The dynamics of the Goldstino mode of spontaneously broken supersymmetry is universal, being fully determined by the non-linearly realized symmetry. We investigate the small-field limit of this theory. This model non-linearly realizes an alternative supersymmetry algebra with vanishing anti-commutators between the fermionic generators, much like an internal supersymmetry. This Goldstino theory is akin to the Galilean scalar field theory that arises as the small-field limit of Dirac-Born-Infeld theory and non-linearly realizes the Galilean symmetry. Indeed, the small-field Goldstino is the partner of a complex Galilean scalar field under conventional supersymmetry. We close with the generalization to extended internal supersymmetry and a discussion of its higher-dimensional origin.
We study field theories with N extended non-linearly realized supersymmetries, describing the couplings of models that contain N goldstini. We review all the known formulations of the N=1 goldstino theories and we generalize them to an arbitrary number N of non-linearly realized supersymmetries. We explicitly prove the equivalence of all these extended supersymmetry breaking models containing N goldstini and reformulate the theory with N supersymmetries in terms of standard N=1 constrained superfields.
Supersymmetric phenomenology has been largely bound to the hypothesis that supersymmetry breaking originates from a single source. In this paper, we relax this underlying assumption and consider a multiplicity of sectors which independently break supersymmetry, thus yielding a corresponding multiplicity of goldstini. While one linear combination of goldstini is eaten via the super-Higgs mechanism, the orthogonal combinations remain in the spectrum as physical degrees of freedom. Interestingly, supergravity effects induce a universal tree-level mass for the goldstini which is exactly twice the …
When supersymmetry is broken in multiple sectors via independent dynamics, the theory furnishes a corresponding multiplicity of "goldstini" degrees of freedom which may play a substantial role in collider phenomenology and cosmology. In this paper, we explore the tree-level mass spectrum of goldstini arising from a general admixture of F-term, D-term, and almost no-scale supersymmetry breaking, employing non-linear superfields and a novel gauge fixing for supergravity discussed in a companion paper. In theories of F-term and D-term breaking, goldstini acquire a mass which is precisely twice th…
We study the 'goldstini' scenario of Cheung, Nomura, and Thaler, in which multiple independent supersymmetry (SUSY) breaking sectors lead to multiple would-be goldstinos, changing collider and cosmological phenomenology. In supergravity, potentially large corrections to the previous prediction of twice the gravitino mass for goldstini masses can arise when their scalar partners are stabilized far from the origin. Considerations arising from the complexity of realistic string compactifications indicate that many of the independent SUSY-breaking sectors should be conformally sequestered or situa…
Within the context of type I strings, we show the equivalence between BPS D9 branes with internal magnetic fluxes H_i in the three torii and non-BPS D3 branes with inverted internal magnetic fluxes 1/H_i. We then construct new supersymmetric examples of Z_2 x Z_2 orientifolds with discrete torsion which in the past had only non-supersymmetric solutions and emphasize the role of new twisted tadpole cancellation conditions, arising in the presence of magnetic fields, in order to get a consistent spectrum. In a second and independent part of the paper, we construct a new nine-dimensional type IIB…
The supersymmetry invariance of flat supergravity (i.e., supergravity in the absence of any internal scale in the Lagrangian) in four dimensions on a manifold with non-trivial boundary is explored. Using a geometric approach we find that the supersymmetry invariance of the Lagrangian requires to add appropriate boundary terms. This is achieved by considering additional gauge fields to the boundary without modifying the bulk Lagrangian. We also construct an enlarged supergravity model from which, in the vanishing cosmological constant limit, flat supergravity with a non-trivial boundary emerges…
We construct the supersymmetric completion of E$_{6(6)}$-covariant exceptional field theory. The theory is based on a $(5+27)$-dimensional generalized space-time subject to a covariant section constraint. The fermions are tensors under the local Lorentz group ${\rm SO}(1,4)\times {\rm USp}(8)$ and transform as weighted scalars under the E$_{6(6)}$ (internal) generalized diffeomorphisms. We present the complete Lagrangian and prove its invariance under supersymmetry. Upon explicit solution of the section constraint the theory embeds full $D=11$ supergravity and IIB supergravity, respectively.
