- Email: [email protected]
Geometry of T-duality
13 ELSEVIER
Nuclear Physics B 514 [PM] (1998) 721-739
Geometry of T-duality Javier Borlaf l Departamento de Ffsica Te6rica, Universidad Aut6noma, 28049 Madrid, Spain Received 2 July 1997; accepted 29 December 1997
Abstract
A "reduced" differential geometry adapted to the presence of abelian isometries is constructed. Classical T-duality diagonalizes in this setting, allowing us to get conveniently the transformation of the relevant geometrical objects such as connections, pullbacks and generalized curvatures. Moreover we can induce privileged maps from the viewpoint of covariant derivatives in the target space and in the world-sheet generalizing previous results, at the same time that we can correct connections and curvatures covariantly in order to have a proper transformation under T-duality. (~) 1998 Elsevier Science B.V. PACS: 11.25; 11.27; 11.30; l l.30.p; 02.40.k; 02.40.H,M Keywords: Strings; T-duality; Geometry; Abelian killings
1. I n t r o d u c t i o n
T-duality is a fundamental tool in the understanding of the fashionable string dualities [ l ]. The elementary statement of the T-duality establishes that the perturbative spectrum of a string theory with a dimension compactified on a circle of radius R, is equivalent to the one compactified in a circle of radius l/R, provided we interchange winding and momentum quantum numbers at the same time that we transform the string coupling constant [8,10]. If we allow the presence of a generic geometry (metric G#~, torsion potential B ~ and dilaton ~/,) having an abelian Killing vector in the compactifled direction X °, the backgrounds resulting to be one-loop (conformally)equivalent are given by Buscher's formulas [2]: I E-mail: [email protected] 0550-3213/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PII S 0 5 5 0 - 3 2 13 ( 9 8 ) 0 0 0 0 4 - 2
722
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739 : I/Coo, Goi =
Boi / Goo,
Boi : Goi / Goo,
Gij = Gij - ( GoiGoj - BoiBoj )/Goo, l~i.i = Bij - ( GoiBoj - BoiGoj) /Goo,
~b=~/, - ~1 In Goo.
(1)
In recent years a non-perturbative usage of T-duality has been made in the context of open strings playing with the map of the Neumann-Dirichlet boundary conditions. The promotion of the hypersurfaces in which strings rest their end points to be dynamical extended objects called Dirichlet branes, allow their identification with the carriers of the RR charges required by the string duality at the same time that it makes doubtful the name of "string theory" for the resulting scenario [4,5]. Many features of this topic of T-duality have been developed extensively in the literature [3,6-8]. Despite that important effort, there does not seem to be a systematic study of the mapping between geometries for the stringy (one-loop) equivalent spacetimes. This gap is related with the nature of the non-linear map (1) which highly complicates the calculations for the transformation of geometrical objects such as the generalized connection and its curvature, privileged maps for the covariant derivatives and pullbacks, and many others. In order to clarify all these points some kind of "parallel differential geometry" that we will call "reduced geometry" is presented, having the property of being adapted to the presence of abelian Killing vectors (in fact it is only defined in that context). We will see how T-duality transformations diagonalize in this setting for the main geometrical objects, including the generalized curvature; for the latter, we found for the first time its complete transformation, which can be expressed in a covariant way in terms of itself and of the Killing vector. Moreover, a "canonical" T-duality transformation is constructed for arbitrary tensors with the property of transforming linearly the covariant derivatives calculated from the generalized connections. These results unify and generalize the map obtained for the p-forms in [6], and it includes the fundamental one of the complex structures for holomorphic Killing vectors [7]. We can extend the result giving above to the "canonical" T-duality transformation of maps from the world-sheet to the tangent space of our target-space manifold. The classical string dynamics will be the most representative example of this "canonical" map. In Section 2 we define and describe the construction of the "reduced geometry" giving the basic map relating "usual" and "reduced" objects (generic tensors, connections and curvatures). In Section 3 we get the "canonical" T-duality map, relating linearly covariant derivatives, and a "non-canonical" one relating linearly "covariant divergences". In Section 4 we obtain the "canonical" map for the classical world-sheet dynamics. Section 5 shows the generalized curvatures' transformation and the minimal correction for them to transform linearly under T-duality. As a straightforward outcome I rederive the
J. B o r l a f / N u c l e a r
Physics B 514 [PM] (1998) 721-739
723
one-loop beta function's transformation. In Section 6 we found a "canonical" covariant derivative commuting with the "canonical" T-duality transformation. It is used to get a set of new T-duality scalars. In the appendices we summarize the basic formulas.
2. Reduced differential geometry In this section a parallel tensor calculus for manifolds with a structure endowed with abelian Killing vectors will be developed. The main objective is to exploit the presence of these Killings in order to get a strongly simplified structure that I will call the "reduced geometry". As we will see, that structure is nicely adapted to T-duality. Let us assume the existence of a set of n commuting vector fields {Ka~} with {/~, v = 0, 1. . . . . D - 1 }, {a, b = 0 . . . . . n - 1 } and D the dimension of the manifold M. We restrict our attention to the space s2 of tensors V in M satisfying £k,,V = 0,
(2)
which means simply that we can choose coordinates {x i, x a} with i = n . . . . . D - 1, called adapted coordinates, in which V does not depend on x ~, i.e. OaV = O. The covariant differentiation is not a mapping in O, or in other words, its commutator with the Lie derivative is in general non-vanishing, m
[£k~, ' ~ p ] V 1)1 m'''''ut ,...,1)n,
V"(£.
= -- ~..~"
1
w~cr Vm,...,u~ 1)l,.,O-,.,p m
ka'ClPb'~
~,-"(£"Ra V ~)pO" V1)1,...,1)m m'''~'~'
~U ~ . . d ~
S=I
(3)
r=l
where I have defined 2 o"
o"
(t
(1"
(£k ~7)7~~ = K~R,~,~ v + V u ~ T v K a + 2~7u(KaT~,,),
(4)
R~1) being the curvature for the connection FSa# and TuP~ the corresponding torsion. It can be checked that (4) reduces to the desired 0 ~ Fo"~ in adapted coordinates. If we are interested in connections preserving the condition (2), we must impose £k,,V = o.
(5)
Then, I have established our framework through the conditions (2) and (5). Moreover, I assume the choice of adapted coordinates to the Killings. There is a freedom for that choice that is reflected in the existence of a subset of diffeomorphisms (adapted diffeomorphisms) relating the different possibilities. Modulo arbitrary changes in the x i transverse coordinates, the relevant adapted ones are X 'i = X i,
X 'a = X a + A a ( x J ) . Tensors in 12 transform linearly under this change, as is expected, 2 The conventions for the curvature can be found in the appendix.
(6)
.1. Borlaf/NuclearPhysicsB 514 [PM] (1998) 721-739
724
'iz,,...,izt = V ( x i ) ,,,...,,,,
j(cga)~l,...,pl~t)~l,...,~;,V(xJ)flall,...,~t,, ........ ~,l,...,p .....
(7)
where I define JI"'. OA "~at'''m;a'''''a" .'~,1........ ;~,,...,~, =- H1 J~[ (Oa) f i r=l
J~,, ,~s ( - a a ) ,
(8)
s=l
/~ a i J ~ ( O A ) = 8~ + 8aOiA t3,,,
(9)
using the short notation V t = J(c~A)V.
(10)
Because of the abelian nature of the diffeomorphism, J provides a representation of U(1) n in the vector space of tensors of the same rank as V satisfying (2); in particular, J(OAj )J(aA2) = J ( a ( A l + A2)) and J ( 0 ) = 1 implying J - l ( a A ) = J ( - a A ) . In the cases we are interested in, it is natural to find a set of "transverse" gauge fields { A a ( x j) } transforming under the adapted diffeomorphism as A~a(x j ) = A a ( x j) - OiAa(xJ).
(11)
Now I can define the "reduced tensor" v associated to V, v~J(A)V,
(12)
which has the property of being invariant under the adapted diffeomorphism (6), u' = J ( A - cTA)J( OA) V = v.
(13)
It makes sense to think of reduced tensors as the ones in a D-dimensional manifold with a dimension locally shrunk to a point. They keep their indices corresponding to the collapsed dimension, which are inert to the adapted diffeomorphisms, but they are sensitive to other transformations, as we will see in the T-duality case. Looking at the explicit form of the matrix J, it is clear that the operation giving "reduced tensors" commutes with linear combinations, tensor products, contraction and permutation of indices. Before following with this logic development let us see the most significant example. If we have a Riemannian manifold with n commuting Killings
G~z~,= ( Gab
Abj
Aai
Gij + AicAjdG cd
)
'
~_.kaG#u=O,
(14)
where G ab is the inverse matrix of Gab. It is well known that the desired "transverse" gauge fields are A ~ ( x j ) = GabAbi(XJ).
(15)
Using the convention of writing the usual tensors in capital letters and the reduced ones in small letters, the reduced metric takes the simple form
725
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
gg~' =
Gij
"
If we restrict ourselves to the conditions (2) and (5), we can repeat the arguments given above and conclude that there exists a corresponding reduced covariant derivative 3 (16)
XTv -- J ( A ) x Y V
Taking account the explicit expression for J in every tensor representation, we can read off the reduced connection y 4 yP~,, = J ( - A ) ~ J'~( - A ) , , J ( A ~) ~ ( F c , P
~ B - cg,~J(A)~).
(17)
In an arbitrary choice of the "transverse" gauge field A"i the resulting reduced connection could not have any advantage, but with a "natural" choice, i.e. (15) in a Riemannian manifold, it is a very simplified version of the usual one. As the most significant example I will write the reduced connection for the Levi-Civita connection for the metric in (14), l^i
Y,~II, = O,
~'I',, = ~'~at i Y.ja
i
y i b = - ~0 Gab, =
!~bc~.~ 2~ vt~ca' 1
^
= "Yaj = 7 G a b F ( A ) j
~ bi
~ ~ PkI J '
(18)
' ')/~j = _ ~ F ( A ) i j a
where hatted objects are the ones calculated with the quotient metric Gij 5, and F(A)~) OgA'] - c~jA~' is the field strength of the gauge fields. The usual Levi-Civita connection
for this case is written in the Appendix and the comparison shows the great advantage of using the reduced one. In the case of an arbitrary covariant derivation, we can get the corresponding reduced connection adding the reduced tensor of the additional one to the Levi-Civita connection (17). Despite its simplicity, even in the simplest case, the reduced connection (18) has a very rich structure because of the presence of torsion at the same time as a nonLevi-Civita symmetric part, both restricted by the necessary covariant constancy of the reduced metric. With the definition given above of the reduced covariant derivative the operation that gives the reduced tensors commutes with the basic operations of tensor calculus: linear combination, tensor product, contraction, permutation of indices and covariant derivation. That feature together with its simplicity is the reason to call the whole setting the "reduced geometry". The generic presence of torsion is the responsible for a little subtlety in calculating the reduced curvature. To see that, let us start with the Riemann-Christoffel curvature. Due to the commutation of reduced covariant differentiation with the reduced mapping, we should write 3 The reduced covariant derivative is denoted by ~ too. The distinction is made by looking at what kind of tensor acts. 4 Details in the appendix.
