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IPhT-T11/151
Double, Double Supertube Bubble
Iosif Bena1, Jan de Boer2,
Masaki Shigemori3 and Nicholas P. Warner4
1 Institut de Physique Théorique,
CEA Saclay, F-91191 Gif sur Yvette, France
2Institute for Theoretical Physics, University of Amsterdam,
Science Park 904, Postbus 94485, 1090 GL Amsterdam, The Netherlands
3Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
Nagoya University, Nagoya 464-8602, Japan
4 Department of Physics and Astronomy,
University of Southern California,
Los Angeles, CA 90089, USA
iosif.bena at cea.fr, J.deBoer at uva.nl,
shige at kmi.nagoya-u.ac.jp, warner at usc.edu
We argue that there exists a new class of completely smooth 18-BPS, three-charge bound state configurations that depend upon arbitrary functions of two variables. These configurations are locally 12-BPS objects in that if they form an infinite flat sheet then they preserve 16 supersymmetries but even with arbitrary two-dimensional shape modes they still preserve 4 supersymmetries. They have three electric charges and can be thought of the result of two successive supertube transitions that involve adding two independent dipole moments and giving rise to the arbitrary two-dimensional shape modes. We further argue that in the D1-D5-P duality frame this construction will give rise to smooth, horizonless solutions, or microstate geometries. We expect these solutions to be extremely important in the semi-classical and holographic descriptions of black-hole entropy.
One of the most interesting brane configuration to have been discovered in the last two decades is probably the supertube. In its original incarnation [1] , the supertube is a moving D2 brane in which D0 branes and F1 strings are dissolved. What makes this D2 brane special is the fact that the D0 and F1 densities satisfy a relation that allows the D2 brane to have an arbitrary shape that can follow any closed curve in the eight dimensions transverse to the F1 and yet remain supersymmetric [2] . The eight Killing spinors preserved by the supertube are exactly the same as the common Killing spinors of its "electric" components: the D0 branes and F1 strings.
The fact that there are supersymmetric string theory configurations determined by arbitrary continuous functions might appear unexpected, especially if one is used to thinking about the supersymmetries preserved by infinite, flat branes at angles. Nevertheless, one can dualize the supertube into a fundamental string that carries an arbitrary left-moving momentum profile, and supertubes of various shapes are simply dual to various ways of putting BPS momentum modes on the fundamental string [3, 4] .
Another interesting feature of supertubes is that, in the duality frame in which the electric charges are those of D1 and D5 branes, the back-reacted supertube solution can be made into a smooth geometry [5, 6, 7, 8] . Moreover, since this solution can be put in an asymptotically AdS3×S3×T4 geometry, the back-reacted supertube geometries can be related to various half-BPS microstates of the D1-D5 CFT, and the entropy of the half-BPS D1-D5 system can be reproduced by counting the smooth horizonless supertube configurations [6, 9, 10, 11] . This has led to the conjecture by Mathur that similar physics will be at work in the three-charge D1-D5-P system, and therefore the entropy of the D1-D5-P black hole will come from string and brane configurations that do not have a horizon and have unitary scattering in the same region of moduli space in which the classical black hole exists (see [12, 13, 14, 15, 16] for reviews). When such configurations are smooth, horizonless solutions of supergravity, they are often referred to as microstate geometries because they represent microstates of the black hole both semi-classically and through the AdS/CFT correspondence. It is reasonable to expect that many of the microstates of the black hole will be dual to geometries that involve Planck-scale details that go beyond the validity of the supergravity approximation. On the other hand, it is hoped that, within the validity of the supergravity approximation, one can find a suitably dense, representative sample of microstate geometries that will not only give a semi-classical picture of the microstates but also yield some of the thermodynamic details of the full system and perhaps even reproduce the entropy of the black hole. Many such configurations have been constructed, both in supergravity and using non-back-reacted branes, but so far the entropy of the back-reacted configurations is not of the same order as that of black holes with similar charges [17, 18] .
One of the common features of the geometries and brane configurations constructed so far is that they depend either on a finite number of parameters, or they come from putting arbitrarily-shaped supertubes in various three-charge geometries and thus depend on several functions of one variable. However, two of the authors have recently proposed [19] that there may be BPS string configurations that depend on functions of two variables, and these potentially have much more entropy than that of the the systems constructed so far. It is our purpose in this paper to present evidence that such brane configurations do indeed exist and that they preserve the same supersymmetries as those of the D1-D5-P black hole and are determined by several functions of two variables. We will refer to such objects as superstrata111For picture of what is intended here, see the Strata Tower http://www.dezeen.com/2008/05/13/strata-tower-by-asymptote/ or Corkscrew Peak http://www.summitpost.org/corkscrew-peak/617471.
In Section 2 we describe exactly how and why superstrata can be constructed as smooth, 18-BPS solutions that depend upon two variables. In Section 3 we summarize how one can obtain and analyze the supersymmetries in a supertube transition. We have also included a much more systematic development of this process in the Appendix. In Section 4 we combine two such supertube transitions to make the "double bubbled" superstratum and examine its supersymmetry structure and substantiate the physical description of Section 2. We then make some final remarks in Section 5.
2 The physical description of superstrata
To understand our approach to establishing the existence of superstrata it is important to recall some of the defining properties of supertubes and the methods by which one can establish the existence of these arbitrary-shaped, supersymmetric configurations.
In string theory it is easy to create many species of two-charge, 14-BPS states by a simple superposition of compatible D-branes, momentum states or other solitons. Each charge component is 12-BPS and compatibility means that the 12-BPS supersymmetry projectors for each charge commute with one another so that the two components do not interact. The resulting object is not really a new fundamental object in string theory, and without string corrections and back-reaction it is really only a marginally bound superposition of the two components. The transition to a supertube fuses these two components into a new fundamental bound state and this is achieved by giving the system an additional dipole moment and angular momentum in a transverse direction, so that the entire configuration follows a new and arbitrary profile transverse to the original progenitor configuration.
The resulting object is still a 14-BPS state but very close to the supertube the supersymmetry is locally enhanced to 12-BPS. In other words, if the supertube profile were straight and the configuration were a flat sheet then it would be exactly 12-BPS with the preserved supersymmetries depending on the orientation of the sheet. For an arbitrary supertube profile the 16 local supersymmetries depend upon the direction of the tangent to the profile, but there is always a set of 8 common supersymmetries that are preserved independent of the profile, and these supersymmetries are precisely those of the original two-charge system before the supertube transition. Thus one of the hallmarks of the supertube transition that distinguishes it from the progenitor two-charge superposition is the emergence of this local 12-BPS structure.
Every two-charge system has a supertube transition, and they can all be related by dualities. However, the physics underlying the supertube transition and the way the local 12-BPS structure emerges is different in different duality frames. One of the simplest supertubes has D0 and F1 "electric" charges dissolved in a rotating D2-brane [1] . The 12-BPS, near-tube limit is simply an infinite, flat D2-brane and the easiest way to understand the emergence of the 16 supersymmetries in this limit is to consider the M-theory uplift, in which the entire object a boosted 12-BPS M2-brane and the D0 and F1 charges correspond to momentum and winding around the eleventh dimension. As the orientation of the supertube changes, the set of 16 supersymmetries varies but there is a common subset of 8 supersymmetries that is preserved, independent of the orientation of the D2-brane. This subset of eight supersymmetries is precisely the common set of supersymmetries of the electric (F1 and D0) charges of the underlying system.
There is another very important feature of the supertube transition: In some duality frames it "puffs up" the brane by adding one dimension to the object but in other duality frames it does not. For example, the original D0-F1 system is intrinsically (1+1)-dimensional but the added dipole charge puffs it up to a (2+1)-dimensional D2 brane. Similarly, the D1-D5 system has codimension 4, but adding the KKM dipole charge "smears it out" into a codimension 3 object. On the other hand, from the perspective of M-theory the puffing up of the D0-F1 system to a D2 brane is simply a matter of tilting and boosting the (2+1)-dimensional M2 brane and there is no gain of dimension: In M-theory it is always a configuration of codimension 8. This is because the D0 charge is actually a momentum charge from the eleven-dimensional perspective, and tilting and boosting the brane momentum simply re-orients the surface and the momentum to create the dipole charge and angular momentum. This seems to be a general pattern: If one of the electric charges is a momentum then the corresponding supertube has the same codimension as the original object but if neither of the electric charges is a momentum then the supertube transition necessarily adds an extra dimension to the object, puffing it up so that the codimension of the brane configuration decreases by one.
There are two basic approaches to establishing the existence of supertubes. First there are direct methods using either the Dirac-Born-Infeld (DBI) action or using supergravity. For example, for D0-F1 supertubes one can use the DBI action of a rotating D2 brane and induce the D0 and F1 charges using world-volume fields, or one can go to the T-dual of this supertube in which it is a D1 string with a momentum profile. Alternatively, one can show that there exists a BPS supergravity solution for any supertube profile. Moreover, in the D1-D5 duality frame, this solution is smooth. Both the DBI and the smooth D1-D5 supergravity descriptions yield the supersymmetry structure described above. However, the problem with these direct methods is that they are usually difficult to implement because they involve analyzing the totality of the supertube or constructing a complicated supergravity solution.
Figure 1: The D1-P or F1-P supertube profile and a typical local neighbourhood.
There are are also indirect, "local" methods that can be used to argue that a certain type of supertube exists. In this approach one imagines cutting the supertube into very small pieces, or zooming in very near some point of the tube. Each such bit of the supertube will look, locally, like an infinite flat brane and will be a 12-BPS fundamental object. However, as the tangent to the supertube profile changes, the set of preserved supersymmetries will also change. The critical question is then whether there is a subset of common supersymmetries within all the sets of locally-preserved supersymmetries. Put differently, the generic situation is that if one takes a brane and tilts it or puffs it up with some dipole charge, then there are no common supersymmetries that are preserved by both the original brane and the tilted or puffed up brane: The supersymmetries at different points on the profile are generically completely incompatible. The remarkable thing that distinguishes the supertube from the generic brane is that there is indeed a subset of common supersymmetries within the sets of locally-preserved supersymmetries and, by definition, this subset of common supersymmetries is independent of direction of the tangent to the supertube profile.