We consider theories consisting of a planar interface with $\N=4$ super-Yang-Mills on either side and varying gauge coupling across the interface. The interface does not carry any independent degrees of freedom, but is allowed to support local gauge invariant operators, included with independent interface couplings. In general, both conformal symmetry and supersymmetry will be broken, but for special arrangements of the interface couplings, these symmetries may be restored. We provide a systematic classification of all allowed interface supersymmetries. We find new theories preserving eight an…
The dynamics of a vortex string, which describes the Nambu-Goldtone modes of the spontaneous breakdown of the target space D=4, N=1 supersymmetry and internal $U(1)_R$ symmetry to the world sheet ISO(1,1) symmetry, is constructed by using the approach of nonlinear realization. The resulting action describing the low energy oscillations of the string into the covolume (super)space is found to have an invariant synthesis form of the Akulov-Volkov and Nambu-Goto actions. Its dual scalar field action is obtained by means of introducing two vectorial Lagrangian multipliers into the action of the st…
The reduced field equations and BPS conditions are derived in Type IIB supergravity for configurations of the Janus type, characterized by an $AdS_4$-slicing of $AdS_5$, and various degrees of internal symmetry and supersymmetry. A generalization of the Janus solution, which includes a varying axion along with a varying dilaton, and has SO(6) internal symmetry, but completely broken supersymmetry, is obtained analytically in terms of elliptic functions. A supersymmetric solution with 4 conformal supersymmetries, SU(3) internal symmetry, a varying axion along with a varying dilaton, and non-tri…
We consider warped AdS$_3\times $S$^2\times$M$_5$ backgrounds in type II supergravity preserving small ${\cal N}=(4,0)$ supersymmetry. We show that imposing ${\cal N}=(4,0)$ supersymmetry imposes between 0 and 3 a priori isometries in the internal M$_5$. In this work we focus on classes of solution where M$_5$ exhibits no a prioi isometry which imposes additional constraints. Solving these in IIA forces M$_5$ to support an SU(2)-structure, a case already studied in arXiv:1908.09851, while in IIB one arrives at two broad new classes with identity structure that we reduce to local expressions fo…
A gauge theory for a superalgebra that includes an internal gauge (G) and local Lorentz algebras, and that could describe the low energy particle phenomenology is constructed. These two symmetries are connected by fermionic supercharges. The system includes an internal gauge connection 1-form $A$, a spin-1/2 Dirac spinor $ψ$, the Lorentz connection $ω$, and the vielbein $e$. The connection one-form is in the adjoint representation of G, while $ψ$ is in the fundamental. In contrast to standard supergravity, the metric is not a fundamental field and is in the center of the superalgebra: it is no…
We consider a novel class of constraints on chiral superfields to obtain supersymmetric nonlinear sigma models in four spacetime dimensions, which strictly combine the internal symmetry breaking with spontaneous supersymmetry breaking. The resultant massless modes can be exclusively Nambu-Goldstone bosons without their complex partners and the goldstino that is charged under the internal symmetry. The massive modes show a peculiar relation among their masses and the scales of symmetry breakings.
