Semichiral Sigma models with 4D hyperkähler geometry

Abstract

Semichiral sigma models with a four-dimensional target space do not support extended N = (4, 4) supersymmetries off-shell [ 1 , 2 ]. We contribute towards the understanding of the non-manifest on-shell transformations in (2, 2) superspace by analyzing the extended on-shell supersymmetry of such models and find that a rather general ansatz for the additional supersymmetry (not involving central charge transformations) leads to hyperkähler geometry. We give non-trivial examples of these models.

Published for SISSA by Springer

Received: August 22, 2012 Accepted: December 11, 2012

Published: January 10, 2013

Semichiral Sigma models with 4D hyperkahler geometry

M. Gteman,a U. Lindstrma and M. Roekb

aDepartment Physics and Astronomy,

Division for Theoretical Physics, Uppsala University, Box 803, SE-751 08 Uppsala, Sweden

bC.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794-3840, U.S.A.

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: Semichiral sigma models with a four-dimensional target space do not support extended N = (4, 4) supersymmetries o -shell [1, 2]. We contribute towards the understanding of the non-manifest on-shell transformations in (2, 2) superspace by analyzing the extended on-shell supersymmetry of such models and nd that a rather general ansatz for the additional supersymmetry (not involving central charge transformations) leads to hyperkahler geometry. We give non-trivial examples of these models.

Keywords: Supersymmetry and Duality, Extended Supersymmetry, Di erential and Algebraic Geometry, String Duality

ArXiv ePrint: 1207.4753v1

c

JHEP01(2013)073

SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP01(2013)073

Web End =10.1007/JHEP01(2013)073

Contents

1 Introduction 1

2 Semichiral sigma models 2

3 On-shell N = (4, 4) supersymmetry 33.1 The ansatz 33.2 No o -shell supersymmetry 43.3 Invariance of action 43.4 Integrability 53.5 On-shell algebra closure 5

4 Hyperkahler solutions 8

5 Summary and conclusions 11

A Linear transformations 11

B Complex structures 12

C Example metric 14

1 Introduction

In a previous paper [2], we presented the general structure of semichiral sigma models with (4, 4) supersymmetry. We found conditions for invariance of the action and interesting geometric structures related to simultaneous integrability (Magri-Morosi concomitants) and a weaker conditions than (almost) complex structures for the transformation matrices (Yano f-structures). This rich mathematical context prompt us to take a closer look at specic models.

In [2] we treated both o -shell and on-shell supersymmetric (4, 4) models with manifest (2, 2) supersymmetry. One particular model where the non-manifest supersymmetry can only close on-shell, is the case of one left and one right semichiral eld corresponding to a four-dimensional target space.1 These models are simple enough that a lot of the calculations can be carried out explicitly. They also enjoy a number of special properties such as carrying an almost (pseudo-) hyperkahler structure and having a B-eld that is governed by a single function.

1The o -shell N = (4, 4) pseudo-supersymmetry for a semichiral sigma model with four-dimensional target space was discussed in detail in [1].

1

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In the present paper we start from the same general ansatz for the extra supersymme-tries as in [2], we then solve the conditions for invariance of the action and discover that on-shell closure of the algebra follows from these, with one additional input from the algebra. The solution leads to a geometry which is necessarily hyperkahler. Note that we are not proving that (4, 4) supersymmetry in four dimensions restricts the target space geometry to be hyperkahler, since there may be more general ansatze combining supersymmetry with central charge transformations. We briey discuss this option in our conclusions.

We relate our solution to geometric conditions from [2] and illustrate our ndings in an explicit (non-trivial) example.

The paper is organized as follows. Section 2 contains background material, denitions and sets the notation for the paper. In section 3 we give the derivation of our conditions for invariance of the action and on-shell closure while section 4 examplies them. Section 5 contains our conclusions and in the appendix we have collected the special case of linear transformations, the explicit form of the complex structures and the metric for the example in section 4.

