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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 947 (2019) 114733

www.elsevier.com/locate/nuclphysb

D-branes and non-Abelian T-duality

Robin Terrisse

, Dimitrios Tsimpis

a,∗

, Catherine A. Whiting

b,c

Institut de Physique Nucléaire de Lyon, Université de Lyon, UCBL, UMR 5822, CNRS/IN2P3, 4 rue Enrico Fermi,

69622 Villeurbanne Cedex, France

National Institute for Theoretical Physics, School of Physics and Mandelstam Institute for Theoretical Physics,

University of the Witwatersrand, Johannesburg, WITS 2050, South Africa

Department of Physics and Astronomy, Bates College, Lewiston, ME 04240, USA

Received 7

February 2019; received in revised form 24 July 2019; accepted 12 August 2019

Available

online 19 August 2019

Editor: Stephan

Stieberger

Abstract

study the effect of non-Abelian T-duality (NATD) on D-brane solutions of type II supergravity.

Knowledge of the full brane solution allows us to track the brane charges and the corresponding brane

conﬁgurations, thus providing justiﬁcation for brane setups previously proposed in the literature and for the

common lore that Dp brane solutions give rise to D(p+1)-D(p+3)-NS5 backgrounds under SU(2) NATD

transverse to the brane. In brane solutions where spacetime is empty and ﬂat at spatial inﬁnity before NATD,

the spatial inﬁnity of the NATD is universal, i.e. independent of the initial brane conﬁguration. Furthermore,

it gives enough information to determine the ranges of all coordinates after NATD. In the more complicated

examples of the D2 branes considered here, where spacetime is not asymptotically ﬂat before NATD, the

interpretation of the dual solutions remains unclear. In the case of supersymmetric D2 branes arising from

M2 reductions to IIA on Sasaki-Einstein seven-manifolds, we explicitly verify that the solution obeys the

appropriate generalized spinor equations for a supersymmetric domain wall in four dimensions. We also

investigate the existence of supersymmetric mass-deformed D2 brane solutions.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

Corresponding author.

E-mail

addresses: rterrisse@ipnl.in2p3.fr (R. Terrisse), tsimpis@ipnl.in2p3.fr (D. Tsimpis), cwhiting@bates.edu

(C.A. Whiting).

https://doi.org/10.1016/j.nuclphysb.2019.114733

The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

2 R. Terrisse et al. / Nuclear Physics B 947 (2019) 114733

1. Introduction

Non-Abelian T-duality (NATD) has recently attracted renewed interest, not least as a tool for

generating new supergravity solutions. A partial list of related literature includes [1–13]. One

problem in this context is that NATD only carries local information: even when the starting

point (the “seed”) is a globally well-deﬁned solution, NA

TD will typically generate a highly

complicated local solution whose global completion, if it exists, is completely obscured. The

main motivation of the present paper is to consider, as seeds of the NATD, full-ﬂedged brane

solutions – as opposed to AdS near-horizon solutions, which has been the case before. The point

is that, as we will see, being able to follo

w the interpolation between the near-horizon and spatial

inﬁnity limit, gives us a better handle on the brane conﬁguration and the global properties of the

dual.

Before NA

TD, standard intersecting brane solutions (i.e. those following the simple harmonic

superposition rule) often interpolate between two asymptotic regions, each of which is an inde-

pendent supergravity solution in its own right: ﬂat space at spatial inﬁnity, and the near-horizon

limit – which in the examples considered here always contains an AdS f

actor. In fact the brane

is not strictly-speaking present in these solutions: the brane backreacts on the ﬂat space in which

it is initially inserted and dissolves into ﬂux, so that the resulting solution is without sources.

the interpolating and near-horizon solutions there is still a remnant of the brane, whose charge

can be computed by integrating the ﬂux.

The NA

TD of the spatial inﬁnity limit of a standard intersecting brane solution is then univer-

sal, i.e. it is the same for all standard intersecting brane solutions: it is simply the NA TD of ﬂat

space. As we will see in detail in section 2.2.2, in this case an SU(2) NATD generates a distrib

tion, whose density can be determined explicitly, of parallel NS5 branes continuously distributed

along a half line. However after NATD the notion of spatial inﬁnity limit and near-horizon limit

are no longer necessarily meaningful. Still, as previously stated, both limits are genuine solu-

tions so that NA TD will generate ne

w solutions out of them. These will be called respectively

the spatial inﬁnity and near-horizon limit of the dual. For the case of the D3 brane, discussed in

section 2, this deﬁnition leads to the commutative diagram of Fig. 1.

