We continue explorations of non-Abelian strings, focusing on the solution of a heterotic deformation of the CP(N-1) model with an extra right-handed fermion field and N=(0,2) supersymmetry. This model emerges as a low-energy theory on the worldsheet of the BPS-saturated flux tubes (strings) in N=2 supersymmetric QCD deformed by a superpotential of a special type breaking N=2 supersymmetry down to N=1. Using large-N expansion we solve this model to the leading order in 1/N. Our solution exhibits spontaneous supersymmetry breaking for all values of the deformation parameter. We identify the Goldstino field. The discrete Z_{2N} symmetry is shown to be spontaneously broken down to Z_2; therefore, the worldsheet model has N strictly degenerate vacua (with nonvanishing vacuum energy). Thus, the heterotic CP(N-1) model is in the deconfinement phase. We can compare this dynamical pattern, on the one hand, with the N=(2,2) CP(N-1) model which has N degenerate vacua with unbroken supersymmetry, and, on the other hand, with nonsupersymmetric CP(N-1) model with split quasivacua and the Coulomb/confining phase. We determine the mass spectrum of the heterotic CP(N-1) model in the large-N limit.
We investigate the coset structures which appear in the dimensional reduction of supergravity theories. Especially we investigate how to recognize the global symmetry groups if the coset is non-split. As an example we apply our analysis to the theories emanating from the dimensional reduction of Heterotic supergravity.
Exact black hole solutions of the five dimensional heterotic $S$-$T$-$U$ model including all perturbative quantum corrections and preserving $1/2$ of $N=2$ supersymmetry are studied. It is shown that the quantum corrections yield a bound on electric charges and harmonic functions of the solutions.
A two dimensional anomaly cancellation argument is used to construct the SO(32) heterotic and type IIB membranes. By imposing different boundary conditions at the two boundaries of a membrane, we shift all of the two dimensional anomaly to one of the boundaries. The topology of these membranes is that of a 2-dimensional cone propagating in the 11-dimensional target space. Dimensional reduction of these membranes yields the SO(32) heterotic and type IIB strings.
We consider four-dimensional charged black-holes occuring in toroidally compactified heterotic string theory, whose ten-dimensional interpretation involves a Kaluza-Klein monopole and a five-brane. We show that these four-dimensional black-holes can be connected to two-dimensional charged heterotic black-holes upon removal of the constants appearing in the harmonic functions associated with the Kaluza-Klein monopole and the five-brane.
The first-order bosonic field equations of the D-dimensional effective low energy theory which describes the massless background coupling of the D-dimensional fully Higgsed heterotic string are derived.
In view of the expectation that the solitonic sector of the lower dimensional world may be originated from the solitonic sector of string theory, various solitonic solutions are reduced from the heterotic fivebrane solutions in the ten-dimensional heterotic string theory. These solitons in principle can appear after proper compactifications, {\it e.g.} toroidal compactifications.
The Euclidean analogues of the sine-Gordon solitons are used as sources of the heterotic fivebrane solutions in the ten-dimensional heterotic string theory. Some properties of these soliton solutions are discussed. These solitons in principle can appear as string-like objects in 4-dimensional space-time after proper compactifications.
Generalizing the work by Cherkis and Schwarz [1], we carry out the double dimensional reduction of supersymmetric M5 brane on K3 to obtain the supersymmetric action of heterotic string in 7-dimensional flat space-time. Motivated by this result, we propose the supersymmetric heterotic action in 10-dimensional flat space-time where the current algebra is realized in a novel way. We explicitly verify the kappa-symmetry of the proposed action.
The equations of motion of the massless sector of the two dimensional string theory, obtained by compactifying the heterotic string theory on an eight dimensional torus, is known to have an affine o(8,24) symmetry algebra generating an O(8,24) loop group. In this paper we study how various known discrete S- and T- duality symmetries of the theory are embedded in this loop group. This allows us to identify the generators of the discrete duality symmetry group of the two dimensional string theory.
We show that a large class of supersymmetric solutions to the low-energy effective field theory of heterotic string theory compactified on a seven torus can have finite energy, which we compute. The mechanism by which these solutions are turned into finite energy solutions is similar to the one occurring in the context of four-dimensional stringy cosmic string solutions. We also describe the solutions in terms of intersecting eleven-dimensional M-branes, M-waves and M-monopoles.