The important point is that if one can establish that each infinitesimal bit of the supertube preserves some set of supersymmetries, and there is a common subset of these supersymmetries that are preserved by all the bits of supertube, then these separate supertube bits are mutually BPS and so do not interact with each other. This strongly suggests that they can be strung together to form a supersymmetric, continuous profile of arbitrary shape. This perspective thus provides a local argument as to why the complicated supergravity configurations can, in fact, exist.
Our purpose in this paper is to apply the supertube process twice in succession (hence "double-bubbling") and then use the supersymmetry analysis in a local argument to show that, in string theory, there exist fundamental bound-state configurations of branes that carry three electric charges, have several dipole moments, preserve 4 supersymmetries, and are determined by a two-dimensional surface arbitrarily-embedded in R4×S1.
These 18-BPS configurations, which we refer to as superstrata, can be thought of as being made of bits of infinite, flat two-dimensional surfaces, each bit preserving 16 supersymmetries, of which 4 supersymmetries are common to all the bits, and are the same as the 4 supersymmetries common to BPS objects carrying each of the the three electric charges of the superstratum. As the local argument implies, the fact that all the bits of the superstratum are mutually 18-BPS means that the force between various bits of different orientation will be zero and so one should be able to assemble them into a superstratum whose shape is given by five arbitrary functions of two variables.
The direct approach to finding these superstrata would be to assemble various types of branes in string or M-theory, find the supersymmetries preserved by these configurations, and vary over all types of branes and all values of the brane densities until one finds a configuration that preserves the same Killing spinors irrespective of two orientations. While this may indeed be possible, it is a technically formidable problem. Our approach, using the doubling of the supertube transition, or double-bubbling, has several advantages in that it first enables us to identify what the charges and dipole charges of such a superstratum should be, and then determine the conditions these must satisfy in order for the superstratum to have an arbitrary shape. Having achieved this, one then has a good starting point for tackling the far more strenuous and difficult supergravity analysis and solution. We will defer the latter to a subsequent paper.
Our starting point will be the D1-D5-P system familiar in the three-charge black hole story. We use this duality frame because it will lead to a smooth configuration. The first supertube transition will involve adding a dipole moment and angular momentum to take original D1-D5-P system to a three-charge two-dipole charge supertube222This is dual to a configuration that was originally constructed in the D4-D0-F1 duality frame [20] as a solution of the Born-Infeld action of a D6 brane of arbitrary shape. The complete solution has fluxes on the world-volume that induce D4, D0 and F1 electric charges and a D2 dipole charge that is related to the other charges. The corresponding supergravity solution is the same as that of a black ring with only two dipole charges and the relation between the dipole and other charges emerges from the requirement that the supergravity solution is free of closed timelike curves [21] .. The shape of this generalized supertube is determined by a set of profile functions of one variable. As one approaches the location of the tube, the supergravity solution has a curvature singularity. If one now zooms in on this three-charge two-dipole charge supertube, or considers the infinite supertube limit, one finds that this infinite tube preserves eight supersymmetries, out of which four are the common supersymmetries associated with the component electric charges. Since the infinite, flat supertube preserves 8 supersymmetries, it should be a superposition of two mutually-BPS branes and this is most easily seen in the D1-D5-P duality frame.
To make the first supertube transition of the D1, D5 and P electric charges one first partitions the momentum between the D1 and D5 systems to obtain separate D1-P and D5-P systems. Each of these then undergoes a supertube transition to objects we will refer to as D1-P and D5-P supertubes. The details of these transitions will be given later, for now it suffices to know that the D1-P and D5-P supertubes still carry the original charges but acquire angular momentum and D1 and D5 dipole moments respectively. A cartoon of the D1-P supertube is given in Fig. 1. If z denotes the original common direction of the D1 and D5 branes, and θ is the coordinate along the new supertube profile in R4 then the entire configuration now lies along a curve in the (z,θ)-plane. We will denote the manifold consisting of the common direction of the D1 and D5 branes and the four transverse dimensions by M5 and to keep things simple this manifold will be either R4×S1 or R5 depending on whether we compactify the original brane direction or not. The three-charge supertube thus has codimension 4 and is defined by a curve, γ, in M5. The D1 (or D5) electric and dipole charges are then simply the z and θ components of the total number of D1 (or D5) branes. This geometric description immediately implies that the dipole and electric charges are related by:
Q1Q5 = d1d5⇔Q1d5 = Q5d1, (2.1)
and this is precisely the relation required by either solving the DBI action or by requiring the absence of closed time-like curves in supergravity.
It is important to remember that in making this supertube transition we have fused some of the momentum with the D1 branes and some of the momentum with the D5 branes. The result is parallel D1-P and D5-P supertubes that are each fundamental locally-12-BPS objects and together preserve eight supersymmetries locally. Indeed, the generalized "supertube bit" is simply a boosted and tilted superposition of D1 and D5 branes. At this point it also becomes clear why the three-charge supertube has a curvature singularity: this is simply the curvature singularity of a solution of superposed D1 and D5 branes in supergravity.
It is also equally evident how to make a second supertube transition that fuses the coincident D1-P and D5-P supertubes described above into a new fundamental object that locally preserves 16 supersymmetries: One applies a second supertube transition that involves adding a KKM dipole charge and angular momentum. Locally, this is the same as the standard supertube transition of the D1-D5 system. It is important to remember that this transition decreases the codimension of the system, and because the D1-D5 common direction shrinks smoothly to zero at the KKM profile, the resulting configuration is smooth [6, 7] . Hence, the puff-up into a codimension-three object completely resolves the singularity of the D1-D5 system.
To be more specific, let ^z denote the common direction of the D1 and D5 branes before puffing up and recall that there is, locally, a patch, U, of R4 transverse to the branes (see Fig. 1). The smooth solution is obtained by introducing a KKM dipole charge along a closed path, ^γ, in U and smearing the D1 and D5 charge along this path. We will parametrize the curve, ^γ, by an angle, ψ, so the puffed up brane is a codimension 3 object that sweeps out the (^z,ψ)-plane. The resulting object is now described by the curve, ^γ, in U and the three-dimensional transverse geometry in U in the neighborhood of a point on ^γ, appears, at first sight, to be singular. However, it is a Kaluza-Klein monopole and if the ^z direction is compactified with the proper periodicity then the KKM fiber shrinks to zero at a certain profile in R4 in such a way that the resulting geometry is smooth.
Figure 2: The double bubbling of the D1-D5-P system. There are two ways to obtain a superstratum: The D1 and P can fuse into a D1-P supertube spiral (red dotted line), and the D5 and P can fuse into a D5-P spiral (blue continuous line). The spirals can then fuse into a superstratum. Alternatively the D1-D5 can fuse into a D1-D5-KKM tube (violet straight supertube), which upon adding momentum can start shaking and become a superstratum.
The second supertube transition thus has two very important effects: First, it completes the fusion of the D1-D5-P system into a true bound state by fusing the D1-P and D5-P supertubes into a single, locally 12-BPS object. Secondly, it resolves the singularity of the generalized three-charge supertube through a KKM puffing at every point, ^z, along the original profile, γ, of the first generalized supertube. This second supertube transition puffs the configuration up by one dimension along another arbitrary curve, ^γ^z, whose profile can depend upon ^z. Thus the resolution of the singularity allows the first (arbitrary-shaped) three-charge supertube profile to be replaced by a freely choosable curve, ^γ^z, at every point of the original profile: In other words, the original three-charge supertube can be puffed up into a two-dimensional sheet, or stratum that has codimension 3 in M5. Moreover, since the profile of the second puff up is freely choosable at every point of the original profile, the resulting sheet, or stratum, is defined by a freely choosable function, →F(v1,v2), of two variables into the M5. This map defines the superstratum and it will be a smooth, 18-BPS configuration that emerges from "double bubbling." The solution is locally 12-BPS but globally has the electric charges of, and the same supersymmetries as, the D1-D5-P system and carries several dipole charges corresponding to transverse D1 and D5 branes and KK-Monopoles.
The foregoing argument and lays out precisely the construction that leads to the superstrata and the process is depicted in Fig. 2. However, to substantiate the claim that superstrata can have a completely arbitrary two-dimensional shape and still preserve four supersymmetries, we need to complete the "local argument" and show that each locally-flat two-dimensional surface bit that makes a superstratum preserves 16 supersymmetries, and that collectively the bits preserve a common subset of four supersymmetries. This will be done in the next two sections. Once this is established, it follows that all the surface bits are mutually BPS and hence non-interacting and therefore they can be combined make a complete two-dimensional superstratum. Of course, to fully establish the existence of a superstratum we either need to find a Born-Infeld-like description or to construct its complete supergravity solution. We leave this somewhat daunting task for future work.
3 Supertube transitions
3.1 Supersymmetries and supertubes in general
We will consider several examples of supertube transitions and their effect upon the structure of the supersymmetries. If Q1, Q2 are the original electric charges corresponding to some branes and d1 and J are the dipole charge and angular momentum of the supertube configuration, then we will use the following canonical notation to denote the supertube construction:
(Q1(x)Q2(y)) → (d1(zψ)J(ψ)), (3.1)
where x,y,z and ψ are some subsets of coordinate directions along which the branes are wrapped or the momentum is directed. The coordinate, ψ, will generically indicate the new direction associated with the supertube.
