Imperial/TP/2009/JG/03

DESY 09-067

Solutions of type IIB and D=11 supergravity

with Schrödinger symmetry

Aristomenis Donos and Jerome P. Gauntlett

DESY Theory Group, DESY Hamburg

Notkestrasse 85, D 22603 Hamburg, Germany

*[.6cm] Theoretical Physics Group, Blackett Laboratory,

Imperial College, London SW7 2AZ, U.K.

*[.6cm] The Institute for Mathematical Sciences,

Imperial College, London SW7 2PE, U.K.

*[.6cm]

Abstract

We construct families of supersymmetric solutions of type IIB and supergravity that are invariant under the non-relativistic Schrödinger algebra for various values of the dynamical exponent . The new solutions are based on five- and seven-dimensional Sasaki-Einstein manifolds, respectively, and include supersymmetric solutions with .

## 1 Introduction

An interesting development in string/M-theory is the possibility of using holographic ideas to study condensed matter systems. Starting with [1, 2], one focus has been on non-relativistic systems with Schrödinger symmetry, a non-relativistic version of conformal symmetry. The corresponding Schrödinger algebra is generated by Galilean transformations, an anisotropic scaling of space () and time () coordinates given by , , and an additional special conformal transformation. More generally, one can consider systems invariant under what we shall call Schrödinger (or Sch) symmetry, where one maintains the Galilean transformations, but allows for other scalings, , , with the “dynamical exponent”, and, in general, sacrifices the special conformal transformations. In this notation the Schrödinger algebra is Sch. The full set of commutation relations for are written down in e.g. [2].

Various solutions of type IIB supergravity and supergravity have been constructed
that are invariant under Sch symmetry, for different values of .
The type IIB solutions of [3]-[8]
can be viewed as deformations of the supersymmetric solutions, where is a five-dimensional Sasaki-Einstein space,
and should be
holographically dual to non-relativistic systems with two spatial dimensions. Similarly, there
are deformations of the solutions^{1}^{1}1Note that deformations of solutions
of supergravity were studied in [9]. of supergravity,
where is a seven-dimensional Sasaki-Einstein space,
that are invariant under Sch and these should be dual to non-relativistic systems with a single spatial dimension
[7, 8].

The type IIB solutions constructed in [3, 4, 5] with , and hence invariant under the larger Schrödinger algebra, are based on a deformation in the three-form flux and do not preserve any supersymmetry [4]. In [6] supersymmetric solutions of type IIB with various values of were constructed which are based on a metric deformation and include supersymmetric solutions with . However, it was argued that these supersymmetric solutions are unstable. On the other hand it was shown that the instability can be removed by also switching on the three-form flux deformation, which then breaks supersymmetry. In a more recent development a rich class of supersymmetric solutions of both type IIB and supergravity were constructed in [8] which have various values of and , respectively (particular examples of the and solutions were first constructed in [4] and [7], respectively).

In this short note, we generalise the constructions in [8] for both type IIB and supergravity, finding new classes of supersymmetric solutions with various values of including .

Note Added: In the process of writing this paper up, we became aware of [10], which also constructs some of the supersymmetric solutions of type IIB supergravity that we present in section 2.

## 2 Solutions of type IIB supergravity

Consider the general ansatz for the bosonic fields of type IIB supergravity given by

(2.1) |

where is the complex three-form and the axion and dilaton are set to zero. Here , are functions, is a one-form and is a complex two-form all defined on the Calabi-Yau three-fold, . Our conventions for type IIB supergravity [11, 12] are as in [13]. One finds that all the equations of motion are satisfied provided that

(2.2) |

where with indices raised with respect to the metric. Observe that when we have the standard D3-brane class of solutions with a transverse space.

If we choose the two-form to be primitive and have no pieces (i.e. just
and/or components),
on then
the solutions generically preserve 2 supersymmetries^{2}^{2}2We note that one can
add a closed, primitive -form on to the three-form while still preserving the same amount of
supersymmetry. This changes two of the equations to
and .
Such solutions will not, in general, admit a scaling symmetry, so we shall not consider them further here, however we note
that solutions with and Galilean symmetry were presented in [14].,
which is enhanced to 4 supersymmetries if the is flat.
More specifically, we introduce the orthonormal frame
etc. and
choose positive orientation to be given by , where is the volume element on .
Consider first the special case that . Then, as usual,
a generic breaks 1/4 of the supersymmetry, while the harmonic function leads
to a further breaking of 1/2, the Killing spinors satisfying the additional projection .
Switching on we find that generically
we need to also impose and .