We explicitly describe, in the language of four-dimensional N=1 supersymmetric field theory, what happens when the moduli of a heterotic Calabi-Yau compactification change so as to make the internal non-Abelian gauge fields non-supersymmetric. At the edge of the region in Kahler moduli space where supersymmetry can be preserved, an additional anomalous U(1) gauge symmetry appears in the four-dimensional theory. The D-term contribution to the scalar potential associated to this U(1) attempts to force the system back into a supersymmetric configuration and provides a consistent low-energy descri…
Scherk-Schwarz compactification in string theory can be defined as orbifolding by an R symmetry, a symmetry that acts differently on bosons and fermions. Such a symmetry can arise in many situations, including toroidal and orbifold compactifications, as well as smooth Calabi-Yau spaces. If the symmetry acts freely then for large radius there are no tachyons in the spectrum. We focus mainly on stabilization by fluxes, and give examples with all moduli stabilized where the coupling is small and the internal manifold is large. Such models appear to be perturbatively stable with supersymmetry brok…
Among the possible superalgebras that contain the AdS$_3$ isometries, two interesting possibilities are the exceptional $F(4)$ and $G(3)$. Their R-symmetry is respectively SO(7) and $G_2$, and the amount of supersymmetry ${\cal N}=8$ and ${\cal N}=7$. We find that there exist two (locally) unique solutions in type IIA supergravity that realize these superalgebras, and we provide their analytic expressions. In both cases, the internal space is obtained by a round six-sphere fibred over an interval, with an O8-plane at one end. The R-symmetry is the symmetry group of the sphere; in the $G(3)$ ca…
Using an effective field theory approach and the language of SU(N)-structures, we study higher derivative corrections to the supersymmetry constraints for compactifications of string or M-theory to Minkowski space. Our analysis is done entirely in the target space and is thus very general, and does not rely on theory-dependent details such as the amount of worldsheet supersymmetry. For manifolds of real dimension n<4 we show that the internal geometry remains flat and uncorrected. For n=4, 6, Kahler manifolds with SU(N)-holonomy can become corrected to SU(N)-structure, while preserving supersy…
We present the supersymmetric extension of the recently constructed E$_{8(8)}$ exceptional field theory -- the manifestly U-duality covariant formulation of the untruncated ten- and eleven-dimensional supergravities. This theory is formulated on a (3+248) dimensional spacetime (modulo section constraint) in which the extended coordinates transform in the adjoint representation of E$_{8(8)}$. All bosonic fields are E$_{8(8)}$ tensors and transform under internal generalized diffeomorphisms. The fermions are tensors under the generalized Lorentz group SO(1,2)$\times$SO(16), where SO(16) is the m…
Nonlinear supersymmetry is used to compute the general form of the effective D-brane action in type I string theory compactified to four dimensions in the presence of internal magnetic fields. In particular, the scalar potential receives three contributions: (1) a nonlinear part of the D-auxiliary component, associated to the Dirac-Born-Infeld action; (2) a Fayet-Iliopoulos (FI) D-term with a moduli-dependent coefficient; (3) a D-auxiliary independent (but moduli dependent) piece from the D-brane tension. Minimization of this potential leads to three general classes of vacua with moduli stabil…
We study in a general way the construction of string bit Hamiltonians which are supersymmetric, We construct several quadratic and quartic polynomials in string bit creation and annihilation operators ${\barϕ}^A_{a_1\cdots a_n}$, $ϕ^A_{a_1\cdots a_n}$,which commute with the supersymmetry generators $Q^a$. Among these operators are ones with the spinor tensor structure required to provide the lightcone worldsheet vertex insertion factors needed to give the correct interactions for the IIB superstring, whenever a closed string separates into two closed strings or two closed strings join into one…
Can global internal and spacetime symmetries be connected without supersymmetry? To answer this question, we investigate Minkowski spacetimes with d space-like extra dimensions and point out under which general conditions external symmetries induce internal symmetries in the effective 4-dimensional theories. We further discuss in this context how internal degrees of freedom and spacetime symmetries can mix without supersymmetry in agreement with the Coleman-Mandula theorem. We present some specific examples which rely on a direct product structure of spacetime such that orthogonal extra dimens…
We consider examples of D=4 string theory vacua which, although globally non-geometric, admit a local description in terms of D=10 supergravity backgrounds. We analyze such backgrounds and find that the supersymmetry spinors vary non-trivially along the internal manifold, reproducing the interpolating supergravity solutions found by Frey and Grana. Finally, we propose a simple, local expression for non-geometric fluxes in terms of the internal spinors of the compactification.