We explore a relation between four-dimensional N=2 heterotic vacua induced by Mirror Symmetry via Heterotic/Type II duality. It allows us to compute the α' corrections to the hypermultiplet moduli space of heterotic compactifications on K3xT^2 in the limit of large base of the elliptic K3. We concentrate on the case of point-like instantons on orbifold singularities leading to low-dimensional hypermultiplet moduli spaces.
The sigma model action described in this paper differs in four important features from the usual sigma model action for the four-dimensional Green-Schwarz heterotic superstring in a massless background. Firstly, the action is constructed on an N=(2,0) super-worldsheet using a Kahler potential and an Ogievetsky-Sokatchev constraint; secondly, the target-space background fields are unconstrained; thirdly, the target-space dilaton couples to the two-dimensional curvature; and fourthly, the action reduces in a flat background to a free-field action. A conjecture is made for generalizing this N=(2,…
We consider the low-energy effective field theory of heterotic string theory compactified on a seven-torus, and we construct electrically charged as well as more general solitonic solutions. These solutions preserve 1/2, 1/4 and 1/8 of N=8, D=3 supersymmetry and have Killing spinors which exist due to cancellation of holonomies. The associated space-time line elements do not exhibit the conical structure that often arises in 2+1 dimensional gravity theories.
Using the heterotic--type II duality of $N=2$ string vacua in four space-time dimensions we study non-perturbative couplings of toroidally compactified six-dimensional heterotic vacua. In particular, the heterotic--heterotic $S$-duality and the Coulomb branch of tensor multiplets observed in six dimensions are studied from a four-dimensional point of view. We explicitly compute the couplings of the vector multiplets of several type II vacua and investigate the implications for their heterotic duals.
We show how the level matching condition in six dimensional, abelian and supersymmetric orbifolds of the E_8 x E_8 heterotic string can be given equivalently in terms of fractional gauge and gravitational instanton numbers. This relation is used to restate the classification of the orbifolds in terms of flat bundles away from the orbifold singularities under the constraint of the level matching condition. In an outlook these results are applied to Kaluza-Klein monopoles of the heterotic string on S^1 in Wilson line backgrounds.
We propose that the ten-dimensional $E_8\times E_8$ heterotic string is related to an eleven-dimensional theory on the orbifold ${\bf R}^{10}\times {\bf S}^1/{\bf Z}_2$ in the same way that the Type IIA string in ten dimensions is related to ${\bf R}^{10}\times {\bf S}^1$. This in particular determines the strong coupling behavior of the ten-dimensional $E_8\times E_8$ theory. It also leads to a plausible scenario whereby duality between $SO(32)$ heterotic and Type I superstrings follows from the classical symmetries of the eleven-dimensional world, just as the $SL(2,{\bf Z})$ duality of the t…
In the covariant lattice formalism, chiral four-dimensional heterotic string vacua are obtained from certain even self-dual lattices which completely decompose into a left-mover and a right-mover lattice. The main purpose of this work is to classify all right-mover lattices that can appear in such a chiral model, and to study the corresponding left-mover lattices using the theory of lattice genera. In particular, the Smith-Minkowski-Siegel mass formula is employed to calculate a lower bound on the number of left-mover lattices. Also, the known relationship between asymmetric orbifolds and cova…
We re-examine the question of heterotic - heterotic string duality in six dimensions and argue that the $E_8\times E_8$ heterotic string, compactified on $K3$ with equal instanton numbers in the two $E_8$'s, has a self-duality that inverts the coupling, dualizes the antisymmetric tensor, acts non-trivially on the hypermultiplets, and exchanges gauge fields that can be seen in perturbation theory with gauge fields of a non-perturbative origin. The special role of the symmetric embedding of the anomaly in the two $E_8$'s can be seen from field theory considerations or from an eleven-dimensional …
We study the constraints of spacetime supersymmetry for perturbative three- and two-dimensional Minkowski vacua of the critical heterotic string. Assuming a standard RNS construction of the spacetime supersymmetry generators and a compact unitary internal superconformal worldsheet theory, we describe the worldsheet structures associated to various spacetime supersymmetries. In three dimensions we show that there are no CFT surprises: each allowed spacetime supersymmetry is realized by a supergravity compactification. As a recent orbifold construction shows, in two dimensions there are more exo…
