Wick Rotation and Supersymmetry

TMR meeting, Paris, 1999

PROCEEDINGS

Wick rotation and supersymmetry

Arthur J. Mountain ∗ The Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, United Kingdom. E-mail: [email protected]

Abstract: Wick rotation of supersymmetric theories is studied. In order to do this a comprehensive prescription for the Wick rotation of spinor fields is presented. This allows the consistent Wick rotation of the supersymmetry algebra and leads eventually to the Wick rotation of supersymmetric field theories. Two complementary descriptions of the Euclidean theory are found. Either it involves “doubled” spinor fields and is manifestly covariant under the Euclidean tangent space group or the spinor fields are subject to a nonlocal projection which recovers the physical degrees of freedom but breaks the tangent space symmetry group. This procedure is then applied to the Wess-Zumino model and eleven-dimensional supergravity.

1. Introduction one can sometimes see different parts of the same theory Wick rotated in different ways. Our aim Wick rotation has its history in the regularisa- here is to present a clear prescription for Wick tion of quantum field theories. The pathR integral rotation in the presence of supersymmetry. weights paths with e−iS where S = d4xL(x)is The difficulty in performing a Wick rotation the action. Paths which correspond to large ac- on a theory containing spinors is that the rep- tion are suppressed as the exponential becomes resentations of spinors change as the signature highly oscillatory. A Wick rotation is an analytic of the spacetime changes. The Wick rotation of continuation of the time co-ordinate spinors was first considered by Osterwalder and t →−it . (1.1) Schrader [2] from the point of view of constructive QFT. Their aim was to directly construct a The aim is to make the action imaginary so that field theory in Euclidean space which reproduced the analytic continuation of e−iS has a real, nega- the Green’s functions of a Minkowski space the- tive exponent so that the path integral converges. ory, that is to look for a Euclidean spinor Ψ such An area of much recent interest is instan- that ton physics. Instantons are defined in Euclidean ∗ 0 space so in order to perform instanton calcula- hΨα(x)Ψβ(y)i = hTψα(x)ψδ(y)iγδβ . (1.2) tions in a “realistic” field theory, one must first Wick rotate the theory. A particular example This is not possible as the right hand side is not which has generated much interest recently is in- Hermitian. To solve this they introduced two stanton cosmology [1]. This describes the cre- Euclidean spinors Ψ(1) and Ψ(2) which are inde- ation of a universe as an instanton which, at a pendent and correspond to the Wick rotation of certain size, continues into a Lorentzian space- ψ and ψ respectively. This is the phenomenon time. It is tempting to try to create this scenario of “fermion doubling” which we return to be- inside M-theory in which case one needs to know low. This work was extended by Nicolai [3] who the Wick rotation of 11-dimensional supergravity used their treatment of Dirac spinors to define to Euclidean space. There is no standard treat- Euclidean Majorana spinors. As supersymmet- ment of Wick rotation in the literature. Indeed, ric theories typically contain Majorana fermions ∗Work done in collaboration with K. S. Stelle this allowed Nicolai to define Euclidean super- TMR meeting, Paris, 1999 Arthur J. Mountain symmetric field theories. nally we apply these ideas to two theories. The There have been several other approaches Wess-Zumino model is used as a simple exam- to Euclideanisation of theories involving spinors. ple of a supersymmetric theory. We also discuss Most of these have at their heart the notion of Wick rotation of 11-dimensional supergravity. A analytically continuing the time co-ordinate in more detailed exposition of the results presented a continuous manner. Thus (1.1) would be re- here will be given in [9]. placed by t → e−iθt, 2. Wick rotation and the tangent space group where θ runs from 0 (Minkowski space) to π/2 (Euclidean space). These days we would describe Scalar field in two dimensions this as a T-duality in a timelike direction, effec- In general we will consider spacetimes with co- tively embedding the D-dimensional Minkowski ordinates x =(t, x). In this section, we will work theory in a space of signature (1,D) and rotating in two dimensions so the co-ordinates are x = to a theory in the D spacelike dimensions. This (t, x1).Themetricis approach was initially described by Mehta [4] and more recently has been studied in [5]. These −10 µ ν 2 2 ηµν = ,ηµν x x = −t + x1 . papers consider the example of N =2super- 01 Yang-Mills in four dimensions, rotating this the- The basic operation of Wick-rotation is to ana- ory to obtain a supersymmetric theory in four lytically continue the time co-ordinate as Euclidean dimensions which involves Dirac spi- →− nors. From the Kaluza-Klein point of view this t it . (2.1) is all well known; both these theories are dimen- Under this, the canonical inner product becomes sional reductions of the N = 1 super-Yang-Mills µ ν − 2 2 → 2 2 µ ν theory in D = 5 + 1 dimensions. This proce- ηµν x x = t + x1 t + x1 =˜ηµν x x , dure is a valid means to arrive at a supersym- which involves the expected “Euclidean” metric, metric Euclidean theory (one which was first pre- η˜µν = diag(1, 1). sented by Zumino [6]). However we would argue The simplest example of Wick rotation, but that the aim of Wick rotation is to produce a one which still has many interesting properties, theory which is a Euclidean-space description of is the real scalar field φ(t, x). Under a Wick ro- Minkowski-space physics. In this respect, these tation this becomes approaches fail. One clear example of this is that in Zumino’s theory there are two scalars which φ(t, x1) → Φ(t, x1)=φ(−it, x1) . (2.2) have kinetic terms of opposite sign. This clearly We will define an operation # which is com- cannot represent Minkowski space physics as the plex conjugation followed by a Euclidean time- vacuum would be unstable with respect to one of reversal. As φ is real, the Euclidean field Φ has these scalars. the following “reality” condition In this paper, we shall investigate the process # ≡ ∗ of Wick rotation in the presence of spinors. In or- [Φ(x)] [Φ(θx)] =Φ(x), (2.3) der to do this we start by introducing some of the where we have introduced the notation θx = ideas in the relatively simple setting of a scalar (−t, x1). This condition follows directly from the field theory. We then proceed to study spinors reality condition in Minkowski space. What hap- in the Wick-rotated theory, following the work of pens if we apply an SO(2) transformation? Con- Osterwalder and Schrader and Nicolai. We focus sider the simple example φ(x)=t+x1.The on Majorana spinors as we are interested eventu- Euclidean field is Φ(x)=−it + x1. Under the ally in the Wick rotation of supersymmetric field infinitesimal SO(2) transformation theories. We analyse the supersymmetry algebra b → in Euclidean space and show how this generates U : x Ux = x+δx, (2.4) supersymmetry in the Wick-rotated theory. Fi- t → t + x1 ,x1→x1−t , (2.5)