2 Semichiral sigma models

Consider a generalized Kahler potential with one left- and one right semichiral eld and their complex conjugates, K(XL, XR), where L = (,

) and R = (r, r). The action,

S = Z

d2xd2d2

JHEP01(2013)073

K(XL, XR) (2.1)

has manifest N =(2, 2) supersymmetry. The supersymmetry algebra is dened in terms of the anti-commutator of the covariant supersymmetry derivatives as

{D, D} = i (2.2)

and the semichiral elds are dened by their chirality constraints as

D+X = 0 , DXr = 0 . (2.3)

The geometry of the model is governed by two complex structures J(+) and J() that

both preserve the metric g

J()tgJ() = g (2.4)

as well as by an anti-symmetric B-eld whose eld strength H enters in the form of torsion in the integrability conditions

0 = ()J() =

J() , (2.5)

where (0) is the Levi-Civita connection. These conditions identify the geometry as bihermitean [3], or generalized Kahler geometry (GKG) [4].

2

+ (0)

1

2Hg1

The fact that our superelds are semichiral species the GKG as being of symplectic type where the metric g and the B-eld take the form2

g = [J(+), J()]

B = {J(+), J()} . (2.6)

The matrix is dened as

= 12

and the submatrix KLR is the Hessian

. (2.8)

An additional condition results from the target space being four-dimensional and reads [5]

{J(+), J()} = 2c11, B = 2c , (2.9)

where c = c(XL, XR). For reference, we rewrite this relation as

(1 c)|Kr|2 + (1 + c)|Kr|2 = 2K

Krr . (2.10)

The condition (2.9) allows us to construct an SU(2) worth of almost (pseudo-) complex structures3 J(1), J(2), J(3), [1, 5, 6],

J(1) := 1

1 c2 J

J(2) := 1

21 c2

3

0 KLR

KRL 0

!

(2.7)

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KLR = Kr KrK

r K

r

!

() + cJ(+)

,

[J(+), J()] ,

J(3) := J(+) . (2.11)

For |c| < 1 the geometry is almost hyperkahler, while for |c| > 1 the geometry is almost

pseudo-hyperkahler [1].

3 On-shell N = (4, 4) supersymmetry

3.1 The ansatz

In this rst subsection we recapitulate some denitions from [2].

Additional supersymmetry transformations on the semichiral elds must preserve the chirality constraints (2.3). We make the following ansatz for the additional supersymmetry,

X =

+ D+f(XL, XR) + g(X) DX + h(X)DX , X = +D+ f(XL, XR) + g(X)DX + h(X) DX,

Xr =

D ~f(XL, XR) +(Xr)+ D+Xr +(Xr)+D+Xr ,Xr = D ~f(XL, XR) + (Xr)+D+Xr + (Xr)+ D+Xr . (3.1)

2This gives the B eld in a particular global gauge as B = B(2,0) + B(0,2) with respect to both complex structures.

3In higher dimensions than four, almost hyperkahler implies hyperkahler [7].

For later convenience, a compact form of these transformations will be useful. We thus introduce transformation matrices

U(+) =

f

0 0 0

f

fr fr

0

fr fr

0 0 0 0 0 0 0

~

h

, V (+) =

,

0 0 0

~

g

(3.2)

U() =

g 0 0

0 h 0

~f ~f

~fr

0 0 0

, V () =

h 0 0

0 g 0

0 0 0

~

f ~f

~fr

.

One column in each of the matrices is arbitrary. Further writing the semichiral elds as a vector Xi where i = (,

, r, r), the transformations read

Xi =

U()ij DXj + V ()ijDXj , (3.3)

where the spinor index takes the values + and .

3.2 No o -shell supersymmetry

Consider supersymmetry closure of the transformations in (3.1). The supersymmetry algebra dened in (2.2) requires that two subsequent transformations commute to a translation, e.g., that [(+)1, (+)2]X = i+[2

+1]X. But the transformations in (3.1) commute to

[1, 2]X = +[2

+1]

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|f|2i++X + (f fr + fr)D+D+Xr + (f fr + fr)D+D+Xr

+[21](gh)i=

X + . . . , (3.4)

where the dots represent the mixed +-terms in the algebra. Since |f|2 6= 1, the +

+-

part of the algebra (3.1) can never close o -shell.4 In section 3.5, we will see that the algebra closes on-shell.