We examine the vacua of the scalar potential of the effective 4-d action, obtained after the dimensional reduction of the 10-d $\mathcal{N}=1$ heterotic supergravity coupled to an $\mathcal{N}=1$ Yang-Mills sector. The (Coset Space) dimensional reduction takes place over the three 6-d nearly-Kähler manifolds, namely the homogeneous 6-d non-symmetric coset spaces, $G_2/SU(3)$, $Sp_4/SU(2)\times U(1)$ and $SU(3)/U(1)\times U(1)$. The current work consists a complete catalogue of the kinds of vacua of theories obtained after the reduction of the heterotic string over the 6-d non-symmetric coset s…
We present a Kahler potential for four dimensional heterotic M-theory which includes moduli describing a gauge five brane living on one of the orbifold fixed planes. This result can also be thought of as describing compactifications of either of the weakly coupled heterotic strings in the presence of a gauge five brane. This is the first example of a Kahler potential in these theories which includes moduli describing background gauge field configurations. Our results are valid when the solitons width is much smaller than the size scale of the Calabi-Yau threefold and can be used to provide a m…
Gaugino condensation is discussed in the context of a consistent new version of low energy heterotic M-theory. The four dimensional reduction of the theory is described, based on simple boson and fermion backgrounds. This is generalised to include gaugino condenstates and various background fluxes, some with non-trivial topology. It is found that condensate and quantised flux contributions to the four-dimensional superpotential contain no corrections due to the warping of the higher dimensional metric.
Our aim is to derive the effective action in four dimensions resulting by reducing dimensionally the ten-dimensional ${\cal N}=1$ heterotic supergravity coupled to ${\cal N}=1$ super Yang-Mills over manifolds admitting a nearly-Kähler structure. Given the fact that all six-dimensional nearly-Kähler manifolds are included in the class of the corresponding non-symmetric coset spaces plus a group manifold, our procedure amounts in applying the Coset Space Dimensional Reduction scheme using these coset spaces as internal manifolds. In our examination firstly the rules of the reduction of the theor…
The effective action for five-dimensional heterotic M-theory in the presence of five-branes is systematically derived from Horava-Witten theory coupled to an M5-brane world-volume theory. This leads to a five-dimensional N=1 gauged supergravity theory on S^1/Z_2 coupled to four-dimensional N=1 theories residing on the two orbifold fixed planes and an additional bulk three-brane. We analyse the properties of this action, particularly the four-dimensional effective theory associated with the domain-wall vacuum state. The moduli Kahler potential and the gauge-kinetic functions are determined alon…
We analyze the possible dynamical emergence of IR conformal field theory describing the low-energy excitations of near-extremal black holes in five-dimensional compactification of heterotic strings. We find that, by tuning the mass and charges in such a way that the extremal black holes have a classically vanishing horizon area, the near-horizon develops an AdS3 throat and when we combine the low-energy limit with vanishing Newton coupling constant, the system has a dual conformal field theory description. We compare our results with c-extremization and the Kerr/CFT predictions.
The effective action in four dimensions resulting from the ten-dimensional N=1 heterotic supergravity coupled to N=1 supersymmetric Yang-Mills upon dimensional reduction over nearly-Kaehler manifolds is discussed. Nearly-Kaehler manifolds are an interesting class of manifolds admitting an SU(3)-structure and in six dimensions all homogeneous nearly-Kaehler manifolds are included in the class of the corresponding non-symmetric coset spaces plus a group manifold. Therefore it is natural to apply the Coset Space Dimensional Reduction scheme using these coset spaces as internal manifolds in order …
It is shown that the heterotic string emerges as a soliton in the type I superstring theory in ten dimensions. The collective coordinates of the soliton are described by a smooth, chiral worldsheet theory. There are eight bosonic and eight right-moving fermionic zero modes that arise from the partially broken supertranslations. In addition, there are 496 charged bosonic zero modes of the gauge field that describe a left-moving WZNW model on a $spin(32)/{Z_2}$ group manifold. Small, stable loops of the solitonic string furnish the massive states required by duality that transform as spinors of …
We present the first example of a Kahler potential for heterotic M-theory which includes gauge bundle moduli. These moduli describe the background gauge field configurations living on the orbifold fixed planes. We concentrate on the bundle moduli describing the size and SU(2) orientation of a gauge five brane - a soliton which is primarily composed of these gauge fields. Our results are valid when the width of this object is small compared to the overall size of the Calabi-Yau threefold. We find that, in general, it is not consistent to truncate away these moduli in a simple manner.
