Supersymmetric heterotic solutions via non- standard embedding
A supersymmetric solution to type II supergravity is constructed by superposing two hyper-Kählers with torsion metrics. The solution is given by a Kähler with torsion metric with holonomy. The metric is embedded into a heterotic solution obeying the Strominger system, together with a Yang–Mills instanton obtained by the standard embedding. T dualities lead to an instanton describing a symmetry breaking from to . The compactification by taking a periodic array yields a supersymmetric domain wall solution of heterotic supergravity.
The Green–Schwarz mechanism GSmechanism () is one of the cornerstones of superstring theory. Its role is twofold: First, of course, is to tell us how to cancel the gauge and gravitational anomalies of ten-dimensional type I and heterotic superstrings, which were apparently considered anomalous and hence unacceptable as consistent theories. With the mechanism, however, it turned out that all the anomalies were canceled out in a miraculous manner if and only if the gauge group was or , for the latter of which heterotic string theory has been constructed heterotic_string ().
The second important role of the Green–Schwarz mechanism is to constrain the background geometry through the modified Bianchi identity of the 3-form field ; the mechanism requires the 2-form field to vary under both the gauge and local Lorentz transformations so that the invariant 3-form field must be of the form
where is the Chern–Simons 3-form associated with the Yang–Mills connection, and is also a Chern–Simons 3-form but made of a particular linear combination of the Levi-Civit connection and the 3-form field:
The equation (1) leads to the Bianchi identity
Note that the combination (2) is different from the one that appears in the supersymmetry(SUSY) variation of the gravitino
where is the covariant derivative associated with the combination
For heterotic string theory on a six-dimensional space without fluxes, the Killing spinor equation arising from the vanishing gravitino variation (4) constrains to have holonomy, that is, to be Calabi–Yau. On the other hand, for the Bianchi identity (3) to be satisfied, the easiest and most common way is to set the connection, which is nothing but the spin (Levi-Civit) connection for , to be equal to a part of the gauge connection. This is called the standard embedding Witten_New_Issues (). In this case, a part of the gauge field background is required to be , and the gauge symmetry is partially broken to the centralizer . This reduction of the gauge symmetry is one of the hallmarks of Calabi–Yau compactifications of heterotic string theory.
If, on the other hand, there is a nonzero field, then the vanishing gravitino variation (4) asserts that the linear combination belongs to but says nothing about the other linear combination Strominger (); BdR (); KimuraYi (). Thus is generically in on the six-dimensional space , and the gauge symmetry is broken to a smaller subgroup , which is more favorable from the point of view of applications to string phenomenology. Note that, in the presence of fluxes, is achieved by the “standard embedding”, that is, by simply equating the modified spin connection with a part of the gauge connection. This is in striking contrast to the Calabi–Yau case, in which one needs the nonstandard embedding that requires complicated mathematical machinery Witten_New_Issues (); GSW () involving the construction of stable holomorphic vector bundles.
However, for the smeared intersecting NS5-brane solution, which is obtained as a superposition of two smeared symmetric 5-brane solutions CHS () and is one of the simplest SUSY heterotic supergravity solutions with fluxes in the six-dimensional space, not only but also happens to be in , and therefore the unbroken gauge symmetry is still . The reason for this can be traced back to the parity invariance of the symmetric 5-brane solution; indeed, the sign of is a matter of convention, and the configuration after the sign flip still remains a solution of the heterotic supergravity.
In this paper, we construct a supersymmetric heterotic supergravity solution such that is in (and hence a SUSY solution) but is not , by superposing two hyper-Kählers with torsion (HKT) geometries. As already pointed out in Ref. CHS (), one can obtain HKT geometries by conformally transforming hyper-Kähler geometries. We choose the Gibbons–Hawking space as the starting point and apply a conformal transformation to obtain a HKT geometry. Since the Gibbons–Hawking space is not parity invariant, the connection of the resulting HKT space is in but not in , though still belongs to .