726
J. BorlaJVNuclear Physics B 514 [PM] (1998) 721-739 = J~pp ( a ) [ ~ a , V~] w,~
(19)
for an arbitrary one-form W belonging to J2, and its reduced version w. Substituting commutators in both sides and taking account of the presence of torsion in the second one, we get R(FL_c)izl,pWo.= ~r J~l,p ;a6~"( a ) (R(yt_c)]8~rwn + 2T(yt-c)]~Vnw~r).
(20)
I denote by Yl-c the reduced Levi-Civita connection (18), by T ( y t - c ) the associated torsion and by F t . - c the Levi-Civita connection. At first glance there seems to be an obstruction to identify the reduced curvature by the presence of the torsion term. The little paradox is solved by realizing that the only non-vanishing torsion is ya. (18) and tJ therefore that contribution does not contain derivatives of w (because 0aw = 0) and can be added to the standard curvature. The resulting reduced curvature is =
- 2T(yt-c)a,~(Yt-c)a~-.
(21)
In the general case we can write the connection as r = FL-C + H. Using the formula
R(F
+ Q)P,o- = R(F)Pu,,,~+ V,,QP,~ - V,,QPoa
p
a
p
a
p
-Quo-Q,,,~ + Q,,o-Qu,~ + 2T(F),,,Q,~.,
(22)
Vu being the covariant derivative calculated from the generic F connection and T ( F ) the associated torsion, and using the, properties of the reduced map, we get for the reduced curvature r~a,~cr= R ( y l - c + h)~a,~cr- 2T(yt-c)aa~(yt-c + h)~arr,
(23)
where hP,. is the reduced tensor corresponding to HP,. In the present work we are interested in the basic U(1) duality. There, the reduced metric gu" =
(G~ 0
0) Gij
has the additional advantage of having an almost trivial transformation under T-duality, (1) g'~=
(') ~
0
.
^
0
Gij
It is not relevant here to make an exhaustive catalogue of the reduced geometry. The reduced exterior differentiation and the reduced Lie-derivative mapping can be obtained writing them in terms of covariant derivatives (for example with the LeviCivita connection); in such a way the reduced expression is manifest across (16) and coincides with the usual one except for the presence of torsion in the reduced Levi-Civita connection. The case of the differential of a two-form, which is relevant to the U( 1 ) T-duality calculations is done.
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
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Given a two-form B, its exterior differential, can be written in terms of the covariant one calculated with the Levi-Civita connection as Hl,~o = l ( V i s B p p 4- V~,Bpp~ + V p B # v ) .
(24)
in those terms, the reduced H, i.e. h, is h ~ p =- ½(V~b. o + V.bpu + V p b ~ ) = I~ ( Olxb~,o 4- o~l,bpu 4- Opb~u) -- T ( y l _ c ) # ov b o p
- T ( Y l - c ) o~,pbo~z -- T ( T l _ c ) oplzbou.
( 25 )
As we see, the expression of the reduced version in terms of ordinary derivatives gives an additional term due to the presence of torsion in the reduced connection. Despite this apparent setback the reduced H has again nice properties from the T-duality point of view: hoi.i : - ½ F ( b ) i j ,
(26)
hijk : hi ik,
(27)
where F ( b ) i j =- F(Bok = bok = (~ok/(~oo = -'~k)ij. 6 I label the/jk component with a hat because it is T-duality self-dual: hoi.i =
-½F(g)ij,
l]ij k = ]li.ik,
(28) (29)
where F ( g ) i j = F ( A~ = Gok/Goo). The main purpose of the remaining sections is to exploit the power of the reduced framework in its application for the study of the classical geometry of the T-duality mapping.
3. Generalized T-duality mapping The natural connections defined in the context of T-duality are F ±, with their reduced partners y±,
r,~,,~ = r ( L - c)~,, + H,~o,
(30)
4-0 = ~.(1 - c ) ~ . + h~., P ~%,
(31)
where the torsion H~,~ = H~vGoo. is as in (24). The explicit expressions for the reduced connection are greatly simplified: 7 6 hi)k = ½(3,bjk + 3kbij + Ojbki) + ¼(f(g)ijeok + F(b)ij(aoi/G~)) + f(g)kieoj + F(O)ki(Goj/Gl~,) + F(g)jkBoi + F(b)jk(Goi/G(~) ) and bij ~ Bij - (GoiBoj - GojBoi)/2Goo is the T-duality invariant bij = I~o transverse torsion potential.
7 ilk = ~ijl~lk.
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
728 3 / ~ ° = 0, 4- i
1 ^i
Yoo = - ~ 0 Goo, ±o
±o
1
Yio = Yo~ = ~?i In G00, +k ^ ~, Ti.i
= Pkij ±
+i = ½ ( G 0 0 ~ ( g ) j i qz f ' ( b ) j i ) , 3/oj
3/±~jo - ½(G00P(g)/ ± P ( b ) / ) , 7 ij+ ° -- - ½ ( F ( g ) i j
± G~olF(b)ij)
(32)
Now it is easy to read off their T-duality transformation, which turns out to be diagonal: ~+/ 3/00 = _
T ~ t,
~±o ~±o ±o ~/iO = 3/0i = --3/i0 ' ~±i
•
-±k
Ti.j
±k
= T(j
1
±i
1
+i
•
(33)
Except for the 3/+0 io component, this can be arranged as y~~+ p = ( - 1 ) g~ ( ~ G o o ) (n°-n°)'v±P "u~'
(34)
where I define go -= 1 if the /x index is contravariant, go = - 1 if it is covariant and gi :- 0 in both cases. Moreover, (n o - no) in front of an object (connection or tensor component) with indices c~ means ~ g~. 8 The ~/i~0 component transforms flipping the sign with respect to (34), which will be relevant in what follows. It will also be important to notice that 3/i~o° acts as a connection for (34)-type transformations: let the generic T-duality transformation in the reduced setting be 6~+ = (±Goo) ao+ O + for a given function O + ( x J ) , there is a natural covariant derivative, di O ± =_ ( 0 i + A O ± T i ~ 0 0 ) O + ,
(35)
transforming as the O + itself 9 c~ ±
=
(-+-Goo)ao~:di 0 + .
(36)
81 call it n o - no because it reduces to the number of contravariant components being zero minus the number of zero covariant ones. 9 Moreover, di satisfies the Leibniz's rule di(A * B) = (diA) * B + A • (diB) and the "covariant constancy" of Gm~, i.e. diGoo = O.
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
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The simplicity of the reduced connection's transformation allow us to realize that there is a diagonal T-duality transformation of reduced tensors that maps (again diagonally in the reduced setting) the usual (target-space) ~7~ covariant derivatives. To see this, let us write them in terms of (35), m k--7±t,~ ,...,#/ vo Vl ..... Vm
1
_ ~ ' ~ .,,4-o-,jZl ,...,tz/ ~
YOMs ~ P l ,...,O',..+,Vm
+ ~
S= l
Y0rr
vl ....,v,,
r= 1
f,~,,~,,,ml
, ' " , a t ~" ~ ~I~MI ........ - -~7 - i -1- b,l/tZl Vl .........
m
Z - - i v , ' - l - O " j ~ L 1 ' t l L ,...,O',..,,Vm I"'" ~V, 21-
S=I/u~4=O
, +juPd,...,#l "YiO vl , . . . , j , . . . , v m
--
)" i j
s=1/v,=O +((n
Z[
lio'%t-l-/tLr/)~Llvl ,... ........... ,Vm ILl
r = l / # ~ v~O
vvl,...,vm
r=-1//x~=O
0 -- nO) -- A .(,_1,/Zl,...,#l j~ .. . +. o.. ~. . ....... ~ LI,I ,...v m f l u Vl ,...,~m
(37)
The last term in (37) transforms with an undesired - 1 with respect to the dominant div, fixing the weight d,,.~,...,.i = (n o - no). Therefore the reduced transformation is vI ,.-,Vm ~+~....,m Vl ,...,Pro
= (±Goo)
(n°-n°) v ~ ' ' ' m
(38)
Pl ,...,Vm '
giving the linear map for the covariant derivatives under T-duality lO ~7-~ ~2tS]J,...... /LI --p Vl ,...~Vm
W+.~,....,~l = ( _ 1 )g' (±G0o) (no-,,o) ' --p ~v, ........ •
(39)
Before writing the transformations in the usual (non-reduced) setting, I express them in a compressed notation as
?9+ =- D+ ( Goo)v,
(40)
V+?~+ = D~,( Goo ) V+v,
(41)
D+ and D ~ being diagonal matrices in every tensor representation. Inverting (12) for v and ~+ taking account Ai = Goi/Goo and/~i = Boi we get for ~'± and V + ~ ' +
f/± = J ( - B o i ) D + ( Goo)J( Goi/Goo) V,
(42)
V+f/+ = J(-Boi)D~(Goo)J(Goi/Goo)V±V.
(43)
Because J, D+ and D ~ factorize, the T-duality mapping does too and can be written in terms of the matrices T + and T~: defined as / v, ........
m
+~r
= --
iX S=]
"'
] "
'~,...,'~,,'
(44)
m It m u s t be stressed that in order not to overload the notation, I always write n o - no, but in every case its value is given b y the tensor's ( c o n n e c t i o n ) c o m p o n e n t in front o f it in the w a y described above.
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
730
1
......
m
±~r r=l
kll'~'~
) - - a ' - i ........ '
\
"
s=l
(45)
with
(
,
±coo
( + Boi - Goi ) / Goo
o)
,
T
~i..1
o=
(.
