BPS Objects Research Articles

Torsion in cohomology and dimensional reduction

Conventional wisdom dictates that ℤN factors in the integral cohomology group Hp(Xn, ℤ) of a compact manifold Xn cannot be computed via smooth p-forms. We revisit this lore in light of the dimensional reduction of string theory on Xn, endowed with a G-structure metric that leads to a supersymmetric EFT. If massive p-form eigenmodes of the Laplacian enter the EFT, then torsion cycles coupling to them will have a non-trivial smeared delta form, that is an EFT long-wavelength description of p-form currents of the (n − p)-cycles of Xn. We conjecture that, whenever torsion cycles are calibrated, their linking number can be computed via their smeared delta forms. From the EFT viewpoint, a torsion factor in cohomology corresponds to a ℤN gauge symmetry realised by a Stückelberg-like action, and calibrated torsion cycles to BPS objects that source the massive fields involved in it.

Anomalies as obstructions: from dimensional lifts to swampland

We revisit the relation between the anomalies in four and six dimensions and the Chern-Simons couplings one dimension below. While the dimensional reduction of chiral theories is well-understood, the question which three and five-dimensional theories can come from a general circle reduction, and are hence liftable, is more subtle. We argue that existence of an anomaly cancellation mechanism is a necessary condition for liftability. In addition, the anomaly cancellation and the CS couplings in six and five dimensions respectively determine the central charges of string-like BPS objects that cannot be consistently decoupled from gravity, a.k.a. supergravity strings. Following the completeness conjecture and requiring that their worldsheet theory is unitary imposes bounds on the admissible theories. We argue that for the anomaly-free six-dimensional theories it is more advantageous to study the unitarity constraints obtained after reduction to five dimensions. In general these are slightly more stringent and can be cast in a more geometric form, highly reminiscent of the Kodaira positivity condition (KPC). Indeed, for the F-theoretic models which have an underlying Calabi-Yau threefold these can be directly compared. The unitarity constraints (UC) are in general weaker than KPC, and maybe useful in understanding the consistent models without F-theoretic realisation. We catalogue the cases when UC is more restrictive than KPC, hinting at more refined hidden structure in elliptic Calabi-Yau threefolds with certain singularity structure.

Categorical Webs and S-Duality in 4d $${\mathcal {N}}$$ = 2 QFT

We review the categorical approach to the BPS sector of a 4d $$\mathcal {N}=2$$ QFT, clarifying many tricky issues and presenting a few novel results. To a given $$\mathcal {N}=2$$ QFT one associates several triangulated categories: they describe various kinds of BPS objects from different physical viewpoints (e.g. IR versus UV). These diverse categories are related by a web of exact functors expressing physical relations between the various objects/pictures. A basic theme of this review is the emphasis on the full web of categories, rather than on what we can learn from a single description. A second general theme is viewing the cluster category as a sort of ‘categorification’ of ’t Hooft’s theory of quantum phases for a 4d non-Abelian gauge theory. The S-duality group is best described as the auto-equivalences of the full web of categories. This viewpoint leads to a combinatorial algorithm to search for S-dualities of the given $$\mathcal {N}=2$$ theory. If the ranks of the gauge and flavor groups are not too big, the algorithm may be effectively run on a laptop. This viewpoint also leads to a clearer view of 3d mirror symmetry. For class $$\mathcal {S}$$ theories, all the relevant triangulated categories may also be constructed in terms of geometric objects on the Gaiotto curve, and we present the dictionary between triangulated categories and the WKB approach of GMN. We also review how the VEV’s of UV line operators are related to cluster characters.

BPS objects in D = 7 supergravity and their M-theory origin

We study several different types of BPS flows within minimal mathcal{N}=1 , D = 7 supergravity with SU(2) gauge group and non-vanishing topological mass. After reviewing some known domain wall solutions involving only the metric and the ℝ+ scalar field, we move to considering more general flows involving a “dyonic” profile for the 3-form gauge potential. In this context, we consider flows featuring a Mkw3 as well as an AdS3 slicing, write down the corresponding flow equations, and integrate them analytically to obtain many examples of asymptotically AdS7 solutions in presence of a running 3-form. Furthermore, we move to adding the possibility of non-vanishing vector fields, find the new corresponding flows and integrate them numerically. Finally, we discuss the eleven-dimensional interpretation of the aforementioned solutions as effective descriptions of M2 − M5 bound states.

