Real symplectic formulation of local special geometry

Physics Letters B 637 (2006) 102–106 www.elsevier.com/locate/physletb

Real symplectic formulation of local special geometry Sergio Ferrara a,b , Óscar Maciá c,∗ a Physics Department, Theory Unit, CERN, CH-1211 Geneva 23, Switzerland b INFN, Laboratori Nazionali di Frascati, Via Enrico Fermi 40, I-00044 Frascati, Italy c Departament de Física Teòrica, Universitat de València and IFIC C/Dr. Moliner, 50, E-46100 Burjassot (València), Spain

Received 23 March 2006; accepted 7 April 2006 Available online 24 April 2006 Editor: L. Alvarez-Gaumé

Abstract We consider a formulation of local special geometry in terms of Darboux special coordinates P I = (p i , qi ), I = 1, . . . , 2n. A general formula for the metric is obtained which is manifestly Sp(2n, R) covariant. Unlike the rigid case the metric is not given by the Hessian of the real function S(P ) which is the Legendre transform of the imaginary part of the holomorphic prepotential. Rather it is given by an expression that contains S, its Hessian and the conjugate momenta SI = ∂SI . Only in the one-dimensional case (n = 1) is the real (two-dimensional) metric proportional to ∂P the Hessian with an appropriate conformal factor. © 2006 Elsevier B.V. All rights reserved.

1. Introduction Local special geometry [1–5], is the underlying geometry of N = 2 supergravity in four dimensions coupled to vector multiplets. It also emerges in any compactification, obtained from a higher-dimensional theory, when eight supersymmetries are preserved in four dimensions or, in a more general setting, when the lower-dimensional theory can be viewed as a “deformation” of a N = 2 locally supersymmetric Lagrangian. These theories include Calabi–Yau compactifications from type II supergravity [3,4] as well as compactifications on more general manifolds with G = SU(3) structures [6–10]. Special geometry also plays an important role in the physics of black holes, attractor equations [11–14], which control the horizon geometry, and more recently it has been used also to connect the black hole entropy to topological partition functions in superstring theory [15–17]. In this framework it is useful to reconsider a formulation of special geometry in real [18–20], rather than complex vari-

ables. Real variables, will be denoted as Darboux coordinates P I = (p i , qi ), i = 1, . . . , n, where n is the complex dimension of the (local) special manifold. One of the reasons to adopt these coordinates is that the attractor equations can be written as real equations [11–14,21–23], which determine the real charge vector Q = (mΛ , eΛ ) (Λ = 0, 1, . . . , n) in terms of the holomorphic sections V = (X Λ , FΛ = ∂Λ F ) of special geometry [1–3,5]; a direct real formulation of the geometry may therefore have a simplifying role. In this Letter we extend a previous investigation [24] on the real formulation of rigid special geometry to the local case. The main result is the derivation of a general formula for the metric in special Darboux coordinates and also comment on the main differences from the rigid case. A very important role is also played here by a 2n × 2n symplectic real symmetric matrix M(fij , fij ) constructed in terms of the holomorphic matrix fij =

∂ 2 f (t) , ∂t i ∂t j i

* Corresponding author.

E-mail addresses: [email protected] (S. Ferrara), [email protected], [email protected] (Ó. Maciá). 0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2006.04.010

X where t i = X 0 are the “special coordinates” of the local special geometry and f (t) is the holomorphic prepotential in special coordinates. As in the rigid case the M matrix (which satisfies MΩM = Ω where Ω is the invariant symplectic form of

S. Ferrara, Ó. Maciá / Physics Letters B 637 (2006) 102–106

Sp(2n, R)) is related to the Hessian ∂ 2S ∂P I ∂P J of a certain Hamiltonian function S(p, q) written in Darboux special coordinates but, unlike the rigid case, the general formula for the metric is given by

HI J =

 1 1  HI J + 2 SI SJ + (H ΩS)I (H ΩS)J . (1) 2S 4S The first term is analogous to the term that is present in the rigid case. The second term, due to the Hodge–Kähler structure of local special geometry, has no rigid analogue and depends, other than S and its Hessian, also on the conjugate momenta ∂S I SI = ∂P I , of the Darboux coordinates P . Note that (H ΩS)I denotes the expression gI J (P ) = −

(H ΩS)I = HI K Ω KL SL ,

Ω KL = −ΩKL .