,
/
,
~j
where p is the column index a n d / z is the row index. These matrices T+ and T +, introduced by Hassan in the study of the T-duality of the extended supersymmetry [7], can be thought of as a sort of "vielbein" relating indices of the initial and dual geometries. In what follows I will call (44) the T-duality canonical transformation for the tensor V. Let us note that (45) is not canonical. This anomaly is responsible for the generalized curvatures of original and dual geometries do not transform simply by changing indices with the T ± vielbeins. We will see this in detail in the last section. It must be stressed there are other tensors whose T-duality transformation is not as in (44); the most appealing cases are the torsion potential B~,~ and its field strength, the torsion H~Br [7]. It seems the natural place for the torsion to take part in the generalized connection; then from ( 4 4 ) ( 4 5 ) or (33) we can extract the whole transformation to be F- ~±,p = T f f a-r±/~rp i u l ± a - -r a±B. + (OuT)~)TP, B"
(46)
For tensors covariantly constant with respect to one of both derivatives V ~ , the Tduality mapping is enforced to be given by (44). This is the case of the metric itself because V ~ G = ~r~(~ = 0. If we have in the manifold two covariantly constant pforms A ± satisfying V~+ A ± = 0, there is a W-algebra in the underlying string sigma model. I I Another dual W-algebra is present in the dual string theory provided the p-forms transform as in (44) [6], -±
1
+
Aoit...ip_~ = - t - - ~ Aoil...ie_~ , ,4~: t, ...t,,• = A ~ '1 ...t,,• +
~--~ ( -I-bois - goi, ) a-~
Ail...o...i,,"
(47)
..;=1
If we have another Killing K (2) (necessarily commuting with the one used to Tdualize) giving rise to a chiral current, we have [9] V ~ K (2) = 0 ( + for an holomorphic current and - for an antiholomorphic one). The T-duality canonical transformation of K (2) gives another dual Killing vector with a chiral current. Instead of the privileged T-duality mapping for the generalized covariant derivative, we could be interested in the one for the "generalized divergence" V A+ '~,~4~am ' ~ ' Again ul ',...,Z'm" the study is very simplified in the reduced framework. l J That includes the mapping of complex structures, although it requires the additional vanishing of the Nijenhuis tensor [7].
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
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After determining the T-duality weight ~,, ,1 ........ following the same procedure as ~¢ ,t+ ~au~.....u~ in the covariant derivatives' map, the required mapping reads ~la~,...,~t ± ~,~,...,v,, = ( - l)ga,'-q-G ~. 001",(n°--no+l)aA#l,...,#l -,~'l,....v,, ' 0
I
f7 pi f ''~±~, p " ' ""~' ........ = ( + G o o ) (n -no) +1 x-'7±..,a,~,,...,~, --a '~,t........ •
(48)
We have learned to read it in the common language as (K 2 = K , K / ~ ) 1
~±vt,...,v,,,-
Z:~p
m
5:Br
I 1 - 're
]~-'al,...,°tr,,
'
S=I l
m
- - a z ± v ~ ........ : K2(1-I r,r ±/3r ) { ~ 1-I i X - vr±o? , ]--p~--,r, ........ \
r=l
\
s=l
•
(49)
/
This section shows how elementary geometrical objects transform under T-duality depending on their defining properties: covariantly constant tensors transforms canonically (44), i.e. changing indices with the "vielbeins" T:~ and T+; "divergenceless" tensors transform under (49) changing the index corresponding to the divergence with T~: instead of T±; the generalized connection transforms as a true connection (46) with respect to the T-duality canonical transformation with the only peculiarity of using TT instead of T± in the index corresponding to the derivation. This anomaly propagates to the T-duality transformation of every index associated to derivation as we have seen in (44), (49), and finally it will be responsible for the inhomogeneous transformation of the generalized curvature, as we will show in Section 5.
4. T-duality classical dynamics In addition to the D-dimensional manifold M representing the target space-time, in the context of strings we have a two-dimensional world embedded in it, the worldsheet X, identified with the dynamical string. 12 At tree level in the string dynamics, X has the topology of the sphere. Choosing light-cone real coordinates o-±, the covariant world-sheet derivatives for mappings Y(o -+, o--) between X and the tangent space of our manifold M in X ~ ( t r + , o - - ) are V± = O+ + F ± , the pull-back being in terms of the string's embedding XU(o-+, o--),
Extending the definition (12) to the Y mappings, 12 l omit here p - b r a n e s , D-branes and any other kinds o f stringy extended objects.
(50)
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
732
y ( o - + , o - - ) -- j ( A ( X ( o - + , o - - ) ) ) y ( o - + , o - - ) ,
(52)
where y is the reduced mapping. With that definition we can prove that the reduced partner o f the pull-back is =,
= , rt~,
(53)
where I call (O±x) ~ - J ( A ) ~ O ± X ~ following (52). 13 In the reduced framework it is easy to convince oneself that the only T-dual change for (O±x) that transforms the pull-back diagonally is nothing but the one responsible for Buscher's formulas:
O±X~ = T ~ v O ± X ~,
(54)
the reduced pullback transformation being =
o,
o = T
¢
,
= TG0o3~°' i,
1 i hJ = Goo'Y±.i,
o,
(55)
which again can be summarized as
-~, ~z = (qzGoo)("°-"o)y~ ~z Y±
(56)
except for the y~: o, in which there is a flip of sign allowing the T-duality covariantization (35) o f the world-sheet covariant derivatives. It is worthwhile to mention that (54) implies KuV+V_X u = KuV_V+X u = 0 ,
~ / ~ u ~ + ~ , _ ~ s , = / ~ u ~ , _ ~ , + ) ~ = 0,
which is automatically valid for the classical string (it is the current conservation corresponding to the isometry). Again the simplicity of Eqs. (55) allows us to build the diagonal mapping for any Y(o -+, o - - ) mapping 14 y q2 {gl'-r- t T - - ~ l ' " " f l 4 \v
, ~
/!]l,...,p
m
~-
(TGoo)(n°-n°)y(cr +, 0 " - ~ ] ~i i1]'l','.'.'.[,~l ~1m
(57)
with the property of transforming linearly the covariant derivatives V i : [3q,.., ,/,Zl ~ ± y ( o -+, ~r- ~ m~,...,~, o , v, ,...,v,, = (qzGo0) (71--n0) V-4-y(o"J- ,o- -- )v, ........ •
(58)
In the usual setting Eqs. (57) and (58) can be written with the help of ( 3 8 ) , (39) and ( 4 4 ) , (45) as 131 write the reduced OX,~ between parentheses to specify that in general it is not the partial derivative of anything. Only when the pull-back of F(A)q vanishes can xt~ be identified with a sort of U(I) invariant embedding coordinates. laThe intermediate tool is now the analogue to (35) in the world-sheet, i.e. d±O(o-+,o--) --= (0± + Aoy° o)O(o-+, ~r-), transforming as O with weight Ao under T-duality.
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
733
(' )(.
r [ 77± a,'~ y ( o -+, o-- )~,, .,~, ,
Vl,...,v,,
£'7 v
~±~,,...,u, =
T
pl,...,l)m
r=l
(
s=l
)
n' T±'rI~r )( fi T±OlV~ "~ ____
)
r=l
yfll,...,fll •
:F c[1 ........
(59)
s=l
Eq. (59) allows us to extract the pull-back's T-duality transformation in the common language:
r~
= T~ aT~: /~F~, a + ( a± T~ t~) T~ e.
(60)
The most relevant example of this mapping is provided by O±X~, for which we have V±Oq:X** = T~:~V±Om X~, giving the classical stringy equivalence between the two different geometries.
5. Generalized curvature's transformation + ± a Defining as usual the generalized curvature as R # w r p = R ( F ) u ~ G a p which has the + o = -Ryder + p - -R/zvp~, + + - R p~rtzv q: and taking account the symmetry properties R~,~ Rtzvp~r generalized connection's transformation (33)15 we get the dual generalized curvature in the reduced framework,
?± oioj roij __ q:
(
1 2 roioj ± (Goo)
'
)
OiGooOj GO0 ,
( r~jk - aiGooy~°),
~± ijko - :t: G ~ (r~k o + &Goo3'~., o), -± _ r ± .±o..mo ijkl _ 2 p~O0)'ij Zk/ •
(61 )
rijkl --
We can convince ourselves that the inhomogeneous part of the transformation can be written in terms o f the Killing vector, giving the compact result
r#vcr = ( --1 )(gu+g~)(±Goo)-n°(_r/~vcrp ± - 2k2--~ V ± Z ".~v--cr ~ m k p ~j ,
(62)
where K 2 = k 2 = K u K ~ = Goo. In the usual setting the transformation (62) reads J6
~± R/~vcrp = T : C ' T v T B T : a T ±Pn { Rt
+ al3&#
22V:K~K,7)
"
(63)
It is important to note that the inhomogeneous part of the transformation only depends on the transverse components G0~, B0i and their first skew-symmetric derivatives. 15See the appendix for explicit formulas. i6 V~ kv only has an antisymmetric part due to the Killing condition.
J. Borlaf/Nuclear Physics B 514 [PAl] (1998) 721-739
734
When an object transforms inhomogeneously, say f = +(Goo)a(r+~,), the involution property of the T-duality transformation T 2 = 1 fixes the ~b transformation to be ~ = 1 T(G00)a~b, allowing us to create the homogeneous w =- r + ~¢, i.e. v~ = +(Goo)~w. Specifically, it means that
aTq:BT+~T+rI(-~V~aKBV~Kn), and therefore we can create the "corrected" generalized curvature W ~ p linearly under T-duality: 4-
+
W~uo-p :-- Rlxvo-p
1
+
(64) transforming
q:
K2 V~ K~V,~ Kp,
ff'~. = T~T:'T) ~T~oW~8~.
(65)
At this point is for free the rederivation of the generalized Ricci-tensor's transformation giving the need of a non-trivial dilaton change under T-duality in order to guarantee the one-loop conformal invariance of the dual string sigma model [2]. We get the dual-reduced Ricci tensor from (61) rg~.+•
~+ = ( - 1)g~(iGoo)-nO(r~p - V~± Vp+ In Goo) . ru.
(66)
It is well known that for the sigma-model coupling between the two-dimensional curvature and a scalar field called dilaton, ¢, it is enough to ensure the vanishing of the dual one-loop beta function, provided the former transforms under T-duality as = 'it' - ½ In a0o.
(67)
In other words, the tensor representing the one-loop beta function for the string sigma ± ± model (bosonic and supersymmetric) [11], i.e. , 8 ~ = R ~ 2 V u V ~ ~/', transforms linearly under T-duality as can be read off from (66) and (67),
: rJ
(68)
Therefore the this approach, minimally the transformation
vanishing of / ? ~ implies the one for ¢),, and vice versa. Following the dilaton one-loop beta function can be obtained trying to complete generalized scalar curvature R + in order to get a T-duality scalar; the for R ± = r ± results in, using Eqs. (61) and (66),
k ± = ~± = R ± - 2~7± icgiIn Goo + 2 l___y~~y~i tl00
(69)
and the desired T-duality scalar is fit, = R + -t-4((O~ib) 2 - ( v + ) z q b) -- 2H2,
(70)
where we define H2~ =- H,~,,RH/~~p. Modulo a constant term it coincides with the one-loop dilaton beta function [ 11 ].
