Extended supersymmetries in topological Yang-Mills theory

Extended supersymmetries in topological Yang-Mills theory

Volume 236, number 1 PHYSICS LETTERS B 8 February 1990 EXTENDED SUPERSYMMETRIES IN TOPOLOGICAL Y A N G - M I L L S T H E O R Y A. G A L P E R I N l...

443KB Sizes 0 Downloads 41 Views

Volume 236, number 1

PHYSICS LETTERS B

8 February 1990

EXTENDED SUPERSYMMETRIES IN TOPOLOGICAL Y A N G - M I L L S T H E O R Y A. G A L P E R I N l Laboratory ofTheoretical Physics. JINR, P.O. Box 79, SU-IOI 000 Moscow, USSR

and O. O G I E V E T S K Y Theoretical Department, P.N. Lebedev Physical Institute, Leninsky prospect 53, SU-117 924 Moscow, USSR

Received 14 November 1989

Extended supersymmetriesin D= 4 topologicalYang-Millstheory are studied. Their existenceimposes constraints on the metric of the manifold, on which the topological Yang-Mills theory is defined. For irreducible manifolds we establish a 1-1 correspondence between extended supersymmetries and covariantly constant complex structures. In particular, the theory possesses one additional supersymmetryon a K~ihlermanifold. By analogywith a general riemannian case, this givesa way for the construction of invariants of complex structures. Ingredients, needed for this construction, such as the Donaldson map, are generalized for K~ihlermanifolds.

1. Introduction A few years ago new invariants of four-manifolds were discovered [ 1 ]. Later on, they were interpreted as expectation values in a q u a n t u m N = 2 , D = 4 euclidean supersymmetric Yang-Mills theory ( S Y M T ) with the SU (2) a u t o m o r p h i s m group of N = 2 supersymmetry identified with the SU (2) L subgroup of the euclidean tangent group S O ( 4 ) - - ~ S U ( 2 ) L × S U ( 2 ) R [2]. It turns Out that on a curved background a singlet rigid supersymmetry survives in spite of the absence of graviton's superpartners. Roughly speaking, the existence of this supersymmetry implies that the expectation values ofsupersymmetric observables do not depend on the choice of r i e m a n n i a n metric and hence give invariants of a smooth structure of the underlying four-manifold. One may wonder whether extended supersymmetry in the topological Yang-Mills theory ( T Y M T ) is possible. The m a i n purpose of the present paper is to study this problem. We restrict ourselves to the most On leave of absence from Institute of Nuclear Physics, SU-702 132 Ulugbek (Tashkent), USSR.

interesting case of four-manifolds which are not locally metric products. We find that the additional supersymmetry requires the existence of a covariantly constant complex structure on the four-manifold. This complex structure reduces the holonomy group of the manifold. In four dimensions there are essentially two nontrivial possibilities: the holonomy group can be reduced to U ( 2 ) or S U ( 2 ) . In the former case there is just one complex structure while in the latter one there are three complex structures. We prove that the additional supersymmetries are in 1-1 correspondence with covariantly constant complex structures. Therefore when the holonomy group is U (2) (that is when the manifold is K~ihler) the T Y M T theory has one additional supersymmetry, and when the holonomy group is SU (2) (hyperK~ihler manifolds) there are three additional supersymmetries. This picture resembles amazingly the situation with D = 2 omodels: the target space is an arbitrary r i e m a n n i a n manifold in the N = l case, while in the N = 2 (or, equivalently N = 1, D = 4) case the target space should be K~ihler, and for N = 4 (or N = 2 , D = 4 ) only hyperK~ihler manifolds are allowed [3]. Here, extended supersymmetry requires the existence of co-

0370-2693/90/$ 03.50 © ElsevierScience Publishers B.V. ( North-Holland )