This Letter is organized as follows. In Section 2 we describe local special geometry in holomorphic and Darboux special coordinates and compute the Kähler metric in real symplectic coordinates. In Section 3 we compute the (real symmetric) metric in the real symplectic Darboux coordinates, and we show the explicit Sp(2n, R) covariance of this metric and its differences from the rigid case. In Section 4 some examples are given. 2. Local special geometry in Darboux coordinates

the Sp(2n + 2, R) covariance is lost (by the Kähler gauge-fixing X 0 = 1) but still a Sp(2n, R) structure is present with respect to the reduced 2n-dimensional holomorphic section   ∂f VR = t i , fi = i . (5) ∂t 2

In particular, in terms of the matrix fij = ∂t∂i ∂tf j , we can construct a real symmetric 2n × 2n symplectic matrix M(fij ) as follows [12,26]:   fij + fik (f −1 )kl flj −fik (f −1 )kj M(fij ) = −(f −1 )ik fkj (f −1 )ij (6) with the obvious property MΩM = Ω. We note at this point that the M(fij ) matrix considered here is not the M(F ) matrix considered in [12,26]. Indeed while M(F) ∈ Sp(2n + 2, R) is a (2n + 2) × (2n + 2) symplectic matrix, the matrix considered here is 2n × 2n and it is Sp(2n, R) symplectic. However, it is this last matrix which plays a role in our considerations. The Kähler potential of special geometry in these coordinates is K = − log Y,     Y = i 2f − 2f¯ − t i − t¯ i (fi + f¯i )

(7) (8)

from which the following expression for the metric holds

We consider local special geometry in special coordinates Λ t Λ = XX0 = (t 0 = 1, t i ). In these coordinates the holomorphic symplectic sections   V = X Λ , ∂Λ F (2) take the simple form   ∂f i i V = 1, t ; f0 = 2f − t fi , fi = i ∂t

103

(3)

in terms of the prepotential f (t) = (X 0 )−2 F (X) where F (X) is homogeneous of degree 2 [1] X Λ ∂Λ F = 2F. From this, it follows that f (t) is a fairly arbitrary holomorphic function of t i . Moreover, t i are “scalar” sections of the Hodge bundle since under a holomorphic gauge transformation     Λ ∂¯i α = 0 X , FΛ −→ eα X Λ , FΛ , (4) the special coordinates t i and the prepotential f (t) are invariant. Property (4) is crucial in defining real coordinates. In fact, while it makes sense to define the real part of the holomorphic special coordinates t i , it would not make sense to define a real part of X Λ since this operation would be incompatible with the Hodge structure of the manifold. Note that this is also the main difference between the local and the rigid cases [14,18,19,24], since in the latter the Hodge structure is absent and this obstruction does not arise. We also note that in local special coordinates

1 1 gi j¯ = ∂i ∂¯j K = − Yi j¯ + 2 Yi Yj¯ . Y Y By explicit computations the two terms in (9) give i Ki j¯ = − (fij − f¯ij ) Y    1  + 2 fi − f¯i + t¯k − t k fki Y     × f¯j − fj + t k − t¯k f¯kj .

(9)

(10)

We now go to Darboux special coordinates exactly as we did in the rigid case [24], i.e., we define t i = p i + iφ i ,

fi = qi + iψi

(11)

and the Legendre transform S(p, q) of the imaginary part of the prepotential L = f.

(12) , φi )

and Then the (qi gate variables for L qi =

∂L , ∂φ i

ψi =

(p i , ψ

i)

real sections are pairs of conju-

∂L ∂p i

(13)

and in terms of S given by   S(p, q) = qi · φ i (p, q) − L p, φ(p, q)

(14)

we have the following set of equations φi =

∂S , ∂qi

ψi = −

∂S . ∂p i

(15)

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By comparing (14) and (8) we realize that [15–17,27] Y = 4S,

(16)

i.e., 1 S = e−K . (17) 4 Eq. (17) is the main difference between rigid and local special geometry. We just recall that in the former the (square of the) distance ds 2 is given by [18,19,24,25] Kij dzi ⊗ d z¯ j = −2HI J dP I ⊗ dP J ,

(18)

∂2S

where HI J = ∂P I ∂P J is the Hessian of the S functional. In the next section we will see how the rigid formula (18) is modified in the local case. The change of variables from the holomorphic sections (t i , fi ) to the Darboux variables (p i , qi ) and (φ i , ψi ) is exactly as in the rigid case it is still true [24] that the H matrix ⎛ 2 ⎞   ∂ S ∂2S j i j i S S ∂p ∂p ∂p ∂qj ij i ⎠= H =⎝ 2 (19) ∂ S ∂2S Sji S ij ∂qi ∂p j