On the oscillation of species

We describe a new class of BPS objects called magnetubes: their supersymmetry is determined by their magnetic charges, while their electric charges can oscillate freely between different species. We show how to incorporate these objects into microstate geometries and create BPS solutions in which the charge densities rotate through different U(1) species as one moves around a circle within the microstate geometry. Our solutions have the same "time-like" supersymmetry as five-dimensional, three-charge black holes but, in various parts of the solution, the supersymmetry takes the "null" form that is normally associated with magnetic charges. It is this property that enables the species oscillation of magnetubes to be compatible with supersymmetry. We give an example in which the species oscillate non-trivially around a circle within a microstate geometry, and yet the energy-momentum tensor and metric are completely independent of this circle: only the amplitude of the oscillation influences the metric.

1/4BPS string junctions andN3problem in 6-dimensional (2,0) superconformal theories

We explore 1/4 BPS objects in the Coulomb phase of the ADE-type 6-dim (2,0) superconformal theories. By using the previous work on the junctions of strings in 5-dim gauge theories and 6-dim superconformal theories, we count the number of 1/4 BPS objects, which are made of waves on selfdual strings and junctions of selfdual strings and show that for all cases the number matches exactly one third of the anomaly constant $c_G=d_G h_G$ which is the product of dimension $d_G$ and dual Coxeter number $h_G$. This suggests the long sought after $N^3$ degrees of freedom are these 1/4 BPS objects at least in the Coulomb phase.

Hopf-Wess-Zumino term in the effective action of the 6d, (2, 0) field theory revisted

We discuss the Hopf-Wess-Zumino term in the effective action of the 6d (2, 0) theory of the type A N−1 in a generic Coulomb branch. For such terms, the supergravity calculation could be trusted. We calculate the WZ term on supergravity side and show that it could compensate the anomaly deficit, as is required by the anomaly matching condition. In contrast with the SYM theory, in which each WZ term involves one root e i − e j , here, the typical WZ term involves two roots e i − e j and e k − e j . Such kind of triple interaction may come from the integrating out of the massive states carrying three indices. A natural candidate is the recently proposed 1/4 BPS objects in the Coulomb phase of the 6d (2, 0) theories. The WZ term could be derived from the field theory by the integration out of massive degrees of freedom. Without the 6d (2, 0) theory at hand, we take the supersymmetric equations for the 3-algebra valued (2, 0) tensor multiplet as the prototype to see how far we can go. The H 3 ∧ A 3 part of the WZ term is obtained, while the A 3 ∧ F 4 part, which is the term accounting for the anomaly matching, cannot be produced by the standard fermion loop integration.

Ab initio wall-crossing

We derive supersymmetric quantum mechanics of n BPS objects with 3n position degrees of freedom and 4n fermionic partners with SO(4) R-symmetry. The potential terms, essential and sufficient for the index problem for non-threshold BPS states, are universal, and 2(n − 1) dimensional classical moduli spaces \( {\mathcal{M}_n} \) emerge from zero locus of the potential energy. We emphasize that there is no natural reduction of the quantum mechanics to \( {\mathcal{M}_n} \), contrary to the conventional wisdom. Nevertheless, via an index-preserving deformation that breaks supersymmetry partially, we derive a Dirac index on Mn as the fundamental state counting quantity. This rigorously fills a missing link in the “Coulomb phase” wall-crossing formula in literature. We then impose Bose/Fermi statistics of identical centers, and derive the general wall-crossing formula, applicable to both BPS black holes and BPS dyons. Also explained dynamically is how the rational invariant ∼ Ω(β)/p 2, appearing repeatedly in wall-crossing formulae, can be understood as the universal multiplicative factor due to p identical, coincident, yet unbound, BPS particles of charge β. Along the way, we also clarify relationships between field theory state countings and quantum mechanical indices.