We study the sub-structure of the heterotic Kahler moduli space due to the presence of non-Abelian internal gauge fields from the perspective of the four-dimensional effective theory. Internal gauge fields can be supersymmetric in some regions of the Kahler moduli space but break supersymmetry in others. In the context of the four-dimensional theory, we investigate what happens when the Kahler moduli are changed from the supersymmetric to the non-supersymmetric region. Our results provide a low-energy description of supersymmetry breaking by internal gauge fields as well as a physical picture …
The non-renormalization theorems of chiral vertex functions are derived on the basis of an algebraic analysis. The property, that the interaction vertex is a second supersymmetry variation of a lower dimensional field monomial, is used to relate chiral Green functions to superficially convergent Green functions by extracting the two supersymmetry variations from an internal vertex and transforming them to derivatives acting on external legs. The analysis is valid in the massive as well as in the massless model and can be performed irrespective of properties of the superpotential at vanishing m…
We consider phase transitions occurring in four-dimensional heterotic orbifold models, when the scale of spontaneous breaking of N=1 supersymmetry is of the order of the string scale. The super-Higgs mechanism is implemented by imposing distinct boundary conditions for bosons and fermions along an internal circle of radius R. Depending on the orbifold action, the usual scalars becoming tachyonic when R falls below the Hagedorn radius may or may not be projected out of the spectrum. In all cases, infinitely many other scalars, which are pure Kaluza-Klein or pure winding states along other inter…
We consider non-Abelian 1/2 BPS flux tubes (strings) in a deformed N=2 supersymmetric gauge theory, with mass terms mu_{1,2} of the adjoint fields breaking N=2 down to N=1. The main feature of the non-Abelian strings is the occurrence of orientational moduli associated with the possibility of rotations of their color fluxes inside a global SU(N) group. The bulk four-dimensional theory has four supercharges; half-criticality of the non-Abelian strings would imply then N=1 supersymmetry on the world sheet, i.e. two supercharges. In fact, superalgebra of the reduced moduli space has four supercha…
Type I string theory in the presence of internal magnetic fields provides a concrete realization of split supersymmetry. To lowest order, gauginos are massless while squarks and sleptons are superheavy. For weak magnetic fields, the correct Standard Model spectrum guarantees gauge coupling unification with \sin^2{θ_W}=3/8 at the compactification scale of M_{\rm GUT}\simeq 2 \times 10^{16} GeV. I discuss mechanisms for generating gaugino and higgsino masses at the TeV scale, as well as generalizations to models with split extended supersymmetry in the gauge sector.
We revisit the backgrounds of type IIB on manifolds with $SU(4)$-structure and discuss two sets of solutions arising from internal geometries that are complex and symplectic respectively. Both can be realized in terms of generalized complex geometry. We identify a map which relates the complex and symplectic supersymmetric systems. In the semi-flat torus bundle setting this map corresponds to T-duality and suggest a way of extending the mirror transform to non-Kähler geometries.
We present a comprehensive classification of supersymmetric vacua of M-theory compactification on seven-dimensional manifolds with general four-form fluxes. We analyze the cases where the resulting four-dimensional vacua have N = 1,2,3,4 supersymmetry and the internal space allows for SU(2), SU(3) or G_2 structures. In particular, we find for N = 2 supersymmetry, that the external space-time is Minkowski and the base manifold of the internal space is conformally Kähler for SU(2) structures, while for SU(3) structures the internal space has to be Einstein-Sasaki and no internal fluxes are allow…
We study a class of compactifications of M-theory to three dimensions that preserve N=2 supersymmetry and which have the defining feature that a probe space-time filling M2 brane feels a non-trivial potential on the internal manifold. Using M-theory/F-theory duality such compactifications include the uplifts of 4-dimensional N=1 type IIB compactifications with D3 potentials to strong coupling. We study the most general 8-dimensional manifolds supporting these properties, derive the most general flux that induces an M2 potential, and show that it is parameterised in terms of two real vectors. W…