Our purpose here is to find the supersymmetries preserved by the supertubes by constructing a one-parameter family of projectors whose null spaces intersect over and define the subspace of supersymmetries that are preserved by the particular supertube configuration.
There is a standard procedure for computing the required supersymmetry projectors: One starts with the projectors appropriate to the electric charges, Q1 and Q2 and uses carefully selected rotation matrices that tilt and boost the brane configuration along the supertube directions. The end result is a projector that satisfies three conditions: (i) It must be a linear combination of projectors for all the underlying branes and momenta, (ii) it must define a 12-BPS state (for fixed x,y,z and ψ) and so have sixteen null vectors, and (iii) it must be a linear combination of the original projectors associated with the electric branes.
The first condition simply stipulates that the supertube has the required brane constituents and the second condition implies that the infinite planar supertube is 12-BPS and preserves 16 supersymmetries. If the orientation, ψ, of the supertube varies then the set of sixteen supersymmetries also varies, however the last condition guarantees that no matter how the orientation varies the supertube will always preserve the original eight common supersymmetries associated with the electric charges and so the supertube can have an arbitrary profile and still be a 14-BPS state.
It turns out that these constraints are enough to determine the requisite projectors for a supertube configuration and so we will use this approach to derive all the projectors we need. A more formal and precise derivation of the validity of our projectors can be found in the appendix where we also summarize the dictionary that defines the 12-BPS projectors associated with each and every type of brane charge.
3.2 A simple example: the F1–P system
To illustrate some basic properties of the supertube transition, we consider the F1–P system. This system is described in considerably more detail in appendix A.2.1.
The starting point is a certain number of fundamental strings stretched in the x1-direction. These fundamental strings preserve half of the supersymmetries, namely those that obey
ΠF1Q=0, (3.2)
ΠF1=12(1+Γ01σ3), (3.3)
and σ3 is the third Pauli matrix which acts on the doublet of Majorana-Weyl supercharges of the type II superstring theory.
To generate the F1-P system, we consider an arbitrary transverse direction, denoted by ψ, and T-dualize the system along a direction in the (x1,ψ)-plane that makes an angle, α, with the x1-axis. Under this T-duality, the system remains 12-BPS. It is not very important in this discussion whether the direction along which we T-dualize is compact or not. If it is not compact, what we are doing is not a symmetry of the theory but it can be viewed as a solution-generating operation.
To find the supercharges that are preserved after the T-duality, we (trivially) rewrite the projector in components parallel and perpendicular to the direction of the T-duality axis:
ΠF1=12(1−sinα(cosαΓ0ψ−sinαΓ01)σ3+cosα(cosαΓ01+sinαΓ0ψ)σ3). (3.4)
T-duality parallel to fundamental strings converts them into momentum, and the supersymmetries preserved by momentum are determined by the same projector as for fundamental strings but without the σ3. Thus the projector we get after the T-duality is simply:
ΠF1-P=12(1−sinα(cosαΓ0ψ−sinαΓ01)σ3+cosα(cosαΓ01+sinαΓ0ψ)) (3.5)
and this describes the projector for the bound state of momentum along the T-duality axis with some fundamental strings in the orthogonal direction. By construction, this projector describes a 12-BPS system.
If we start with N coincident fundamental strings, the resulting system has (F,P)-charges given by (w,n)=(Nsin2α,Ncos2α) in the x1-direction, and (F,P)-charges (d,J)=(Nsinαcosα,Nsinαcosα) in the ψ-direction.
The remarkable feature of this new system is that it preserves a fixed set of eight supercharges regardless of the choice of the direction ψ and the angle α. For α=0 we have momentum in the x1-direction, and for α=π/2 we have fundamental strings in the x1 direction, so the eight supercharges are the same as the eight supercharges preserved by parallel fundamental strings and momentum in the x1-direction. Indeed, we can write
ΠF1-P=sinα(sinα−cosαΓ1ψ)ΠF1+cosα(cosα+sinαΓ1ψ)ΠP, (3.6)
which clearly demonstrates that the common supersymmetries of a marginal bound state of fundamental strings and momentum in the x1-direction are always preserved.
For all this to work it is crucial that the amount of F1-string charge and momentum is correlated with the angle α. Without this correlation the configuration would have no remaining supersymmetry.
By gluing together pieces of fundamental strings and momentum that locally look like the above F1–P system, we can make a 14-BPS F1–P supertube which is the S-dual of the D1-P supertube shown in Fig. 1.
3.3 Bubbling the D1–D5 system
Consider the bubbling
(D1(0z)D5(01234z)) → (KKM(01234ψ;z)P(ψ)). (3.7)
In this configuration the special direction of the KKM lies along the common direction, z, of the D1–D5 system and after bubbling the charges and KKM dipole are distributed along the closed curve parametrized by ψ. This is therefore a true "puffing up" in that the configuration has gained an extra dimension defined by ψ.
Before bubbling, the electric projectors are (see the appendix for details):
ΠD1 = 12(to0.0pt1l + Γ0zσ1),ΠD5 = 12(to0.0pt1l + Γ01234zσ1), (3.8)
and these two projectors commute. Bubbling combines these and adds a momentum part, Γ0ψ, and a KKM part, Γ01234ψ.
One can then show that the bubbled projector can be written in either of the three following ways
Π = 12(to0.0pt1l+Γ0z(cosβσ1to0.0pt1l+sinβΓzψ)(cosβto0.0pt1l−sinβσ1Γ1234zψ)) (3.9)
= 12(to0.0pt1l+cos2βΓ0zσ1+sin2βΓ01234zσ1 + sinβcosβ(Γ0ψ−Γ01234ψ)) (3.10)
= cosβ(cosβto0.0pt1l−sinβσ1Γzψ)ΠD1 + sinβ(sinβto0.0pt1l+cosβσ1Γzψ)ΠD5. (3.11)
The first expression, (3.9), shows the underlying rotations and middle expression, (3.10), shows that the projector is, indeed, a combination of the projectors for component branes. The second term in (3.9) squares to to0.0pt1l and is traceless, and hence Π preserves sixteen supersymmetries. The third expression, (3.11), shows that these sixteen supersymmetries include the eight supersymmetries in the common nullspace of ΠD1 and ΠD5. Note that these expressions are very similar to the expressions obtained for the F1–P system described above, as they should be because the two systems are dual to one another.
Thus this projector has a sixteen-dimensional null space that depends upon the orientation of the supertube through the appearance of Γψ. The projectors, ΠD1 and ΠD5, are independent of Γψ and so their eight-dimensional common null space is independent of the supertube orientation and shape. As a result, if the supertube is an infinite flat sheet then ψ has a constant orientation and it is 12-BPS but if the supertube has a varying orientation, or shape, then it is still a 14-BPS configuration.
3.4 Bubbling the D1–P and D5–P system
In the D1–P and D5–P systems, one of the electric charges is a momentum and so the bubbling to a supertube does not "puff up" the supertube because the final supertube configuration has the same dimension as the original electric configuration. The supertube is rather a "superhelix," and the tilt and boost of the electric charges along a transverse direction, θ, convert some of the D-brane charge into dipole charge and some of the momentum into angular momentum. Thus we have:
(D1(0z)P(z)) → (d1(0θ)J(θ)),(D5(01234z)P(z)) → (d5(01234θ)J(θ)). (3.12)
Indeed, there is an underlying tilt angle, α, that determines how the charges are realigned after tilting:
Q1 =QD1,z = QD1cosα, d1 = QD1,θ = QD1sinα, (3.13)
QP =QP,z = Psinα, Jθ = Pθ = Pcosα. (3.15)
Note that (3.15) differs from the first two equations essentially because the momentum is perpendicular to the branes just as we had in the F1–P system. Notice that Pθ here should be thought of as the momentum in the negative θ-direction.
The fundamental projectors associated with this system are:
ΠD1 = 12(to0.0pt1l + Γ0zσ1),ΠD5 = 12(to0.0pt1l + Γ01234zσ1),ΠP = 12(to0.0pt1l + Γ0z). (3.16)
The projectors associated with these two supertube transitions are
ˆΠD1 = (3.17)
= 12(to0.0pt1l+cos2αΓ0zσ1+sin2αΓ0z + sinαcosαΓ0θ(to0.0pt1l−σ1)) (3.18)
= (3.19)
ˆΠD5 = 12(to0.0pt1l+Γ0z(cosαΓ1234σ1+sinαΓzθ)(cosαto0.0pt1l−sinαΓzθ)) (3.20)
= 12(to0.0pt1l+cos2αΓ01234zσ1+sin2αΓ0z + sinαcosαΓ0θ(to0.0pt1l−Γ1234σ1)) (3.21)
where ΠD1, ΠD5 and ΠP are given in (3.13), (3.14) and (3.15).
In both of these equations, the middle expressions show that the projectors are a combination of the appropriate component parts. The second term in each of (3.17) and (3.20) squares to to0.0pt1l and is traceless, and hence each projector preserves sixteen supersymmetries. The expressions (3.19) and (3.22) show that each of these new projectors can be expressed in terms of the original projectors of the D1–P system or D5–P systems respectively. The projectors ΠD1, ΠD5 and ΠP all commute with one another and are independent of Γθ and so their nullspaces are independent of the supertube orientation and shape. As a result, if the supertube is an infinite flat sheet then θ has a constant orientation and the supertube is 12-BPS but if the supertube has a varying orientation, or shape, then it is still a 14-BPS configuration.
4 Double Bubbling
4.1 The transition to the three-charge supertube
We now consider the D1–D5–P system and consider it to be a superposition of the D1–P and the D5–P systems considered above with the momentum partitioned into two parallel parts, P=P(1)+P(2) associated with the two different sets of branes. The first supertube is then obtained by tilting and boosting both sets of branes in exactly the same manner:
(D1(0z)P(1)(z)) → (d1(0θ)J(1)(θ)),(D5(01234z)P(2)(z)) → (d5(01234θ)J(2)(θ)). (4.1)
The charges of the system are then given by (3.13)–(3.15) and the angular momentum is similarly decomposable into two parts, J=J(1)+J(2).
