On Black Attractors in 8d and Heterotic/Type IIA Duality
Motivated by the study of black attractors in 8D supergravity with 16 supersymmetries, we use the field theory approach and 8D supersymmetry with non trivial central charges to shed light on the exact duality between heterotic string on T and type IIA on real connected and compact surfaces . We investigate the two constraints that should be obeyed by and give their solutions in terms of intersecting 2-cycles as well their classification using Dynkin diagrams of affine Kac-Moody algebras. It is shown as well that the moduli space of these dual theories is given by where stands for the rank of the gauge symmetry of the 10D heterotic string on . The remarkable cases as well as other features are also investigated.
In the last few years, there has been a growing interest in the
study the attractor mechanism of black brane configurations in
higher dimensional supergravity theories with typical moduli space
given by the coset group manifold G/H -.
This mechanism was first
discovered in the context of supersymmetric black holes in 4D supergravity coupled to vector supermultiplets -; and was extended for black branes living in
diverse dimensions; for reviews see - and
refs therein. The attractor mechanism tells that the values of the moduli at the horizon geometry of
the black attractor are independent of the asymptotic values ; and, up on solving the
attractor equations, they are completely determined by the electric
and magnetic charges respectively
given by the fluxes and
of the various massless abelian gauge fields of the supergravity
theory including p-branes.
A remarkable class of the supergravity theories that have attracted much interest in the recent literature concerns the ultra-violet (UV) finite ones arising from superstrings and M-theory compactifications having interpretation in terms of D- and M- branes wrapping cycles of the internal space. There, the Maxwell type gauge fields are associated with semi- realistic string models with multitude of gauge factors that survive in the typical gauge symmetry breaking
where can be thought of as the or
gauge symmetries of the 10D heterotic
string, or a subsymmetry of them. The s are rank
subgroups of with the property
and associated 2-form gauge
field strengths having non trivial fluxes along the directions and playing a central role in
the study of black attractors in these supergravity theories. In
this issue, several results have been obtained on attractor
mechanism of extremal (vanishing temperature for non-zero entropy)
BPS and non BPS black branes
living in 4D, 5D, 6D, 7D and 8D supergravities with various numbers of conserved supersymmetries [8, 9] and refs therein.
On the other hand, in completing missing results on intersecting attractors in non chiral supergravity in eight dimensions  arising from 10D heterotic string on 2-torus and involving 1-form gauge fields together with real scalars parameterizing the moduli space
we have noticed an interesting property concerning their type IIA interpretation using D- branes. This property, which is investigated in this paper, concerns the two following things:
(1) the geometric engineering of the exact duality between the 10D heterotic superstring on with some gauge symmetry ; and the type IIA superstring compactified on real compact surfaces with D- branes wrapping cycles of .
(2) The heterotic/ type IIA interpretation of the singular limit describing the coupling of the 8D supergravity multiplet to one Maxwell gauge supermultiplet. The moduli space of this special supersymmetric gauge theory is given by the particular real 3-dimensional coset manifold,
Moreover, because of the heterotic/type IIA duality in 8D, we also show that the real compact surfaces , used in the compactification scenario
should obey two basic consistency constraints. The first one, which is familiar in string compactification, deals with the number of conserved supersymmetries after space time dimension reduction. The second one however concerns the connected 2-cycle homology of that requires a vanishing self intersection; i.e : . This constraint turns out to capture two remarkable issues:
(a) it allows to explain the stringy origin of the non chiral 8D supergravity models with moduli spaces and which miss in the listing (2) and respectively recovered by roughly setting .
(b) it captures data on superconformal invariance in 8D that underly known results on 4D conformal models. Recall that 4D supersymmetric conformal models were first obtained in , see also  for an extension, from 10D type II superstrings on Calabi-Yau threefolds with affine ADE singularities; they may be re-derived from non chiral 8D supergravity by a further compactification on K3 complex surfaces that reduce the 16 conserved supersymmetries down to 8 [26, 27, 28].