2 TMR meeting, Paris, 1999 Arthur J. Mountain the scalar field transforms into gives us an SO(D-1) symmetry group. In the next section we will show how, by introducing 0 b Φ (x)=UΦ(x)=Φ(Ux) extra degrees of freedom, this can be extended = −i(t+x1)+(x1 −t) . (2.6) to the full SO(D) group required for a covariant Euclidean theory. We shall now work in four di- We see that our naive “reality” condition (2.3) is mensions, although the generalisation to higher not consistent with the action of SO(2) in that dimension is clear. # [Φ0(x)] =Φ6 0(x). 3. Spinor fields At a general point on the SO(2) orbit the only “reality” condition we can impose is The added complexity when dealing with spinors is that one has to be careful which representa- [Ub −1Φ0(x)]# =[Ub−1Φ0(x)] . (2.7) tions are allowed. In 3+1 dimensions, the mini- mal spinor can be chosen to be Majorana or Weyl In section 3 we will see how to form a covariant while in four Euclidean dimensions we only have version of this condition. a Weyl spinor. See the Appendix for Clifford algebra and spinor conventions. Wick rotation: general prescription We shall work in D dimensions. The co-ordinates Dirac spinors are (t, x) and we shall analytically continue t as To define Dirac spinors in the Euclidean theory, in (2.1). This procedure requires a choice of t we start from the Minkowski spinor ψ and its and breaks Lorentz covariance. Hence the only conjugate ψ = ψ†γ0. Using properties of the part of the SO(1,D-1) tangent space group which γ-matrices we can see that the SO(1,D-1) sym- commutes with Wick rotation is the SO(D-1) metry of the Minkowski theory acts on these as group which preserves t. The implication of this 1 µ ν 1 µ ν is that the process of Wick-rotation should be δψ = ωµν γ γ ψ,δψ=−ψωµν γ γ . 4 4 viewed in the following way: (3.1) Let us consider the analytic continuation of ψ • Start from a theory in D-dimensional Min- and ψ separately: kowski space with co-ordinates xµ, SO(1,D-1) symmetry and metric ψ(t, x) → Ψ(1)(t, x)=ψ(−it, x) , ψ(t, x) → Ψ(2)(t, x)=[ψ(it, x)]†γ0 . ηµν = diag(−1, 1,...,1). Before applying a Lorentz transformation we have • Choose a preferred time co-ordinate t such h i# h i† k that we have co-ordinates (t, x ). This is Ψ(2)(x)= Ψ(1)(x) γ0 = Ψ(1)(θx) γ0 . a gauge-fixing up to the SO(D-1) group (3.2) k which acts on the x . Following the treatment of the scalar field, we have defined the action of # on spinors as Her- • Perform the Wick-rotation t →−it. mitian conjugation followed by a Euclidean time- • The result is a theory in D-dimensional Eu- reversal. This relationship is not SO(4)-covariant. clidean space but a theory which is gauge- In order to proceed to an SO(4) covariant theory, fixed. By this we mean that the only al- let us define the operator I by lowed tangent space rotations are the h i† (2) (1) 0 SO(D-1) group inherited from the Minkow- IΨα (x)= Ψβ (θx) γβα , ski theory. h i† (1) (2) 0 IΨα (x)= Ψβ (θx) γβα . However, this is clearly not the full story. The in- µ ν µ ν (2) ner productη ˜µν x x = δµν x x is invariant un- I is simply an operator which rewrites Ψ in der SO(D) whereas the prescription above only terms of Ψ(1) and vice versa. At a general point