We do not get a contradiction from the

-part of the algebra, however; it closes if

and only if

gh = 1 . (3.5)

3.3 Invariance of action

The action in (2.1) is invariant under the

(+)-transformations if and only if the Lagrangian satises the following partial di erential equations [2]

(KiU(+)i[j)k] D+Xj D+Xk = 0 , (3.6)

4From the derivation we see that the full statement is that we cannot have a left (or right) supersymmetry o -shell.

4

together with the corresponding equations for the other transformation matrices. Explicitly

frK

f

Kr +K

r = 0 ,

frK

f

Kr + K

r = 0 ,

(

~

h)Krr frKr + frKr = 0 , (3.7)

and

~fKrr

~frKr + gKr = 0 , ~f

Krr + hKr

~frKr

= 0 ,

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r = 0 . (3.8)

In addition we have the relations complex conjugate to those in (3.7) and (3.8). Further, a useful relation between and may be derived from the previous;

hg =

~

h

=

(g

h)K

~f

Kr + ~fK

K

Krr |Kr|2

K

Krr |Kr|2

. (3.9)

3.4 Integrability

In [2] we discuss on-shell closure of the (4, 4) algebra in terms of the SU(2) set of complex structures J(A)() known to exist from the (1, 1) reduction. We show that all the (4, 4) closure conditions are satised on-shell by relating them to expressions involving the J(A)()s. In other words, we prove that the (4, 4) conditions needed for additional supersymmetry of the (2, 2)

model are equivalent to the (4, 4) conditions needed for extra (on-shell) supersymmetry of the corresponding (1, 1) model.

Here we follow a di erent route. We assume that the systems of linear partial di erential equations (3.7) and (3.8) are integrable and show that this, together with g,, h and all being constant, is su cient to prove that all the (2, 2) closure conditions are satised on-shell.

Integrability of (3.7) and (3.8) in turn, may be discussed in terms of the usual machinery for analyzing systems of linear partial di erential equations. We do not include such an analysis, but solve the equations in examples below.

A nal comment on the relation to the analysis in [2] is relegated to appendix B. There we show the (2, 2) relation

1

2

J(1)() iJ(2)() = U()() , (3.10)

found in [2] and relating the abstract J(A)s to the transformation matrices (3.2), indeed holds for the two additional complex structures J(1)(+) and J(2)(+) explicitly constructed in (2.11).

3.5 On-shell algebra closure

Before discussing how to close the algebra on-shell, we point to some consequences of imposing the full algebra.

5

X closes if and only if gh = 1. The same is true for the (+)-part of the algebra for

In (3.5), we have seen that the ()-part of the algebra for

= 1. From (3.9) we then deduce that

det(KLR) 6= 0 . (3.11)

This is a familiar condition which, amongst other things, ensures that a non-degenerate geometry can be extracted from the action [6]. A further consequence of = 1 = gh in

conjunction with (3.7) and (3.8) is that (2.10) is satised with

c = 1 |g|2

1 + |g|2

+ [21]

, (3.13)

where the transformation matrices U() and V () are dened in (3.2). The Nijenhuis tensor

N and the Magri-Morosi concomitant M are dened as

N(I)ijk = Il[jIik],l + IilIl[j,k] ,

M(I, J)ijk = IljJik,l JlkIij,l + JilIlj,k IilJlk,j . (3.14)

We will now show that each of the terms in (3.13) close to a supersymmetry algebra on-shell and discuss the geometric interpretation. In section 3.2 we have seen that the transformations in (3.1) cannot close to a supersymmetry o -shell. Hence we have to go on-shell. The eld equations that follow from the action in (2.1) are

D+K = 0 ,

DKr = 0 , (3.15)

6

Xr; it closes if and only if

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= 1 ||2

1 + ||2

. (3.12)

Since we have assumed that g = g(X) and =(Xr), the relations (3.12) tell us that g,, h and are all constant.