We study AdS$_3\times \text{S}^2$ solutions in massive IIA that preserve small ${\cal N}=(4,0)$ supersymmetry in terms of an SU(2)-structure on the remaining internal space. We find two new classes of solutions that are warped products of the form AdS$_3\times \text{S}^2\times \text{M}_4\times \mathbb{R}$. For the first, M$_4$=CY$_2$ and we find a generalisation a D4-D8 system involving possible additional branes. For the second, M$_4$ need only be Kahler, and we find a generalisation of the T-dual of solutions based on D3-branes wrapping curves in the base of an elliptically fibered Calabi-Ya…
We start developing a formalism which allows to construct supersymmetric theories systematically across space-time signatures. Our construction uses a complex form of the supersymmetry algebra, which is obtained by doubling the spinor representation. This allows one to partially disentangle the Lorentz and R-symmetry group and generalizes symplectic Majorana spinors. For the case where the spinor representation is complex-irreducible, the R-symmetry only acts on an internal multiplicity space, and we show that the connected groups which occur are SO(2), SO(1,1), SU(2) and SU(1,1). As an applic…
In the presence of internal magnetic fields, a D9 brane can acquire a D5 (or anti-D5) R-R charge, and can therefore contribute to the corresponding tadpole. In the resulting vacua, supersymmetry is generically broken and tachyonic instabilities are present. However, suitable choices for the magnetic fields, corresponding to self-dual configurations in the internal space, can yield new chiral supersymmetric vacua with gauge groups of reduced rank, where the magnetic energy saturates, partly or fully, the negative tension of the O5+ planes. These models contain Green-Schwarz couplings to untwist…
String and M-theory realizations of brane world supersymmetry breaking scenarios are considered in which visible sector Standard Model fields are confined on a brane, with hidden sector supersymmetry breaking isolated on a distant brane. In calculable examples with an internal manifold of any volume the Kahler potential generically contains brane--brane non-derivative contact interactions coupling the visible and hidden sectors and is not of the no-scale sequestered form. This leads to non-universal scalar masses and without additional assumptions about flavor symmetries may in general induce …
We investigate the transmutation of D-branes into Abelian magnetic backgrounds on the world-volume of higher-dimensional branes, within the framework of global models with compact internal dimensions. The phenomenon, T-dual to brane recombination in the intersecting-brane picture, shares some similarities to inverse small-instanton transitions in non-compact spaces, though in this case the Abelian magnetic background is a consequence of the compactness of the internal manifold, and is not ascribed to a zero-size non-Abelian instanton growing to maximal size. We provide details of the transitio…
We examine the notion of symmetry in quantum field theory from a fundamental representation theoretic point of view. This leads us to a generalization expressed in terms of quantum groups and braided categories. It also unifies the conventional concept of symmetry with that of exchange statistics and the spin-statistics relation. We show how this quantum group symmetry is reconstructed from the traditional (super) group symmetry, statistics and spin-statistics relation. The old question of extending the Poincare group to unify external and internal symmetries (solved by supersymmetry) is reexa…
We generalise the study of constraints imposed by supersymmetry on the Berry connection to transformations with component fields in representations of an internal symmetry group G. Since the fields act as co-ordinates of the underlying space one finds a non-trivial extension to its structure and, correspondingly, there are new non-abelian constraints on the Berry connection. The specific case of G=SU(2) is shown to constrain the connection to behave as a magnetic monopole over su(2), its Lie algebra.