3 TMR meeting, Paris, 1999 Arthur J. Mountain on the SO(4) orbit given by x0 = Ux, the rela- by (3.6), the statement of Osterwalder-Schrader tionship (3.2) must be generalised to positivity of the Euclidean action is h i # L L # L † Ub −1Ψ(2)(x0)= Ub−1Ψ(1)(x0) γ0 . (3.3) Θ (x)=[ (x)] =[ (θx)] . (3.7) As an example of this, we know that, in Minkow- Bearing this in mind, we define an involution op- ski space, the combination ψφψ is real and could erator Θ by appear as a Yukawa interaction term in a La- Θ ≡ Ub ◦ I ◦ Ub −1 . (3.4) grangian. In the Euclidean theory, we have the corresponding result In terms of this, we have the discrete symmetry h i Θ Ψ(2)(x)Φ(x)Ψ(1)(x) h i# (2) (1) 0 h i ΘΨ (x)= Ψ (x) γ , (3.5) † = Ψ(2)(θx)Φ(θx)Ψ(1)(θx) (3.8). where θ(t, x)=(−t, x). The operator Θ is the Of course, all this discussion is using the doubled, Osterwalder-Schrader involution operator. The SO(4)-covariant spinor fields. If we are prepared property Θ2 = 1 is evident from the definition to break SO(4) covariance then, after imposing (3.4). Let us write an SO(4) element U which 0 0 the nonlocal projection on all fields, we have the rotates x into x by U (x ,x). The action of Θ is # reality condition [L(x)] = L(x). ΘU(x0,x)=U(θx0,θx), (3.6) # ∗ Majorana spinors ΘΦ(x)=[Φ(x)] =[Φ(θx)] , h i# We wish eventually to perform Wick rotations (1) (2) 0 ΘΨα (x)= Ψβ (x) γβα , on supersymmetric field theories. Such theories h i# typically contain Majorana spinors. These are (2) (1) 0 ΘΨα (x)= Ψβ (x) γβα . defined by equation (A.4) which can be written (in Minkowski space) as Included above is the covariant treatment of the real scalar field which was mentioned in section 2. ψ†γ0 = ψT C. (3.9) We now have two different descriptions of Wick rotated spinor fields. On one hand, we can work Now we need to define our Wick-rotated spinor with doubled spinors, where Ψ(1) and Ψ(2) are fields. Using the spinor fields at imaginary time the Wick rotation of ψ and ψ and have a the- defined above, we see that the Wick rotation of ory which is manifestly SO(4) covariant in the the Majorana condition as originally given by manner described above. On the other hand, Nicolai [3] is we can impose the involution (3.6) as a nonlo- (2) (1) Ψ (x)=Ψ (x)Cβα . (3.10) cal projection on the fields, for example setting α β h i# (1) (2) 0 Ψα (x)= Ψβ (x) γβα. This halves the de- The Majorana condition is a reality condition on grees of freedom (recovering the degrees of free- spinor fields. Following the discussion for Dirac dom of the Minkowski spinors) but breaks SO(4) spinors, we see that the action of Θ is given by covariance. (1) 0 † (1) ΘΨα (x)=γαβCβγΨγ (θx) . (3.11) Hermiticity of the action This is the covariant version of the reality con- In general, a theory in Minkowski space will have dition one would obtain by simply performing a a Hermitian action. We have seen above that Wick rotation of (3.9). when a theory is mapped into Euclidean space using a Wick rotation, Hermiticity is lost. The 4. Supersymmetry algebras corresponding symmetry of the Euclidean action is Osterwalder-Schrader positivity, generated by Having defined the image of spinors under Wick the involution Θ. Using the action of Θ defined rotation, we are now in a position to investigate