Classification of BPS objects in $ \mathcal{N} = 6 $ Chern-Simons matter theory

We investigate BPS conditions preserving n/12 (n = 1,...,6) supersymmetries in the Aharony-Bergman-Jafferis-Maldacena (ABJM) model. The BPS equations are classified in terms of the number of preserved supercharges and remaining subgroups of the SU(4)_R symmetry. We study structures of a map between projection conditions for the supercharges in eleven dimensions and those in the ABJM model. The BPS configurations in the ABJM model can be interpreted as known BPS objects in eleven-dimensional Mtheory,such as intersecting M2, M5-branes, M-waves, KK-monopoles and M9-branes. We also show that these BPS conditions reduce to those in N = 8 super Yang-Mills theory via the standard D2-reduction procedure in a consistent way with the M-theory interpretation of the BPS conditions.

Classification of BPS Objects in N=6 Chern-Simons Matter Theory

Classification of BPS Objects in N=6 Chern-Simons Matter Theory

Mass-deformed Bagger-Lambert theory and its BPS objects

We find a 16-supersymmetric mass-deformed Bagger-Lambert theory with $SO(4)\ifmmode\times\else\texttimes\fi{}SO(4)$ global $R$ symmetry. The $R$ charge plays the role of the noncentral term in the superalgebra. This theory has one symmetric vacuum and two inequivalent broken sectors of vacua. Each sector of the broken symmetry has $SO(4)$ geometry. We find the $1/2$ BPS domain walls connecting the symmetric phase and any broken phase, and $1/4$ BPS supertubelike objects, which may appear as anyonic $q$-balls in the symmetric phase or vortices in the broken phase. We also discuss mass deformations, which reduce the number of supersymmetries.

The supersymmetric solutions and extensions of ungauged matter-coupledN= 1,d= 4 supergravity

We find the most general supersymmetric solutions of ungauged N=1,d=4 supergravity coupled to an arbitrary number of vector and chiral supermultiplets, which turn out to be essentially pp-waves and strings. We also introduce magnetic 1-forms and their supersymmetry transformations and 2-forms associated to the isometries of the scalar manifold and their supersymmetry transformations. Only the latter can couple to BPS objects (strings), in agreement with our results.

BPS objects inN=2supersymmetric gauge theories

We explore BPS soliton configurations in N=2 supersymmetric Yang-Mills theory with matter fields arising from parallel D3 branes on D7 branes. Especially we focus on two parameter family of 1/8 BPS equations, dyonic objects, and 1/8 BPS objects and raise a possibility of absence of BPS vortices when the number of D3 branes is larger than that of D7 branes.

AdS5black holes with fermionic hair

The study of new BPS objects in AdS_5 has led to a deeper understanding of AdS/CFT. To help complete this picture, and to fully explore the consequences of the supersymmetry algebra, it is also important to obtain new solutions with bulk fermions turned on. In this paper we construct superpartners of the 1/2 BPS black hole in AdS_5 using a natural set of fermion zero modes. We demonstrate that these superpartners, carrying fermionic hair, have conserved charges differing from the original bosonic counterpart. To do so, we find the R-charge and dipole moment of the new system, as well as the mass and angular momentum, defined through the boundary stress tensor. The complete set of superpartners fits nicely into a chiral representation of AdS_5 supersymmetry, and the spinning solutions have the expected gyromagnetic ratio, g=1.