To fix the ideas, let us briefly describe the two constraints that should be obeyed by the real compact surface used in type IIA compactification:
it should preserve only 16 supercharges amongst the initial 32 existing supersymmetries living in 10D type IIA superstring. A far as duality is concerned, this implies, amongst others, that heterotic string on and type IIA string on should have the same underlying non chiral 8D supersymmetric algebra, the same types of central charges and the same extremal black attractors.
the 2-cycle homology of should lead to the same fields content one obtains in heterotic string on with some gauge invariance . This means that, on both sides, we should have the same number of supersymmetric representations (supergravity and Yang-Mills multiplets). In particular, we should have the same number of 8D gauge fields and the same number of 8D scalars that parameterize the moduli space (2) of these theories.
The presentation is as follows: In section 2, we give general features as well as constraints on non chiral 8D, supergravity arising from superstrings compactification. In section 3, we revisit the duality between heterotic string on T and type IIA on and show that exact duality between the massless spectrum of the two string theories requires, amongst others, the vanishing of the self intersection of the compact surface; i.e: . In section 4, we give three examples of non chiral 8D supergravity models; the first one aims to illustrate the general result of the paper; and the two others correspond to two extreme situations including the case where the 2-torus degenerate. In section 5, we give our conclusion.
2 Non chiral 8d supergravity
We begin by recalling that in eight space time dimensions; one has two kinds of non chiral supergravity theories: the maximal one arising from M-theory on 3-torus; it has 32 supersymmetries and a moduli space,
The non chiral 8D supergravity with a moduli space (2); it may be viewed as describing the physics at Plank energy following from two dual superstring models ; either from the 10D heterotic superstring compactified on with some invariance; or from 10D type IIA superstring on some compact surface together with D- branes wrapping cycles of . Recall that in 10D space time, the bosonic massless spectrum of these two theories is as follows
where are respectively the graviton, the NS-NS antisymmetric field and the dilaton; and where the gauge fields are given by the trace of with generating the invariance contained in or gauge symmetry. The other gauge fields are associated with the D0- and D2- branes of type IIA superstring.
2.1 Compactification to 8d
Under compactification, the resulting massless spectrum following from heterotic superstring on is familiar from the view of standard quantum field theory and is easily derived. However, the field content following from type IIA string on , and which should be the same as in the case of heterotic on T, needs a careful treatment as it implies D-branes wrapping cycles of the compact manifold. From group theoretical view, these extended objects are manifested through a generalized supersymmetric algebra going beyond the usual Haag-Lopozanski-Sohnius one. The latter has central charges carrying no Lorentz space time charges and predicts only black holes. The generalized superalgebra involves however operators with non trivial charges having interpretation in terms of p-branes; see eqs(19) for details.
2.1.1 Deriving the homology of
Focussing on non chiral 8D supergravity with 16 supercharges, one may use the duality in 6D space time between heterotic string on ; and the type IIA string on in order to study the geometric structure of the real compact and connected surfaces that are needed for the brane interpretation of the extremal BPS and non BPS black attractors in 8D. By using the adiabatic argument between the non chiral supergravities with 16 supersymmetries in 6D and 8D arising from superstring compactifications; and roughly thinking about the 4-dimensional compact manifold (complex surface) K3 as given by the local description ; one ends with the following possibility
This is a simple solution that agrees with the well known result that the compactification on the 2-sphere preserves half of the initial supersymmetries namely 16. But eq(8) let also understand that generally speaking the compact surface may be also thought of as given by the intersection of several copies of 2-spheres as given below
where s are positive integers that will be interpreted later on as Dynkin weights of affine Kac-Moody algebras. As we will see, the duality between heterotic on T and type IIA on puts a strong constraint on the intersection matrix between the 2-spheres of the reducible 2-cycle (9). It happens that should have a vanishing self intersection
with intersection matrix
Below, we first focus on the case (8) and determine the gauge fields as well as the moduli space associated with type IIA superstring on . The investigation of the general possibility (9) will be done in next section. The reason for splitting this study into two cases: and comes from the fact that the first case corresponds to a singular limit which deserves to be treated separately.
2.1.2 Type IIA string on
Starting from the spectrum (7) and computing the fields content that follow from the compactification of the 10D IIA supergravity multiplet on , we get various 8D fields and too particularly the following real 3- dimensional moduli space
Here the factor is associated with the 8D dilaton while the 2- dimensional coset group has to do with S. The two real moduli parameterizing this non compact manifold correspond to the Kahler parameter of the 2-sphere and the value of the B field on .