We then smear the harmonic functions to those of two dimensions and take a superposition of two such geometries. Because of our superposition ansatz, we are forced to set some of the entries of the metric to zero in order to satisfy the equations of motion. Consequently, we find that the holonomy of the superposed solution remains to be . We also show that by T duality this solution turns into one with or holonomy.
We also take a two-dimensional periodic array of the “intersecting HKT” solutions to get a compact six-dimensional solution. We find that the fundamental parallelogram of the two-dimensional periodic array is separated into distinct smooth regions bordered by codimension-1 singularity hypersurfaces, hence the name “supersymmetric domain wall.” This novel solution has some interesting properties, as we will see below.
This paper is organized as follows. In Sec. II, we give a brief review of HKT geometries obtained by conformal transformations acting on four-dimensional hyper-Kähler spaces. In Sec. III, we consider a superposition of HKT spaces to construct a six-dimensional Kähler with torsion (KT) space with special properties which serves as a supersymmetric solution of type II supergravity. In Sec. IV, we embed this geometry into heterotic supergravity theory and take T dualities. In Sec. V, we compactify this six-dimensional space by taking a periodic array and study some of its properties. The final section presents the summary and conclusion.
Ii HKT geometry as a conformal transform
We start with a four-dimensional HKT metric obtained as a conformal transform of a hyper-Kähler metric, where for the latter we specifically consider the Gibbons–Hawking (GH) metric ,
The GH metric is given by GH ()
where and are scalar functions of the coordinates of obeying the relation
is a scalar field of which the properties will be described shortly. We define the orthonormal basis
so that the hypercomplex structure is given by the three complex structures satisfying the quaternionic identities,
where are the ’t Hooft matrices. The corresponding fundamental 2-forms are
In the present case, we have
in terms of dual vector fields to the 1-forms (9),
The exterior derivative is calculated as
with the vector fields . Therefore, if is chosen to be a harmonic function with respect to the GH metric (7), then the torsion becomes a closed 3-form.
Using this , we introduce the two types of connections ,
where is a Levi-Civitá connection. The corresponding connection 1-forms are defined by
and the curvature 2-forms are written as
The torsion curvature satisfies the holonomy condition
On the other hand, if the torsion is a closed 3-form, that is, is a harmonic function, then the curvature becomes an anti self dual 2-form, which may be regarded as a Yang–Mills instanton with the gauge group .
Iii Intersecting HKT metrics
In the previous section we have seen that the HKT metrics obtained by a conformal transformation have in but in strictly larger than as long as the original GH space is not a flat Euclidean space. In this section we construct their six-dimensional analogs by superposing two such HKT metrics embedded in different four-dimensional subspaces. This construction is motivated by that used in constructing intersecting brane solutions AEH (); O ()444The term “intersecting” in the (commonly used) name is misleading since they are smeared and hence do not have intersections with larger codimensions. See, e.g., Ref. MR () for recent developments in constructing localized intersecting brane solutions in supergravity.; namely, we assume the form of the metric as
The HKT metric that we have considered in the previous section is characterized by a triplet on obeying (8). So at first it might seem that or could to be functions of or , and or could be replaced with a more general form or , respectively. However, it turns out that such a more general ansatz does not lead to a metric with holonomy even in the case . Thus we are led to consider the metric of the form (III), assuming the following:
and are harmonic functions on the two-dimensional flat space .
and , of which the components are harmonic functions on satisfying the Cauchy–Riemann conditions
Under these assumptions, we will show that a six-dimensional space with the metric (III) has the following KT structure:
We first introduce an orthonormal basis
The space has a natural complex structure defined by
Indeed, it is easy to see that the Nijenhuis tensor associated with vanishes under the condition and . Then, the metric (III) becomes Hermitian with respect to the complex structure , and the fundamental 2-form takes the form
The Bismut torsion is uniquely determined by
Explicitly we have
It should be noticed that in our case the Bismut torsion is a closed 3-form, . We shall refer to and as the Bismut connection and Hull connection, respectively, according to Ref. MS (). The Lee form is a 1-form defined by IP (), which becomes a closed 1-form,
We will identify the Bismut torsion with 3-form flux, , and the function with a dilaton. It is shown that the Ricci form IP () of the Bismut connection vanishes, which is equivalent to the condition
so that the holonomy of is contained in and admits two independent Weyl Killing spinors obeying in type II theory. Thus the triplet gives rise to a supersymmetric solution to the type II supergravity theory.