33

Volume 236, number 1

PHYSICS LETTERS B

variantly constant complex structures and hence reduces the holonomy group of the corresponding target space. The only difference from the TYMT is that in the TYMT all supersymmetry generators anticommute. In the same way as the original TYMT [2] gives invariants of a smooth structure, in our case the theory gives invariants of complex or hypercomplex structures. We show that the properties of the lagrangian and the energy-momentum tensor, needed for the expectation values of observables to be invariants, are satisfied in the K/ihler and hyperK~ihler cases. The basic observables in the TYMT on a general riemannian manifold are connected with the Donaldson map which relates the de Rham cohomology groups of the riemannian manifold with the de Rham cohomology groups of the modulus space of instantons on this manifold. On the K~ihler manifold the de Rham groups are decomposed into the Dolbeault groups. We propose a generalization of the Donaldson map which relates the Dolbeault groups of the K/ihler manifold with the Dolbeault groups of its modulus space of instantons. The paper is organized as follows. In section 2 we establish the basic relation between holonomy groups and extended supersymmetries. In section 3 we discuss properties of the TYMT on K/ihler manifolds, generalize the Donaldson map for this case and make several comments about the hyperK~ihler case. This paper is a short version of a text which will appear elsewhere.

2. Extended supersymmetries One of the proposed possibilities in ref. [ 2 ] to construct the TYMT starts with the N = 2 , D = 4 euclidean SYMT. The Yang-Mills N = 2 multiplet consists ~ of a gauge field A~a, spinor fields ~,~, ~'a~ and *~ We use the following notations. The Lorentz group of the euclidean four-space is locally isomorphic to the group S U ( 2 ) L X S U ( 2 ) R. W e d e n o t e i n d i c e s o f f u n d a m e n t a l representation of SU (2) L by a, fl, 7..... and of SU (2) R by &,/~, p..... The automorphism group of N = 2 euclidean supersymmetry is SU (2)A X R*, where ~* is the multiplicative group of nonzero real numbers. It replaces the U( l ) subgroup of the automorphism group of N = 2 supersymmetry in Minkowski space. We denote indices of the fundamental representation of SU(2)A by i,j, k .....

34

8 February 1990

scalar fields M, N. For the closure of supersymmetry algebra it is necessary to introduce an auxiliary field T i j = Tji. All these fields are antihermitean and take values in the Lie algebra of some compact gauge group G. The supersymmetry algebra is graded by dimension and by ~*-charge. The TYMT can be obtained from the N = 2 SYMT by changing the action of the Lorentz group, namely by replacing SU(2)L by the diagonal subgroup in SU (2) LX SU (2) A (or, in other words, identifying the indices o~, fl.... with the indices i,j .... ) and minimally coupling the theory to euclidean gravity. This amounts to considering the action S = f d 4 x x/g L, where

L=tr{ ~F ~,F + u"+ i ( %,w.)z*'"- ½i~,~ [ ~,~, N] - ½i~,[ ~,, M]

-~[M,N]2+~i[x*`v, Mlz*`V-~T*`vTUV}.

(I)

Here /z, v.... are four-dimensional world indices, Fu+ is the self-dual part of the YM field strength, + Fu~ = ½( F v , + P u , ) , the fields Tv, and Z,,, are selfdual. The original N=2 SYMT was invariant under supersymmetry transformations with parameters ~,, ~'i. Witten observed [2] that the singlet supersymmetry with parameter ~= ½~'f~j survives on an arbitrary gravitational background g*`,. We want now to study what happens with other supersymmetries of the N=2 SYMT. The supersymmerry p a r a m e t e r ~ = ½(~i + ~i~) becomes a self-dual tensor. Since we have no preferred coordinate system we cannot take it to be a constant. So ~ is replaced by an arbitrary self-dual tensor ~v,. The transformation law of the N= 2 SYMT fields takes the form

8A~=i~j~u.,

8N=½i~*`~X*`.,

8M=0, I

*`tJ

+

~Xltu-!'y-pT 2~11 a p t P - - %;-P~+ l t - - p u +I~u.[M, N I - ( t 2 ~ v ) aT,,, = l i (~uP~p~,,-~P C¢p~,u) + -I- ½i&[zp., M ] -(i~

v),

(2)

Here Z,,+ stands for the self-dual part of a tensor Zu~. (Note that for self-dual tensors Xu~, Y,,, the expression Xu"Y,¢-X,~Y,~, is also self-dual, so the RHS of 6Z,., and 6Tu, have the correct symmetry.) A straightforward though slightly cumbersome calculation shows that the lagrangian ( I ) is invariant under