We discuss the predictions of S-duality for the monopole spectrum of four-dimensional heterotic string theory resulting from toroidal compactification. We discuss in detail the spectrum of "H-monopoles", states that are magnetically charged with respect to the U(1) groups arising from the dimensional reduction of the ten dimensional antisymmetric tensor field. Using an assumption concerning the correct treatment of collective coordinates in string theory we find results which are consistent with S-duality.
We present a class of heterotic compactifications where it is possible to lower the string unification scale down to the GUT scale, while preserving the validity of the perturbative analysis. We illustrate this approach with an explicit example of a four-dimensional chiral heterotic vacuum with N=1 supersymmetry.
We derive the precise relation between level matching condition and fractional instanton numbers in six dimensional, abelian and supersymmetric orbifolds of E8 x E8 heterotic string theory. The fractional part of the two E8 instanton numbers is explicitly calculated in terms of the gauge twist. This relation is then used to show that the classification of these orbifolds can be given in terms of flat bundles away from the orbifold singularities under the only constraint that the sum of the fractional parts of the gauge instanton numbers match the fractional part of the gravitational instanton …
Progress towards the classification of the meromorphic $c=24$ conformal field theories is reported. It is shown if such a theory has any spin-1 currents, it is either the Leech lattice CFT, or it can be written as a tensor product of Kac-Moody algebras with total central charge 24. The total number of combinations of Kac-Moody algebras for which meromorphic $c=24$ theories may exist is 221. The next step towards classification is to obtain all modular invariant combinations of Kac-Moody characters. The presently available results are sufficient to obtain a complete list of all ten-dimensional …
We study the symmetries of the two dimensional Heterotic string theory by following the approach of Kinnersley et al for the study of stationary-axially symmetric Einstein-Maxwell equations. We identify the finite dimensional groups $G'$ and $H'$ for the Einstein-Maxwell equations. We also give the constructions for the infinite number of conserved currents and the affine $\hat{o}(8, 24)$ symmetry algebra in this formulation. The generalized Ehlers and Harrison transformations are identified and a parallel between the infinite dimensional symmetry algebra for the heterotic string case with $\h…
We derive the ten-dimensional effective action of the strongly coupled heterotic string as the low energy limit of M-theory on S^1/Z_2. In contrast to a conventional dimensional reduction, it is necessary to integrate out nontrivial heavy modes which arise from the sources located on the orbifold fixed hyperplanes. This procedure, characteristic of theories with dynamical boundaries, is illustrated by a simple example. Using this method, we determine a complete set of R^4, F^2R^2, and F^4 terms and the corresponding Chern-Simons and Green-Schwarz terms in ten dimensions. As required by anomaly…
Recent developments in string theory have reinforced the notion that the space of stable supersymmetric and non-supersymmetric string vacua fills out a ``landscape'' whose features are largely unknown. It is then hoped that progress in extracting phenomenological predictions from string theory -- such as correlations between gauge groups, matter representations, potential values of the cosmological constant, and so forth -- can be achieved through statistical studies of these vacua. To date, most of the efforts in these directions have focused on Type I vacua. In this note, we present the firs…
We consider heterotic string solutions based on a warped product of a four-dimensional domain wall and a six-dimensional internal manifold, preserving two supercharges. The constraints on the internal manifolds with SU(3) structure are derived. They are found to be generalized half-flat manifolds with a particular pattern of torsion classes and they include half-flat manifolds and Strominger's complex non-Kahler manifolds as special cases. We also verify that previous heterotic compactifications on half-flat mirror manifolds are based on this class of solutions.
We classify all warped product de-Sitter backgrounds in heterotic supergravity, up to two loops. We find that warped $dS_n$ backgrounds, with $n\ge 3$, are $\mathbb{R}^{1,n}\times M_{9-n}$, where $M_{9-n}$ is a $(9-n)$-dimensional Riemannian manifold. Moreover, we establish that warped $dS_2$ backgrounds are $AdS_3\times M_7$, where $M_7$ is a 7-dimensional Riemannian manifold.