Locally, the picture of these supertubes is one where D1 and D5 branes are tilted and have some momentum in the transverse direction. The momentum has to be distributed in such a way that the D1 and D5-branes remain parallel, so locally the split in momenta between the D1 and D5-branes is not arbitrary but determined by the ratio of the tensions of the D1 and wrapped D5-branes.
This produces a three-charge, two-dipole-charge supertube that follows the trajectory defined by θ. It preserves the supersymmetries defined by the projectors ˆΠD1 and ˆΠD5 defined in (3.17)–(3.22). Note that, unlike the projectors in (3.8), ˆΠD1 and ˆΠD5 do not commute. However their commutator is proportional to ˆΠD1−ˆΠD5 and so they commute on their common null space and thus define eight compatible supersymmetries333In section A.3 of the Appendix we construct a commuting set of projectors for this three-charge, two-dipole-charge supertube configuration and find that using these instead of ˆΠD1 and ˆΠD5 leads to the same final result.. However, these expressions depend upon Γθ and so the eight-dimensional common nullspace depends upon the supertube orientation. On the other hand, (3.19) and (3.22) show that these projectors can be expressed in terms of the projectors of the D1–D5–P system, (3.16), and the common nullspace of ˆΠD1 and ˆΠD5 includes the four supersymmetries of the D1–D5–P system that lie in the common nullspace of the projectors (3.16). Thus the generic configuration is still 18-BPS.
4.2 A basis change
It is convenient to define new gamma matrices:
Γ^z = cosαΓz−sinαΓθ,Γ^θ = sinαΓz+cosαΓθ. (4.2)
In this new basis one has:
ˆΠD1 = 12(to0.0pt1l+cosαΓ0^zσ1+sinαΓ0^θ) (4.3)
ˆΠD5 = 12(to0.0pt1l+cosαΓ01234^zσ1+sinαΓ0^θ). (4.4)
This shows that the projectors of the three-charge, two dipole charge supertube are simply a combination of the fundamental brane projectors along the (^z,^θ) directions.
4.3 The double-bubbled superstratum
The goal is now to combine the transition in Section 3.3 with that described in Sections 3.4 and 4.1. The "quick and dirty" way to achieve this is to essentially replace ΠD1 and ΠD5 in (3.11) with ˆΠD1 and ˆΠD5. Since the fundamental branes associated with three-charge, two-dipole-charge supertube are tilted and lie along the ^z-direction, one must also replace the Γz's in (3.11) with Γ^z's. The resulting candidate projector is:
ˆΠ = cosβ(cosβto0.0pt1l−sinβσ1Γ^zψ)ˆΠD1 + sinβ(sinβto0.0pt1l+cosβσ1Γ^zψ)ˆΠD5. (4.5)
Note that β and α are independent rotation angles.
While well motivated, this form of the projector has not been rigorously established, because we have applied the equation for the new projector of a supertube transition (given in equation (A.39)) to two non-commuting projectors ˆΠD1 and ˆΠD5, whereas up to now the projectors were built out of commuting projectors. In the Appendix we present a more detailed analysis of the system in which we find its commuting projectors and obtain the superstratum projector rigorously; the final result is exactly the one in (4.5).
One can now expand and simplify in a number of ways. One such instructive form is:
Again the second term in this equation squares to to0.0pt1l and is traceless, and hence ˆΠ preserves sixteen supersymmetries. In addition, (4.5) shows that the nullspace of ˆΠ contains the common nullspace of ˆΠD1 and ˆΠD5 while (3.19) and (3.22) show that this common nullspace contains the common nullspace of ΠD1, ΠD5 and ΠP. In other words, the sixteen supersymmetries preserved by ˆΠ contain the four supersymmetries of the D1–D5–P system and these four supersymmetries are independent of the tilt angles, (α,β), and of coordinates, (θ,ψ).
The superstratum is therefore a two-dimensional sheet swept out by (^z,ψ) and locally preserves the sixteen supersymmetries defined by ˆΠ. However as the directions θ and ψ vary there is always a set of four supersymmetries common to all the local pieces of the superstratum and so all these pieces are mutually BPS and non-interacting and so one should be able to assemble them into a complete 18-BPS superstratum that has an arbitrary two-dimensional shape.
We have thus completed the local argument that strongly suggests that our conjectured superstratum should exist as a regular solution in string theory.
We have shown that there should exist a completely new set of 18-BPS bound states of D-branes, namely, superstrata. These have three electric charges and globally preserve 4 supersymmetries while locally appear to be 12-BPS objects preserving 16 supersymmetries. Moreover, they have a KKM dipole charge whose world-volume wraps a codimension-three surface, and hence their back-reaction should yield smooth supergravity solutions. The shape of the superstrata in five dimensions is determined by five functions of two variables, and since superstrata have the same charges as the D1-D5-P black hole their back-reaction will describe microstates of this black hole in the same regime of parameters where the black hole exists. The fact that these microstate geometries depend on functions of two variables leads us to expect that they will be able to account for vastly more entropy than ordinary supertubes, whose shapes only depend upon functions of one variable.
The new bound states should describe particular degrees of freedom of the D1-D5-P system, which one may hope to be able to see either by studying the appropriate microscopics or by relating normalizable modes of the supergravity solution, when it is constructed, to the various expectation values of the dual theory (as was done for simpler systems in [22, 23] ). While the second method may indeed yield interesting physical information, it is unlikely that one will be able to describe microscopically the superstrata using the non-Abelian degrees of freedom of the D1-D5-P system, essentially because one of the dipole charges corresponds to a KK-monopole, and describing systems with dipole charges whose tension scales like 1/g2s (KK-monopoles of NS5 branes) using brane non-Abelian degrees of freedom is equivalent to proving confinement [24] .
While the arguments in this paper strongly suggest that superstrata exist, there are still some serious calculations to be done to prove their existence. Of course, ideally one should construct a fully-back-reacted supergravity solution that depends on five functions of two variables and has the charges and the dipole charges indicated in this paper. In fact, two special limits of this would-be solution have already been constructed in the literature. As we explained in Section 2 and illustrated in Fig. 2, the superstrata can be thought as coming from a smooth D1-D5 supertube with a KKM dipole charge to which one adds momentum-carrying modes that break the isometry along the common D1-D5 direction. A perturbative solution in which one unit of momentum is added to the smooth D1-D5 supertube was constructed in [25] , and the rather non-trivial matchings that insured the existence of that solution make us confident that more complicated superstratum solutions exist.
A second highly-non-trivial supergravity solution that can be thought of as a limit of a superstratum was obtained in [26] by spectrally-flowing a supertube of arbitrary shape. The resulting solution depends non-trivially both on the common D1-D5 direction444Consequently this represents a supersymmetric solution of six-dimensional ungauged supergravity [27] that does not descend to five dimensions. and on one of the angles in the base, but this dependence is "the same" in that this solution is a superstratum whose function of two variables only depends on their sum but not on their difference. Given the existence of these non-trivial limit solutions, and given that the physical description provided in this paper gives a rather precise description of its charges and dipole moments, we believe the complete construction of the supergravity superstratum solution to be within reach.
There are, however, some potential issues that might arise in this construction. First, our local argument is based upon the fact that BPS bits of the stratum will not interact, since they are mutually BPS, and so can be assembled at will into the superstratum. This is not exactly true: multi-charge BPS configurations do interact if they are not mutually local and must satisfy bubble equations or integrability conditions that constrain their locations. However, we do not expect this to be a problem because there is still freedom to adjust some of the electric charge and angular momentum densities so as to accommodate the freedom to adjust the relative locations of the bits of superstratum. Put differently, we expect any such conditions not to emerge as restrictions on the shape but to constrain, for a given shape, certain integrals of the charge and angular-momentum densities (much as one finds for wiggly supertubes in bubbling solutions [18] ).
Another delicate issue that might constrain the superstrata is the fact that one must compactify the common D1-D5 direction, ^z, and have a properly quantized coefficient of the potential along this U(1) fiber in order to smooth out the geometry using the KK monopole. This works beautifully for the usual D1-D5 configuration where ^z is simply the coordinate along a compactification circle. For superstrata, the coordinate ^z parametrizes a curve of arbitrary shape and so there may be an issue in making the KKM construction smooth on such a geometry. Once again we suspect that this will not present a problem because the process of adding charges and smearing them out includes some choices of charge density functions and this geometric issue should simply amount to selecting the KKM density distribution so that it is a fixed integer along the curve defined by ^z.
We raise these issues to show that the complete proof of the existence of superstrata as microstate geometries still requires some further work. Indeed, the supergravity solutions corresponding to superstrata will be extremely interesting. One should recall that, from the six-dimensional perspective, the smooth geometry created by the D1-D5 supertube is simply a non-trivial cycle in the three-dimensional homology of the space-time and that the usual supertube profile represents fluctuations of this 3-cycle that depend upon functions of one variable. The superstratum will thus represent the much richer and more extensive space of two-dimensional fluctuations of this 3-cycle. We are thus very optimistic about these new solutions and the role that we expect them to play in black-hole thermodynamics.
