A potential for Generalized Kahler Geometry

Abstract

We show that, locally, all geometric objects of Generalized Kahler Geometry can be derived from a function K, the "generalized Kahler potential''. The metric g and two-form B are determined as nonlinear functions of second derivatives of K. These nonlinearities are shown to arise via a quotient construction from an auxiliary local product (ALP) space. Comment: 12 pages, contribution to "Handbook of pseudo-Riemannian Geometry and Supersymmetry"

Full text

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arXiv:hep-th/0703111v1 12 Mar 2007
UUITP-04/07
HIP-2007-12/TH
YITP-SB-07-09
A potential for Generalized K¨ ahler
Geometry
Ulf Lindstr¨ oma,b, Martin Roˇ cekc, Rikard von Unged, and Maxim Zabzinea
aDepartment of Theoretical Physics Uppsala University,
Box 803, SE-751 08 Uppsala, Sweden
bHIP-Helsinki Institute of Physics, University of Helsinki,
P.O. Box 64 FIN-00014 Suomi-Finland
cC.N.Yang Institute for Theoretical Physics, Stony Brook University,
Stony Brook, NY 11794-3840,USA
dInstitute for Theoretical Physics, Masaryk University,
61137 Brno, Czech Republic
Abstract
We show that, locally, all geometric objects of Generalized K¨ ahler Geometry can be derived
from a function K, the “generalized K¨ ahler potential”. The metric g and two-form B are
determined as nonlinear functions of second derivatives of K. These nonlinearities are
shown to arise via a quotient construction from an auxiliary local product (ALP) space.