Actually this is a remarkable result; since heterotic/type IIA duality predicts at least four scalars; that is two Maxwell multiplets. To shed light on this property, we study here below the moduli space of the heterotic string on a regular .
2.1.3 Heterotic string on
Starting from (6) and doing the same thing by using the dual description, given by the heterotic superstring on with abelian gauge subsymmetry, we find that the moduli space of the resulting non chiral 8D, supergravity is generally given by
Here is as in eq(11) and the integer stands for the number of Maxwell gauge multiplets in the non chiral 8D supergravity that results from the compactification of the 10D SYM sector of the heterotic string; in other words it is the rank of the gauge symmetry group
Notice that for the case ; that is in the case of pure 10D supergravity on and SYM sector ignored, the above moduli space reduces to
where the factor is a 4-dimensional non compact manifold parameterized by 4 real moduli with the following interpretation:
3 of the four real scalars namely those given by the symmetric term are given by the value of the metric on ; they describe the Kahler and the complex structure of the 2-torus. For later use, we denote by R and R the radii of the two 1-cycles of T and set
, . (16)
respectively describing the Kahler and the shape (complex structure) parameters of the 2-torus.
the fourth parameter is given by the value of the field on i.e: . This modulus combine with to give the complex Kahler modulus .
Under compactification of the 10D supergravity multiplet (6) on ; we also have 4 gauge fields and that respectively follow from the metric and the NS-NS fields; two of these gauge fields; say , are involved in the non chiral 8D supergravity multiplet while the two s combine with to make two 8D Maxwell multiplets.
with internal indices taking the values .
2.2 Singular limits
By comparing the moduli spaces (11) and (15), it follows that the non chiral 8D supergravities following from the type IIA superstring on and the (gravity sector of the) heterotic superstring on are not exactly identical. As such, one concludes that they are not exactly dual
but still have:
the same gravity supermultiplet with the bosonic fields ,
the same non chiral 8D superalgebra generated by the fermionic generators and .
Recall that, in absence of central charges, the supercharges and obey the typical relation ; but in general they satisfy the following extended anticommutation relations,
where the bosonic operators are ”central charges” carrying non trivial charges and having interpretation in terms of electric and magnetic charges of p-branes. The ’s are antisymmetric products of the Pauli- Dirac - matrices in 8D.
Notice also that to get the exact duality
one has to put 2 constraint relations on the 4 real fields ( 2 complex ones and ) in order to equate the moduli spaces (11) and (15). This may be achieved either by fixing the complex structure of the 2-torus or its complexified Kahler parameter. In the last case; this corresponds to a singular limit where one (or both) of the two circles of T shrinks to zero,
and happens whenever one of the two radii is reduced to zero; i.e in eq(18). The singular case where both and is associated, on the side of type IIA compactification side, with the shrinking of down to zero. This special situation corresponds precisely to dealing with pure non chiral 8D supergravity with moduli space .
3 Heterotic/type IIA duality in 8d revisited
We start by recalling the degrees of freedom of non chiral 8D supergravity by focussing on those used in the study
of black attractors namely the abelian gauge fields and the scalars.
Then we derive the constraint relation (10) on the compact
surfaces that lead to the exact heterotic/type IIA
duality in 8D.
Restricting the investigation to bosons, it is interesting to notice that the fields of the non chiral 8D supergravity organize into two basic supermultiplets:
the gravity multiplet: it contains, in addition to the gravitino and graviphotino capturing 40+8 on shell degrees of freedom, the usual graviton one antisymmetric field, 2 graviphotons denoted as and the dilaton . The on shell degrees of freedom of these fields are respectively as follows:
N Maxwell multiplets: they contain gauge fields and real scalar fields and . The number of gauge supermultiplets depends on the gauge symmetry of the theory. In the case of non chiral 8D supergravity arising from heterotic string on regular 2-torus, this number should be as
The gauge fields and as well as the real scalar moduli have geometric and stringy meaning in the framework of superstring compactifications; but with different interpretations depending on whether they are following from heterotic string on T or from type IIA string on .