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
735
6. Canonical connection a n d c u r v a t u r e The anomalous transformation of the index corresponding with first covariant derivations propagates in an annoying way to objects constructed from higher derivations, such as the generalized curvatures and the one-loop beta functions. In other words, the covariant derivation V + does not commute with the canonical map defined in (44). Looking at the generalized connections' transformation (33) we notice that there is a minimal covariant subtraction giving a connection, the canonical connection, which commutes with the canonical T-duality, -± p -_±p
F~
± p
1__~_K VmKp
= r 2±a T ;±~TP , ± o - f.±~ +
(O~T~¢)T o,
(71)
Because the anomaly mentioned above is located in the y ± ( r components, the whole effect of the subtraction is to cancel against them, giving 9~:"n O# = 0 and 9i~ p -- - ") ' i u± p 17 This fact allows the commutation between V ~ and the canonical T-duality map, 1 b'l,"',/",,,
=
m ± B r 1.][ -b'¢
]
(n)(m
' (£1,"',¢2"m
s=l
--p
...... , , = ,', ........
T ±~r
r= 1
1-1 1 1 T± ~ ]--a s= 1
,~,........ -
(72)
"
Another consequence is V0± = 0, implying R0,,,~p = 0. What is more, the new connection is compatible with the metric provided that K ~ is a Killing vector. In a certain sense the barred connection seems to be the most natural associated to the presence of a Killing. If we think of the Killing as a vector field indicating the direction in which nothing changes, we would expect that parallel transport is really insensitive to displacements in that direction. This happens with the canonical connection but not with the Levi-Civita (or its torsionfull generalizations) connection. The commutation with the canonical T-duality implies that the curvature for V ± transforms canonically (the same as the Ricci tensor and the scalar curvature) -±
-
±"
±~ ±~T ± n g ' ±
(73)
I will list five independent T-duality scalars Ii = R ± - ( V ± ) e l n
12 = H 2 -
K 2-
~-~H~K'~K ~,
~K--~H ~ K ' ~ K B,
17 These conditions guarantee that if a tensor is covariantly constant with respect to V ~ , it does too with respect to ~ ' ~ . The converse is in general not true. Therefore, the metric commutes with V ~
736
J. BorlaJTNuclear Physics B 514 [PM] (1998) 721-739
1
2
13 = - K S ( ~ 7 u K ~ 7 U K ~ + H ~ K
l
14 = ~ K ~ O ~ K a H
cL 13
K ),
a'a
~" ,
70K22 I5 = \ K2 ,] ,
(74)
built with the help of R+ and V~ being the Levi-Civita covariant derivation. In particular, R+ = Il T 2/4. Finally, with this covariant derivation, the T-duality canonical map commutes with the basic geometrical operations: linear combinations, tensor products, permutation of indices, contractions and covariant derivations.
7. Conclusions This work shows how the reduced geometry is a privileged framework for the study of the T-duality's geometry and possibly of many other different issues related with abelian Killing vectors. The T-duality transformation diagonalizes in the reduced setting, allowing us to get in a straightforward way results pursued for a long time, such as the generalized curvatures' map, the canonical map for the covariant derivatives in the target space and in the world-sheet, the minimal correction to connections and curvatures in order to transform linearly and T-duality scalars. The introduction of the dilaton can be seen as the minimal modification needed to map the one-loop beta functions preserving conformal invariance, but from the geometrical view, the dilaton is completely insufficient to build in a systematic way T-duality tensors (i.e. tensors transforming canonically under T-duality). The object serving to "covariantize" under T-duality is the Killing vector itself across the canonical connection, as shown in the last section. In connection with that, new T-duality scalars have been found without the help of the dilaton. Future work could comprise the higher-loop corrections to Buscher's formulas, the map for the invariants characterizing the geometry and the topology of the manifold, the global questions in the reduced setting, the non-abelian generalizations of this procedure, and a deeper study of the geometry of the canonical connection and its relation with the string sigma model.
Acknowledgements I wish to thank E. Alvarez for suggesting, reading and commenting on this work. Also I want to thank P. Meessen for useful discussions. This work has been partially supported by a C.A.M. grant.
J. B o r l a f / N u c l e a r P h y s i c s B 5 1 4 [ P M ] (1998) 7 2 1 - 7 3 9
737
Appendix A. The reduced connection and Riemannian curvature In order to have the opportunity of getting an idea of the operativity of the reduced framework, I show here the usual Levi-Civita connection for the generic metric with n commuting Killing vector fields G I~,,
( Gab Aai "~ Abj Caij + AicAjclG ca J '
(A.I)
\
Fa 1aa~if, bc = 2"~ti o " J b c , r [ t b = - 513 ^i G ~ b ,
F~i,i = F)(, : ~' ' ~~,U a b r~ "i J i-el~i
--
Abi3JGa,,)
,
= Fi~b, -_ ~I ( G bc OiGac - a, ub~t -a7c , ~,c.j i q- A icA j Ob ^ j Gac),
F a
ij = 21{ ( X g^ i A j a + ~jA~/) + A icA j Ab k Oa
^k
Gbc
c a t'^ck + Al'Pyk)}. + G ab (AjO~Gbc + ACc~iGi, c) _ AkG~b(AjF~
(A.2)
Eqs. (A.2) must be compared with (18) to appreciate the advantages. Now I would like to make explicit the relation between a generic connection F and its reduced version y, i i Tab = Fah,
.yc
_
c
i
c
ob -- Fab q- FabAi '
"Y[*i= F~l*i - FJbA~ ~, y~,, = I ,~, - F[,aAai ,
el',= <,- F~.A~"+ h
b
b
Yai = Fai - F a c A i
c
Tfj = F~j - Fajk Aia
FJiaA.~j
b
F{oAfA}, fj
acab
+ FaiAj - - ac'', "'j' pk aaab F(~ aa-g q---abZa, ta.j, --~a"
a a Fa ab a b F.a a b a c "Yij = F i j - - - - b j ' ~ i - - F i b A j + - - b c ' ~ i n j
- 3 i a ; ! + I'~ia ~.
r,k a b a a r,k A b a a ilk abAcaa - - - - b j n i n k - - ,~ ibraj n k -}- * b c t a i n j n k
(a.3)
Paradoxical as it may seem, the task to calculate the reduced Levi-Civita connection is shorter than the usual one, because the knowledge of its invariance under the adapted diffeomorphisms (6) drops out of the terms proportional to A~, being non-derivatives (we can call it A = 0 projection). Therefore (A.3) must be understood as TP~ = (rL, - O,,J,~)Ia=0. One just needs to know Fla= 0 instead of F itself, which makes it easier to get the reduced version. The usual curvature, torsion and Ricci tensor's expression for a generic connection FPv are
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
738
P = I ( F P u - F~,~),
R( F) t,,, = R( F)Pp~.
(A.4)
The reduced Riemannian curvature for the U(1)" is given by 1
^i
rabcd = ~ ( 3 GacOiGbd -- Oi GbcOiGad ) , d -- GebOC~a) rabci = -l Fije ~"]Gcd ( Gea ¢~b I cd ^ck d rijab = ~ ( G cgiGacajGbd ÷ GacGbdFi F'.ik - ( i j < 1 ^
^
l
raibj = ~ V i V jGab - ~G
cd
l
>j i ) ) ,
^d
OiGbdo3jGac -- "~GacGbdF'i kFfk,
^ b rakij = - ~1 Vk ( GabFijb ) + ~1 ( F~iOj Gab --
b V;,jOiG.b),
rijkl=~ilkl ÷ I G a b ( F,Ik a F)tb __ F)kFila b ÷ 2FijF~t) , ab
(A.5)
where the hatted objects are the ones calculated with the quotient metric Gij, Fi~~ =3iA.~] - OjA'~, and A~~= GabAib. The Riemannian curvature is obtained using (12), (8), (9), i.e. R = J ( - A ) r .
Appendix B. The generalized curvature In this appendix I will write the reduced generalized curvature needed for the calculations in Section 5. To do it in a T-duality suitable way means the introduction of t h e / 3 ~ O = ~7~ + ~-3i ln Goo)O T-duality covariant derivative. As in (35), Ao is O's T-duality weight. In terms of G0o and F/~: = Yij+ o the relevant components are -4-
roioj : {½biiOiGoo + G~oP,~kF~ } + -G-~ai in Gooaj In Goo, FOijk =
GooOi Fjk ÷
( Fi~ aj ln aoo -F/jOklnGoo) +
r i±j k l -- {Rijkl ^± ÷ G°°( Fi~ ± Fit± --
+ ~GOOF~: jkOi lnGoo,
+ ± )} + Goo(Y~+ Tijm ) )'kl, m F)kF.
(B.1)
R~,, = R ( P + h)i,k • The other comoonents are related by the symmetry properties where = The terms in brackets are the ones transR ~c~p '~'R~,~p = - R ~ , ~ and R ~ p = R ~ . forming with the dominant ( - 1 )(g-+g")(+Goo)-no, while the remaining terms transform with - ( - 1) (g~+g~)(+Goo) -~° giving the inhomogeneous part - ~2 w+~v ~ . . . .w:F ~ Kp.
References [1 | J. Polchinski, S. Chaudhuri and C.V. Johnson, hep-th/9602052; C.V. Johnson, hep-th/9606196. [2] T.H. Buscher, Phys. Lett. B 159 (1985) 127; B 194 (1987) 51; B 201 (1988) 466. [ 3] O. Alvarez, hep-th/9511024; O. Alvarez and Chien-Hao Liu, hep-th/9503226; E. Alvarez, L. Alvarez-Gaum6, J . L E Barb6n and Y. Lozano, Nucl. Phys. B 415 (1994) 71.