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1Introduction
Generalized K¨ ahler Geometry [1] is a special and particularly interesting example of Gen-
eralized Geometry [2, 1]. It succinctly encodes the bihermitean geometry of Gates, Hull
and Roˇ cek [3] in terms of a pair of commuting generalized complex structures (J1,J2).
Here we use the equivalent bihermitean data to give a complete (local) description of
Generalized K¨ ahler Geometry. We show that there exists a function K with the interpre-
tation of a generating function for symplectomorphisms between two sets of coordinates in
terms of which all geometric quantities may be described. In particular, the combination
E =1
terms of second derivatives of K. When further analyzed, this expression has the structure
of a quotient from some higher dimensional space. We find this auxiliary space, which
we call an ALP space, and display the quotient structure. Finding a Generalized K¨ ahler
Geometry is thus reduced to a linear construction in terms of the corresponding K on the
ALP.
The bihermitean geometry was first derived as the target space geometry of N = (2,2)
supersymmetric nonlinear sigma models [3]. Correspondingly, all our constructions have a
sigma model realization, and many of the proofs were derived in that setting. In particular,
the “generalized K¨ ahler potential” K has an interpretation as the superspace sigma model
Lagrangian1. Detailed descriptions of this approach may be found in the two articles on
which the presentation is based [5, 6].
2(g+B) of the metric and antisymmetric B-field is given as a nonlinear expression in
2 Generalized Complex Geometry (GCG)
This section contains a brief recapitulation of the salient features of GCG. The references
for the mathematical aspects are chiefly [2, 1], while a presentation more accessible to
physicists (containing, e.g., coordinate and matrix descriptions) may be found in [7, 8].
A generalized almost complex structure is an algebraic structure on the sum of the
tangent and cotangent bundles of a manifold M defined via
J ∈ End(T ⊕ T∗)(2.1)
such that
J2= −1 (2.2)
and
JtIJ = I
(2.3)
1A similar situation arises in the projective superspace description of hyperk¨ ahler sigma models [4].
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where I is the metric corresponding to the natural pairing of elements X+η,Y +λ ∈ T⊕T∗:
< X + η,Y + λ >=1
2(ıXλ + ıYη) .
(2.4)
The generalized almost complex structure is a (twisted) generalized complex structure iff
it is integrable with respect to the (twisted) Courant bracket
[X + η,Y + λ]C:= [X,Y ]L+ LXλ − LYη −1
2d(ıXλ − ıYη) + ıXıYH
(2.5)
where [X,Y ]Lis the Lie-bracket and “twisted” refers to inclusion of the last term involving
the closed three-form H. Integrability is defined by the requirement that the distributions
defined by the projection operators
1
2(1 ± iJ)(2.6)
are involutive with respect to the Courant-bracket. Of great importance to physical ap-
plications is the fact that the automorphisms of the Courant-bracket, in addition to dif-
feomorphisms, also contain the b-transform:
eb(X + η) = X + η + ıXb
(2.7)
where b is a closed two-form: db = 0. When acting on a generalized complex structure
J it produces an equivalent generalized complex structure Jb. In a matrix representation
this reads;
?
b 1
Jb=
1 0
?
J
?
10
−b 1
?
(2.8)
We now restrict our attention to a subset of generalized complex geometries: generalized
K¨ ahler geometries (GKG). These are GCG’s with two commuting generalized complex
structures J1,2whose product defines a positive definite metric G on T ⊕ T∗that squares
to the identity [1];
[J1,J2] = 0
G = −J1J2
G2= 1(2.9)
This is the proper setting for the bihermitean geometry of [3], which we now describe.
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3Bihermitean Geometry
The data found in [3] to be necessary and sufficient to describe the target space geometry
of a supersymmetric nonlinear sigma model with (2,2) supersymmetry consists of two
(ordinary) complex structures J+ and J−, a metric g, hermitean with respect to both
complex structures, and a closed three-form H with a (local) two-form potential B.
The structures (J±,g,B) satisfy
J2
±= −1
Jt
±gJ±= g
∇(±)J±= 0
H = dc
+ω+= −dc
−ω−
(3.10)
where ω±are the two-forms gJ±, dc
derivatives have (Bismut) torsion [9]
±is dcwith respect to J±respectively, and the covariant
∇(±):= ∇ ± g−1H,dH = 0 ,
(3.11)
where ∇ is the Levi-Civita connection.
When J±commute, it was found in [3] that there exist coordinates z, ¯ z and z′, ¯ z′that
coordinatize ker(J+∓J−), respectively, and for which the metric and three-form are given
as derivatives of a function K(z, ¯ z,z, ¯ z′). Explicitly:
ω±= (dc
∓d + dc
Πdc
±)K
⇔
gz¯ z= ∂z∂¯ zK , gz′¯ z′ = −∂z′∂¯ z′K ,
H = ddc
+dc
−K
⇔
Hz,¯ z,z′ = ∂z∂¯ z∂z′K , Hz,¯ z,¯ z′ = −∂z∂¯ z∂¯ z′K ,
Hz′,¯ z′,z= −∂′
z∂¯ z′∂zK , Hz′,¯ z′,¯ z= ∂z′∂¯ z′∂¯ zK
(3.12)
where dc
differentials dc
A complete description for geometries with a nonvanishing coker[J+,J−] was an open
problem until our paper [5]. Before describing the solution to the problem in sections 5
and 6, we recall Gualtieri’s map of the bihermitean data (J±,g,B) into Generalized K¨ ahler
Geometry [1]:
Πis the dcoperator defined with respect to the product Π := J+J− and the
+,dc
−,dc
Πanticommute.
J1,2=
?
1
B 1
0
??
J+± J−
ω+∓ ω−
−(ω−1
−(Jt
+∓ ω−1
+± Jt
−)
−)
??
10
−B 1
?
.
(3.13)
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G =
?
1
B 1
0
??
0 g−1
g
0
??
10
−B 1
?
(3.14)
Note that what looks like a b-transform is not, since dB = H ?= 0.
It is an interesting fact that if we introduce the usual GCS’s corresponding to J±;
J±=
?
J±
0
0
−Jt
±
?
(3.15)
the map (3.13) may be summarized as2
J1,2= Π+J+B± Π−J−B,
(3.16)
with
Π±:=1
2(1 ± G) .
(3.17)
4Poisson-structures
There are three Poisson-structures relevant to our discussion. We first consider two real
ones:
π±:= (J+± J−)g−1= −g−1(J+± J−)t.
(4.18)
These were introduced in [10], where they were used essentially as described below, sim-
plifying an earlier derivation in [11]. They ensure the existence of coordinates adapted to
ker(J+− J−) ⊕ ker(J++ J−).
In a neighborhood of a regular point x0of π−we may choose coordinates xAwhose
tangents lie in the kernel of π−:
πAµ
− = 0,⇒ JA
+µ= JA
−µ.
(4.19)
Similarily, in a neighborhood of a regular point of π+we have coordinates xA′such that
πA′µ
+
= 0,⇒ JA′
+µ= −JA′
−µ
(4.20)
It then follows from the nondegeneracy of π+± π−that the Poisson brackets defined by
π+and by π−cannot have common Casimir functions. In other words, the directions A
and A′cannot coincide. The result is that we may write
J±=