3.1 Heterotic string on T
In the case of heterotic string on T with a abelian gauge subsymmetry, the real scalars of the non chiral 8D supergravity decomposes like
and are interpreted as follows:
the real 4 scalars: they split as 3+1 having respectively a geometric and stringy origin; the three scalars describe the Kahler and complex structure of the 2-torus; the fourth is given by the B- field on T. These are precisely the complex fields moduli and encountered previously.
real scalars : they come from the gauge fields of the SYM sector of the heterotic string along the 8-th and 9-th compact directions as given below
, . (25)
Notice that is just the Cartan subgroup of the gauge symmetry or . Notice also that the directions fill the space coordinates .
3.2 Type IIA superstring on
Here we use a field theoretical method and requires the exact heterotic/type IIA duality to derive the 2-cycle homology of the compact surfaces as well as the constraint relation (10).
3.2.1 Field theory method
In non chiral 8D supergravity with a generic moduli space , the 1-form gauge fields and the scalars are as follows:
1-form gauge fields: there are gauge fields that may be written collectively like
where for later use, it is useful to think about the two graviphotons as follows
These gauge fields transform differently under the symmetry of the moduli space. We have:
, . (28)
In the type IIA picture on , the above gauge fields are associated with two kinds of D0- branes as described below:
From the second property as well as the fact that half of the 32 supersymmetries should be preserved, we learn that the compact surface should have irreducible 2-spheres as shown below.
where the positive integers are as mentioned before.
Using this 2-cycle homology, one clearly sees that the gauge fields are given by
where are complex coordinates that
parameterize the 2-sphere and is the gauge 3- form associated with D2- brane wrapping
However to determine the intersection matrix between the irreducible 2- spheres; we need an extra information that turns out to be exactly given by matching the number of the real scalar fields in the supergravity. This feature is discussed below.
matching the scalar fields: Besides the dilaton , there are real scalars transforming under the symmetry of the moduli space as follows,
Since each 2-sphere contributes with two real scalars and ; one associated with the Kahler parameters and the other given by the value of the B field , we have the following.
A complex scalar given by
and N similar others as follows:
, , (33)
where stands for the complexified Kahler form and where we have set .
From this construction, we learn that there are two real (one complex) undesired scalars since Maxwell supermultiplets need only real scalars as in eq(31). This means that the (complex) scalars should obey one complex (two real) condition. A natural constraint equation corresponds to express in terms of the others as given by the following linear relation,
that lead to the appropriate number of scalars. To get the explicit expression of the coefficients , we use a well known result [29, 30] giving a correspondence between (irreducible) 2-spheres and (simple) roots of affine Kac-Moody algebras,
In the second relation of the above eqs, which should be put in 1:1 with eq(34), the is the simple root that is
with affine node in the Dynkin diagram of affine Kac-Moody algebras, are the usual simple
roots of finite dimensional Lie algebras and the Dynkin
weights. is an imaginary root; we will ignore it
Using this correspondence, we end with an intersection matrix given by minus the generalized Cartan matrix of affine Kac-Moody algebras. More precisely, we have
With this intersection matrix and the identities and , we end with the following self intersection property .
The exact duality between heterotic string on with some gauge symmetry and type IIA string on requires:
real compact and connected surfaces type
with the properties:
have vanishing self intersection
preserve 16 of the 32 supersymmetries
surfaces classified by the affine Kac-Moody extension of the since:
the intersection matrix between 2-spheres of is given by minus the generalized Cartan matrix of affine Kac-Moody algebras
Since affine Kac-Moody algebras are classified; it follows that the ’s are given by the Dynkin diagrams of the affine Kac- Moody- Lie algebras. In the case symmetric generalized Cartan matrices , these surfaces are given by the Dynkin diagrams of the following simply laced Kac-Moody algebras
A graphic representation of examples of are depicted in figure 1 where the 2-spheres are given by the nodes of the quivers and the intersection by lines joining the nodes.
These surfaces capture also
the nature of the non abelian gauge symmetry that may live in the
8D supergravity theory. Since affine Lie algebras are built out of ordinary Lie algebras ; one can
recover the non abelian gauge invariance by switching off the vevs
Below we discuss three examples of dual models namely those involving compact and connected surfaces given by the Dynkin diagrams of , and .