∂qi ∂qj

is equal to the (negative of the) M matrix defined in (6), then it follows that H ΩH = Ω, i.e., Sik Skj − Sjk Ski = 0, j

j

dt i ⊗ d t¯j  ∂ 2S ∂ 2S j = δki δl + k ∂qi ∂p ∂qj ∂p l  2  ∂ S j ∂ 2S i +i δ − δ dp k ⊗ dp l ∂qi ∂p k l ∂qj ∂p l k  2 ∂ 2S ∂ 2S ∂ S ∂ 2S + + ∂qi ∂ql ∂qj ∂pk ∂qi ∂pk ∂qj ∂ql  2  ∂ S j ∂ 2S i +i δ − δ dp k ⊗ dql ∂qi ∂ql k ∂qj ∂ql k  2  ∂ 2S ∂ S + dqk ⊗ dql . ∂qi ∂qk ∂qj ∂ql

(20)

By comparing the M ij and Mji entries of M with H it also follows that    fij = − Sik + iδik S −1 kj , (21) where (S −1 )kj = (S kj )−1 is the inverse of S kj defined by (19). We can now write Eq. (10) in Darboux special coordinates by means of Eqs. (11), (13), and (21) and we finally obtain  1  ij −1 1 ∂S ∂S ∂S ∂S Ki j¯ = − + 2 + fki f¯lj S 2S ∂qk ∂ql 4S ∂p i ∂p j  ∂S ∂S ∂S ∂S ¯ + f + (22) f ki kj , ∂qk ∂p j ∂qk ∂p i

(24)

By explicit multiplication of (22) with (24) we finally obtain, from the first term in (22) −

j

S ik Sk − S j k Ski = 0, Sik S kj − Sik Sk = δi .

We just mention that the computation of dt i ⊗ d t¯j is exactly the same as in the rigid case [24] since the Legendre transform and the conjugate variables are the same. Indeed only Eqs. (19)–(21) which are the same as in rigid special geometry really matter. The change of variables in the differentials was given in Eq. (50) of Ref. [24] and it is reproduced here for the benefit of the reader:

1  ij −1 i 1 dt ⊗ d t¯j = − HI J dP I ⊗ dP J . S 2S 2S

(25)

For the second term of (22) (by multiplying by dt i ⊗ d t¯j ) we find  1  − 2 S K HKI SJ − S K HKJ SI 8S   × dPL H LI dP J − dPL H LJ dP I , (26) where S K = Ω KI SI ,

H LJ = (HLJ )−1 ,

dPL = ΩLI dP I

and Ω KI = (ΩKI )−1 = −ΩKI . By multiplying the four terms in (26) we finally obtain

where fij is given by (21). We just note that while in the rigid case Ki j¯ = −2(S ij )−1 , in the local case (22) also depends on S and the canonical mo∂S menta SI = ∂P I . The second term in (22) is the main difference between the rigid and local case.

 1  SI SJ + (H ΩS)I (H ΩS)J , 2 4S where

3. The real symplectic metric

The final expression for the real symmetric metric in Darboux coordinates, for local special geometry, is therefore given by

In order to compute the real metric we have to transform the “distance” from holomorphic to Darboux special coordinates

gI J (P ) = −

ds 2 = Ki j¯ dt i ⊗ d t¯j = gI J dP I ⊗ dP J .

(23)

The result is obtained by computing the differentials dt i ⊗ dt j in Darboux coordinates and then comparing the two expressions in Eq. (23).

(27)

(H ΩS)I = HI K Ω KL SL .

 1 1  HI J + 2 SI SJ + (H ΩS)I (H ΩS)J . (28) 2S 4S We note at this point that a major simplification occurs for I = 1, 2 which corresponds to one-dimensional complex special geometry. Indeed in this case  KL SL , S K HKI SJ − S K HKJ SI = −ΩI J SK H −1 (29)

S. Ferrara, Ó. Maciá / Physics Letters B 637 (2006) 102–106

dPL H LI dP J − dPL H LJ dP I = −Ω I J dP L HLK dP K

and then (30)

and then (26) becomes  KL 1 SK H −1 SL HI J dP I ⊗ dP J . 2 4S In this case, the metric therefore is    1 1  −1 SH S HI J . + gI J = − 2S 4S 2

(31)

(32)

4. Some examples In this section we compute the real metric in some examples of special geometry. The two examples in question correspond to a cubic and quadratic prepotential, for one complex scalar: 1 f (t) = t 3 , (33) 3  i f (t) = t 2 − 1 . (34) 4 Let us first consider the cubic case. The S functional is 8 4S = Y = − (t)3 , (35) 3 where S = − 23 (t)3 so that t < 0. Note that in this case ftt = 2t < 0 and therefore the Hessian will be positive definite. Going to Darboux coordinates we find  1/2 q = p 2 − φ 2 −→ φ = ± p2 − q (36) so we have p 2 > q and also the negative root must be chosen such that S is positive. From (36) and (35) we obtain 3/2 2 2 . p −q 3 The Hessian matrix is given by   −1/2 2(2p 2 − q) −p  2 . H = p −q 1 −p 2