Volume 236, number 1

PHYSICS LETTERS B

this transformation iff ~,. is covariantly constant,

%~=0. Let us consider a one parameter transformation with ~u,= gd~/~,,,where ~ is a Grassmann constant. One may check that the conditions H~,=/Tu, and (/pHi,,=0 imply that Hu"Hf=-226P,, where 2 is a real constant. Thus, I j = ( 1 / 2 ) H S is a covariantly constant complex structure. Moreover, since H j is self-dual, it follows that the metric is hermitean with respect to this complex structure. We conclude that an additional supersymmetry of the form (2) defines uniquely a K/ihler structure on the manifold. The N = 2 SYMT on the flat background possesses also supersymmetries with parameter ~ . Later we will briefly discuss the effects of their presence in the curved space. The existence of a covariantly constant complex structure on the manifold reduces its holonomy group. In dimension four there are, in essence, two possibilities for irreducible manifolds: the holonomy group reduces to U (2) or to SU (2). In the first case there is only one covariantly constant complex structure (and the manifold is called K~ihler), while in the second case there are three covariantly constant complex structures (and the manifold is called hyperK~ihler). Hence we have come to the following statement: the TYMT has one additional supersymmetry operator on a K~ihler manifold and three additional supersymmetry operators on a hyperK~ihler manifold. A comment. Our original motivation for the existence of extended supersymmetries stemmed from N = 2 conformal supergravity arguments, like those proposed by Karlhede and Ro~ek [4] in the case of the singlet supersymmetry. They considered the coupling of N = 2 SYMT to conformal N = 2 supergravity (not Einstein N = 2 supergravity, since SU(2)A becomes local after twisting and the local SU (2) Agroup is specific for the conformal case). In TYMT only the vierbein is left from the whole supergravity multiplet. Therefore TYMT has only those supersymmetries which preserve this configuration of supergravity multiplet. After twisting and identifying the spin SU (2)L connection with SU (2)A connection we have for the left-handed gravitino field 8q/u = 0u~ ,

(3)

8 February 1990

where ~ = ~ " . The requirement 8~,u = 0, 8q)~a = 0 leads to supersymmetries of TYMT. One possibility is obvious ~"a=0,

~=const.

This gives precisely the singlet supersymmetry of TYMT. Eqs. ( 3 ), (4) show moreover that additional supersymmetries lead to the existence of covariantly constant ~"a. This proves once again our statement. Remark. We could have tried to look for supersymmetrics from the right sector (i.e. with parameters ?,a'i before twisting). Then we would have had to worry about the gravitino field q/ua, or, after twisting, q/uap. Its transformation law is 6 ~ p = c-/u~p. Demanding that 8~ua p = 0, we find that the existence of such supersymmetry leads to the existence of a covariantly constant vector ~,=euaP~-p on the manifold, cJu~,= 0. One can show that this equation implies that the holonomy group is SO(3). So in this case we are back in three dimensions which was the starting point of the whole story for topological YangMills theory [2,5 ] and where the Floer groups arose [6]. However, we are concerned in this paper with the irreducible case (i.e. with the case when the manifold is not a local metric product) and so we will not pursue this line further.

3. Topological Yang-Mills theory on K~ihler manifolds Let us now discuss the K~ihler case in more detail. Let z m, m = 1, 2 be local complex coordinates. The self-dual tensor Zu, decomposes into Zm,,, Z,~,, and Z,,,,=Zgm,,, and analogously for Tu,,. Let Q be the singlet supersymmetry operator in the general case ~2, and Q' be the additional supersymmetry operator arising from the K~ihler structure (so that 8X= ~Q' X in formulas (2) ). It is convenient to introduce fields B=~'--ZI2, C = ~u+Zl2 and use operators q = ½( Q Q' ), #= ½( Q + Q ' ) . One easily checks that with our

~2 We recall that QAu=i~u,

QM=O,

Q~u = - ~u M ,

QN=2i~',

Q~u=½[M,N],

Q Z u . - Tu,, + 2 F u,, + ,

QTj,.=-2i(~u~U.-~Auu)+-i[xu.,M],

Qgu. = 0 •

35

Volume 236, number 1

PHYSICS LETTERS B

reality conditions the operators q and c7are complex conjugated. We have

8 February 1990

where W = tr(½Xm.Z m" - ½BC+ iFN) .