We now turn to the on-shell closure. Two subsequent transformations dened in (3.1) acting on a left-semichiral eld commute to

[1, 2]X =

+[2+1]

hM(U(+), V (+))jk D+XjD+Xk

(U(+)V (+))j D+D+Xj (V (+)U(+))jD+ D+Xj

i

+[21]

hM(U(), V ())jk DXjDXk

(U()V ())j DDXj (V ()U())jD DXj

i

++[2

1]

hM(U(+), U())jkD+Xj DXk [U(), U()]jD+ DXj

i

++[21]

hM(U(+), V ())jkD+XjDXk [U(+), V ()]j D+DXj

i

+[2

1]

hN (U())jkDXj DXk

i

hN (V ())jk DXjDXk

i

++[2

+1]

hN (U(+))jkD+Xj D+Xk

i

and their complex conjugates. To investigate on-shell closure, we use the the rst equation to solve for, e.g., D+X:

D+X =

1 K

(Kr

D+Xr + KrD+Xr) . (3.16)

Using the expressions for the transformation functions in (3.7)(3.9) and the on-shell relation (3.16), the last term in the algebra for X, (3.13), becomes

[

(+)1,

(+)2]X =

+[2

+1]

hN (U(+))jkD+Xj D+Xk

i

=

+[2

+1]

(fr,fr fr,fr + fr,r fr,r~h)D+Xr D+Xr

+(fr,f

f

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,fr f

,r)D+X D+Xr

+(fr,f

f

,fr f

,r)D+X D+Xr

=

+[2

+1]f

Krr

K

2c 1

,D+Xr D+Xr

= 0, (3.17)

where in the last line we used the fact that and c are constants. The vanishing of the Nijenhuis tensor for an almost complex structure means that the structure is integrable, hence a complex structure. Here we see that the relevant parts of the Nijenhuis tensor for the transformation matrix U(+) vanish. The same is true for the relevant parts of the Nijenhuis tensor for U() and V ().

We now move over to the terms in the algebra (3.13) involving the Magri-Morosi concomitant. For clarity, we dene the following combinations of the parameter functions,

= f

fr + fr, = f

fr + fr ,

h) fr

~f

, = frg

~frfr . (3.18)

To investigate on-shell closure of the (+)-supersymmetry for X, we use the conjugate version of the eld equation (3.15) to solve for D+Xr. The algebra becomes

[(+)1,

(+)2]X =

+[2+1]

hM(U(+), V (+))jkD+XjD+Xk + (U(+)V (+))j D+D+Xj i

=

+[2+1] D+

|f|2D+X

D+Xr D+Xr

=

+[2+1] D+

K

= f

(g

K

r

|f

|2

D+X+

KrKr

D+Xr

=

+[2+1] D+

|

f

|2

|f

|2

D+X + K

lr

K

(

g

)D+Xr

~

+[2+1]i++X , (3.19)

where in the last line we used that = 1. We already know that the ()-part of the

algebra for X closes to a supersymmetry if and only if and satises this constraint.

7

=

The vanishing of the Magri-Morosi concomitant for two commuting complex structures is equivalent to the statement that the structures are simultaneous integrable [8]. Here we see that the relevant parts of the Magri-Morosi concomitant of U(+), V (+) combines with

products of the transformation matrices to vanish on-shell, such that the algebra closes to a supersymmetry.

Now we focus on the mixed

+

-terms in the algebra. Again using the eld equations (3.15) to write DXr in terms of DX and DX, together with the expressions for the transformation functions in (3.7)(3.8), the algebra closes to

[

(+)1,

()2]X =

+[2

1]

hM(U(+), U())jkD+Xj DXk [U(+), U()]jD+ DXj i

=

+[2

1] D+

h DX + DXr + (fr ~f)DX

i

=

+[2

1] D+

K

r

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Krr

DX+

fr ~f

Krr

DX

Kr

h)DX

= 0 , (3.20)

where in the last line we use that g and are constants. A similar derivation can be done for the

+-term in the algebra, which also vanishes when using the eld equations. The closure of the algebra on the right semichiral eld follows in exactly the same way.