∗ ∗
∗ ∗
0 0 Ic
0 0
∗
∗
∗
∗
0
0
±It




,
(4.21)
2This structure is related to how the map is derived in chapter 6 of [1]
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in coordinates adapted to
ker(J+− J−) ⊕ ker(J++ J−) ⊕ coker[J+,J−] ,
(4.22)
where we write a canonical complex structure as
I =
?
i
0 −i
0
?
(4.23)
and c labels the A and t labels the A′directions.
A third Poisson structure σ, related to the real Poisson structures, was introduced in
[12].
σ := [J+,J−]g−1= ±(J+∓ J−)π±= ∓(J+± J−)π∓.
(4.24)
The relation to π±implies that
kerσ = kerπ+⊕ kerπ−.
(4.25)
Tangents to the symplectic leaf for σ lie in coker[J+,J−]. Using (4.25), we can use σ to
investigate the remaining directions in (4.21). We need that
J±σJt
±= −σ ⇒ σ = σ(2,0)+ ¯ σ(0,2),
(4.26)
with the decomposition with respect to both J+and J−. Further σ(2,0)is holomorphic [12]:
¯∂σ(2,0)= 0 ,
(4.27)
We now investigate the structure of the cokernel.
5The structure of coker[J+,J−]
It is convenient to first treat the case ker[J+,J−] = ∅. Then we have at our disposal a
symplectic form Ω:
Ω := σ−1,dΩ = 0,Jt
±ΩJ±= −Ω .
(5.28)
Choosing complex coordinates with respect to J+,
J+=
?
Is
0
0
Is
?
,
(5.29)
with I as in (4.23), we have the decomposition
Ω = Ω(2,0)
+
+¯Ω(0,2)
+
.
(5.30)
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and
∂Ω(2,0)
+
= 0 ,
¯∂Ω(2,0)
+
= 0 ,
(5.31)
which means in particular that Ω(2,0)
coordinates such that
+
is holomorphic. We may then choose Darboux
Ω(2,0)
+
= dqa∧ dpa,
¯Ω(0,2)
+
= d¯ q¯ a∧ d¯ p¯ a.
(5.32)
Alternatively, we can choose complex coordinates with respect to J− and similarily
derive:
Ω(2,0)
−
= dQa′∧ dPa′,
¯Ω(0,2)
−
= d¯Q¯ a′∧ d¯P¯ a′.
(5.33)
The transformation {q,p} → {Q,P} is a canonical transformations (symplectomor-
phism) and can thus be specified by a generating function3K(q,P). Thus in a neighbor-
hood, the canonical transformation is given by the generating function K(q,P)
p =∂K
∂q
,Q =∂K
∂P
.
(5.34)
We calculate all our geometic structures J+, J−, Ω and g in the “mixed” coordinates
{q,P}, using the transformation matrices
∂(q,p)
∂(q,P)=
?
1
∂p
∂q
0
∂p
∂P
?
=
?
10
∂2K
∂q∂q
∂2K
∂P∂q
?
:=
?
10
KLL KLR
?
(5.35)
and
∂(Q,P)
∂(q,P)=
?
∂Q
∂q
0
∂Q
∂P
1
?
=
?
∂2K
∂q∂P
0
∂2K
∂P∂P
1
?
:=
?
KRL KRR
01
?
,
(5.36)
where the labels L and R are shorthand for the coordinates {qa, ¯ q¯ a} and {Pa′,¯P¯ a′} in
(5.32) and (5.33) above. The expressions for J±are nonlinear and can be read off from the
formulae given below for the general case with ker[J+,J−] ?= ∅. The symplectic structure
Ω is the only linear function of K
Ω = KAA′dqA∧ dPA′,
(5.37)
where we use the collective notation A = {a,¯ a}, A′= {a′,¯ a′}. We write Ω as a matrix
Ω =
?
0
KLR
0
−KRL
?
.
(5.38)
and find the metric and B-field using this matrix as follows [14] (cf. 4.24)
g = Ω[J+,J−] ,B = Ω{J+,J−} .
(5.39)
3There always exists at least one polarization such that any symplectomorphism, can be written in
terms of such a generating function [13].
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6The general case
In the general case when both ker[J+,J−] and coker[J+,J−] are nonempty, we combine the
discussion leading to (4.21) in section 4 with that of the previous section. The formulas
that we compute are rather complicated, and we have not found a suitable coordinate free
way to express them.
We assume that in a neighborhood of x0, the ranks of π±are constant, and as result,
the rank of σ is constant. We work in coordinates {q,p,z,z′} adapted to the symplectic
foliation of σ as well as to the description of ker[J+,J−] given in section 4. In such
coordinates J+may be taken to have the canonical form4
J+=