qAm=O,

qAm=i~ll,.,

q;6.. = 2F; ....

qM=O,

qX,,,.=T,~.,

qB=-½T-F+½[M,N], q~',, = - C-/mM ,

qC=O,

qvl,~ = 0 ,

qT,,,,, = 2i c-/~6,,- 2i %.~,. - i [z ..... M ] , qT,.. = 0 ,

qT=-2ig""%.q/,~--i[C,M],

(5)

where Fm~= [ elm, 9~], F,~n= [ C~m, 9 . ] , F = g m " [ %., % ] . Formulas for c7are obtained by complex conjugation. The lagrangian of the theory takes the following form in complex coordinates:

:/m~.).l""-iz,,,.:/mq/"+i(:-:,,,~/m)B

+i(9'"~u,.)C-]i[z

.... Z ' n " I M - i [ v / . ,

(6)

(7)

where tr(agv.z

+½~u. c z v U - ~ g [ M , U ]

-~T..z"").

(8)

In the K~ihler case, these formulas can be refined: L=qglW, 36

1

o-r

1

8g~.~Z/,.- ~e,,..zpw/g g

2o-

5g~z

vo

,

8 T ~ . - 'a g ~ 8 g ~ ( T u . + 2 F u . )

T;'"=Q2U.,

The careful reader may ask where the K~ihler condition for the metric comes into play, or how it is seen in complex notation that the lagrangian (6) is not invariant under transformations (5) on the background of an arbitrary hermitean metric. The answer is that in the lagrangian (6) one meets expressions like %,,~m=~m~,m+Fmk"~k-F,,g"~,E , and the supersymmetry transformation of the term containing F,,,v"~,Zdoes not cancel. The detailed analysis shows that this is the only obstruction. So the condition ensuring the invariance of the lagrangian (6) is g'""F,,,,k=O (and complex conjugated). One can show that in four dimensions this equation is precisely equivalent to the K~ihler condition. Now we turn to the invariants of a complex structure. On a general riemannian manifold the lagrangian ( 1 ) can be written in the form L=QV.

~/~,,--- ag

and all other fields do not vary. Now

~u"lN

- ½iM[B, C] -- ~ [M, /V]2 1 T TmnA",--a*mn--~r2}.

For constructing topological invariants in the general case it was very important to show that the energym o m e n t u m tensor T u" of the theory can also be represented as Q acting on some combination fields. To show this one must first define variations of the fields under an infinitesimal change of the metric. It is natural to require that the operators Q and 8/Sgu. commute, [Q, 6/8g/,.] = 0 . One may check that this is so if

- ½e..aox/g g;°Sgo~( T~P- Z g - w ) ,

L = t r { F m ~ F m " - ½F2-f - ( ~ m C / m M ) N + ½F[ M, N]

+i(

( 1O)

qN=iC,

(9)

(11)

with 2/'"=

~gOg/,. ( x / g

V).

(12)

(In ref. [2], Q c o m m u t e d with 8 / 8 g . . only on shell, so eq. ( 11 ) needed a tedious verification which we have avoided by using the auxiliary field Tu. and the appropriate transformation law for this field. ) In the K~ihler case we have to worry only about the variation of the action with respect to a K~ihler potential K. Let g t = 8S/8K, g = d e t g .... We require that all fields except T do not vary, while the variation of T is as follows: 6 T = - 2F,.. 8g "~ . Then one can check that 8 / 8 K c o m m u t e s with the operators q, g This together with (9) implies that t can be represented in the form t=qCl2. The construction of topological invariants proposed in ref. [2] used path integrals of the action ( 11 ) which is not positively defined. So this construction is more suggestive than rigorous. However, playing freely with ill-defined objects gives self-consistent resuits which can be verified by different methods. Witten shows that for an observable A which by definition is invariant under supersymmetry, QA = 0, and under changes of the metric, 8A/Sg""=O, it follows