As a summary, we see that the transformations dened in (3.1) close to a supersymmetry algebra on the semichiral elds on-shell,

[1, 2]Xi =

+[2+1]i++Xi +

[21]i=Xi , (3.21)

and that the action is invariant under the same transformations, if and only if the transformation functions take the expressions in (3.7)(3.9) and

|g|2 = ||2 =

K

=

+[2

1] D+(g

Krr |Kr|2

K

(3.22)

Krr |Kr|2

is a constant.

4 Hyperkahler solutions

The four-dimensional target space geometry is hyperkahler if c in (2.10) is a constant with absolute value less than one. We see from (3.12) that this is the case at hand. In this section we explore some additional properties of this hyperkahler geometry. We notice that the structures in (2.11) are now integrable and give us an SU(2) of complex structures,

[J(A), J(B)] = AB + ABCJ(C) . (4.1)

To describe the hyperkahler (HK) geometry, a generalized potential K must satisfy (2.10) with constant |c| < 1. An additional requirement is that the the determinant

8

of the matrix KLR is nonvanishing (3.11). The transformations functions are then found from (derivatives of) this K as solutions of (3.7)(3.9).

As discussed in the appendix, there are many quadratic actions which satisfy (2.10) and (3.11), e.g.,

K = (X

X)(Xr

Xr) + (Xr + Xr + X)(Xr + Xr + X) . (4.2)

Solutions to (3.7)(3.9) are easily found for this K since all the coe cients are constants.

The supersymmetry transformations are linear. More on this in the appendix.

There are a number of nontrivial examples of HK geometries written in semichiral coordinates. In particular in [9], the relation between the Kahler potential and the generalized Kahler potential is discussed, and HK geometries in semichiral coordinates are generated from Kahler potentials with certain isometries amongst their chiral and twisted chiral coordinates (see also [7] and [10] for further discussions of semichiral formulations of hyperkahler geometries). The method is an adaption of the Legendre transform construction of [11, 12]. In brief, the construction yields a semichiral description of a four-dimensional HK manifold from any function F (x, v, v) which satises Laplaces equation

Fxx + Fvv = 0 . (4.3)

Here x and v correspond to certain expressions in the chiral and twisted coordinates that need not concern us here. The T-duality between a chiral and twisted chiral model with N = (4, 4) supersymmetry and its semichiral dual counterpart is investigated in detail in [13]. The generalized Kahler potential is obtained via the Legendre transform

K(X

X

X + X + 2Xr) =: 1

2 y . (4.5) The resulting K satises (2.10) with5 c = 0 and has det(KLR) 6= 0.

We now use this construction to generate a nontrivial example illustrating the discussion in the previous sections. Since we do not need the connection to a Kahler potential, we can start from any F satisfying (4.3). A convenient example is6

F (x, v, v) = r xln(x + r) , r2 := x2 + 4vv . (4.6)

5The restriction to c = 0 is not necessary for this kind of construction-see [7].

6When v and x are chiral and real linear superelds, respectively, F is the superspace Lagrangian of the improved tensor multiplet [14], which in turn is dual to at space. In our construction, v and x can can be dualized to chiral and twisted chiral elds, but as far as we can see there is no transformation that relates them to a at metric neither in these nor in chiral and complex linear elds. Hence we see no reason why our semichiral example should be at space in disguise. The detailed properties of the metric remain to be investigated, however.

9

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X, X + X + 2Xr, X + X + 2Xr)

= F (x, v, v)

1

2v(

1

2 v(

1

2x(

X + X + 2Xr)

X

X + X + 2Xr) i

X) , (4.4)

with

X) =: i12z ,

Fv = 12(

X + X + 2Xr) =: 12y ,

Fv = 12(

Fx = i1

2(

Solving the relations (4.5) and plugging into (4.4) results in the generalized Kahler potential

K = 1

2e

1

2 iz

1 1 4yy

, (4.7)

which indeed satises all the relevant requirements. Note that it is written in coordinates invariant under the Abelian symmetry

X = , Xr = ,

R . (4.8)

To nd the transformation functions, we calculate the various second derivatives of K and insert into (3.7)(3.9). The resulting partial di erential equations may then be solved to yield

f = 2i( )ln(2 iy) + 2i( +)ln(2 iy) + z , ~f = i ln(y) + ig

1 8y2

1

2( + g)z , (4.9)

where and are integration constants. The appearance of these constants may seem surprising, since we expect the transformations to be unique. Below we shall see how they are determined.