The local construction of the superstrata has revealed some particularly satisfying properties. First, they seem to be the most natural fundamental bound-state constituent of a three-charge black hole. The two-charge supertube is a simple fusion of two electric charges to form a fundamental bound object and much of the work of Mathur and collaborators has shown how these objects naturally carry the entropy of the two-charge system. Once one has a three-charge system we now find that there is a very natural ''double bubbling'' that leads to a fundamental, bound object that carries all three charges and by very good fortune these configurations depend upon functions of two variables. While supertubes and generalized supertubes may account for some of the entropy of the three-charge black hole, they are not really fundamentally three-charge objects whereas the new superstratum is precisely such an object and should carry far more entropy555A generalization of the "free supergravity estimate" of [17] to the D1-D5-P system suggests that the entropy that one can maximally obtain from supergravity should behave as ∼c3/4L1/40 for large L0 (using two-dimensional CFT notation), which would still be smaller than the Cardy result ∼c1/2L1/20. It will be interesting to see whether superstrata respect this entropy bound, or can evade it because of their non-perturbative dipole charges.. We find it remarkable that such an object emerges precisely when the three-charge entropy seems to require it and its doubly remarkable that this object can be represented in terms of a microstate geometry.
In this paper, we studied the possibility of a particular two-fold supertube transition of the D1-D5-P system. Namely, at the first stage of the transition, the D1 and P charges are transformed into a tilted D1-P system and the D5 and P charges into a tilted D5-P system. Then, at the second stage, these tilted D1-P and D5-P systems are puffed up into a superstratum. However, since the D1, D5 and P charges can be dualized into each other, at the first stage we could have also considered the possibility of the D1 and D5 charges forming a supertube with KKM dipole charge, which in turn could have also participated in the second transition. It is not clear whether such different patterns of two-fold supertube transitions give the same final result, and if not, whether the resulting configuration could still source a smooth geometry. We leave further investigations into such dynamical issues for future research.
We would like to thank S. Giusto, K. Papadodimas, S. Raju and C. Ruef for helpful discussions. The work of IB was supported in part by the ANR grant 08-JCJC-0001-0, and by the ERC Starting Independent Researcher Grant 240210 - String-QCD-BH. The work of NPW was supported in part by DOE grant DE-FG03-84ER-40168. This work was partly supported by the Foundation of Fundamental Research on Matter (FOM). MS is very grateful to the ITFA, University of Amsterdam for hospitality. MS and NPW are also very grateful to the IPhT, CEA-Saclay for hospitality while much of this work was done.
Appendix A Detailed analysis of the projectors for double bubbling
In this appendix, we develop and study the projectors for double bubbling in detail. Most of the arguments about the supersymmetry of supertubes in various duality frames can be found in the original papers [1, 2, 3] , but here we try to give a general picture that is independent of the duality frame used to describe the supertube. In A.1, we summarize the 12-BPS projectors for various branes and solitons and explain, via examples, how to combine them to construct BPS states with given charges. In A.2, we derive the projector (A.43) for generic supertube transitions where a combination of two electric charges transforms into a new configuration with new dipole charge. In A.3, we use this result to construct the projector for the double bubbling, namely the two-fold supertube transition. In A.4, we explain the relation between the projector derived here and the one used in the main text.
Our convention for the ten-dimensional Clifford algebra are that {Γμ,Γν}=2ημν, wih ημν=(−+⋯+) and we define Γμ1…μk≡Γ[μ1⋯Γμk]. We also define the ten-dimensional helicity operator by Γ∗≡Γ0…9, with Γ2∗=to0.0pt1l.
In type II superstring theory, there are two ten-dimensional Majorana-Weyl supercharges, Q and ˜Q, each of which will be described in terms of 32 component spinors satisfying Majorana and helicity constraints. In type IIA they have opposite helicity, Γ∗Q=Q and Γ∗˜Q=−˜Q, while in type IIB they have the same helicity, Γ∗Q=Q and Γ∗˜Q=˜Q. We will also think of these supersymmetries as belonging to a doublet: Q=(Q˜Q) that has 2×32=64 components and the Pauli σ-matrices will be thought of as acting on the doublet label Q. This means that the helicity projector, Γ∗, is equivalent to the action of σ3 in the type IIA theory and to0.0pt1l2 in type IIB. We will understand that the σ matrices and the identity matrix, to0.0pt1l2, are implicitly tensored with the action of the gamma matrices, Γμ, and that to0.0pt1l32 is implicitly tensored with the action of the σ-matrices.
a.1 Supersymmetry algebra and projectors for branes
The supercharges preserved by various fundamental branes and solitons satisfy
ΠQ =0,Π=12(1+P), (A.1)
where the matrices, P, are given by [28, 29, 30] :
PP =Γ01 PF1 =Γ01σ3
PIIANS5 =Γ012345 PIIBNS5 =Γ012345σ3
PIIAKKM(12345;6) =Γ012345σ3=Γ6789 PIIBKKM(12345;6) =Γ012345=Γ6789
PD0 =Γ0iσ2 PD1 =Γ01σ1
PD2 =Γ012σ1 PD3 =Γ0123iσ2
PD4 =Γ01234iσ2 PD5 =Γ012345σ1
PD6 =Γ0123456σ1 (A.2)
From these conditions, one can reverse-engineer the supersymmetry algebra. For example, one can show that the F1 condition in (A.2) means that we have the following terms on the right hand side of the Q,Q† anticommutator:
12{Q,Q†}=PμΓ0μ+τF1QF1μΓ0μσ3,τF1=12πα′, (A.3)
where QF1μ corresponds to the charge of fundamental strings.
One can show this as follows. If we have straight fundamental strings at rest, the supersymmetry algebra (A.3) becomes
12{Q,Q†}=M+qiΓ0iσ3,τF1QF1μ≡(0,q) (A.4)
The charge vector q measures the tension of the fundamental string, including the direction and multiplicity. Now, assume that the supercharge Q also satisfies:
ΠQ=0,Π=12(1+qi|q|Γ0iσ3), (A.5)
corresponding to fundamental strings along the direction q|q|. Equivalently, one has:
Π′Q=Q,Π′≡1−Π=12(1−qi|q|Γ0iσ3). (A.6)
The superchage Q satisfies the following relation:
12{Q,Q†} =12{Π′Q,(Π′Q)†}=12Π′{Q,Q†}Π′
=14(1−qi|q|Γ0iσ3)(M+qiΓ0iσ3)(1−qi|q|Γ0iσ3)=(M−|q|)Π′, (A.7)
which vanishes for a BPS configuration of mass M=|q|. So, the supercharge, Q, satisfying (A.6) is preserved in this F1 configuration. As one can see from (A.5), half the eigenvalues of Π are 1 and the other half are 0 and so half the components of Q survive the projection (A.5), and hence the state satisfying (A.3) is 12-BPS.
The essential point is that (A.4) becomes the projector (A.5) precisely on the BPS states and, conversely the supersymmetry algebra must be compatible with the projectors that define 12-BPS states.
In this manner, one can determine the supersymmetry algebra for more general configurations to be:
12{Q,Q†} ={PμΓ0μ+τF1QF1μΓ0μσ3+τD1QD0Γ0iσ2+τD2QD2μ1μ2Γ0μ1μ2σ1+⋯(IIA)PμΓ0μ+τF1QF1μΓ0μσ3+τD1QD1μΓ0μσ1+τD3QD3μ1μ2μ3Γ0μ1μ2μ3iσ2+⋯(IIB) (A.8)
where τDp=(2π)−p(α′)−(p+1)/2g−1s.
Another instructive exercise is to derive the linear combination of supersymmetries that vanishes on states with particular combinations of charges. This then describes the BPS bound states with those charges. For a bound state of F1(1) and D1(1) at rest666As in the main body of this paper, the parentheses after a configuration label denotes the spatial directions they wrap, for example, F1(i) means a fundamental string wrapped along xi., the anticommutator, (A.8), is simply:
12{Q,Q†}=M+τF1QF11Γ01σ3+τD1QD11Γ01σ1. (A.9)
Just as we saw in (A.7), it is easy to show that (A.9) vanishes if Q satisfies
ΠF1D1Q=0,ΠF1D1=12[1+Γ01(cosβσ3+sinβσ1)],tanβ=τD1QD11τF1QF11, (A.10)
provided that the mass is equal to M=√(τF1QF11)2+(τD1QD11)2. The angle β is a "mixing angle" between F1 and D1. It is again easy to see that this F1-D1 bound state is 12-BPS.
Similarly, a D1 brane with charge vector q, boosted transverse to its world-volume with momentum p, such that q⋅p=0, is also 12-BPS, and satisfies
(1+piMΓ0i+qiMΓ0iσ1)Q=0,M=√p2+q2. (A.11)
The mass, M, obtained from this BPS condition is indeed the mass of an object with rest mass |q| boosted to have momentum p.
We can also derive this result by directly boosting a D1-brane in a transverse direction. Starting with the projector ΠD1=12(1+Γ01σ1) for a D1-brane in the x1 direction, boosting in the x2-direction amounts to replacing
Γ0→coshξΓ0−sinhξΓ2. (A.12)
The resulting projector does not look like (A.11) but can be brought in that form as follows. First multiply ΠD1 on the left by Γ0, then perform the substitution (A.12), and finally multiply the projector once more on the left by −Γ0/coshξ. The result is precisely of the form (A.11) with p/M=tanhξ and q/M=1/coshξ.