4 TMR meeting, Paris, 1999 Arthur J. Mountain supersymmetry. In four dimensional Minkowski sets Z(0), Z(3) and Z(4) to zero, giving the al- space, the most general supersymmetry algebra gebra is (1)T µ (2) µ ν (1) [δ1,δ2]=2 C 2˜γ ∂µ + Zµν γ˜ γ˜ 1 . { } µ µ ν Qα,Qβ =2γαβ∂µ + Zµν (γ γ )αβ , (4.1) (4.3) This now looks very similar to (4.1) and indeed where the central charge Zµν is real and antisym- has 10 real degrees of freedom on the right hand metric. There are 10 real degrees of freedom on side. However, the supersymmetry parameter is each side of (4.1). a Dirac spinor (1) which has twice as many de- In Euclidean space, supersymmetry will be grees of freedom as a (Majorana) supersymme- generated by a “Euclidean Majorana spinor”. - try parameter in Minkowski space. Only half Before we impose the Osterwalder-Schrader pro- (1) (2) of these degrees of freedom can be physical but, jection, this is a Dirac spinor Q with Q = (1)T as mentioned above, there is no SO(4)-covariant Q C. We now have a familiar story. Clearly, (1) way of identifying the physical supersymmetry only half of the degrees of freedom in Q can generators. Following the familiar argument, we be physical but there is no SO(4)-covariant way would like to impose a non-local projection on to identify these degrees of freedom. This means the spinors to recover the degrees of freedom of that given a physical state |χi there is no SO(4)- the original Minkowski theory. At ﬁrst sight, this covariant way to decompose the 8 supersymm- looks inconsistent; the supersymmetry parame- etry generators as ter is constant so the Osterwalder-Schrader pro- Q →{QM,QN},M,N =1...4 jection is simply