As in the derivation of the supersymmetry transformations in [15], we identify part of (4.9) as eld equation symmetries, that is symmetries of a Lagrangian L() of the form

i = Aij Lj , (4.10)

with Aij anti-symmetric (or symmetric for spinorial indices). These transformations will leave the action invariant and vanish on-shell. When inserted into (3.1) the part of f gives an expression that vanishes due to the X eld equation,

(2 iy)(2 iy)

D+z + 2(2 iy)

D+y + 2(2 iy)

D+y = 0 , (4.11)

and the part of ~f gives an expression that vanishes due to the Xr eld equation,

yDz + 2iDy = 0 . (4.12)

This means that the transformations (4.9) reduce to

f = 2i( ln(2 iy) + ln(2 iy)) , ~f = g

i18y2 1 2z

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. (4.13)

According to (3.22), and g are phases when c = 0, and from (3.1) we see that they may be absorbed by an R-transformation of the charges Q for the extra supersymmetries:

=: ei , g =: ei ,

+ei

+ ,

ei

. (4.14)

10

The nal form of the functions f and ~f thus becomes

f = 2i ln 2 iy 2 iy

,

~f = i1

2 iz + 14y2 . (4.15)

Inserting this in (3.1) yields the transformations

X =

+(2 iy)(2 iy)

2 i(y y)D+X + 2(2 iy)D+Xr 2(2 iy) D+Xr

+ DX

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DX ,

(2 iy)DX (2 + iy)DX + + D+Xr +D+Xr , (4.16)

and their complex conjugates.

5 Summary and conclusions

We have found that our ansatz (3.1) for additional supersymmetries of a semichiral sigma model with four-dimensional target space corresponds to hyperkahler geometry on the target space. For this case we have provided the form of the transformation functions, related them to previous general discussions in the literature and described generalized Kahler potentials satisfying the invariance conditions.

The existence of four-dimensional examples of semichiral sigma models with non-trivial B-eld [16] indicates that the ansatz (3.1) has to be modied for on-shell algebras. It is well known, e.g., from four-dimensional sigma models with chiral elds (,

) as coordinates, that extra supersymmetries may come together with central charge transformations [11, 17] in the form

i = D2( i) , (5.1)

where the scalar transformation supereld contains the supersymmetry at the level. Central charge transformations vanish on-shell, but will have e ect, e.g., on the conditions that follow from invariance of the action. Such a generalization of the transformations (3.1) will presumably cover the dB 6= 0 case.

Acknowledgments

The research of UL was supported by VR grant 621-2009-4066. MR acknowledges NSF Grant phy 0969739.

A Linear transformations

In [1] we were able to extract interesting information from a semichiral sigma model with four-dimensional target space and additional non-manifest linear pseudo-supersymmetry.

11

Xr =

4

In this section we show that linear supersymmetry transformations are only possible for quadratic potentials (at geometry) or when the metric is degenerate.

There is a very direct argument why linear transformations lead to quadratic actions. If the transformation on a eld X is linear and the algebra closes on-shell, we have, schematically,

X = QX, Q2X = [, ]X = X + F (A.1) where F is (derivatives of) a eld equation. The latter then reads

F = OX = (Q2 )X = 0 , (A.2)

which means that the Lagrangian must be quadratic. In the present case, it still turns out to be instructive to explicitly consider the linear case.

The transformations are the same as in the general case (3.1), with the di erence that the transformation functions are now all constant. Again, o -shell closure of the algebra cannot occur, since |f|2 6= 1. On-shell closure, however, can be obtained just as in the

general case.