Is
0
0
0
0
Is
0
0
0
0
Ic
0
0
0
0
It




,
(6.40)
where {q,p} are Darboux coordinates for a symplectic leaf of σ and {z,z′} parametrize the
kernels of π∓. Similarily, there are coordinates {Q,P,z,z′} where J−takes a diagonal form
(identical to that of J+in (6.40) except for a change of sign in the last entry; It→ −It).
For every leaf separately we may now apply the arguments of section 5. There thus exists
a generating function K(q,P,z,z′) for the symplectomorphisms between the two sets of
coordinates. The transformation matrices to the coordinates {q,P,z,z′} are given by the
obvious extensions of (5.35) and (5.36) to include z,z′. The expression for J± in the
“mixed” coordinates are then5:
J+=





Is
000
K−1
RLCLL K−1
0
0
RLIsKLR K−1
0
0
RLCLc K−1
Ic
0
RLCLt
0
It




,
(6.41)
and
J−=





K−1
LRIsKRL K−1
0
0
0
LRCRR −K−1
−Is
0
0
LRCRc
0
Ic
0
K−1
LRARt
0
0
−It




,
(6.42)
where
K−1
LR
= (KRL)−1,
4For historical reasons related to the origin in sigma models involving chiral and twisted chiral super-
fields, we again use the labels c and t for the z and z′directions.
5The rows and columns of this matrix correspond to the directions along {q,P,z,z′}.
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C
= IK − KI =
?
02iK
0
−2iK
?
,
A = IK + KI =
?
2iK
0
0
−2iK
?
,
(6.43)
where we suppress the indices in the last two entries6. The relations (5.39) no-longer hold
when ker[J+,J−] ?= ∅. The definition (4.24) of the Poisson structure σ still determines the
metric, except along the kernel. However, the additional relation H = ±dc
provides us with an equation for the remaining components of g and allows us to find the
B-field. From the sigma model we already know the solution; the sum E =1
the metric g and B-field takes on the explicit form:
±ω±from (3.10)
2(g + B) of
ELL = CLLK−1
ELR = IsKLRIs+ CLLK−1
ELc = KLc+ IsKLcIc+ CLLK−1
ELt = −KLt− IsKLtIt+ CLLK−1
ERL = −KRLIsK−1
ERR
= −KRLIsK−1
ERc
= KRc− KRLIsK−1
ERt
= −KRt− KRLIsK−1
EcL = CcLK−1
EcR = IcKcRIs+ CcLK−1
Ecc = Kcc+ IcKccIc+ CcLK−1
Ect = −Kct− IcKctIt+ CcLK−1
EtL = CtLK−1
EtR
= ItKtRIs+ CtLK−1
Etc
= Ktc+ ItKtcIc+ CtLK−1
Ett
= −Ktt− ItKttIt+ CtLK−1
LRIsKRL
LRCRR
LRCRc
LRARt
LRIsKRL
LRCRR
LRCRc
LRARt
LRIsKRL
LRCRR
LRCRc
LRARt
LRIsKRL
LRCRR
LRCRc
LRARt
(6.44)
In view of this last relation (6.44) as well as (6.41), (6.42) and (5.38) it seems appro-
priate to call K the generalized K¨ ahler potential.
6In (6.43), the rows and columns correspond to {A,¯ A}, where {A} are any one of the {q,P,z,z′}
directions in (6.41),(6.42),(6.44).
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7 Linearization of Generalized K¨ ahler geometry
As can be seen in (6.44), the expressions for the metric and the B-field in terms of the
generalized K¨ ahler potential are highly nonlinear in contrast to what is the case in ordinary
K¨ ahler geometry. The nonlinearities all stem from the cokernel of [J+,J−]: when [J+,J−] =
0, the metric and B-field can be expressed as linear functions of the Hessian of K. We
note that the structure of the nonlinearities are such as one might expect from a quotient
construction. One is therefore faced with a natural question: Is there a space in which all
the geometric data is encoded linearly with respect to the generalized K¨ ahler potential and
which gives the nonlinear Generalized K¨ ahler geometry through a quotient construction?
Inspired by superspace sigma-models, we have found the following local prescription
for generating such a space: In local coordinates, we simply make the substitution
q → zL+ z′
L, P → zR+ ¯ z′
R
(7.45)
in the Generalized K¨ ahler potential K and use the naive commuting complex structures J±
in which the z-coordinates are biholomorphic and the z′-coordinates are J+holomorphic
and J−antiholomorphic. This introduces isometries of the metric (3.12) corresponding