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
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141 J. Dai, R.G. Leigh and J. Polchinski, Mod. Phys. Lett. A 4 (1989) 2073; J. Polchinski, Progr. Theor. Phys. Suppl. 123 (1996) hep-th/9511157; J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724 hep-th/9510017; E. Alvarez, J.L.F. Barb6n and J. Borlaf, Nucl. Phys. B 479 (1996) 218. 151 C.V. Johnson, N. Kaloper, R.R. Khuri and R.C. Myers, Phys. Lett. B 368 (1996) 71. 161 B. Kim, Phys. Lett. B 335 (1994) 51. 171 S.F. Hassan, Nucl. Phys. B 460 (1996) 362, hep-th/9504148; Nucl. Phys. B 454 (1995) 86, hepth/9408060. [ 81 A. Giveon, M. Porrati and E. Rabinovici, Phys. Rep. 244 (1994) 77; E. Alvarez, L. Alvarez-Gaum6 and Y. Lozano, Nucl. Phys. B (Proc. Suppl.) 41 (1995) 1. [91 M. Rocek and E. Verlinde, Nucl. Phys. B 373 (1992) 630. I I0t E. Alvarez and M.A.R. Osorio, Phys. Rev. D 40 (1989) 1150. [ 11 [ C.G. Callan, D. Friedan, E.J. Martinec and M.J. Perry, Nucl. Phys. B 262 (1985) 593. G. Aldazfibal, E Hussain and R. Zhang, ICTP preprint IC/86/400.
Nuclear Physics B 514 [PM] (1998) 721-739
Geometry of T-duality Javier Borlaf l Departamento de Ffsica Te6rica, Universidad Aut6noma, 28049 Madrid, Spain Received 2 July 1997; accepted 29 December 1997
Abstract
A "reduced" differential geometry adapted to the presence of abelian isometries is constructed. Classical T-duality diagonalizes in this setting, allowing us to get conveniently the transformation of the relevant geometrical objects such as connections, pullbacks and generalized curvatures. Moreover we can induce privileged maps from the viewpoint of covariant derivatives in the target space and in the world-sheet generalizing previous results, at the same time that we can correct connections and curvatures covariantly in order to have a proper transformation under T-duality. (~) 1998 Elsevier Science B.V. PACS: 11.25; 11.27; 11.30; l l.30.p; 02.40.k; 02.40.H,M Keywords: Strings; T-duality; Geometry; Abelian killings
1. I n t r o d u c t i o n
T-duality is a fundamental tool in the understanding of the fashionable string dualities [ l ]. The elementary statement of the T-duality establishes that the perturbative spectrum of a string theory with a dimension compactified on a circle of radius R, is equivalent to the one compactified in a circle of radius l/R, provided we interchange winding and momentum quantum numbers at the same time that we transform the string coupling constant [8,10]. If we allow the presence of a generic geometry (metric G#~, torsion potential B ~ and dilaton ~/,) having an abelian Killing vector in the compactifled direction X °, the backgrounds resulting to be one-loop (conformally)equivalent are given by Buscher's formulas [2]: I E-mail: [email protected] 0550-3213/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PII S 0 5 5 0 - 3 2 13 ( 9 8 ) 0 0 0 0 4 - 2
722
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739 : I/Coo, Goi =
Boi / Goo,
Boi : Goi / Goo,
Gij = Gij - ( GoiGoj - BoiBoj )/Goo, l~i.i = Bij - ( GoiBoj - BoiGoj) /Goo,
~b=~/, - ~1 In Goo.
(1)
In recent years a non-perturbative usage of T-duality has been made in the context of open strings playing with the map of the Neumann-Dirichlet boundary conditions. The promotion of the hypersurfaces in which strings rest their end points to be dynamical extended objects called Dirichlet branes, allow their identification with the carriers of the RR charges required by the string duality at the same time that it makes doubtful the name of "string theory" for the resulting scenario [4,5]. Many features of this topic of T-duality have been developed extensively in the literature [3,6-8]. Despite that important effort, there does not seem to be a systematic study of the mapping between geometries for the stringy (one-loop) equivalent spacetimes. This gap is related with the nature of the non-linear map (1) which highly complicates the calculations for the transformation of geometrical objects such as the generalized connection and its curvature, privileged maps for the covariant derivatives and pullbacks, and many others. In order to clarify all these points some kind of "parallel differential geometry" that we will call "reduced geometry" is presented, having the property of being adapted to the presence of abelian Killing vectors (in fact it is only defined in that context). We will see how T-duality transformations diagonalize in this setting for the main geometrical objects, including the generalized curvature; for the latter, we found for the first time its complete transformation, which can be expressed in a covariant way in terms of itself and of the Killing vector. Moreover, a "canonical" T-duality transformation is constructed for arbitrary tensors with the property of transforming linearly the covariant derivatives calculated from the generalized connections. These results unify and generalize the map obtained for the p-forms in [6], and it includes the fundamental one of the complex structures for holomorphic Killing vectors [7]. We can extend the result giving above to the "canonical" T-duality transformation of maps from the world-sheet to the tangent space of our target-space manifold. The classical string dynamics will be the most representative example of this "canonical" map. In Section 2 we define and describe the construction of the "reduced geometry" giving the basic map relating "usual" and "reduced" objects (generic tensors, connections and curvatures). In Section 3 we get the "canonical" T-duality map, relating linearly covariant derivatives, and a "non-canonical" one relating linearly "covariant divergences". In Section 4 we obtain the "canonical" map for the classical world-sheet dynamics. Section 5 shows the generalized curvatures' transformation and the minimal correction for them to transform linearly under T-duality. As a straightforward outcome I rederive the
J. B o r l a f / N u c l e a r
Physics B 514 [PM] (1998) 721-739
723
one-loop beta function's transformation. In Section 6 we found a "canonical" covariant derivative commuting with the "canonical" T-duality transformation. It is used to get a set of new T-duality scalars. In the appendices we summarize the basic formulas.
2. Reduced differential geometry In this section a parallel tensor calculus for manifolds with a structure endowed with abelian Killing vectors will be developed. The main objective is to exploit the presence of these Killings in order to get a strongly simplified structure that I will call the "reduced geometry". As we will see, that structure is nicely adapted to T-duality. Let us assume the existence of a set of n commuting vector fields {Ka~} with {/~, v = 0, 1. . . . . D - 1 }, {a, b = 0 . . . . . n - 1 } and D the dimension of the manifold M. We restrict our attention to the space s2 of tensors V in M satisfying £k,,V = 0,
(2)
which means simply that we can choose coordinates {x i, x a} with i = n . . . . . D - 1, called adapted coordinates, in which V does not depend on x ~, i.e. OaV = O. The covariant differentiation is not a mapping in O, or in other words, its commutator with the Lie derivative is in general non-vanishing, m
[£k~, ' ~ p ] V 1)1 m'''''ut ,...,1)n,
V"(£.
= -- ~..~"
1
w~cr Vm,...,u~ 1)l,.,O-,.,p m
ka'ClPb'~
~,-"(£"Ra V ~)pO" V1)1,...,1)m m'''~'~'
~U ~ . . d ~
S=I
(3)
r=l
where I have defined 2 o"
o"
(t
(1"
(£k ~7)7~~ = K~R,~,~ v + V u ~ T v K a + 2~7u(KaT~,,),
(4)
R~1) being the curvature for the connection FSa# and TuP~ the corresponding torsion. It can be checked that (4) reduces to the desired 0 ~ Fo"~ in adapted coordinates. If we are interested in connections preserving the condition (2), we must impose £k,,V = o.
(5)
Then, I have established our framework through the conditions (2) and (5). Moreover, I assume the choice of adapted coordinates to the Killings. There is a freedom for that choice that is reflected in the existence of a subset of diffeomorphisms (adapted diffeomorphisms) relating the different possibilities. Modulo arbitrary changes in the x i transverse coordinates, the relevant adapted ones are X 'i = X i,
X 'a = X a + A a ( x J ) . Tensors in 12 transform linearly under this change, as is expected, 2 The conventions for the curvature can be found in the appendix.
(6)
.1. Borlaf/NuclearPhysicsB 514 [PM] (1998) 721-739
724
'iz,,...,izt = V ( x i ) ,,,...,,,,
j(cga)~l,...,pl~t)~l,...,~;,V(xJ)flall,...,~t,, ........ ~,l,...,p .....
(7)
where I define JI"'. OA "~at'''m;a'''''a" .'~,1........ ;~,,...,~, =- H1 J~[ (Oa) f i r=l
J~,, ,~s ( - a a ) ,
(8)
s=l
/~ a i J ~ ( O A ) = 8~ + 8aOiA t3,,,
(9)
using the short notation V t = J(c~A)V.
(10)
Because of the abelian nature of the diffeomorphism, J provides a representation of U(1) n in the vector space of tensors of the same rank as V satisfying (2); in particular, J(OAj )J(aA2) = J ( a ( A l + A2)) and J ( 0 ) = 1 implying J - l ( a A ) = J ( - a A ) . In the cases we are interested in, it is natural to find a set of "transverse" gauge fields { A a ( x j) } transforming under the adapted diffeomorphism as A~a(x j ) = A a ( x j) - OiAa(xJ).
(11)
Now I can define the "reduced tensor" v associated to V, v~J(A)V,
(12)
which has the property of being invariant under the adapted diffeomorphism (6), u' = J ( A - cTA)J( OA) V = v.
(13)
It makes sense to think of reduced tensors as the ones in a D-dimensional manifold with a dimension locally shrunk to a point. They keep their indices corresponding to the collapsed dimension, which are inert to the adapted diffeomorphisms, but they are sensitive to other transformations, as we will see in the T-duality case. Looking at the explicit form of the matrix J, it is clear that the operation giving "reduced tensors" commutes with linear combinations, tensor products, contraction and permutation of indices. Before following with this logic development let us see the most significant example. If we have a Riemannian manifold with n commuting Killings
G~z~,= ( Gab
Abj
Aai
Gij + AicAjdG cd
)
'
~_.kaG#u=O,
(14)
where G ab is the inverse matrix of Gab. It is well known that the desired "transverse" gauge fields are A ~ ( x j ) = GabAbi(XJ).
(15)
Using the convention of writing the usual tensors in capital letters and the reduced ones in small letters, the reduced metric takes the simple form
725
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
gg~' =
Gij
"
If we restrict ourselves to the conditions (2) and (5), we can repeat the arguments given above and conclude that there exists a corresponding reduced covariant derivative 3 (16)
XTv -- J ( A ) x Y V
Taking account the explicit expression for J in every tensor representation, we can read off the reduced connection y 4 yP~,, = J ( - A ) ~ J'~( - A ) , , J ( A ~) ~ ( F c , P
~ B - cg,~J(A)~).