4 Heterotic/type IIA dual models
In this section, we study three models using duality between type IIA string a on compact real surface and heterotic string on T with gauge symmetry. These are:
the model, which we use to illustrate the general idea,
the model since the group is one of the two largest gauge symmetries of the 10D heterotic string theory,
the model as it corresponds to a singular limit for heterotic string on T.
In this supergravity model, the compact surface is associated with affine Kac-Moody algebra having three simple roots , with . Using the correspondence (35), we then have
with intersection matrix like
The 1-form gauge fields as well as the moduli of this model are as follows
4 Maxwell gauge fields partitioned as follows:
the graviphoton associated with the D0-brane (27);
3 Maxwell gauge fields given by
These 3 fields correspond to D2-brane wrapping the three 2-spheres .
4 real scalars:
A priori there are 3 complex ( 6 real) moduli coming from the values of the complexified Kahler form on the 2-spheres of
But these field are not independent; they obey the condition
associated with Lie algebraic relation .
As such one is left with 4 real scalars parameterizing the moduli space .
This model is dual to the 10D heterotic string on T where the fluxes of the gauges fields in the SYM sector and the VEVs of the corresponding scalars completely vanish.
with the field strength 2-forms , their Hodge duals and the adjoint matter are valued in the Lie algebra or gauge symmetry with generators .
In this 8D supergravity model, the compact real surface , whose quiver is depicted in figure 2, is given by the intersection of seventeen 2-spheres
with Dynkin weights and intersection matrix given by minus the generalized Cartan matrix of the affine Kac-Moody .
This model involves 18 Maxwell gauge fields; 2 of them (graviphotons) belong to the gravity multiplet and the 16 others are valued in the Cartan subalgebra of the gauge symmetry. There are also 32 scalar fields parameterizing the moduli space .
This is a simple and special model that follows from type IIA superstring on a compact real surface given by the intersection of two 2-spheres
with intersection matrix as
The resulting non chiral 8D supergravity has, amongst others, 3 gauge fields and one complex scalar in addition to the dilaton . These fields are as follows:
3 Maxwell gauge fields: they are given by:
the graviphoton associated with the D0-brane (27);
the graviphoton associated with the D2-brane wrapping and reading as
with the RR gauge 3-form of the 10D type IIA spectrum.
the Maxwell gauge fields associated with the D2-brane wrapping ; it is given by
one free complex scalar (2 real scalars) solving the condition the constraint relation with and given by the complexified Kahler parameters
,. , (51)
Recall that the constraint relation is associated with the affine Kac-Moody algebraic relation between the simple roots namely .
This involves 2 real moduli rather than the 4 scalars that we have in the 10D heterotic string on whose moduli space is given by as shown on eq(15). It corresponds therefore to a singular configuration where the complexified Kahler structure
and the values of the the ten dimensional field along the two 1-cycles , generating the periods of the 2-torus T,
vanish identically. But the torus still has a complex structure described by a complex moduli that should be put in one to one correspondence with the of (51).
Motivated by the study of black attractors in non chiral 8D supergravity with gauge 2-form , Maxwell gauge fields and real scalars parameterizing the moduli space
and using an explicit field theoretic method, we have studied in this paper two main things: First, we have investigated the exact duality between the heterotic string on a regular , with abelian symmetry contained in some rank r gauge group , and the type IIA string on a connected compact real surface . Because of the fact that should preserve 16 supersymmetries; that is the half of the 32 existing supercharges in the 10D type IIA superstring and because of heterotic/type IIA duality in 8D, we have shown that connected and compact should have a vanishing self intersection,
that is having the same homology as the 2-torus used in the
heterotic side. Moreover using known results on geometric
engineering of QFTs embedded in string theory, we have shown that
2-cycle homology of the surface is given by
intersecting 2-spheres with intersection
matrices classified by the generalized Cartan matrices of the affine Kac-Moody Lie
Second, we have studied some special dual models; in particular the non chiral 8D supergravity theory having 3 Maxwell gauge fields and 1+2 real scalar parameterizing the moduli space
This real 3- dimensional manifold does not belong to the well known family with . We have shown that, from heterotic side, this supergravity model follows from compactification on a singular torus with vanishing complexified Kahler structure. In the type IIA picture, the model fo