  −3/2 3  2 −3/2 1 3 9  2 SH −1 S = − = p −q + + p −q 2 2S 4 8 8 4S (40) so that the metric is   −2 2(2p 2 − q) −p 3 2 gI J (p, q) = p − q (41) . 1 8 −p 2 −

We now turn to the second example (34) where now

We finally note that the metric given by (28) has a manifest Sp(2n, R) covariant structure. The simplification occurring in (29) is due to the fact that for Sp(2, R) the antisymmetric representation is the identity while the traceless part is nonvanishing for n > 1; and this is the reason why (28) and (32) are equivalent only for n = 1. It is obvious that in special coordinates the Sp(2n + 2, R) structure of the geometry is lost and only a Sp(2n, R) structure is manifest. Nevertheless we have shown that this is what is needed to obtain a general expression for the metric in terms of the Darboux data of local special geometry. The Sp(2n + 2, R) covariance will be considered elsewhere.

S=

105

Y = 4S = 1 − t t¯ = 1 − p 2 − φ 2

(42)

therefore we must take p 2 + φ 2 < 1. Since  1 2 p − φ2 − 1 = L 4 we have f =

q=

1 ∂L =− φ ∂φ 2

−→

φ = −2q.

It follows that Y = 4S = 1 − p 2 − 4q 2 ,

S=

1 p2 − − q 2. 4 4

The Hessian matrix is   0 − 12 H= 0 −2

(43)

(44)

and it is negative definite. The conjugate momenta are 1 ∂S = − p, ∂p 2

∂S = −2q. ∂q

We then have  1 2 p + 4q 2 2 and the conformal factor becomes SH −1 S = −

−2

1 . − 4q 2 )2

(1 − p 2

The local metric is then finally given by   1 1 0 gI J (p, q) = . (1 − p 2 − 4q 2 )2 0 4

(45)

Acknowledgements (37)

(38)

Since Det(H ) = 1 and also (p 2 − q)1/2 Tr(H ) = 2(2p 2 − q) + 1 2 > 0, H is indeed positive definite. We can now compute the metric by using (32). First, by computing the second term, we obtain  3/2 SH −1 S = 2 p 2 − q (39)

We would like to thank M.A. Lledó for enlightening discussions. S.F. would like to thank the Theoretical Physics Department at the University of Valencia where part of this work was performed. S.F. was partially supported by funds of the INFN-CICYT bilateral agreement. The work of S.F. has also been supported in part by the European Community Human Potential Program under contract MRTN-CT-2004-005104 “Constituents, fundamental forces and symmetries of the Universe”, in association with INFN Frascati National Laboratories and by the DOE grant DE-FG03-91ER40662, Task C.

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O.M. wants to thank the Department of Physics, Theory Division at CERN for its kind hospitality during the realization of this work. The work of O.M. has been supported by an FPI fellowship from the Spanish Ministerio de Educación y Ciencia (MEC), through the grant FIS2005-02761 (MEC) and EU FEDER funds, by the Generalitat Valenciana, contracts GV04B-226, GV05/102 and by the EU network MRTN-CT2004-005104 “Constituents, fundamental forces and symmetries of the Universe”. References [1] B. de Wit, P.G. Lauwers, A. Van Proeyen, Nucl. Phys. B 255 (1985) 569. [2] A. Strominger, Commun. Math. Phys. 133 (1990) 163. [3] S. Ferrara, A. Strominger, N = 2 space–time supersymmetry and Calabi– Yau moduli space, Presented at Strings ’89 Workshop, College Station, in: Coll. Station Workshop, 1989, pp. 0245–0252; P. Candelas, X. de la Ossa, Nucl. Phys. B 355 (1991) 455, in: Coll. Station Workshop, 1990, pp. 0401–0429; L. Castellani, R. D’Auria, S. Ferrara, Phys. Lett. B 241 (1990) 57. [4] S. Cecotti, S. Ferrara, L. Girardello, Int. J. Mod. Phys. A 4 (1989) 2475. [5] L. Andrianopoli, M. Bertolini, A. Ceresole, R. D’Auria, S. Ferrara, P. Fré, T. Magri, J. Geom. Phys. 23 (1997) 111, hep-th/9605032. [6] S. Gurrieri, J. Louis, A. Micu, D. Waldram, Nucl. Phys. B 654 (2003) 61, hep-th/0211102. [7] S. Kachru, M.B. Schulz, P.K. Tripathy, S.P. Trivedi, JHEP 0303 (2003) 061, hep-th/0211182.

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