QM|χi=0, QM|χi6=0. (1)T (1)† 0 C = γ , (4.4) We are again faced with a choice between SO(4) but this projection is not consistent with the ac- covariance and doubled spinors or physical spin- tion of SO(4) given by (3.1). However, we know ors which break this symmetry. that after imposing the Osterwalder-Schrader pro- The most general supersymmetry algebra ge- jection we arrive at a gauge-fixed theory (that is a nerated by a Dirac spinor in Euclidean space is theory which explicitly breaks SO(4) covariance (omitting the spinor component labels) by involving a reflection in a specific direction † µ (0) (2) µ ν labelled by t). The residual tangent space sym- Q, Q =2˜γ ∂µ+iZ 1 + Zµν γ˜ γ˜ metry is the SO(3) group which preserves t. This +iZ(3) γ˜µγ˜νγ˜ρ + Z(4)γ˜5 , (4.2) µνρ is generated by the transformations (2) where all the central charges are real and Z 1 j k (3) δψ = ωjkγ γ ψ,j,k=1,2,3. (4.5) and Z are antisymmetric. There are 16 real de- 4 grees of freedom on each side of (4.2). Translat- These transformations are consistent with the ing into Osterwalder-Schrader spinors gives the projection (4.4). Hence we see that this projec- commutator of two Euclidean supersymmetry (1) (1) tion is consistent as long as we remember that we transformations with parameters 1 and 2 as end up in a gauge-fixed theory where the only al- (c.f. equation 4.1 of [3]) lowed tangent group transformations are those of (1)T µ (0) (2) µ ν the form (4.5). [δ1,δ2]=2 C 2˜γ ∂µ + iZ 1 + Zµν γ˜ γ˜ After performing an Osterwalder-Schrader (3) µ ν ρ (4) 5 (1) +iZµνργ˜ γ˜ γ˜ + Z γ˜ 1 . projection, the supersymmetry algebra (4.3) has the correct number of degrees of freedom (Majo- This algebra looks rather different to the super- rana spinors on the left, 10 real degrees of free- symmetry algebra in Minkowski space (4.1). dom on the right). However, as this is supposed to be a represen- tation of supersymmetry in a Wick-rotated the- 5. The Wess-Zumino model ory, we must impose positivity under the Oster- walder-Schrader involution Θ. This condition The Wess-Zumino model is defined in Minkowski

5 TMR meeting, Paris, 1999 Arthur J. Mountain space by the Lagrangian • The Lagrangian is supersymmetric under the Euclidean transformations. In fact, de- 1 2 2 µ 2 2 L = (∂µA) +(∂µB) +ψγ ∂µψ − F − G monstrating invariance under these trans- 2 formations is equivalent algebraically to de- + g F(A2 − B2)+2GAB + ψ(A − γ5Bψ monstrating supersymmetry of the Mink- 1 + m FA+GB + ψψ . (5.1) owski Lagrangian. This is because the fun- 2 damental relation ψ = ψ among Majo- rana spinors holds as The spinor is Majorana, that is ψ = ψT C = † 0 ψ γ . This is invariant under the supersymmetry Ψ(2)(1) = (2)Ψ(1) , transformations among Euclidean Majorana spinors. µ δA = ψ, δF = −γ ∂µψ, 5 5 µ • The Lagrangian is manifestly SO(4)-cov- δB = γ ψ,δG=−γ γ ∂µψ, ariant and maps as L˜(x) → [L˜(θx)]† under − 5 µ − 5 δψ = ∂µ(A γ B)γ (F + γ G). Θ.