Noncommutative Tachyon Kinks as D(p-1)-branes from Unstable Dp-brane

We study noncommutative (NC) field theory of a real NC tachyon and NC U(1) gauge field, describing the dynamics of an unstable D$p$-brane. For every given set of diagonal component of open string metric $G_{0}$, NC parameter $\theta_{0}$, and interpolating electric field ${\hat E}$, we find all possible static NC kinks as exact solutions, in spite of complicated NC terms, which are classified by an array of NC kink-antikink and topological NC kinks. By computing their tensions and charges, those configurations are identified as an array of D0${\bar {\rm D}}$0 and single stable D0 from the unstable D1, respectively. When the interpolating electric field has critical value as $G_{0}^{2}={\hat E}^{2}$, the obtained topological kink becomes a BPS object with nonzero thickness and is identified as BPS D0 in the fluid of fundamental strings. Particularly in the scaling limit of infinite $\theta_{0}$ and vanishing $G_{0}$ and ${\hat E}$, while keeping $G_{0}\theta_{0}={\hat E}\theta_{0}=1$, finiteness of the tension of NC kink corresponds to tensionless kink in ordinary effective field theory. An extension to stable D$(p-1)$ from unstable D$p$ is straightforward for pure electric cases with parallel NC parameter and interpolating two-form field.

D-brane Bound States from Charged Macroscopic Strings

We construct new D-brane bound states using charged macroscopic type IIB string solutions.A generic bound state solution, when dimensionally reduced, carries multiple gauge charges. Starting with D=9 charged macroscopic strings, we obtain solutions in D=10, which are interpreted as carrying (F, D0, D2) charges as well as nonzero momenta. The masses and charges are also explicitly shown to satisfy the non-threshold bound of 1/2 BPS objects. Our solutions reduce to the known D-brane bound state solutions with appropriate restrictions in the parameter space. We further generalize the results to (Dp- D(p+2)) bound state in IIA/B theories, giving an explicit example with p=1.

SOLITONIC BRANE WORLD WITH COMPLETELY LOCALIZED (SUPER)GRAVITY

We construct a solitonic three-brane solution in the five-dimensional Einstein–Hilbert–Gauss–Bonnet theory. This solitonic brane is δ-function-like, and has the property that gravity is completely localized on the brane. That is, there are no propagating degrees of freedom in the bulk, while on the brane we have purely four-dimensional Einstein gravity. Thus, albeit the classical background is five-dimensional, the quantum theory (perturbatively) is four-dimensional. Our solution can be embedded in the supergravity context, where we have completely localized supergravity on the corresponding solitonic brane, which is a BPS object preserving 1/2 of the original supersymmetries. By including a scalar field, we also construct a smooth domain wall solution, which in a certain limit reduces to the δ-function-like solitonic brane solution (this is possible for the latter breaks diffeomorphisms only spontaneously). We then show that in the smooth domain wall background the only normalizable mode is the four-dimensional graviton zero mode, while all the other (including massive Kaluza–Klein) modes are not even plane-wave normalizable. Finally, we observe that in compactifications of Type IIB on five-dimensional Einstein manifolds other than a five-sphere the corresponding dual gauge theories on D3-branes are not conformal in the ultraviolet, and at the quantum level we expect the Einstein–Hilbert term to be generated in their world volumes. We conjecture that in full string theory on Type IIB side this is due to higher curvature terms, which cannot be ignored in such backgrounds. A stronger version of this conjecture also states that (at least in some cases) in such backgrounds D3-branes are solitonic objects with completely localized (super)gravity in their world volumes.

Green-Schwarz string in AdS5 × S5: semiclassical partition function

A systematic approach to the study of semiclassical fluctuations of strings in AdS_5 x S^5 based on the Green-Schwarz formalism is developed. We show that the string partition function is well defined and finite. Issues related to different gauge choices are clarified. We consider explicitly several cases of classical string solutions with the world surface ending on a line, on a circle or on two lines on the boundary of AdS. The first example is a BPS object and the partition function is one. In the third example the determinants we derive should give the first corrections to the Wilson loop expectation value in the strong coupling expansion of the n=4 SYM theory at large N.

D0-branes in Gepner models and N = 2 black holes

In this paper D-brane boundary states constructed in Gepner models are used to analyze some aspects of the dynamics of DO-branes in Calabi-Yau compactifications of type II theories to four dimensions. It is shown that the boundary states correspond to BPS objects carrying dyonic charges. By analyzing the couplings to closed string fields a correspondence between the DO-branes and extremal charged black holes in N = 2 supergravity is found.