We consider compactifications of heterotic supergravity on anti-de Sitter space, with a six-dimensional nearly K"ahler manifold as the internal space. Completing the model proposed by Frey and Lippert with the particular choice of SU(3)/U(1)xU(1) for the internal manifold, we show that it satisfies not only the supersymmetry constraints but also the equations of motion with string corrections of order alpha'. Furthermore, we present a non-supersymmetric model. In both solutions we find confirmed a recent result of Ivanov on the connection used for anomaly cancellation. Interestingly, the volum…
We derive novel solutions of flux compactification with D7-branes on the resolved conifold in type IIB String Theory and later extend this solution to allow for non-zero temperature. At zero temperature, we find that adding D7-branes via the Ouyang embedding contributes to the supersymmetry-breaking (1,2) imaginary-self-dual flux, without generating a bulk cosmological constant. We further find that having D7-branes and a resolved conifold together give rise to a non-trivial D-term on the D7-branes. This supersymmetry-breaking term vanishes when we take the singular conifold limit, although su…
In his twistor-particle programme of the 1970's, Roger Penrose introduced a representation of the massive particle phase space in terms of a pair of twistors subject to an internal symmetry group. Here we use this representation to introduce a chiral string whose target is a complexification of this space, extended so as to incorporate supersymmetry. We show that the gauge anomalies associated to the internal symmetry group vanish only for maximal supersymmetry, and that correlators in these string models describe amplitudes involving massive particles with manifest supersymmetry. The models a…
We study ten-dimensional supersymmetric vacua with NSNS non-geometric fluxes, in the framework of $β$-supergravity. We first provide expressions for the fermionic supersymmetry variations. Specifying a compactification ansatz to four dimensions, we deduce internal Killing spinor equations. These supersymmetry conditions are then reformulated in terms of pure spinors, similarly to standard supergravity vacua admitting an SU(3)xSU(3) structure in Generalized Complex Geometry. The standard d-H acting on the pure spinors is traded for a generalized Dirac operator D, depending here on the non-geome…
New local solutions in type II supergravity that are foliations of AdS$_3\times S^3\times S^3$ over an interval and preserve at least large $\mathcal{N}=(4,0)$ supersymmetry are found. Some cases have compact internal space, some not and one experiences an enhancement to $\mathcal{N}=(4,4)$. We present two new globally compact solutions with D brane and O plane sources explicitly, one in each of IIA and IIB. The former is part of an infinite family of solutions with D8/O8s back reacted on AdS$_3\times S^3\times S^3\times S^1$. In the latter the the algebra degenerates to small $\mathcal{N}=(4,…
The emergent gauge theories are reconsidered in light of supersymmetry and an appropriate emergence conjecture is formulated. Accordingly, it might be expected that only global symmetries are fundamental symmetries of Nature, whereas local symmetries and associated massless gauge fields could solely emerge due to spontaneous breaking of underlying spacetime symmetries involved, such as relativistic invariance and supersymmetry. We further argue that this breaking, taken in the form of the nonlinear sigma-model type pattern for vector fields or superfields, puts essential restrictions on geomet…
We propose a (3+1)D linear set of covariant vector equations, which unify the spin 0 ``new Dirac equation'' with its spin 1/2 counterpart, proposed by Staunton. Our equations describe a spin (0,1/2) supermultiplet with different numbers of degrees of freedom in the bosonic and fermionic sectors. The translation-invariant spin deegres of freedom are carried by two copies of the Heisenberg algebra. This allows us to realize space-time supersymmetry in a bosonized form. The grading structure is provided by an internal reflection operator. Then the construction is generalized by means of the Major…
We continue the investigation into non-maximally symmetric compactifications of the heterotic string. In particular, we consider compactifications where the internal space is allowed to depend on two or more external directions. For preservation of supersymmetry, this implies that the internal space must in general be that of a $Spin(7)$ manifold, which leads to a $1/4$-BPS four-dimensional supersymmetric perturbative vacuum breaking all but one supercharge. We find that these solutions allow for internal geometries previously excluded by the domain-wall-type solutions, and hence the resulting…
It might be expected that only global symmetries are fundamental symmetries of Nature, whereas local symmetries and associated massless gauge fields could solely emerge due to spontaneous breaking of underlying spacetime symmetries involved, such as relativistic invariance and supersymmetry. This breaking, taken in the form of the nonlinear sigma-model type pattern for vector fields or superfields, puts essential restrictions on geometrical degrees of freedom of a physical field system that makes it to adjust itself in such a way that its global internal symmetry G turns into the local symmetr…
Supersymmetry Breaking with Fields, Strings and Branes
CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris（巴黎高等理工学院）；APC, UMR 7164-CNRS, Université Paris Cité（巴黎西岱大学）；Scuola Normale Superiore and INFN（比萨高等师范学院和意大利国家核物理研究所）
The heterotic--string models in the free fermionic formulation gave rise to some of the most realistic string models to date, which possess N=1 spacetime supersymmetry. Lack of evidence for supersymmetry at the LHC instigated recent interest in non-supersymmetric heterotic-string vacua. We explore what may be learned in this context from the quasi--realistic free fermionic models. We show that constructions with a low number of families give rise to proliferation of a priori tachyon producing sectors, compared to the non--realistic examples, which typically may contain only one such sector. Th…