The examples above were all 12-BPS states. Configurations with less supersymmetry can be studied in precisely the same manner. For example, if we have D1(1) and D5(16789), the supersymmetry algebra has:
12{Q,Q†}=M+q1Γ01σ1+q5Γ016789σ1. (A.13)
The preserved supersymmetry, Q, satisfies
Π1Q=Π5Q=0 (A.14)
Π1=12(1+Γ01σ1),Π5=12(1+Γ016789σ1),[Π1,Π5]=0 (A.15)
and we assumed that q1,q5>0. Because Π1 and Π5 commute, the two conditions in (A.14) are equivalent to the single condition
Π15Q=0,Π15=1−(1−Π1)(1−Π5)=14(3+Γ01σ1−Γ6789+Γ016789σ1), (A.16)
Π′15Q=Q,Π′15=1−Π15=Π′1Π′5=14(1−Γ01σ1+Γ6789−Γ016789σ1). (A.17)
Now, from the supersymmetry anticommutator, (A.13), one can derive
12{Q,Q†}=12Π′15{Q,Q†}Π′15 =(M−q1−q5)Π′15. (A.18)
This vanishes for the BPS mass, M=q1+q5. The fact that tr(Π′15)=14tr(to0.0pt1l) means that this is a 14-BPS state.
a.2 The generic supertube transition
A general supertube transition takes the form:
(Q1(x)Q2(y)) → (d1(zψ)J(ψ)), (A.19)
where x,y,z and ψ are some subsets of coordinate directions along which the branes are wrapped or the momentum is directed. The coordinate ψ indicates the new direction associated with the supertube. The supersymmetry preserved by the configuration after the transition depends upon the orientation of the ψ direction as one goes along the brane. If we zoom in on a point on the supertube, the tube can be locally thought of as straight and we can determine the supersymmetry preserved at that point by the methods above. The supertube has the following properties: (i) 16 supersymmetries are preserved at each point along ψ, and (ii) these 16 symmetries differ from point to point but they all share a common subset of eight supersymmetries and these are the same eight supersymmetries that are preserved by the original charge configuration before the transition. From these requirements, it is straightforward to derive a general formula for the supersymmetry projector after the supertube transition for general charges Q1,Q2 in (A.19).
a.2.1 An example: the F1-P system
In order to derive the supersymmtry projector for the general supertube transition (A.19), we begin with a simple example, the F1-P system, where we know the detailed physics of the supertube transition [3] and can construct the projector from the knowledge of the configuration. We will also take a different approach to the one in the main body of this paper.
Consider the F1-P system in which w fundamental strings are wound along z and n units of momentum are directed along the same z direction. We take z to be a compact direction with periodicity 2πRz. The projectors associated with these charges are:
ΠF1(z)=12(1+Γ0zσ3),ΠP(z)=12(1+Γ0z),[ΠF1(z),ΠP(z)]=0. (A.20)
It is easy to see that supercharge Q annihilated by ΠF1(z) and ΠP(z) describes a 14-BPS state.
We now make wish to make a supertube transition:
wn(F1(z)P(z)) → dJ(f1(ψ)J(ψ)). (A.21)
Figure 3: The zoom-in near an F1-P supertube profile
For this system, the fundamental bound state is a string with a momentum wave on it [4] , which can be thought of as the result of a supertube transition that adds an extra F1 dipole charge and an extra angular momentum along the ψ direction [3] . In the new bound state the string world-sheet extends along a curve parametrized by ψ as a function of the original world-sheet direction, z, and carries J units of momentum along ψ. In particular, if ψ=ztanα the shape of the string is a helix moving up (or down, depending the signs of charges) along its axis, just like the barber's pole, as depicted in Fig. 3a. By studying such circular traveling waves on strings using the Nambu–Goto action or by looking at the corresponding supergravity solutions [3] , one finds that the charges w,n and the dipole charges d,J satisfy the relation:
nw=Jd, (A.22)
and that the angular momentum J and the radius of the ψ circle are related by:
J=2πR2ψτF1d. (A.23)
Note that this relation implies that the local "angular momentum density" (proportional to J/R2ψ)777Technically the angular momentum density is J/R while the quantity J/R2 is the linear momentum density around the supertube, or simply the angular speed. Thus what we refer to as "angular momentum density," ρJ, is really the momentum density along the tube and should perhaps be more consistently denoted by ρp, but this could lead to notational confusion with ρP and so we will persist with the mild abuse of terminology in calling this quantity an angular momentum density and denoting it by ρJ., must be equal to the local dipole charge density, and hence be constant along the profile; we discuss this in more detail below (around equation (A.36)). This relation can also be understood as coming from requiring the Killing spinors of the supertube bit to be the same as those of its electric (F1 and P) charges. Adding dipole charges shifts these Killing spinors, and so does adding angular momentum; however if (A.23) is satisfied the shifts cancel each other and the Killing spinors become again those of the electric charges. This can also be seen from supergravity analysis of the near-supertube solution [21] .
Consider a very small part of the helix, as shown in Fig. 3b. Let the size of the square we are focusing on be (2πRψ/M)×(2πRz/N) for some very large integers M and N. We can think of the strings in this small square as straight lines in a small region on the (ψ,z)-plane. Let the angle between this string and the z axis be α (see Fig. 3b). Because the string is wrapped d/N times along the ψ direction of length 2πRψ/M and w/M times along the z direction of length 2πRψ/N, the angle α satisfies
tanα=dN⋅2πRψMwM⋅2πRzN=dRψwRz=√n/Rz2πwRzτF1, (A.24)
where in the third equality we used the relations (A.22) and (A.23).
Let the angle between the momentum vector carried by the string and the ψ axis be β (see Fig. 3b). The (ψ,z) components of the momentum carried by the entire helix is (−J/Rψ,n/Rz). So, the momentum carried by the piece of strings in the small square is (pψ,pz)=(−JMNRψ,nMNRz). So, the angle, β, between the momentum vector carried by the string bit and the ψ axis satisfies:
tanβ=n/(MNRz)J/(MNRψ)=nRψJRz=√n/Rz2πwRzτF1. (A.25)
In the last equality we used the relations required by the dynamics of the string, (A.22) and (A.23). Comparing this with (A.24) we see that β=α and hence these dynamical conditions require that the momentum be perpendicular to the direction of the string, i.e., the string is boosted transverse to its world-sheet.
Figure 4: Boosting and rotating to obtain the desired F1-P configuration
Based on this physical picture, we will construct the supersymmetry projector for this configuration. Starting from strings sitting at rest and extending along z (Fig. 4a), we boost them along the negative ψ-direction (Fig. 4b). We then rotate them in the (ψ,z) plane by angle −α to get to the desired configuration (Fig 4c). The projector for F1(z) boosted along the negative ψ direction (Fig. 4b) is given by:
ΠF1(z)-P(ψ)=12(1−sinγΓ0ψ+cosγΓ01σ3). (A.26)
Here, γ is related to the ratio of the P(ψ) charge and the F1(z) charge as
tanγ =(momentum P(ψ))(F1(z) charge)for the configuration of Fig.\ ???b. (A.27)
The magnitudes of F1 and momentum charges in Fig. 4b are equal to those in the configuration shown in Fig. 4c, because they are related to each other by rotation. Therefore,
tanγ (A.28)
Using the relations (A.22) and (A.23), it is easy to show that this is equal to (A.24) and thus γ=α=β.
By rotating the projector (A.26) (with γ=α) by angle −α in the (ψ,z)-plane, we get the projector after the supertube transition:
Πst =12[1−s(cΓ0ψ−sΓ01)+c(cΓ01+sΓ0ψ)σ3], (A.29)
where s=sinα and c=cosα. The fact that tr(Πst)=12tr(to0.0pt1l) means that this projector preserves 16 supersymmetries. These supersymmetries depend upon the angle α as well as the ψ direction in space and hence the position along the string. However, note that (A.29) can also be written as:
Πst=c(c+sΓzψσ3)ΠF1(z)+s(s−cΓzψσ3)ΠP(z). (A.30)
This implies that the projector Πst always preserves the set of eight supersymmetries that are preserved by the original two projectors ΠF1(z) and ΠP(z), independent of position along the ψ curve.
Although we started by working with a circular ψ curve, our local analysis is valid also for an arbitrary curve parametrized by ψ embedded in the R8 transverse to the original direction, z. Such a curve in R8 can be parametrized by seven functions. Furthermore, we can change the angle α along the curve. Note that α can be expressed in terms of the local charge densities, as follows. The momentum along z carried by the part of the string in our small square is
pz=nRz⋅1MN=nMNRz. (A.31)
So, the local momentum density per unit area is
ρP=pz(2πRz/N)(2πRψ/M)=n(2πRz)2Rψ (A.32)
Similarly, the F1(z) charge carried by the same string bits and the local F1(z) charge density are computed as
qF1=τF1⋅2πRzw⋅1MN=2πwτF1RzMN,ρF1=qF1(2πRz/N)(2πRψ/M)=wτF12πRψ. (A.33)
Comparing these with (A.24), it is easy to see that the angle α can be written as
tanα=√ρPρF1.


text:
Book > Academic Books > Development Studies
Inclusive Economics
Gandhian Method and Contemporary Policy
Narendar Pani - National Institute of Advanced Studies, Bengaluru, India
Development Economics | Development Studies
October 2001 | 206 pages | SAGE India
The book develops an inclusive method for economic planning from Gandhi's voluminous writings, a method which addresses many of the concerns of contemporary policy making. With a loss of faith on the part of the policy makers in large ideological models, and a rise in pragmatism, it has become necessary to develop a more systematic method of putting together the knowledge that economists have developed about different components of the economy. In this book, Ganghian ideas are used to put together such a method.
The Case for Inclusiveness
Primacy of Action
Economic Action
Methodological Outposts
Departure from Convention
Narendar Pani
A post graduate in Economics from JNU, with an M Phil from CDS, Trivandrum, and a PhD from ISEC, Prof. Pani was a research fellow at IIM Bangalore from 1980 to 1982. From 1982 to 1987 he was the Economics Correspondent with Deccan Herald, Bangalore. From 1987 to 2007 he was with The Economic Times, Bangalore in various capacities. From 2007 till date he has been with the School of Social Sciences, National Institute of Advanced Studies (NIAS), Bangalore. He has written and published in various books and peer reviewed journals, including from SAGE. More About Author
PaperbackElectronic Version
For bulk purchase or shipping within South Asia write to
customerservicebooks@sagepub.in

redpajama_set_name: RedPajamaCommonCrawl


Miguel Fernández eta Dolores Ureta Muskizko Pobeña herrian bizi izan zirenean.