(17)
In an arbitrary choice of the "transverse" gauge field A"i the resulting reduced connection could not have any advantage, but with a "natural" choice, i.e. (15) in a Riemannian manifold, it is a very simplified version of the usual one. As the most significant example I will write the reduced connection for the Levi-Civita connection for the metric in (14), l^i
Y,~II, = O,
~'I',, = ~'~at i Y.ja
i
y i b = - ~0 Gab, =
!~bc~.~ 2~ vt~ca' 1
^
= "Yaj = 7 G a b F ( A ) j
~ bi
~ ~ PkI J '
(18)
' ')/~j = _ ~ F ( A ) i j a
where hatted objects are the ones calculated with the quotient metric Gij 5, and F(A)~) OgA'] - c~jA~' is the field strength of the gauge fields. The usual Levi-Civita connection
for this case is written in the Appendix and the comparison shows the great advantage of using the reduced one. In the case of an arbitrary covariant derivation, we can get the corresponding reduced connection adding the reduced tensor of the additional one to the Levi-Civita connection (17). Despite its simplicity, even in the simplest case, the reduced connection (18) has a very rich structure because of the presence of torsion at the same time as a nonLevi-Civita symmetric part, both restricted by the necessary covariant constancy of the reduced metric. With the definition given above of the reduced covariant derivative the operation that gives the reduced tensors commutes with the basic operations of tensor calculus: linear combination, tensor product, contraction, permutation of indices and covariant derivation. That feature together with its simplicity is the reason to call the whole setting the "reduced geometry". The generic presence of torsion is the responsible for a little subtlety in calculating the reduced curvature. To see that, let us start with the Riemann-Christoffel curvature. Due to the commutation of reduced covariant differentiation with the reduced mapping, we should write 3 The reduced covariant derivative is denoted by ~ too. The distinction is made by looking at what kind of tensor acts. 4 Details in the appendix.
726
J. BorlaJVNuclear Physics B 514 [PM] (1998) 721-739 = J~pp ( a ) [ ~ a , V~] w,~
(19)
for an arbitrary one-form W belonging to J2, and its reduced version w. Substituting commutators in both sides and taking account of the presence of torsion in the second one, we get R(FL_c)izl,pWo.= ~r J~l,p ;a6~"( a ) (R(yt_c)]8~rwn + 2T(yt-c)]~Vnw~r).
(20)
I denote by Yl-c the reduced Levi-Civita connection (18), by T ( y t - c ) the associated torsion and by F t . - c the Levi-Civita connection. At first glance there seems to be an obstruction to identify the reduced curvature by the presence of the torsion term. The little paradox is solved by realizing that the only non-vanishing torsion is ya. (18) and tJ therefore that contribution does not contain derivatives of w (because 0aw = 0) and can be added to the standard curvature. The resulting reduced curvature is =
- 2T(yt-c)a,~(Yt-c)a~-.
(21)
In the general case we can write the connection as r = FL-C + H. Using the formula
R(F
+ Q)P,o- = R(F)Pu,,,~+ V,,QP,~ - V,,QPoa
p
a
p
a
p
-Quo-Q,,,~ + Q,,o-Qu,~ + 2T(F),,,Q,~.,
(22)
Vu being the covariant derivative calculated from the generic F connection and T ( F ) the associated torsion, and using the, properties of the reduced map, we get for the reduced curvature r~a,~cr= R ( y l - c + h)~a,~cr- 2T(yt-c)aa~(yt-c + h)~arr,
(23)
where hP,. is the reduced tensor corresponding to HP,. In the present work we are interested in the basic U(1) duality. There, the reduced metric gu" =
(G~ 0
0) Gij
has the additional advantage of having an almost trivial transformation under T-duality, (1) g'~=
(') ~
0
.
^
0
Gij
It is not relevant here to make an exhaustive catalogue of the reduced geometry. The reduced exterior differentiation and the reduced Lie-derivative mapping can be obtained writing them in terms of covariant derivatives (for example with the LeviCivita connection); in such a way the reduced expression is manifest across (16) and coincides with the usual one except for the presence of torsion in the reduced Levi-Civita connection. The case of the differential of a two-form, which is relevant to the U( 1 ) T-duality calculations is done.
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
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Given a two-form B, its exterior differential, can be written in terms of the covariant one calculated with the Levi-Civita connection as Hl,~o = l ( V i s B p p 4- V~,Bpp~ + V p B # v ) .
(24)
in those terms, the reduced H, i.e. h, is h ~ p =- ½(V~b. o + V.bpu + V p b ~ ) = I~ ( Olxb~,o 4- o~l,bpu 4- Opb~u) -- T ( y l _ c ) # ov b o p
- T ( Y l - c ) o~,pbo~z -- T ( T l _ c ) oplzbou.
( 25 )
As we see, the expression of the reduced version in terms of ordinary derivatives gives an additional term due to the presence of torsion in the reduced connection. Despite this apparent setback the reduced H has again nice properties from the T-duality point of view: hoi.i : - ½ F ( b ) i j ,
(26)
hijk : hi ik,
(27)
where F ( b ) i j =- F(Bok = bok = (~ok/(~oo = -'~k)ij. 6 I label the/jk component with a hat because it is T-duality self-dual: hoi.i =
-½F(g)ij,
l]ij k = ]li.ik,
(28) (29)
where F ( g ) i j = F ( A~ = Gok/Goo). The main purpose of the remaining sections is to exploit the power of the reduced framework in its application for the study of the classical geometry of the T-duality mapping.
3. Generalized T-duality mapping The natural connections defined in the context of T-duality are F ±, with their reduced partners y±,
r,~,,~ = r ( L - c)~,, + H,~o,
(30)
4-0 = ~.(1 - c ) ~ . + h~., P ~%,
(31)
where the torsion H~,~ = H~vGoo. is as in (24). The explicit expressions for the reduced connection are greatly simplified: 7 6 hi)k = ½(3,bjk + 3kbij + Ojbki) + ¼(f(g)ijeok + F(b)ij(aoi/G~)) + f(g)kieoj + F(O)ki(Goj/Gl~,) + F(g)jkBoi + F(b)jk(Goi/G(~) ) and bij ~ Bij - (GoiBoj - GojBoi)/2Goo is the T-duality invariant bij = I~o transverse torsion potential.
7 ilk = ~ijl~lk.
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
728 3 / ~ ° = 0, 4- i
1 ^i
Yoo = - ~ 0 Goo, ±o
±o
1
Yio = Yo~ = ~?i In G00, +k ^ ~, Ti.i
= Pkij ±
+i = ½ ( G 0 0 ~ ( g ) j i qz f ' ( b ) j i ) , 3/oj
3/±~jo - ½(G00P(g)/ ± P ( b ) / ) , 7 ij+ ° -- - ½ ( F ( g ) i j
± G~olF(b)ij)
(32)
Now it is easy to read off their T-duality transformation, which turns out to be diagonal: ~+/ 3/00 = _
T ~ t,
~±o ~±o ±o ~/iO = 3/0i = --3/i0 ' ~±i
•
-±k
Ti.j
±k
= T(j
1
±i
1
+i
•
(33)
Except for the 3/+0 io component, this can be arranged as y~~+ p = ( - 1 ) g~ ( ~ G o o ) (n°-n°)'v±P "u~'
(34)
where I define go -= 1 if the /x index is contravariant, go = - 1 if it is covariant and gi :- 0 in both cases. Moreover, (n o - no) in front of an object (connection or tensor component) with indices c~ means ~ g~. 8 The ~/i~0 component transforms flipping the sign with respect to (34), which will be relevant in what follows. It will also be important to notice that 3/i~o° acts as a connection for (34)-type transformations: let the generic T-duality transformation in the reduced setting be 6~+ = (±Goo) ao+ O + for a given function O + ( x J ) , there is a natural covariant derivative, di O ± =_ ( 0 i + A O ± T i ~ 0 0 ) O + ,
(35)
transforming as the O + itself 9 c~ ±
=
(-+-Goo)ao~:di 0 + .
(36)
81 call it n o - no because it reduces to the number of contravariant components being zero minus the number of zero covariant ones. 9 Moreover, di satisfies the Leibniz's rule di(A * B) = (diA) * B + A • (diB) and the "covariant constancy" of Gm~, i.e. diGoo = O.
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
729
The simplicity of the reduced connection's transformation allow us to realize that there is a diagonal T-duality transformation of reduced tensors that maps (again diagonally in the reduced setting) the usual (target-space) ~7~ covariant derivatives. To see this, let us write them in terms of (35), m k--7±t,~ ,...,#/ vo Vl ..... Vm
1
_ ~ ' ~ .,,4-o-,jZl ,...,tz/ ~
YOMs ~ P l ,...,O',..+,Vm
+ ~
S= l
Y0rr
vl ....,v,,
r= 1
f,~,,~,,,ml
, ' " , a t ~" ~ ~I~MI ........ - -~7 - i -1- b,l/tZl Vl .........
m
Z - - i v , ' - l - O " j ~ L 1 ' t l L ,...,O',..,,Vm I"'" ~V, 21-
S=I/u~4=O
, +juPd,...,#l "YiO vl , . . . , j , . . . , v m
--
)" i j
s=1/v,=O +((n
Z[
lio'%t-l-/tLr/)~Llvl ,... ........... ,Vm ILl
r = l / # ~ v~O
vvl,...,vm
r=-1//x~=O
0 -- nO) -- A .(,_1,/Zl,...,#l j~ .. . +. o.. ~. . ....... ~ LI,I ,...v m f l u Vl ,...,~m
(37)
The last term in (37) transforms with an undesired - 1 with respect to the dominant div, fixing the weight d,,.~,...,.i = (n o - no). Therefore the reduced transformation is vI ,.-,Vm ~+~....,m Vl ,...,Pro
= (±Goo)
(n°-n°) v ~ ' ' ' m
(38)
Pl ,...,Vm '
giving the linear map for the covariant derivatives under T-duality lO ~7-~ ~2tS]J,...... /LI --p Vl ,...~Vm
W+.~,....,~l = ( _ 1 )g' (±G0o) (no-,,o) ' --p ~v, ........ •
(39)
Before writing the transformations in the usual (non-reduced) setting, I express them in a compressed notation as
?9+ =- D+ ( Goo)v,
(40)
V+?~+ = D~,( Goo ) V+v,
(41)
D+ and D ~ being diagonal matrices in every tensor representation. Inverting (12) for v and ~+ taking account Ai = Goi/Goo and/~i = Boi we get for ~'± and V + ~ ' +
f/± = J ( - B o i ) D + ( Goo)J( Goi/Goo) V,
(42)
V+f/+ = J(-Boi)D~(Goo)J(Goi/Goo)V±V.