Invariance of the action (2.1) under the linear transformations implies the same system of partial di erential equations (3.7)(3.8), but with constant parameters. The equations imply that the Lagrangian K must satisfy

Kr =

K

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r , (A.3)

among other relations. The parameters and are dened in (3.18) and are here constant. Taking derivative with respect to Xr on both sides implies that (||2 ||2)Krr = 0. This

implies that either ||2 = ||2, leading to degenerate metric, as we have seen in (3.11), or

that Krr = 0. Similar results can be derived from the other PDEs and as a result we draw the conclusions that linear supersymmetry transformations imply either degenerate metric, or a generalized Kahler potential with vanishing third derivatives, i.e. a quadratic potential.

For a quadratic potential K, everything works as in the general case. From the equation (2.10) we again nd that

c = 1 |g|2

1 + |g|2 |

g|2 =

1 c

1 + c . (A.4)

This again implies that |c| < 1 and thus the geometry is necessarily hyperkahler.

As a nal comment we note that there are a large set of non-trivial (non-quadratic) generalized potentials invariant under the linear transformations with det(KLR) = 0. As

mentioned, this makes it impossible to extract a metric, so they do not correspond to sigma models. One may speculate that these have an application in models where the background is in some sense topological.

B Complex structures

In [2] it is concluded on general grounds that the relation (3.10) holds on-shell

1

2

J(1)() iJ(2)() = U()() , (B.1)

12

where

() := 12(11 + iJ()) , (B.2)

J(A)() are the complex structures obeying an SU(2) algebra with J(3)() := J() and is a phase.7 Here we verify this relation by explicitly constructing J(1) and J(2) as in (2.11).

From [6], J() takes the form

J(+) = i

D

D 0 0 0 0 D 0 0

2K

Kr 2K

K

r S 2KrK

r

2KKr 2K K r

2KrK r

,

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S

S 2K

rK

r 2KrrK

r 2KrrK

r

2KrKr S 2KrrKr 2KrrKr 0 0 D 00 0 0 D

J() = i

D

. (B.3)

where we have dened the sum and di erence

S := |Kr|2 + |Kr|2 , D := |Kr|2 |Kr|2 . (B.4)

Inserting the expressions for J() in (2.11) yields

J(1) = 2iD1 c2

K

Krr K

rK

r KrrK

r KrrK

r

KrKr KKrr KrrKr KrrKr

cK

Kr cK

K

r

1

2 (D + cS) cK

rKr

, (B.5)

cKKr cK K r

cKrKr12(D + cS)

and

J(2) =

2 D1 c2

K

Krr K

rK

r KrrK

r KrrK

r

. (B.6)

KrKr K

Krr KrrKr KrrKr

KKr K K r

KKrr K rKr

KKr K K r

KrK r

K Krr

Finally, we have

D

p1 c2 U(+)(+) =

(frKrfrKr)K

f

D(frK

rfrK

r)K

(frKrfrKr)K

r (frKrfrKr)K

r

0 0 0 0 K

Kr K

K

r |Kr|2 K

rKr

~

hK

Kr K

K

r KrK

r |Kr|2

. (B.7)

Using these expressions for the HK models at hand, we verify explicitly that (3.10) is satised with i = =

~

h due to the invariance conditions (3.7)(3.9).

7This phase is not included in [2], but represents an ambiguity in the choice of related to R-symmetry.

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C Example metric

The metric8 that follows from the potential (4.7) may be found from the formulae in [6]. It is

g = gLL gLR gRL gRR

!

= iei

1

2 z

!

(C.1)

2(y y)

A B

Bt C

with

A = 1

16(4 + yy)

4 yy 2i(y + y) 4 + yy

4 + yy 4 yy + 2i(y + y)

!

JHEP01(2013)073

B = 1

(2 iy)[4 + yy i(y + y)] (2 iy)[4 + yy + i(y + y)]

4 (2 + iy)[4 + yy i(y + y)] (2 + iy)[4 + yy + i(y + y)]

!

C = 4 + y2 4 + yy

4 + yy 4 + y2

!

(C.2)

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14

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