Volume 236, number 1

PHYSICS LETTERS B

from ( 11 ) that the expectation value ( A ) does not depend on the choice of a metric and hence carries only topological information. Moreover, (7) implies that ( A ) does not depend on the coupling constant. Thus we may work in the limit e ~ 0 . In this limit the path integral is concentrated on the superspace ,7/of configurations on which the action is equal to zero. Witten argued that the bosonic part .,,/! of ~7/is the modulus space of self-dual connections, while the fermionic part is described by the tangent sheaf to ~//. The space 77/is Q-invariant and the singlet supersymmetry operator Q reduces to the wedge derivative on • tl considered as a vector field on o,7/. Witten constructs from the fields of the theory forms HI,.,, i= 0 ..... 4, which obey the property dWj=QWj+~,

(13)

where for j = 4 we put Ws=0. Now let co be an arbitrary given j-form o n M 4 which does not depend on fields. Assume that co is closed, dco=0. Then consider the observable ~b= fM" A4_~ ^ O9. We have Q~b =fQA4_j^ ~o=fdA3-jA 09=fd(A3_j^ 0~) =0. Thus ~b satisfies the conditions needed for constructing topological invariants. Moreover, if c o = d a then d,~= Q [ ( - 1 )J+ l f A s _ j ^ a]. The expectation values of observables A such that A = Q B vanish. Therefore we may obtain nontrivial observables only from nonzero de Rham cohomology classes of M 4. By construction, functions on ,7! are differential forms on ,//. Moreover, the ~*-grading corresponds to the grading of differential forms by their degrees. The recipe for the construction of the Donaldson invariants proposed by Witten [ 2 ] is as follows. Each time when A~contains M we replace it by ( M ) where ( M ) is the solution of the equation ~ u ~ u ( M ) = i [ ~/u, ~uu ]. (Here c/~ is a self-dual connection and ~uu is a zeromode. ) Let A~ be the result of this substitution. Now let



fA'4_j^o .

(14)

M

Witten showed that q~o~is a well-defined closed differential j-form on ,~#. Thus, closed j-forms o n M 4 correspond to closed j-forms on Jg and exact forms correspond to exact ones. This gives a map

HJ(M4,~)--~.HJ(~#,[R),

o9v-~ ¢1)o~,

(15)

which is called the Donaldson map. (We used here

8 February 1990

cohomology classes instead of cycles, used in ref. [2], for future convenience in the discussion of the Dolbeault groups in the K~ihler case.) The space J [ consists of components ~/[', enumerated by the topological charge. If we take arbitrary closed forms o91, ..., oak of degrees jl . . . . . Jk SO that Zj~ = dim , # , for some n then q)~l ^ ... ^ q~,o, is a volume form on ~¢[nand one can check that ((~)l'"(~k) = f ~ml A"'A(1)mk" .¢t,

(16)

Topological invariants obtained as expectation values of observables of the form o51...~Sk precisely reproduce the Donaldson invariants. As we have seen above, the TYMT on a Kahler manifold has two supersymmetry operators, q and q. They are complex conjugated and i Q = iq+ iq. This is reminiscent of the wedge differential on complex manifolds ( d = 0 + 0 ) . In fact it shows that the modulus space J# ofinstantons on a Kahler manifold has a natural complex structure. Therefore the space of differential forms on ~¢! is bigraded. In fact, in the K~ihler case the whole space of fields of the theory is bigraded. This bigrading is given by the formulas deg' A,, = deg' A m = (0, 0 ) , deg'q/m=(0,1),

deg'q/m=(1,0),

deg' M = ( 1 , 1 ) ,

deg' N = ( - 1 ,

deg'Zm,=(-1,0),

-1),

deg'zm"=(0,-1),

deg'B=(-l,0),

deg'C=(0,-1),

deg' T i n , = ( - 1 , 1 ) ,

deg' T i n ' = ( 1 , - 1 ) ,

deg' T = (0, 0 ) , deg' q= ( 1 , 0 ) ,

deg' # = (0, 1 ) ,

(17)

where deg' X is the bidegree of X. Note that the degree of X given by ~*-charge is simply the sum of bidegrees of X. It is easy to see that the bigrading on the space of differential forms on ~/! is inherited from the bigrading (1 7). In the K~ihler case we should consider forms B,,s instead of Wi (see (13) ), where B o . o = t r M 2,