(43)
Because J, D+ and D ~ factorize, the T-duality mapping does too and can be written in terms of the matrices T + and T~: defined as / v, ........
m
+~r
= --
iX S=]
"'
] "
'~,...,'~,,'
(44)
m It m u s t be stressed that in order not to overload the notation, I always write n o - no, but in every case its value is given b y the tensor's ( c o n n e c t i o n ) c o m p o n e n t in front o f it in the w a y described above.
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
730
1
......
m
±~r r=l
kll'~'~
) - - a ' - i ........ '
\
"
s=l
(45)
with
(
,
±coo
( + Boi - Goi ) / Goo
o)
,
T
~i..1
o=
(.
,
/
,
~j
where p is the column index a n d / z is the row index. These matrices T+ and T +, introduced by Hassan in the study of the T-duality of the extended supersymmetry [7], can be thought of as a sort of "vielbein" relating indices of the initial and dual geometries. In what follows I will call (44) the T-duality canonical transformation for the tensor V. Let us note that (45) is not canonical. This anomaly is responsible for the generalized curvatures of original and dual geometries do not transform simply by changing indices with the T ± vielbeins. We will see this in detail in the last section. It must be stressed there are other tensors whose T-duality transformation is not as in (44); the most appealing cases are the torsion potential B~,~ and its field strength, the torsion H~Br [7]. It seems the natural place for the torsion to take part in the generalized connection; then from ( 4 4 ) ( 4 5 ) or (33) we can extract the whole transformation to be F- ~±,p = T f f a-r±/~rp i u l ± a - -r a±B. + (OuT)~)TP, B"
(46)
For tensors covariantly constant with respect to one of both derivatives V ~ , the Tduality mapping is enforced to be given by (44). This is the case of the metric itself because V ~ G = ~r~(~ = 0. If we have in the manifold two covariantly constant pforms A ± satisfying V~+ A ± = 0, there is a W-algebra in the underlying string sigma model. I I Another dual W-algebra is present in the dual string theory provided the p-forms transform as in (44) [6], -±
1
+
Aoit...ip_~ = - t - - ~ Aoil...ie_~ , ,4~: t, ...t,,• = A ~ '1 ...t,,• +
~--~ ( -I-bois - goi, ) a-~
Ail...o...i,,"
(47)
..;=1
If we have another Killing K (2) (necessarily commuting with the one used to Tdualize) giving rise to a chiral current, we have [9] V ~ K (2) = 0 ( + for an holomorphic current and - for an antiholomorphic one). The T-duality canonical transformation of K (2) gives another dual Killing vector with a chiral current. Instead of the privileged T-duality mapping for the generalized covariant derivative, we could be interested in the one for the "generalized divergence" V A+ '~,~4~am ' ~ ' Again ul ',...,Z'm" the study is very simplified in the reduced framework. l J That includes the mapping of complex structures, although it requires the additional vanishing of the Nijenhuis tensor [7].
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
731
After determining the T-duality weight ~,, ,1 ........ following the same procedure as ~¢ ,t+ ~au~.....u~ in the covariant derivatives' map, the required mapping reads ~la~,...,~t ± ~,~,...,v,, = ( - l)ga,'-q-G ~. 001",(n°--no+l)aA#l,...,#l -,~'l,....v,, ' 0
I
f7 pi f ''~±~, p " ' ""~' ........ = ( + G o o ) (n -no) +1 x-'7±..,a,~,,...,~, --a '~,t........ •
(48)
We have learned to read it in the common language as (K 2 = K , K / ~ ) 1
~±vt,...,v,,,-
Z:~p
m
5:Br
I 1 - 're
]~-'al,...,°tr,,
'
S=I l
m
- - a z ± v ~ ........ : K2(1-I r,r ±/3r ) { ~ 1-I i X - vr±o? , ]--p~--,r, ........ \
r=l
\
s=l
•
(49)
/
This section shows how elementary geometrical objects transform under T-duality depending on their defining properties: covariantly constant tensors transforms canonically (44), i.e. changing indices with the "vielbeins" T:~ and T+; "divergenceless" tensors transform under (49) changing the index corresponding to the divergence with T~: instead of T±; the generalized connection transforms as a true connection (46) with respect to the T-duality canonical transformation with the only peculiarity of using TT instead of T± in the index corresponding to the derivation. This anomaly propagates to the T-duality transformation of every index associated to derivation as we have seen in (44), (49), and finally it will be responsible for the inhomogeneous transformation of the generalized curvature, as we will show in Section 5.
4. T-duality classical dynamics In addition to the D-dimensional manifold M representing the target space-time, in the context of strings we have a two-dimensional world embedded in it, the worldsheet X, identified with the dynamical string. 12 At tree level in the string dynamics, X has the topology of the sphere. Choosing light-cone real coordinates o-±, the covariant world-sheet derivatives for mappings Y(o -+, o--) between X and the tangent space of our manifold M in X ~ ( t r + , o - - ) are V± = O+ + F ± , the pull-back being in terms of the string's embedding XU(o-+, o--),
Extending the definition (12) to the Y mappings, 12 l omit here p - b r a n e s , D-branes and any other kinds o f stringy extended objects.
(50)
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
732
y ( o - + , o - - ) -- j ( A ( X ( o - + , o - - ) ) ) y ( o - + , o - - ) ,
(52)
where y is the reduced mapping. With that definition we can prove that the reduced partner o f the pull-back is =,
= , rt~,
(53)
where I call (O±x) ~ - J ( A ) ~ O ± X ~ following (52). 13 In the reduced framework it is easy to convince oneself that the only T-dual change for (O±x) that transforms the pull-back diagonally is nothing but the one responsible for Buscher's formulas:
O±X~ = T ~ v O ± X ~,
(54)
the reduced pullback transformation being =
o,
o = T
¢
,
= TG0o3~°' i,
1 i hJ = Goo'Y±.i,
o,
(55)
which again can be summarized as
-~, ~z = (qzGoo)("°-"o)y~ ~z Y±
(56)
except for the y~: o, in which there is a flip of sign allowing the T-duality covariantization (35) o f the world-sheet covariant derivatives. It is worthwhile to mention that (54) implies KuV+V_X u = KuV_V+X u = 0 ,
~ / ~ u ~ + ~ , _ ~ s , = / ~ u ~ , _ ~ , + ) ~ = 0,
which is automatically valid for the classical string (it is the current conservation corresponding to the isometry). Again the simplicity of Eqs. (55) allows us to build the diagonal mapping for any Y(o -+, o - - ) mapping 14 y q2 {gl'-r- t T - - ~ l ' " " f l 4 \v
, ~
/!]l,...,p
m
~-
(TGoo)(n°-n°)y(cr +, 0 " - ~ ] ~i i1]'l','.'.'.[,~l ~1m
(57)
with the property of transforming linearly the covariant derivatives V i : [3q,.., ,/,Zl ~ ± y ( o -+, ~r- ~ m~,...,~, o , v, ,...,v,, = (qzGo0) (71--n0) V-4-y(o"J- ,o- -- )v, ........ •
(58)
In the usual setting Eqs. (57) and (58) can be written with the help of ( 3 8 ) , (39) and ( 4 4 ) , (45) as 131 write the reduced OX,~ between parentheses to specify that in general it is not the partial derivative of anything. Only when the pull-back of F(A)q vanishes can xt~ be identified with a sort of U(I) invariant embedding coordinates. laThe intermediate tool is now the analogue to (35) in the world-sheet, i.e. d±O(o-+,o--) --= (0± + Aoy° o)O(o-+, ~r-), transforming as O with weight Ao under T-duality.
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
733
(' )(.
r [ 77± a,'~ y ( o -+, o-- )~,, .,~, ,
Vl,...,v,,
£'7 v
~±~,,...,u, =
T
pl,...,l)m
r=l
(
s=l
)
n' T±'rI~r )( fi T±OlV~ "~ ____
)
r=l
yfll,...,fll •
:F c[1 ........
(59)
s=l
Eq. (59) allows us to extract the pull-back's T-duality transformation in the common language:
r~
= T~ aT~: /~F~, a + ( a± T~ t~) T~ e.
(60)
The most relevant example of this mapping is provided by O±X~, for which we have V±Oq:X** = T~:~V±Om X~, giving the classical stringy equivalence between the two different geometries.
5. Generalized curvature's transformation + ± a Defining as usual the generalized curvature as R # w r p = R ( F ) u ~ G a p which has the + o = -Ryder + p - -R/zvp~, + + - R p~rtzv q: and taking account the symmetry properties R~,~ Rtzvp~r generalized connection's transformation (33)15 we get the dual generalized curvature in the reduced framework,
?± oioj roij __ q:
(
1 2 roioj ± (Goo)
'
)
OiGooOj GO0 ,
( r~jk - aiGooy~°),
~± ijko - :t: G ~ (r~k o + &Goo3'~., o), -± _ r ± .±o..mo ijkl _ 2 p~O0)'ij Zk/ •
(61 )
rijkl --
We can convince ourselves that the inhomogeneous part of the transformation can be written in terms o f the Killing vector, giving the compact result
r#vcr = ( --1 )(gu+g~)(±Goo)-n°(_r/~vcrp ± - 2k2--~ V ± Z ".~v--cr ~ m k p ~j ,
(62)
where K 2 = k 2 = K u K ~ = Goo. In the usual setting the transformation (62) reads J6
~± R/~vcrp = T : C ' T v T B T : a T ±Pn { Rt
+ al3&#
22V:K~K,7)
"
(63)
It is important to note that the inhomogeneous part of the transformation only depends on the transverse components G0~, B0i and their first skew-symmetric derivatives. 15See the appendix for explicit formulas. i6 V~ kv only has an antisymmetric part due to the Killing condition.
J. Borlaf/Nuclear Physics B 514 [PAl] (1998) 721-739
734
When an object transforms inhomogeneously, say f = +(Goo)a(r+~,), the involution property of the T-duality transformation T 2 = 1 fixes the ~b transformation to be ~ = 1 T(G00)a~b, allowing us to create the homogeneous w =- r + ~¢, i.e. v~ = +(Goo)~w. Specifically, it means that
aTq:BT+~T+rI(-~V~aKBV~Kn), and therefore we can create the "corrected" generalized curvature W ~ p linearly under T-duality: 4-
+
W~uo-p :-- Rlxvo-p
1
+
(64) transforming
q:
K2 V~ K~V,~ Kp,
ff'~. = T~T:'T) ~T~oW~8~.
(65)
At this point is for free the rederivation of the generalized Ricci-tensor's transformation giving the need of a non-trivial dilaton change under T-duality in order to guarantee the one-loop conformal invariance of the dual string sigma model [2]. We get the dual-reduced Ricci tensor from (61) rg~.+•
~+ = ( - 1)g~(iGoo)-nO(r~p - V~± Vp+ In Goo) . ru.