Now let us Wick rotate this theory. We will use Following the familiar argument, we can impose a the “Euclidean Majorana spinors” Ψ(1) and (1), non-local reality condition (Majorana condition) (2) (1) (2) (1) with Ψα = CαβΨβ and α = Cαββ .There on the scalars (spinors) to recover the degrees of is no reality condition on Ψ(1) or (1).TheLa- freedom of the original Minkowski theory. grangian becomes Finally we should look at representations of the supersymmetry algebra in the Wick-rotated 1 2 2 L˜ = (∂µA) +(∂µB) theory. Each Euclidean field has double the de- 2 i grees of freedom of its Minkowski counterpart. (2) µ (1) 2 2 +Ψ γ˜ ∂µΨ − F − G As this is true for the supersymmetry parame- (1) 1 ter we would expect the representations of + m FA+GB + Ψ(2)Ψ(1) 2 supersymmetry to be twice as large in the Eu- + g F (A2 − B2)+2GAB clidean case. This happens in a rather unusual i way; every state in the supersymmetry algebra is (2) 5 (1) +Ψ (A − γ B)Ψ . (5.2) “complexified” so that the whole algebra is con- structed in terms of complex degrees of freedom. whereγ ˜µ =(−iγ0,γk) are the Euclidean Dirac matrices. This Lagrangian is invariant under 6. Eleven-dimensional supergravity (2) (1) (2) µ (1) δA = Ψ ,δF=−γ˜ ∂µΨ , We will now briefly discuss the application of this (2) 5 (2) 5 µ (1) δB = γ˜ ,δG=−γ˜ γ˜ ∂µΨ , work to 11 dimensional supergravity [7]. We have (1) 5 µ (1) 5 (1) δΨ = ∂µ(A − γ˜ B)˜γ − (F +˜γ G) . seen above that Majorana spinors can be consis- tently treated under a Wick rotation. With this Now there are some important points to make in mind, we leave the details to [9] and concen- trate for now on the bosonic sector. The bosonic • There are twice as many degrees of free- sector of eleven dimensional supergravity is dom in the “Euclidean Majorana spinors” (1) (1) e e Ψ and as in a Majorana spinor in L − − µ1...µ4 = 2 R Fµ1...µ4 F Minkowski space. 4κ 48 2κ µ1...µ11 + 4 Fµ1 ...µ4 Fµ5...µ8 Aµ9µ10µ11 .(6.1) • In Euclidean space, the supersymmetry va- (12) riations of A, B, F , G are not real so it In this, e is the determinant of the elfbein, A is makes more sense to regard these as com- a three-form field and plex scalar fields. Hence all fields have been

“doubled” under the Wick rotation. Fµ1...µ4 =4∂[µ1Aµ2µ3µ4].