In this paper, we continue the analysis of heterotic string compactifications on half-flat mirror manifolds by including the 10-dimensional gauge fields. It is argued, that the heterotic Bianchi identity is solved by a variant of the standard embedding. Then, the resulting gauge group in four dimensions is still E6 despite the fact that the Levi-Civita connection has SO(6) holonomy. We derive the associated four-dimensional effective theories including matter field terms for such compactifications. The results are also extended to more general manifolds with SU(3) structure.
We review aspects of N=1 duality between the heterotic string and F-theory. After a description of string duality intended for the non-specialist the framework and the constraints for heterotic/F-theory compactifications are presented. The computations of the necessary Calabi-Yau manifold and vector bundle data, involving characteristic classes and bundle moduli, are given in detail. The matching of the spectrum of chiral multiplets and of the number of heterotic five-branes respectively F-theory three-branes, needed for anomaly cancellation in four-dimensional vacua, is pointed out. Several e…
We consider a class of extremal and non-extremal four-dimensional black-hole solutions occuring in toroidally compactified heterotic string theory, whose ten-dimensional interpretation involves a Kaluza-Klein monopole and a five-brane. We show that these four-dimensional solutions can be connected to extremal and non-extremal two-dimensional heterotic black-hole solutions through a change in the asymptotic behaviour of the harmonic functions associated with the Kaluza-Klein monopole and with the five-brane. This change in the asymptotic behaviour can be achieved by a sequence of S and T-S-T du…
We simplify the classification of supersymmetric solutions with compact holonomy of the Killing spinor equations of heterotic supergravity using the field equations and the additional assumption that the 3-form flux is closed. We determine all the fractions of supersymmetry that the solutions preserve and find that there is a restriction on the number of supersymmetries which depends on the isometry group of the background. We examine the geometry of spacetime in all cases. We find that the supersymmetric solutions of heterotic supergravity are associated with a large number of geometric struc…
Brane actions with chiral bosons present special challenges. Recent progress in the description of the two main examples -- the M theory five-brane and the heterotic string -- is described. Also, double dimensional reduction of the M theory five-brane on K3 is shown to give the heterotic string.
This paper constructs the reduction of heterotic $M$-theory in eleven dimensions to a supergravity model on a manifold with boundary in five dimensions using a Calabi-Yau three-fold. New results are presented for the boundary terms in the action and for the boundary conditions on the bulk fields. Some general features of dualisation on a manifold with boundary are used to explain the origin of some topological terms in the action. The effect of gaugino condensation on the fermion boundary conditions leads to a `twist' in the chirality of the gravitino which can provide an uplifting mechanism i…
This paper considers eleven dimensional supergravity on a manifold with boundary and the theories related to heterotic $M$-theory, in which the matter is confined to the boundary. New low energy actions and boundary conditions on supergravity fields are derived. Previous problems with infinite constants in the action are overcome. The new boundary conditions are shown to be consistent with supersymmetry, and their role in the ten dimensional reduction and gaugino condensation is briefly discussed.
We discuss type I -- heterotic duality in four-dimensional models obtained as a Coulomb phase of the six-dimensional U(16) orientifold model compactified on T^2 with arbitrary SU(16) Wilson lines. We show that Kahler potentials, gauge threshold corrections and the infinite tower of higher derivative F-terms agree in the limit that corresponds to weak coupling, large T^2 heterotic compactifications. On the type I side, all these quantities are completely determined by the spectrum of N=2 BPS states that originate from D=6 massless superstring modes.
We review recent results on near-horizon static black hole solutions and entropy in R^2-corrected N=2 SUGRA in D=5, focusing on actions connected to heterotic string compactified on K3\times S^1. Comparison with α'-perturbative results, results obtained by using simple Gauss-Bonnet R^2-correction, OSV conjecture and microscopic stringy description (for small black holes) shows that situation in D=5 is, in a sense, even more interesting then in D=4.