Iv Embedding into heterotic string theory and T duality
We study supersymmetric solutions describing heterotic flux compactification. The bosonic part of the string frame action, up to the first order in the ’ expansion, is given by
It is assumed that ten-dimensional spacetimes take the form , where is a six-dimensional space admitting a Killing spinor ,
This system together with the anomaly cancellation condition
is known as the Strominger system Strominger ().
Now, we turn to the heterotic solution obeying the Strominger system. If the curvature in the anomaly condition (32) is given by the Hull connection , we can choose a non-Abelian gauge field as since the 3-form flux (III) is closed by the identification . This is a form of the usual standard embedding. Combining the well-known identity
Apparently, seems to take values in , which would describe a symmetry breaking from to . However, for generic choices of the harmonic functions , , , and , it is not ensured that the metric (III) can remain non-negative, and the dilaton (28) can remain real valued. Therefore, to get a meaningful solution we are forced to impose
With this condition, the holonomy of remains , but the instanton reduces to a proper Lie subalgebra of , and the centralizer is .
The metric (35) has isometries generated by Killing vector fields . Therefore, we can T dualize the type II solution along directions of these isometries. It is easy to see that the solution is inert under the T duality along and ; the T dualities along the remaining directions give nontrivial deformations of the solutions, preserving one-quarter of supersymmetries.555See, e.g., Ref. P () for the classification of supersymmetric solutions to heterotic supergravity.
We first T dualize the solution along . The resulting solution is given by
Here, the orthonormal basis is defined by
Then, we have a deformed complex structure ,
with and . The associated fundamental two-form takes the same form as (25), and the Bismut connection has an holonomy. In this case, it turns out that the Hull connection is in , which is still smaller than .
Thus, we further T dualize the solution once more along and finally obtain :
The orthonormal basis is defined by
In this basis the complex structure is given by
with and .
It can be verified that this solution has an Bismut connection and
Hull connection as desired.
V SUSY domain wall metric
The last topic concerns the construction of type II/heterotic supersymmetric solutions on a compact six-dimensional space with the Hull connection not being in . Since the triples obtained in the previous section depend only on and , we can compactify the , , and spaces on by simply identifying periodically, whereas we consider a periodic array of copies of the solution along the and directions.
Let us consider a periodic array of [Eqs. (35), (36), and (37)], [(38), (39), and (40)], or [(IV), (IV), and (45)], which are characterized by a pair of harmonic functions and . In two dimensions both the real and imaginary parts of any holomorphic function are harmonic. Thus we can take to be, say, the real part of any doubly periodic, holomorphic function. In this case, may be taken to be the imaginary part of the same doubly periodic function.
Since the only nonsingular holomorphic function on is a constant function, we need to allow some pole singularities in the fundamental parallelogram of the periodic array, which may be seen to be in accordance with the no-go theorems against smooth flux compactifications GKP (); KimuraYi (). The doubly periodic meromorphic functions are known as elliptic functions. It is well known that, for a given periodicity, the field of elliptic functions is generated by Weierstrass’s function and its derivative . In the following, we consider, as a typical example, the compactification of , , and on a square torus of side by taking
where is of modulus or and . Our solutions are determined entirely by Weierstrass’s function without any reference to because of the choice that causes the rhs of (32) to be closed. Note that they solve the heterotic equations of motion up to .
The real and imaginary parts of are shown in Fig. 1. We see that may take negative as well as positive values, but note that the metric (35), (38), or (38) depends on through as we designed, so the solution is only singular where vanishes (as well as diverges). Also, negative causes no problem as long as is nonzero.