6 TMR meeting, Paris, 1999 Arthur J. Mountain

This LagrangianR is Hermitian and so is the ac- of freedom of the Minkowski theory but breaks tion S = d11xL(x). This action would appear the SO(4) symmetry. We have shown how the inside a path integral as exp(−iS). We now per- supersymmetry algebra is consistent with this form a Wick rotation t →−it as described above. prescription for Wick rotation. We have pre- In the non-covariant formulation (after the non- sented a Wick rotated version of 11 dimensional local projection has been carried out) we have a supergravity. It would be interesting to examine Euclidean Lagrangian L˜ which satisfies the role, discussed in the introduction, that this h i† might play in cosmology. L˜(θx) = L˜(x) . (6.2) Under the Wick rotation we find Acknowledgments Z ∞ Z −iS →−S˜=−i (−idt) d10xL˜(x) −∞ We thank Tim Evans, Hermann Nicolai and Dan Z ∞ Z Winder for discussions. AJM thanks the Royal 10 = − dt d xL˜(x) . Commission of 1851 for a Research Fellowship. −∞ This work was supported in part by PPARC un- Given that L˜ satisfies (6.2) we see that whether a der the SPG grant PPA/G/S/1998/00613. given term of L˜ gives a real or purely imaginary contribution to S˜ depends on its properties under A. Spinors in four dimensions time-reflection. Let us look closely at two terms µνστ in the action. The term Fµνστ F is positive Much of this paper draws on examples in four under time-reversal and hence its contribution to dimensions. In this Appendix, we list briefly our S˜ is real. The Chern-Simons term conventions for spinors in both Minkowski and µ1...µ11 Fµ1...µ4 Fµ5...µ8 Aµ9µ10µ11 Euclidean four dimensional space. We use the Clifford algebra conventions of [8] and refer the is negative under a time-reversal and hence its reader to this paper for more information. In contribution to S˜ is imaginary. This has an im- all four-dimensional calculations, Greek indices portant physical meaning; we want gauge trans- µ,ν,... runfrom0to3(1to4)whenthespace- formations to be quantised in both Lorentzian time is Minkowski (Euclidean). Latin indices and Euclidean settings. This corresponds to the i,j,k,... run from 1 to 3. Chern-Simons term contributing to the path in- −iSCS tegral as e where SCS is real. We have seen Minkowski space that this is indeed the case. Whereas the kinetic term picks up a factor of −i under Wick rota- The Clifford algebra in four-dimensional Mink- tion, the Chern-Simons term in the exponent of owski space is e−iS is purely imaginary in both Lorentzian and {γµ,γν}=2ηµν = 2diag(+1, −1, −1, −1)µν . Euclidean cases. (A.1) The Dirac matrices γµ may be chosen such that 7. Conclusion γ0† = γ0 ,γk†=−γk,k=1,2,3. (A.2) We have shown how to treat supersymmetry under a Wick rotation. The key to this is the cor- In a theory with Majorana spinors, there exist rect treatment of spinors. There are two possible matrices B and C such that Euclidean-space descriptions of the Wick rotated ∗ theory. One has explicit SO(4) covariance but γµ = BγµB−1,B†B=1,BT=B, involves doubled spinors. A Lagrangian which is µT µ −1 † T γ =−Cγ C ,CC=1,C=−C. Hermitian in Minkowski space has Osterwalder- Schrader positivity when Wick rotated. The oth- and BT = Cγ0. The Dirac equation is er is obtained by halving the spinors by a nonlo- µ cal projection. This recovers the physical degrees (iγ ∂µ + m) ψ(x)=0. (A.3)

7 TMR meeting, Paris, 1999 Arthur J. Mountain

We can see from the above definitions that if ψ is [5] P. van Nieuwenhuizen, A. Waldron, On Eu- a solution of this equation then so is B−1ψ∗.We clidean spinors and Wick rotations, Phys. can then equate B−1ψ∗ and ψ via the Majorana Lett. B 389 (1996) 29[hep-th/9608174]; A condition continuous Wick rotation for spinor fields ∗ and supersymmetry in Euclidean space, ψ = Bψ. (A.4) [hep-th/9611043]. This can be rewritten as an equality between the A. Waldron, A Wick rotation for spinor fields: Dirac and Majorana conjugates the canonical approach, Phys. Lett. B 433 (1998) 369[hep-th/9702057]. D † 0 T M ψ = ψ γ = ψ C = ψ . (A.5) [6] B. Zumino, Euclidean supersymmetry and the many-instanton problem, Phys. Lett. B69 Imposing the Majorana condition clearly halves (1977) 369 the number of degrees of freedom of a spinor. [7] E. Cremmer, B. Julia, J. Scherk, Supergrav- ity theory in 11 dimensions, Phys. Lett. B76 Euclidean space (1978) 409 The Clifford algebra in four-dimensional Euclid- [8] T. Kugo, P. K. Townsend, Supersymmetry ean space is generated by {γ˜µ} where and the division algebras. Nucl. Phys. B 221 (1983) 357 k k 4 0 γ˜ = γ for k =1,2,3, γ˜ =iγ . (A.6) [9] A. J. Mountain, K. S. Stelle, To appear.

The algebra is

{γ˜µ, γ˜ν} = −2δµν . (A.7)

We haveγ ˜ µ† = −γ˜ µ for all µ. The matrixγ ˜5 = iγ˜1γ˜2γ˜3γ˜4 is diagonal and antiHermitian. The charge conjugation matrix is the same as that for Minkowski space and hence satisﬁes C†C = 1, CT = −C andγ ˜µT = −Cγ˜µC−1. The Majorana condition is not consistent in Eu- clidean space as B∗B = −1.

References

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