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Zayn releases new song from upcoming album, "Icarus Falls"
Zayn asks if he's wasted all of his 'good years' in the latest track from his upcoming album, "Icarus Falls."
I'd rather be anywhere
I close my eyes and see a crowd of a thousand tears
I pray to God I didn't waste all my good years
"Icarus Falls" will feature 27 tracks and will drop on December 14th.
Tagged: lyric video, new music post, zayn malik
pretensetious
The reason I brought it up was because he still doesn't think he can publicly admit he had an ED. When he did, he immediately retracted his statement afterwards. It's nothing on his character/his mindset but about how people approach his situation. People still believe eds are a women only problem (and they largely are) and that therapy is a very "chicken" thing to do when it's about being strong enough to admit you need a little help.
I hope that's why he stopped going but I also think it seems like he's struggling to cope with being famous. Idk I get a very uncomfortable feeling when I think about his situation. It feels like he's unhappy with his status. I hope he finds something that helps, too :)
ileliberte
That's true about the eating disorder, I just assumed he didn't want to call it an ED since it wasn't medically diagnosed but it's definitely still seen more as a thing that happens to women, and even though it's not like it's easy for women to talk about it either. I just don't think it's a masculinity issue since he talks about his mental issues openly and doesn't say no one needs it, just thinks it's something that isn't for him and says that talking about it with family is his form of therapy. Idk if that's the right thing to do, to be honest. Hopefully whatever it is, works for his life.
I think though that he has enough money to withdraw as much as he wants from the public eye while still making music. It's possible that there are bad things happening beyond the label shit that's making him unhappy with where he is right now and what he's able to do, but it's also possible that he's finding his own sweet spot to be the one where he just makes and shares music and doesn't really participate in any of the more publicly interactive parts of a popular entertainment music career.
that would make sense too! and ita re: hoping he's doing what works for him.
yeah! i think that's probably how he is but it's so far from what we normally expect pop stars to do so it's a little weird to see it play out lol


If the slogans and themes of the various Nineteenth Celebrations are any indication, then the Garifuna Nation has prospered greatly in this country.
The truth of the matter is that we have not prospered as the propaganda of our leaders would make the world believe and in recent times we have descended into such decadence that no one respects us anymore. Our leaders are, for the most part, men of straw who look out for their friends and family only and preach of a unity that they themselves have helped to destroy.
I am of the considered opinion that there is nothing to celebrate except our resilience, (what with a somnambulant economy and projects that we do not participate in) and if I had my way I would silence the drums of my fathers until there comes a time when we are once again a proud and enterprising people.
In the two hundred years that we have been in Belize, we have kept our side of the bargain that we made with the British. And in truth, we became model citizens and the country benefited from our labours.
There has never been a civil unrest that began with us. We have neither plundered nor pillaged but rather we have built towns and villages complete with farms that sustained us. We hired out our labour, became entrepreneurs, traded with Belize City and Mexico before other Belizeans discovered Chetumal. Belize, however, has not kept its promise to us.
The British treated us with suspicion from the get-go and did not allow us to participate in the affairs of the colony in any meaningful way. Their propaganda became our history and their slaves disrespected us at every turn.
They grudgingly allowed us to enter the police force and the nursing profession but our men and women were not promoted as fast as the Creole members of these organizations. Our candidates were routinely cheated out of attaining the Baton of Honor when they graduated from the Police training School. And it took a long time before we had a Garifuna as Commissioner of Police.
The Catholic Church (and indeed other religions that we gravitated towards) took away our religion and we descended into hell. They replaced our gods or our devils if you will, with the God of Israel whose attention seems to be taken up with the survival of the State of Israel that he has had no time to listen to our prayers, much less answer them.
Since we found religion we have had our butts kicked whereas before we accepted Christ as our lord and saviour we used to kick butts.
The Church made us teachers and sent us to the ends of the country to educate the Mayas and the Mestizos. It took the flower of the Garifuna Nation and made them priests and nuns and thereafter isolated them from their own people so that we were left without good, compassionate leaders.
The foreign priests and nuns for the most part educated us but held sway over us and emphasized our differences and shortcomings repeatedly that we began to hate each other. They denigrated our culture and imposed their way of thinking on us. Over time we lost our pride in order to conform to the dictates of the Church and we became tractable for the greater glory of God.
But the irony of it all is that the people who were considered to be the best teachers in the country now have the worst schools. AND STILL WE GO OUR MERRY WAY.
We did not fare much better under the Nationalist Movement led by the Hon. George Cadle Price. Looking back over the years after self-government, I can say categorically that Mr. Price had only two aims and that was to achieve independence and to return this country to the Mayas. He did not care which came first.
We did not fit in the general scheme of things as far as Mr. Price was concerned even though the Garinagu voted overwhelmingly for him in the early days. He was quite content for us to remain as teachers so that we can educate his people who were earmarked to lead the new nation of Belize.
Mr. Cornelius Cachos competence was discounted by Mr. Price and Mr. Rafael Fonseca became the Financial Secretary instead. This has been well documented. Some politicians, it is alleged, got very wealthy from their association with elective politics, but even obvious means of becoming wealthy escaped Garifuna Politicians.
Hon. David McKoy, who served for over thirty-six years in the House of Representatives, is still dancing punta, at 73 years of age, trying to sell insurance to ensure his continued survival. His pension from the House obviously cannot sustain him.
We have not benefited from the recent UNDP/GEF grant program as well despite the large number of proposals that were submitted by the Garifuna Business Association and other groups in Hopkins.
When Prime Minister Musa comes to us on the 13th November, he will be bringing a National Hero Award for the late T. V. Ramos while the son of this national hero languishes in poverty, as did the other son when he was alive.
The Prime Minister will participate in the ceremony for our dead; while he disrespects the living and duly elected area representative of Dangriga. And the non-elected leaders of the Garifuna Nation will dance with him.
The next two hundred years then has got to be different! Although our resilience has seen us through before, and I do celebrate this toughness; there has got to be some meaningful and substantive improvements in the lives of our people.
So I shall not dance! I shall not sulk either! But I shall be forced to do things differently. I will begin by seeking out those devils that the Church said we worshipped and worship them! Historical evidence seems to suggest that they were more attuned to our needs and may just be waiting for our return.
I shall honour the memory of the eleven-year-old Garifuna boy who was part of the garrison on Roatan in 1795.
I shall honour the memory of the 150 Garifuna men who stepped ashore in Belize Town in 1802, walked down its streets and put the British in a tizzy.
I shall tie a black ribbon on my arm to register my disgust at the death of Garifuna Manhood and attend the ceremonies to honor T. V. Ramos; but I shall not dance.
I shall attend the meeting the Prime Minister will hold for his followers and I shall dialogue with him.
To our leaders I say forget the fighting. Roy has gotten his museum money and Augustine his passage money. Let us build it, but let Dr. Aranda use the monument and stop embarrassing him in this very public way. Invite him to the official ceremonies and bury the hatchet in a very public manner. Then we can join together in planning the economic development of the Garifuna Nation.
1. I was born here; not in St. Vincent or Roatan.
2. I am better read, than T. V. Ramos ever was.
3. I am angrier than T. V. Ramos ever was!
4. I am meaner than T. V. Ramos ever was!


text:
Another teachers' strike is likely coming to Chicago
Timothy Mclaughlin,
Sep. 26, 2016, 6:55 PM
Teachers picket outside Morgan Park High School in Chicago.
M. Spencer Green/AP Photo
The Chicago Teachers Union (CTU) said on Monday that its members voted overwhelmingly to authorize a strike, setting the stage for a potential work stoppage as soon as mid-October.
The CTU, which represents nearly 27,000 teachers and educational support workers in the country's third largest public school system, said in a statement that 95.6 percent of votes cast were in favor of a strike, with just over 90 percent of teachers voting.
"This should come as no surprise to (the Chicago Board of Education), the mayor or parents because educators have been angry about the school-based cuts that have hurt special education students, reduced librarians, counselors, social workers and teachers' aides, and eliminated thousands of teaching positions," the union said in a statement.
The CTU's governing body will vote on Wednesday on whether to issue a 10-day strike notice to the Board of Education. If the CTU were to move forward, the first possible date for the strike would be Tuesday, Oct. 11.
The union's last contract expired in July 2015 and its bargaining unit unanimously rejected a contract offer from the financially struggling district in February.
A strike by the Chicago Teachers Union continued into its second week.
"A strike is a very serious step that affects the lives of thousands of parents and children, and we hope that before taking the final steps toward a strike, the CTU leadership works hard at the bargaining table to reach a fair deal," Emily Bittner, a spokeswoman for the Chicago Public Schools (CPS) said in a statement.
Speaking in Springfield, the state capital, Republican Governor Bruce Rauner - a frequent critic of both CPS and CTU - said on Monday that a strike would be a "tragedy."
A strike would be the city's third since 2012. A seven-day strike in 2012 was the district's first in 25 years and a one-day strike in early April of this year affected some 400,000 students.
The prospects for additional teacher layoffs grew on Monday, when the CPS that it had experienced "steeper than expected enrollment drops," with preliminary enrollment showing a decline of around 13,800 students this year.
CPS estimated that this drop could lead to approximately 300 job cuts, including both teachers and support staff.
The district said that it will release $5.7 million to help blunt the impact for some schools experiencing enrollment declines.
(Additional reporting by Karen Pierog, editing by G Crosse)
More: Reuters Chicago Teachers Union Strike Education


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text:
Opinion / Blogs
Gil Lackey Outdoors
Rural Viewpoints
Barefoot Farmer
The Mommy Memoirs
RBS City Council Fires City Clerk
Published: Tuesday, October 24,2017
By Debbie Gregory
By a vote of 4-1, the Red Boiling Springs City Council fired City Clerk Tessa Davis at a special called meeting on Thursday, October 19.