to shifts of zL,z′
respect to these isometries leads to the original nonlinear model. We now attempt to make
mathematical sense of this prescription. We begin with some definitions:
Lthat preserve the sum, and similarly for zR, ¯ zR; taking a quotient with
Definition 1 (BiLP). A Generalized K¨ ahler Geometry with commuting complex structures
will be called a Bihermitean Local Product space (BiLP).
We also give a special name to a BiLP with the isometries sketched above:
Definition 2 (ALP). A BiLP with these additional isometries (further characterized be-
low) we will call an Auxilliary Product Space (ALP).
The isometries are generated by 2n complex Abelian Killing vectors that preserve all
the BiLP geometric data. They can be separated into two groups of n Killing vectors
{kLA} and {kRA}, A = 1...n, satisfying the further requirement
ıkLAH = dαLA,ıkRA′H = −dαRA′ ,
(7.46)
where the one-form α is the dual of the killing vector k: α = g(k); this is equivalent
to saying that the Killing vectors are covariantly constant with respect to the covariant
derivative with torsion (3.11). Such Killing vectors are called Kac-Moody Killing vectors.
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We require that each group of Killing vectors kLand kRby themselves span maximally
isotropic subspaces, i.e., that
g(kLA,kLB) = 0 , g(kRA′,kRB′) = 0 ,
(7.47)
but that the inner product
hA′B= g(kRA′,kLB) (7.48)
is nondegenerate.
Theorem. Locally, any generalized K¨ ahler manifold M is a quotient of an ALP by its
Kac-Moody isometries.
The quotient is performed by going to the orbits of the action of the Killing vector and
choosing a horizontal subspace by specifying a connection. In the particular case with left
and right Kac-Moody Killing vectors there is a corresponding left and right connection
given by
(θR)A′
µ= gµνkν
LBhBA′,
(θL)A
µ= hAB′kν
RB′gνµ,
(7.49)
where hAB′is the inverse of the nondegenerate matrix (7.48). The connections (7.49)
satisfy the following properties
θL(kL) = 1 ,θR(kR) = 1 ,θL(kR) = 0 ,θR(kL) = 0 .
(7.50)
Vectors in the horizontal subspaces are defined as the vectors lying in the kernel of both
θLand θR.
All geometric structures defined on the ALP can now be defined on the quotient space.
As usual, forms are projected using the connection so that the contraction with any
vertical vector is zero, whereas vectors are pushed forward to the quotient using the bundle
projection p. When we project the complex structures
˜J±(X) := p∗
?
J±(X) − θL(X) · J±(kL) − θR(X) · J±(kR)
?
∀X ,
(7.51)
it is clear that the projected complex structures˜J±do not commute even though J±do.
Projecting the metric and B-field, we find
˜ g = g − h(θL,θR) − h(θR,θL) ,
˜B = B − h(θL,θR) + h(θR,θL) .
(7.52)
It is straightforward to see that ˜ g gives zero when contracted with any vertical vector
while this is not so for˜B. This (apparent) mystery is resolved by showing that˜H = d˜B
can be written as
˜H = H − θL∧ ıkLH − θR∧ ıkRH − θL∧ θR∧ ıkLıkRH
(7.53)
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so that the geometrically meaningful object˜H defined by d˜B is indeed well defined on the
quotient. The quotients ˜ g and˜H agree with the expressions for the original generalized
K¨ ahler manifold.
Acknowledgement:
We are grateful to the 2006 Simons Workshop for providing the stimulating atmosphere
where this work was initiated. The work of UL was supported by EU grant (Superstring
theory) MRTN-2004-512194 and VR grant 621-2003-3454. The work of MR was supported
in part by NSF grant no. PHY-0354776. The research of R.v.U. was supported by Czech
ministry of education contract No. MSM0021622409. The research of M.Z. was supported
by VR-grant 621-2004-3177
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