For any case of , , or , some of the components of the metric vanish where , and hence the solution is singular. Also, the “string coupling” (= exponential of the dilaton) vanishes there. The curves are shown in Fig. 2 for the cases and . For both cases, we see that the fundamental parallelogram (shown by the shaded region) is separated into two distinct smooth regions bordered by the codimension-1 singularity hypersurfaces. The two singularity hypersurfaces intersect at , where the function has a unique double pole; its real and imaginary parts rapidly fluctuate at . More details about the solution will be reported elsewhere.
In this paper we have shown that two HKT metrics given by and can be superposed and lifted to a six-dimensional smeared intersecting solution of type II supergravity if the functions , and are restricted to harmonic functions on the two-dimensional flat space , together with and satisfying the Cauchy–Riemann conditions. The simplest geometry that we have considered has an connection that leads to the unbroken gauge symmetry if it is embedded to heterotic string theory as an internal space. By T-duality transformations we have obtained one having an or holonomy. We have also compactified this six-dimensional KT space by taking a periodic array to find a supersymmetric domain wall solution of heterotic supergravity in which the fundamental parallelogram of the two-dimensional periodic array is separated into distinct smooth regions bordered by codimension-1 singularity hypersurfaces. It would be interesting to solve the gaugino Dirac equation on this background and compare the spectrum with the corresponding -type supersymmetric nonlinear sigma model IrieYasui (), similarly to what has been done in the case MY ().
Y.Y. is supported by the Grant-in-Aid for Scientific Research Grant No. 23540317, and S.M. is supported by Grant No. 25400285 from The Ministry of Education, Culture, Sports, Science and Technology of Japan.
- (1) M. B. Green and J. H. Schwarz, Phys. Lett. B 149. 177 (1984).
- (2) D. J. Gross, J. A. Harvey, E. Martinec and R. Rohm, Phys. Rev. Lett. 54, 502 (1985); Nucl. Phys. B256, 253 (1985); Nucl. Phys. B267, 75 (1986).
- (3) M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology, Cambridge Monographs on Mathematical Physics (Cambridge University Press, New York, 2012).
- (4) E. A. Bergshoeff and M. de Roo, Nucl. Phys. B328, 439 (1989).
- (5) T. Kimura and P. Yi, J. High Energy Phys. 07 (2006) 030.
- (6) D. Martelli and J. Sparks, Adv.Theor.Math.Phys. 15, 131 (2011).
- (7) E. Witten, Nucl. Phys. B268, 79 (1986).
- (8) A. Strominger, Nucl. Phys. B274, 253 (1986).
- (9) C. G. Callan, J. A. Harvey and A. Strominger, Nucl. Phys. B359, 611 (1991); Nucl. Phys. B367, 60 (1991).
- (10) G. W. Gibbons and S. W. Hawking, Phys. Lett. 78B, 430 (1978).
- (11) P. S. Howe and G. Papadopoulos, Phys. Lett. B379, 80 (1996).
- (12) G. Grantcharov and Y. S. Poon, Commun. Math. Phys. 213, 19 (2000).
- (13) A. Argurio, F. Englert and L. Houart, Phys. Lett. B 398, 61 (1997).
- (14) N. Ohta, Phys. Lett. B 403 (1997) 218 [arXiv:hep-th/9702164].
- (15) J. McOrist and A. B. Royston, Nucl. Phys. B849 573 (2011).
- (16) S. Ivanov and G.Papadopoulos, Classical Quantum Gravity 18, 1089 (2001).
- (17) G. Papadopoulos, Classical Quantum Gravity 27, 125008 (2010).
- (18) S. B. Giddings, S. Kachru and J. Polchinski, Phys. Rev. D 66, 106006 (2002).
- (19) S. Irie and Y. Yasui, Z. Phys. C 29, (1985) 123.
- (20) S. Mizoguchi and M. Yata, Prog. Theor. Exp. Phys. 2013, 53B01 (2013); T. Kimura and S. Mizoguchi, Classical Quantum Gravity 27, 185023 (2010); J, High Energy Phys. 04 (2010) 028.