The issue had been placed on the agenda following an incident at City Hall that led to the arrest of a city employee's wife, Mrs. Jamie Owens, who had confronted Ms. Davis for allegedly having an inappropriate relationship with her husband, Chad Owens.
When Mayor Joel Coe called the meeting to order at 6:00 p.m. he said, "we are going to be civil, there is going to be no voice raising, and if there is any disorderly conduct you will be escorted out of the council room.
The following councilmen were present: George McCrary, Joe Hill, Cynthia Smalling, Joseph Reardon, and Lee Butram.
"As everyone knows we had an incident here at City Hall recently, involving City Clerk Tessa Davis," said Mayor Coe. "There are several ways we can handle this. Ms. Davis is on a one year probation period as a new city employee and if we think there is disorderly conduct and disruption in city business going on, we need to do something about it."
"I am going to give Tessa Davis a chance to discuss the situation if she wants to," said the Mayor.
Ms. Davis' attorney stood up and said, "Ms. Davis is represented by my firm and my name is Jason Denton. I haven't seen any written documentation of the allegations against her at this time of this disorderly conduct. So, till we have some more details as to exactly what you are alleging was the disorderly conduct, she is not going to speak."
"Number one, sir, I am not alleging," said Mayor Coe.
"Then what the allegations are then sir," said Ms. Davis attorney.
Mayor Coe, "Number two, I am going to ask for depositions."
"That's fine," said Denton.
When Mayor Coe asked is anyone there wanted to submit depositions against Ms. Tessa Davis, three women spoke up. Ms. Davis' attorney asked if these women could be identified for the record. They were Karen Tuck, Gina Bohanan, and Jamie Owens.
"Right now I would like to have an open discussion about this issue," said Mayor Coe.
Councilman George McCrary spoke, "Mr. Mayor since we are not represented by council, I am a little leery of getting into this matter until we have proper council. Since there is going to be be some controversy here, I just don't feel comfortable with the fact that if we took some sort of action, it might lead to some further action down the road, until we are represented."
(RBS City Attorney Jimmy White was not present at the meeting.)
Councilwoman Smalling said, "She is on probation, I thought."
Councilman McCrary, "Well, we can take the action, but I'm just saying…"
"She is on probation," interrupted Smalling.
McCrary, "I'm just saying…"
Mayor Coe, "If you are on probation you don't have…"
"You don't have to have a reason to get rid of her," said Councilman Smalling.
"It's your call Mayor," pointed out Councilman McCrary.
Councilwoman Smalling, "I make a motion we get rid of her tonight. Let's get it settled and over with. I make a motion to get rid of her."
"We have a motion on the table," said Mayor Coe.
Councilwoman Smalling, "Joe Hill, speak up."
Councilman Hill, "I second the motion."
Roll call vote: Councilman Joe Hill, yes; Councliwoman Smalling, yes; Councilman Reardon, yes; Councilman Butram, no; Councilman McCrary, yes.
Mayor Coe commented, "All new full-time employees are on a 12 month probation period and if their job performance is not up to standard, they can be terminated for any reason. This passed in the minutes of 2009."
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The Sharks XV suffered an agonising last-second loss to the Cheetahs in Ballito on Saturday in the SuperSport Rugby Challenge, with the visitors nailing a penalty on the final hooter to claim a 19-21 victory.
Paul Anthony and Ricardo Loubscher would have loved to build on a huge victory over the Leopards the previous week, but with no fewer than six key squad members called up to tour Singapore with the Sharks, were left with no option but to reshuffle their team. That lack of continuity hampered the side in a scrappy game, but one in which the Sharks XV battled bravely all the same.
Free State fullback Marco Mason did most of the damage for the visitors, scoring a try, conversion and penalty in the first half to see the Cheetahs take a 10-0 lead. Courtney Winnaar nailed a penalty for the Sharks, before Hyron Andrews scored a try to close the gap further. Two more penalties from Winaar saw the Sharks edge ahead, only for Mason bring the Cheetahs back within a single point via a late penalty (14-13).
Very few points were scored in the second half, due mainly to mistakes from both sides. Andrews scored a second, to take the Sharks out to a 6-point lead twelve minutes from the end, but Cheetahs flank Tienie Burger soon undid that with a score of his own. It then fell to Mason to nail a pressure kick to overturn a 1-point deficit, when the Sharks conceded a penalty off the final scrum of the match.
« A bonus point! At last!
Great to see Andrews playing consistently well. With Ettas going and Lewies looking comfortable at 4, Andrews will be key as 5 lock in CC and SR next season.
Who is next in line at 4?
@FireTheLooser (Comment 2) : We always thought we got the better deal out of swoppong PSDT and Botha…..
Botha needs two seasons injury free and will be in the Bok side.
@Kabouter (Comment 3) : @FireTheLooser (Comment 2) : the steamers will never admit but I also think at worst it was a equal swop!!!
@pastorshark (Comment 5) : This team is ultimately there for the purpose of supporting the SR team, keep the guys conditioned and in the system.
So it's nice when they win, but that is ultimately not really their purpose (although winning does the player psyche a world of good).


Therapeutic symptoms of homeopathic remedy Conium Maculatum, described by E.B. Nash in his book, Leaders in Homeopathic Therapeutics, published in 1898.
Vertigo, especially worse on turning the head, or looking around sidewise, or turning in bed.
Swelling and induration of glands, after contusions or bruises.
Cancerous and scrofulous persons with enlarged glands.
Urine flows, stops and flows again intermittently, prostatic or uterine affections.
Breasts sore, hard and painful during menses.
This is another of so called spinal remedies. I will not, as I did under COCC., quote what authorities say from a pathological standpoint. All seem to agree that it paralyses from below upwards, and the poisoning of Socrates with it is adduced in illustration. It ought to be a remedy for locomotor ataxia. The strongest characteristic I know from a homoeopathic., standpoint, is its peculiar vertigo, which is much worse by TURNING THE HEAD SIDEWISE.(COLOC., turning head to left). Turning over in bed is the same. Some say LYING DOWN IN BED and TURNING OVER. I have found that it is not so much the LYING DOWN as it is the turning of the HEAD SIDEWISE, whether in an upright or horizontal posture.
The patient had been slowly loosing the use of his legs, could not stand in the dark; and when he walked along the street would make his wife walk either ahead of him, or behind him, for the act of looking sidewise at her or in thee least turning head or eyes that way would cause him to stagger or fall.
CONIUM cured him. It would always worse at first, but he would greatly improved after stopping the remedy. The worse was just as invariable after taking a dose of Fincke's C.M. potency as from anything lower, but the improvement lasted longer after it.
Taking an occasional dose from a week to four weeks apart completely cured him in about a year. It was a bad case, of years' standing, before I took him.
I have often verified this symptom in the vertigo of old people, where it is most frequently found; but it also often accompanies various affections in all ages, and especially is found in ovarian and uterine affections. I know of no remedy that has this symptom so strongly.
There is a form of ophthalmia in strumous subjects which calls for CONIUM in preference to any other remedy, and the peculiar, prominent and uncommon (as Hahnemann says, organon, paragraph 153). symptom is, PHOTOPHOBIA, INTENSE, OUT OF ALL PROPORTION TO THE OBJECTIVE SIGNS of inflammation in the eye. The pains are worse at night and terribly worse by the least ray of light, relieved in dark room and by pressure.
There may or may not be ulcers on the cornea. CONIUM is also one of our best remedies for falling of the eyelids, as are three other remedies, viz.: GELSEMIUM, CAUSTICUM and SEPIA. "Swelling and induration of glands, with tingling and stitches after CONTUSIONS OR BRUISES." Many cases of lumps or swellings in the breasts (for which CON., seems to have a particular affinity) have disappeared under the action of this remedy. Even cancerous affections of breast (ASTER.,)uterus and stomach have been helped or cured especially if the trouble seems to have originated in a BLOW OR INJURY to the part. It is perhaps the first remedy to be thought of in all cases of tumors, scirrhous, or otherwise, coming on after contusions, especially if they are of stony hardness and heavy feeling. CON., and sil., both have hardness of mammae, CON., right, SIL., left nodules (CARB-AN., CON., SIL.,); acute lancinating pains.(ASTER.,)Again it is to be especially considered if at every menstrual period the breasts become LARGE SORE and PAINFUL, worse by the LEAST JAR OR WALKING.
In all scirrhous affections of the breast, womb or other parts the pains of CON., are burning, stinging, or darting, and may make one think of APIS., The other symptoms must then decide between them.
CON., has marked action upon the sexual organs. In the male there is great weakness of the organs. He has intense desire and amorous thoughts, but is unable to perform. He has emissions at the very thought or the presence of a women. The erections are insufficient, last only a short time,"or go back" on him in the act of embrace, and he suffers with weakness and chagrin afterwards. This affects the mind and hypochondriasis of the bluest blue takes possession of him. This condition of mind may obtained in both sexes; as a result of too free, and also especially too INFREQUENT indulgence; or, excessive abstemiousness. Hence CON., becomes a good remedy for old bachelors and old maids. If the vertigo is also present in such cases CON., is sure to be of great benefit.
Dr. Adolph Lippe once made a splendid cure of a complete one sided paralysis in a man 80 years of age with this remedy, and was guided to it by this symptom. I think it would be rather a difficult task to give a correct pathological explanation of such a symptom; but there is a reason, and whether we can give it or not we can cure it if we have a corresponding one appearing under a remedy; where a cure is at all possible.
It is interesting to follow out the connections of symptoms. Take for instance the single prominent symptom of CON., VERTIGO.
A good understanding of such connection often starts the prescriber on the short cut route to the remedy in a case.