Furthermore, all e

xamples considered here will be seen to be consistent with the common

lore that Dp brane solutions give rise to D(p+1)-D(p+3)-NS5 backgrounds under SU(2) NATD

transverse to the brane [14].

More generally, for nonstandard brane solutions, such as the D2 branes considered here in

sections 4, 5, 6, the geometry may not asymptote to ﬂat space even before NATD. In that case the

original solution and its dual will in general contain non-vanishing ﬂuxes even at spatial inﬁnity.

Nevertheless, by zooming in near the locus of the NS5, one can still see the presence of a contin-

uous distrib

ution of NS5 branes in accordance with the harmonic superposition prescription for

intersecting branes.

Having the full interpolating solution can f

acilitate reading off the possible global completions

of the geometry after NA TD. In particular the topology of the slices r = constant can be studied

more easily by taking the spatial inﬁnity limit r →∞, where the various expressions simplify.

Nevertheless the harmonic superposition rule allows us to trace the original source (i.e. before backreaction) as a

delta function deﬁned in ﬂat space.

Some early examples where this has been hinted at through speciﬁc examples are in [8,10], but it was proposed

generally in [14].

4 R. Terrisse et al. / Nuclear Physics B 947 (2019) 114733

existence of supersymmetric mass-deformed D2 brane solutions we have ruled out a large class

of Ansätze.

The outline of the paper is as follo

ws. We revisit the case of the D3 brane in section 2. Various

D2 brane solutions are examined in sections 4, 5, 6. These are obtained by reduction from eleven

dimensional solutions of M2 branes transverse to cones over S

or Y

p,q

reviewed in section 3.

In section 7 we examine massive deformations of the supersymmetric massless IIA D2 brane

solutions of section 4. We start by casting the supersymmetry equations of the branes in the

formalism of generalized geometry in section 7.1. Massive deformations of the resulting pure

spinor equations are examined in section 7.2. We conclude in section 8. In the appendix we

xplain our various conventions and compare them to the literature.

2. D3 brane

The metric describing a stack of parallel D3 branes is given by,

=H(r)

−1/2

1,3

) +H(r)

1/2

[dr

)] , (1)

where H(r) =(1 +

). The S

is parameterized as follows,

) =dα

+sin

αdθ

cos

α(σ

+σ

), (2)

where α ∈[0,

], θ ∈[0, 2π], and σ

are left-invariant SU(2) Maurer Cartan one-forms given

by,

=−sin ψ

dθ

+cosψ

sin θ

dφ

=cos ψ

dθ

+sinψ

sin θ

dφ

=cos θ

dφ

+dψ

, (3)

with ranges ψ

∈[0, 4π ], θ

∈[0, π ], φ

∈[0, 2π ]. This background is supported by a constant

dilaton and an F

ﬂux given by,

=(1 +)dx

∧dx

∧dH(r)

−1

=dx

∧dx

∧dH(r)

−1

−4L

d

. (4)

Upon quantization of the ﬁve-form ﬂux, one obtains the well-known relation between the con-

stant L in the harmonic function and the number N

of D3 branes: L

=4πα

2

The D3 branes lie along the R

1,3

directions. This can be seen in a probe approach. Consider

the same expression as (4)for a ﬂux living now in R

1,9

. The coordinates now refer to the metric

(1), but with H = 1. Since in spherical coordinates the S

collapses at r = 0, its volume form

d

is ill-deﬁned. However F

is a well-deﬁned current (i.e. a distribution-valued form) and we

can compute:

=d F

=4L

δ(r)dr ∧d

(5)

This means that a brane is inserted in r = 0. In this coordinate system this is a codimension 6

space, and thus a D3 lying along R

1,3

. In the transverse space R

the brane looks like a point.

The D3 no

w acts as a source for the ﬂux F

, which backreacts on the metric through Ein-

stein’s equations to give (1). The global geometry has changed, and the S

no longer collapses.

The supergravity equations are solved without sources and the brane cannot be seen anymore.

Nevertheless the information about the brane is still present in the charge carried by the ﬂux.

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