Bl.o=-2tr(M~%)dz',

Bo, l = 2 tr(M~,~)

dg m ,

B2,o=tr(~'mqG) d z ' d z ' ,

Bo,2 = t r ( - - ~ m ~ , , + ~iMT,,,,) d g " d2",

(18) 37

Volume 236, number 1

PHYSICS LETTERS B

Bl,l = 2 tr( ~mq/n + iMFmn) d z " dg n ,

where F, nn = [ 9m, ~ ]

,

B2, i = 2i tr ( ~UmFn~) dz " dz ~ dg k , B~,2 = tr ( 2i~¢Fme + ½i M C/mXe~ -- ½i~'mTn~ ) dz m dg ~ dg k ' B2.2 = tr [ ½Fm~F~r+ F, n r F ~

- ½i~.(~%Z~r) ] dzm dz" dZ~ d U .

( 18 c o n t ' d )

Note that deg' Br.s: ( 2 - p , 2 - q ) ,

[gr,s] : c m

-2-(l/2)(p+q)

We have 4nr,s:OB .... 1,

(19)

where 0 = d Z m O/OZ m. (We put Br_~ = 0 in the RHS of (19) when s = 0. ) The forms Br,s do not have a good behavior with respect to q since in general forms Br,s are not equal to Bs,, Formula (14) now takes the form q~o~= ~ B'2 - - r , 2 - - s A O ) ,

Acknowledgement

where (o is a 0-closed (r, s)-form on M, OoJ=O. Finally, we obtain the map (21)

generalizing for the K/ihler case the Donaldson map. In conclusion we note that in the K~ihler case invariants of the complex structure can be constructed starting not only from observables A such that QA = O. It is sufficient to d e m a n d the invariance of A under any nonzero linear combination o f q and q. The proof goes along the same lines as in ref. [2 ], so we omit it. A c o m m e n t about the hyperKiihler case. The T Y M T on a hyperK~ihler manifold has supersymmetries Q, Q.p and the lagrangian can be written in the form L = QQ.pQ~Qy~ Y ,

(22)

where Y=½ t r N 2. The formulas (7), (9), (22) are suggested by the superspace form of the lagrangian

38

for N = 2 Yang-Mills theory in flat space. The action contains an integration over Grassmann variables which is equivalent to the application of supersymmetry transformations. Let 6gu~ be a variation o f the hyperK~ihler metric compatible with all three complex structures, 6X be the corresponding variation o f a field X. Fixing any complex structure from the S2-family we come back to the K~ihler case where we know that [ 8, Q' ] = 0 for Q' corresponding to this complex structure. This argument proves that [8, Q~p] = 0 for all Q,p. Hence, similarly to the K~ihler case, the variation of the action can be represented as QQ,pQPyQ"y acting on some field combination. Now, analogously to the K~ihler case, the modulus space o f instantons on a hyperK~ihler manifold has a natural hypercomplex structure. Hence the whole story with invariants goes as well for hyperK~ihler manifolds. One can generalize the Donaldson map for the hyperK~ihler case and obtain invariants of a hypercomplex structure from observables which are invariant under an arbitrary combination of Q, Q,a.

(20)

M

H"~(M)---,H~'S(,//) ,

8 February 1990

We are grateful to S. Krivonos for explanations of the basics of TEX and to V.I. Ogievetsky and K.S. Stelle for useful comments.

References [1] S.K. Donaldson, Polynomial invariants for smooth 4manifolds, Oxford preprint (1987). [2] E. Witten, Commun. Math. Phys. 117 (1988) 353. [3] J. Bagger, in: Proc. of the Bonn-NATO Advanced Study Institute on Supersymmetry (Plenum, New York, 1985) p. 213. [ 4 ] A. Karlhede and M. Ro6ek, Phys. Lett. B 212 ( 1988 ) 51. [ 5 ] M.F. Atiyah, in: The mathematical heritage ofHermann Weyl, Proc. Symp. on Pure mathematics, Vol. 48, ed. R. Wells (American Mathematical Socity, Providence, RI, 1988 ). [6] A. Floer, Commun. Math. Phys. 118 (1988) 215. [7] M. Itoh, J. Math. Soc. Japan 40 (1988) 9.