The compactification from the 11-dimensional Horava-Witten orbifold to 5-dimensional heterotic M-theory on a Schoen Calabi-Yau threefold is reviewed, as is the specific $SU(4)$ vector bundle leading to the "heterotic standard model" in the observable sector. A generic formalism for a consistent hidden sector gauge bundle, within the context of strongly coupled heterotic M-theory, is presented. Anomaly cancellation and the associated bulk space 5-branes are discussed in this context. The further compactification to a 4-dimensional effective field theory on a linearized BPS double domain wall is…
In this thesis, we consider heterotic string vacua based on a warped product of a four-dimensional domain wall and a six-dimensional internal manifold preserving only two supercharges. Thus, they correspond to half-BPS states of heterotic supergravity. The constraints on the internal manifolds with SU(3) structure are derived. They are found to be a generalization of half-flat manifolds with a particular pattern of torsion classes and they include half-flat manifolds and Strominger's complex non-Kahler manifolds as special cases. We also verify that heterotic compactifications on half-flat mir…
We determine the two--dimensional Weyl, Lorentz and $κ$--anomalies in the $D=10$ Green--Schwarz heterotic string sigma--model, in an $SO(1,9)$-Lorentz covariant background gauge, and prove their cancellation.
In this paper we study $T^2$ compactification of 6-dimensional massive type IIA supergravity in presence of Ramond-Ramond background fluxes. The resulting theory in four dimensions is shown to possess $SL(2,R)\times SL(2,R)\times O(4,20)$ duality symmetry. It is shown that specific elements of this symmetry relate massive type IIA compactified on $K3\times T^2$ (with fluxes along $K3$) to the ordinary type IIA compactified on $K3\times T^2$ (with fluxes along $T^2$). In turn, this relationship is exploited to relate Romans theory to heterotic strings. The D8-brane (domain-wall) wrapped on $K3\…
The low-energy D=4, N=1 effective action of the strongly coupled heterotic string is explicitly computed by compactifying Horava-Witten theory on the deformed Calabi-Yau three-fold solution due to Witten. It is shown that, to order kappa^{2/3}, the Kahler potential is identical to that of the weakly coupled theory. Furthermore, the gauge kinetic functions are directly computed to order kappa^{4/3} and shown to receive a non-vanishing correction. Also, we compute gauge matter terms in the Kahler potential to the order kappa^{4/3} and find a nontrivial correction to the dilaton term. Part of tho…
We construct an octonionic instanton solution to the seven dimensional Yang-Mills theory based on the exceptional gauge group $G_2$ which is the automorphism group of the division algebra of octonions. This octonionic instanton has an extension to a solitonic two-brane solution of the low energy effective theory of the heterotic string that preserves two of the sixteen supersymmetries and hence corresponds to $N=1$ space-time supersymmetry in the (2+1) dimensions transverse to the seven dimensions where the Yang-Mills instanton is defined.
Higher derivative couplings of hypermultiplets to $6D, N=(1,0)$ supergravity are obtained from dimensional reduction of 10D heterotic supergravity that includes order $α'$ higher derivative corrections. Reduction on $T^4$ is followed by a consistent truncation. In the resulting action the hyperscalar fields parametrize the coset $SO(4,4)/(SO(4)\times SO(4))$. While the $SO(4,4)$ symmetry is ensured by Sen's construction based on string field theory, its emergence at the field theory level is a nontrivial phenomenon. A number of field redefinitions in the hypermultiplet sector are required to r…
In this paper we summarize our recent work about perturbative and non-perturbative effects in four-dimensional heterotic strings with $N=2$ space-time supersymmetry.
We study quantum corrections for a family of 24 non-supersymmetric heterotic strings in two dimensions. We compute their genus two cosmological constant using the hyperelliptic formalism and the genus one two-point functions for the massless states. From here we get the mass corrections to the states in the massless sector and discuss the role of the infrared divergences that appear in the computation. We also study some tree-level aspects of these theories and find that they are classified not only by the corresponding Niemeier lattice but also by their {\it hidden} right-moving gauge symmetr…
We construct cosmological solutions of four-dimensional effective heterotic M-theory with a moving five-brane and evolving dilaton and T modulus. It is shown that the five-brane generates a transition between two asymptotic rolling-radii solutions. Moreover, the five-brane motion always drives the solutions towards strong coupling asymptotically. We present an explicit example of a negative-time branch solution which ends in a brane collision accompanied by a small-instanton transition. The five-dimensional origin of some of our solutions is also discussed.
We point out that in heterotic string theory compactified on a 6-torus, after a consistent truncation of the 10-d gauge fields and the antisymmetric tensor fields, 4-dimensional black holes of Kaluza-Klein theory on a 6-torus constitute a subset of solutions.