We now specialise to the case that the is a metric cone over a five-dimensional Sasaki-Einstein manifold , . In order to get solutions with Sch symmetry we now set

(2.3) |

where is a function, and are, respectively, a real and a complex one-form on , and are constants which we will take to be positive. The full solution now reads

(2.4) |

Generically, when , solutions with will be Sch invariant. In particular, the scaling acts on the coordinates via (for other transformations see [2]). Observe that if we set then we have the standard solution of type IIB. Generically, when , we still need to impose the projections mentioned above in order to preserve supersymmetry. Note in particular that, generically, half of the Poincaré supersymmetries of the solution are preserved, while none of the special conformal supersymmetries are. It would be interesting to explore special subclasses of solutions with enhanced supersymmetry, which occur, for example, when the is flat.

In [6], supersymmetric solutions with , were constructed with

(2.5) |

and give rise to solutions with , with the bound only achievable for . In particular supersymmetric solutions with were found, but, because the solutions have the metric component positive in some regions of the , the solutions were argued to be unstable. In [8], supersymmetric solutions with , were constructed with

(2.6) |

where is the Hodge-deRahm operator on , and give rise to solutions with , with the bound achievable for any space. More specifically, the bound is achieved when is a one-form dual to a Killing vector on the space; the class of such solutions using the one-form dual to the Reeb vector on the space were first constructed in [4]. It was also shown in [8] that one can combine these classes of solutions with (still with ), and providing that one can solve for , so that then the solutions have dynamical exponent .

We now consider . This implies that and we need to set . In addition to (2.6) we also need to solve

(2.7) |

The solutions for which are invariant under Schwith . If then since , necessarily we have .

If we set , which is needed to obtain supersymmetric solutions with for example, then we just need to solve (2). The first equation implies that , with the bound being saturated when is a one-form dual to a Killing vector on the space. A simple solution is obtained by taking for some constant , where is the canonical one-form dual to the Reeb vector on and . This solution has and was first constructed in [3, 4, 5]. Observe that for this solution . Thus while is it is not primitive and so this solution does not preserve any supersymmetry as previously pointed out in [4]. On the other hand it is straightforward to construct solutions with that are supersymmetric. For example, we can take any Killing vector on the space that leaves invariant the Killing spinors on . It is straightforward to construct such solutions explicitly when the metric for the is known explicitly as it is for the , [15], [16] and [17] spaces. For the case of it is also easy to construct explicit solutions for all values of using spherical harmonics. It is worth noting that the solutions for the case can have constant and negative and hence do not suffer from the instability discussed in [6]. This is easy to see since must be a constant linear combination of the 15 harmonic two-forms on , , or, if we demand supersymmetry, of the eight primitive forms and three forms. Then, in general, will be the sum of a negative constant with a scalar harmonic on with eigenvalue 12. It would be interesting to investigate the issue of stability further for all of the new solutions we have constructed. Some additional comments about the solutions are presented in appendix A.

## 3 Solutions of supergravity

We consider the ansatz for the bosonic fields of supergravity given by

(3.1) |

where , are functions, is a one-form and is a three-form all defined on^{3}^{3}3It is straightforward to also consider other
eight-dimensional special holonomy manifolds, but for simplicity we shall restrict our attention to . the
Calabi-Yau four-fold, . Our conventions for supergravity
[18] are as in [19].
One finds that all the equations of motion are satisfied provided that

(3.2) |

where with indices raised with respect to the metric. When we have the standard M2-brane class of solutions with a transverse space.

If we choose the three-form to only have plus pieces and be primitive on the
then the solutions generically preserve
2 supersymmetries^{4}^{4}4As an aside, we note that we can also
add a closed, primitive -form on to the four-form flux while still preserving the same amount of
supersymmetry. This changes two of the equations to and .,
which is enhanced to
4 supersymmetries if the us flat.
More specifically, we introduce the orthonormal frame
etc. and
choose positive orientation to be given by , where is the volume element on .
Consider first the special case that . Then, as usual,
a non-flat breaks 1/8 of the supersymmetry, and the harmonic function can be added “for free”
(the projection on the Killing spinors arising from the automatically imply the projection ).
Switching on we find that generically we need to also impose and .
Note as an aside that we can “skew-whiff” by changing the sign of the four-form flux and
obtain solutions that generically don’t preserve any supersymmetry (apart from the special case when ).