(66)
It is well known that for the sigma-model coupling between the two-dimensional curvature and a scalar field called dilaton, ¢, it is enough to ensure the vanishing of the dual one-loop beta function, provided the former transforms under T-duality as = 'it' - ½ In a0o.
(67)
In other words, the tensor representing the one-loop beta function for the string sigma ± ± model (bosonic and supersymmetric) [11], i.e. , 8 ~ = R ~ 2 V u V ~ ~/', transforms linearly under T-duality as can be read off from (66) and (67),
: rJ
(68)
Therefore the this approach, minimally the transformation
vanishing of / ? ~ implies the one for ¢),, and vice versa. Following the dilaton one-loop beta function can be obtained trying to complete generalized scalar curvature R + in order to get a T-duality scalar; the for R ± = r ± results in, using Eqs. (61) and (66),
k ± = ~± = R ± - 2~7± icgiIn Goo + 2 l___y~~y~i tl00
(69)
and the desired T-duality scalar is fit, = R + -t-4((O~ib) 2 - ( v + ) z q b) -- 2H2,
(70)
where we define H2~ =- H,~,,RH/~~p. Modulo a constant term it coincides with the one-loop dilaton beta function [ 11 ].
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
735
6. Canonical connection a n d c u r v a t u r e The anomalous transformation of the index corresponding with first covariant derivations propagates in an annoying way to objects constructed from higher derivations, such as the generalized curvatures and the one-loop beta functions. In other words, the covariant derivation V + does not commute with the canonical map defined in (44). Looking at the generalized connections' transformation (33) we notice that there is a minimal covariant subtraction giving a connection, the canonical connection, which commutes with the canonical T-duality, -± p -_±p
F~
± p
1__~_K VmKp
= r 2±a T ;±~TP , ± o - f.±~ +
(O~T~¢)T o,
(71)
Because the anomaly mentioned above is located in the y ± ( r components, the whole effect of the subtraction is to cancel against them, giving 9~:"n O# = 0 and 9i~ p -- - ") ' i u± p 17 This fact allows the commutation between V ~ and the canonical T-duality map, 1 b'l,"',/",,,
=
m ± B r 1.][ -b'¢
]
(n)(m
' (£1,"',¢2"m
s=l
--p
...... , , = ,', ........
T ±~r
r= 1
1-1 1 1 T± ~ ]--a s= 1
,~,........ -
(72)
"
Another consequence is V0± = 0, implying R0,,,~p = 0. What is more, the new connection is compatible with the metric provided that K ~ is a Killing vector. In a certain sense the barred connection seems to be the most natural associated to the presence of a Killing. If we think of the Killing as a vector field indicating the direction in which nothing changes, we would expect that parallel transport is really insensitive to displacements in that direction. This happens with the canonical connection but not with the Levi-Civita (or its torsionfull generalizations) connection. The commutation with the canonical T-duality implies that the curvature for V ± transforms canonically (the same as the Ricci tensor and the scalar curvature) -±
-
±"
±~ ±~T ± n g ' ±
(73)
I will list five independent T-duality scalars Ii = R ± - ( V ± ) e l n
12 = H 2 -
K 2-
~-~H~K'~K ~,
~K--~H ~ K ' ~ K B,
17 These conditions guarantee that if a tensor is covariantly constant with respect to V ~ , it does too with respect to ~ ' ~ . The converse is in general not true. Therefore, the metric commutes with V ~
736
J. BorlaJTNuclear Physics B 514 [PM] (1998) 721-739
1
2
13 = - K S ( ~ 7 u K ~ 7 U K ~ + H ~ K
l
14 = ~ K ~ O ~ K a H
cL 13
K ),
a'a
~" ,
70K22 I5 = \ K2 ,] ,
(74)
built with the help of R+ and V~ being the Levi-Civita covariant derivation. In particular, R+ = Il T 2/4. Finally, with this covariant derivation, the T-duality canonical map commutes with the basic geometrical operations: linear combinations, tensor products, permutation of indices, contractions and covariant derivations.
7. Conclusions This work shows how the reduced geometry is a privileged framework for the study of the T-duality's geometry and possibly of many other different issues related with abelian Killing vectors. The T-duality transformation diagonalizes in the reduced setting, allowing us to get in a straightforward way results pursued for a long time, such as the generalized curvatures' map, the canonical map for the covariant derivatives in the target space and in the world-sheet, the minimal correction to connections and curvatures in order to transform linearly and T-duality scalars. The introduction of the dilaton can be seen as the minimal modification needed to map the one-loop beta functions preserving conformal invariance, but from the geometrical view, the dilaton is completely insufficient to build in a systematic way T-duality tensors (i.e. tensors transforming canonically under T-duality). The object serving to "covariantize" under T-duality is the Killing vector itself across the canonical connection, as shown in the last section. In connection with that, new T-duality scalars have been found without the help of the dilaton. Future work could comprise the higher-loop corrections to Buscher's formulas, the map for the invariants characterizing the geometry and the topology of the manifold, the global questions in the reduced setting, the non-abelian generalizations of this procedure, and a deeper study of the geometry of the canonical connection and its relation with the string sigma model.
Acknowledgements I wish to thank E. Alvarez for suggesting, reading and commenting on this work. Also I want to thank P. Meessen for useful discussions. This work has been partially supported by a C.A.M. grant.
J. B o r l a f / N u c l e a r P h y s i c s B 5 1 4 [ P M ] (1998) 7 2 1 - 7 3 9
737
Appendix A. The reduced connection and Riemannian curvature In order to have the opportunity of getting an idea of the operativity of the reduced framework, I show here the usual Levi-Civita connection for the generic metric with n commuting Killing vector fields G I~,,
( Gab Aai "~ Abj Caij + AicAjclG ca J '
(A.I)
\
Fa 1aa~if, bc = 2"~ti o " J b c , r [ t b = - 513 ^i G ~ b ,
F~i,i = F)(, : ~' ' ~~,U a b r~ "i J i-el~i
--
Abi3JGa,,)
,
= Fi~b, -_ ~I ( G bc OiGac - a, ub~t -a7c , ~,c.j i q- A icA j Ob ^ j Gac),
F a
ij = 21{ ( X g^ i A j a + ~jA~/) + A icA j Ab k Oa
^k
Gbc
c a t'^ck + Al'Pyk)}. + G ab (AjO~Gbc + ACc~iGi, c) _ AkG~b(AjF~
(A.2)
Eqs. (A.2) must be compared with (18) to appreciate the advantages. Now I would like to make explicit the relation between a generic connection F and its reduced version y, i i Tab = Fah,
.yc
_
c
i
c
ob -- Fab q- FabAi '
"Y[*i= F~l*i - FJbA~ ~, y~,, = I ,~, - F[,aAai ,
el',= <,- F~.A~"+ h
b
b
Yai = Fai - F a c A i
c
Tfj = F~j - Fajk Aia
FJiaA.~j
b
F{oAfA}, fj
acab
+ FaiAj - - ac'', "'j' pk aaab F(~ aa-g q---abZa, ta.j, --~a"
a a Fa ab a b F.a a b a c "Yij = F i j - - - - b j ' ~ i - - F i b A j + - - b c ' ~ i n j
- 3 i a ; ! + I'~ia ~.
r,k a b a a r,k A b a a ilk abAcaa - - - - b j n i n k - - ,~ ibraj n k -}- * b c t a i n j n k
(a.3)
Paradoxical as it may seem, the task to calculate the reduced Levi-Civita connection is shorter than the usual one, because the knowledge of its invariance under the adapted diffeomorphisms (6) drops out of the terms proportional to A~, being non-derivatives (we can call it A = 0 projection). Therefore (A.3) must be understood as TP~ = (rL, - O,,J,~)Ia=0. One just needs to know Fla= 0 instead of F itself, which makes it easier to get the reduced version. The usual curvature, torsion and Ricci tensor's expression for a generic connection FPv are
J. Borlaf/Nuclear Physics B 514 [PM] (1998) 721-739
738
P = I ( F P u - F~,~),
R( F) t,,, = R( F)Pp~.
(A.4)
The reduced Riemannian curvature for the U(1)" is given by 1
^i
rabcd = ~ ( 3 GacOiGbd -- Oi GbcOiGad ) , d -- GebOC~a) rabci = -l Fije ~"]Gcd ( Gea ¢~b I cd ^ck d rijab = ~ ( G cgiGacajGbd ÷ GacGbdFi F'.ik - ( i j < 1 ^
^
l
raibj = ~ V i V jGab - ~G
cd
l
>j i ) ) ,
^d
OiGbdo3jGac -- "~GacGbdF'i kFfk,
^ b rakij = - ~1 Vk ( GabFijb ) + ~1 ( F~iOj Gab --
b V;,jOiG.b),
rijkl=~ilkl ÷ I G a b ( F,Ik a F)tb __ F)kFila b ÷ 2FijF~t) , ab
(A.5)
where the hatted objects are the ones calculated with the quotient metric Gij, Fi~~ =3iA.~] - OjA'~, and A~~= GabAib. The Riemannian curvature is obtained using (12), (8), (9), i.e. R = J ( - A ) r .
Appendix B. The generalized curvature In this appendix I will write the reduced generalized curvature needed for the calculations in Section 5. To do it in a T-duality suitable way means the introduction of t h e / 3 ~ O = ~7~ + ~-3i ln Goo)O T-duality covariant derivative. As in (35), Ao is O's T-duality weight. In terms of G0o and F/~: = Yij+ o the relevant components are -4-
roioj : {½biiOiGoo + G~oP,~kF~ } + -G-~ai in Gooaj In Goo, FOijk =
GooOi Fjk ÷
( Fi~ aj ln aoo -F/jOklnGoo) +
r i±j k l -- {Rijkl ^± ÷ G°°( Fi~ ± Fit± --
+ ~GOOF~: jkOi lnGoo,
+ ± )} + Goo(Y~+ Tijm ) )'kl, m F)kF.
(B.1)
R~,, = R ( P + h)i,k • The other comoonents are related by the symmetry properties where = The terms in brackets are the ones transR ~c~p '~'R~,~p = - R ~ , ~ and R ~ p = R ~ . forming with the dominant ( - 1 )(g-+g")(+Goo)-no, while the remaining terms transform with - ( - 1) (g~+g~)(+Goo) -~° giving the inhomogeneous part - ~2 w+~v ~ . . . .w:F ~ Kp.
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