersymmetric particle content and the appearance of a soft breaking Lagrangian thus ap…
This is the third paper in a series of four papers which introduce cybersusy, which is a new mechanism for supersymmetry breaking in the supersymmetric standard model (SSM). In this paper we display some solutions to the constraint equations of BRS cohomology in the SSM. In particular we discuss the leptonic dotspinor pseudosupermultiplets that were used in Cybersusy I to calculate leptonic supersymmetry breaking in the supersymmetric standard model. We also introduce examples of hadronic dotspinor pseudosupermultiplets that will induce baryonic supersymmetry breaking in the SSM for the baryon…
A mechanism of supersymmetry breaking in two or four-dimensions is given, in which the breaking is related to the Fermat's last theorem. It is shown that supersymmetry is exact at some irrational number points in parameter space, while it is broken at all rational number points except for the origin. Accordingly, supersymmetry is exact {\it almost everywhere}, as well as broken {\it almost everywhere} on the real axis in the parameter space at the same time. This is the first explicit mechanism of supersymmetry breaking with an arbitrarily small change of parameters around any exact supersymme…
For several classes of BPS vacua, we find a procedure to modify the PDEs that imply preserved supersymmetry and the equations of motion so that they still imply the latter but not the former. In each case we trace back this supersymmetry-breaking deformation to a distinct modification of the pure spinor equations that provide a geometrical interpretation of supersymmetry. We give some concrete examples: first we generalize the Imamura class of Mink6 solutions by removing a symmetry requirement, and then derive some local and global solutions both before and after breaking supersymmetry.
As a contribution to the current efforts to understand supersymmetry-breaking by meta-stable vacua, we study general properties of supersymmetry-breaking vacua in Wess-Zumino models: we show that tree-level degeneracy is generic, explore some constraints on the couplings and present a simple model with a long-lived meta-stable vacuum, ending with some generalizations to non-renormalizable models.
It is shown that the auxiliary field in the low-energy effective theory of the supersymmetric QCD (SU(N_c) gauge symmetry with flavors N_f < N_c) can be understood as a physical degree of freedom, once the supersymmetry is explicitly broken. Although the vacuum expectation value of the auxiliary field is just a measure of the supersymmetry breaking in the perturbative treatment of the supersymmetry breaking, it can be the vacuum expectation value of the quark bilinear operator in the non-perturbative treatment of the supersymmetry breaking. We show that the vacuum expectation value remains fin…
A formulation of new six-dimensional theories with (1,0) supersymmetry and E_8 global symmetry is proposed. The model is based on the large n theory describing n D-strings interacting with parallel D-fivebranes in Type I string theory.
Generic partial supersymmetry breaking of $N=2$ supergravity with zero vacuum energy and with surviving unbroken arbitrary gauge groups is exhibited. Specific examples are given.
When supersymmetry is spontaneously broken at tree level, the spectrum of the heterotic string compactified on orbifolds of tori contains an infinite number of potentially tachyonic modes. We show that this implies instabilities of Minkowski spacetime, when the scale of supersymmetry breaking is of the order of the string scale. We derive the phase space structure of vacua in the case where the tachyonic spectrum contains a mode with trivial momenta and winding numbers along the internal directions not involved in the supersymmetry breaking.
These lectures provide a simple introduction to supersymmetry breaking. After presenting the basics of the subject and illustrating them in tree-level examples, we discuss dynamical supersymmetry breaking, emphasizing the role of holomorphy and symmetries in restricting dynamically-generated superpotentials. We then turn to mechanisms for generating the MSSM supersymmetry-breaking terms, including ``gravity mediation'', gauge mediation, and anomaly mediation. We clarify some confusions regarding the decoupling of heavy fields in general and D-terms in particular in models of anomaly-mediation.
We show explicitly by the heuristic and practical arguments that for $N = 2$ supersymmetry (SUSY) a SUSY invariant relation between component fields of a vector supermultiplet of linear SUSY and Nambu-Goldstone fermions of the Volkov-Akulov model of nonlinear SUSY is written by using only three arbitrary dimensionless parameters, which can be recasted as the vacuum expectation values of auxiliary fields in the vector supermultiplet.
It is proved that, even if the gauge symmetry has been broken spontaneously at tree level, supersymmetry would never break through any finite orders of perturbation if it is not broken classically.
We discuss supersymmetry breaking in some supersymmetric quantum mechanical models with periodic potentials. The sensitivity to the parameters appearing in the superpotential is more acute than in conventional nonperiodic models. We present some simple elliptic models to illustrate these points.