We now specialise to the case that the is a metric cone over a seven-dimensional Sasaki-Einstein manifold , . In order to get solutions with Sch symmetry we now set

(3.3) |

where is a function, and are, respectively, a one-form and a two-form on , and are constants which we will take to be positive. The full solution now reads

Generically, when , solutions with will be Sch invariant. In particular, the scaling now acts as . Note that if we set then we have the standard solution. Generically, when , we still need to impose the projections mentioned above in order to preserve supersymmetry. Thus, generically, half of the Poincaré supersymmetries of the solution are preserved, while none of the special conformal supersymmetries are. It would be interesting to explore special subclasses of solutions with enhanced supersymmetry, which occur, for example, when the is flat.

In [8], supersymmetric solutions with , were constructed with

(3.5) |

and give rise to solutions with , with the bound only achievable for . In particular supersymmetric solutions with were found, but they suffer from a similar instability to that found for the analogous type IIB solutions in [6]. In [8], supersymmetric solutions with , were constructed with

(3.6) |

and give rise to solutions with , with the bound achievable for any space. More specifically, the bound is achieved when is a one-form dual to a Killing vector on the space; and one can always choose the one-form dual to the Reeb vector on the space. It was also shown in [8] that one can combine these classes of solutions with , (still with ), and providing that one can choose then they have dynamical exponent again with .

We now consider . This implies and we need to set . In addition to (2.6) we also need to solve

(3.7) |

The solutions for which are invariant under Sch with . If then necessarily we have and hence .

If we set then we just need to solve (3). Let us illustrate with some simple solutions when . In fact it is easiest to directly solve (3). For example, if we let be standard complex coordinates on , with Kähler form we can take , where is constant, which obviously has only and pieces and is primitive, and (setting a possible solution of the homogeneous equation in (3) to zero). This gives a supersymmetric solution with and hence . In particular we note that the metric component is always negative. A simple solution with is obtained by splitting and considering a sum of terms which are and primitive on one factor with a factor on the other:

Solving for (and setting to zero a solution of the homogeneous equation in (3)) we get

For this solution, the metric component is again always negative. Clearly there are many additional simple constructions for the case that could be explored as well as for the more general class of other explicit metrics.

### Acknowledgements

JPG is supported by an EPSRC Senior Fellowship and a Royal Society Wolfson Award.

## Appendix A Comments on solving (2)

Here we make a few further comments concerning solving (2) (which also have obvious analogues for solving (3)). To solve (2), we first solve the first line for and then substitute into the second. It is illuminating to expand out the source term in the right hand side of the equation in the second line using a complete set of scalar harmonics on the space:

(A.1) |

where , corresponding to the harmonic function on the cone. We then find

(A.2) |

In this expression we have allowed for the possibility of an arbitrary solution to the homogeneous equation, , assuming it exists. The point is that the relevant putative eigenvalue for is fixed by the eigenspectrum of the Laplacian acting on one-forms. For the special case when , for example, there is always such a possibility of adding a solution to the homogeneous equation. Another point to notice about (A.2) is that it only makes sense providing that the coefficient whenever .

For the special case when , not only is this coefficient zero but the sum appearing in (A.2) is a finite sum terminating at . To see this we observe that

(A.3) |

which can be recast as an integral on the flat cone

(A.4) |

where for , and

(A.5) |

with defining the scalar harmonics on . To proceed we write as

(A.6) |

where define the vector spherical harmonics on . In carrying out the integral (A.4) we will get all possible contractions of the indices of the scalar spherical harmonic with some of the indices

(A.7) |

In particular, since the tensor defining the scalar harmonic is traceless, we conclude that the are zero for all with .

Let us now consider this issue for a general space, but in the special case when is a one-form dual to a Killing vector on corresponding to and hence . As above, we have (A.4). Write

(A.8) |

and observe that on the cone and that . We then compute

(A.9) | |||||

In getting to the last line one needs to take into account the factor in the measure and use

(A.10) |

We thus conclude from (A.9) that the problematic coefficient in (A.2) when again vanishes for this class of solutions.

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