manifolds sasakieinstein and holes blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...c...
TRANSCRIPT
������� �
Black Holes and Sasaki�Einstein Manifolds
阪口 真 �岡山光量子科学研究所�
�������� KEK理論研究会����
collaboration with
橋本 義武氏�� 木原 裕充氏 y � 安井 幸則氏�
��大阪市大理� y大阪市大数研�
������� �
based on
�Hashimoto�Yasui�MS������ �New in�nite series of Einstein metrics on sphere bundles
from AdS black holes��
Commun� Math� Phys� �� �� ���� �arXiv�hep�th����������
�Hashimoto�Yasui�MS����� �Sasaki�Einstein twist of Kerr�AdS black holes��
Phys� Lett� B ��� ���� ��� �arXiv�hep�th����������
�Yasui�MS������� �Seven�dimensional Einstein manifolds from Tod�Hitchin geometry��
J� Geom� Phys� �arXiv�hep�th����������
�Yasui�MS����� �Notes on �ve�dimensional Kerr black holes��
Int� J� Mod� Phys� A �arXiv�hep�th����������
�Kihara�Yasui�MS���� �Scalar Laplacian on Sasaki�Einstein manifolds Yp�q��
Phys� Lett� B ��� ���� ��� �arXiv�hep�th����������
������� �
Introduction and Summary
‘‘BPS’’
R�� � �g��
������� �
Black Holes Compact Einstein metrics
inhomogeneous
��d Kerr�A dS �Carter ����� �M�J���Page limit
�� a metric on S�e�S� �Page �����
Kerr with n J �s �Myers�Perry ����� �M�J�� ��� Jn� n � �d���
�
Kerr�A dS �Hawking et al ����� �Hashimoto�Yasui�MS ������
��dim� �M�J�� J����Page limit
�� A in�nite series of metrics on S��S� and S�e�S�
d�dim� �M�J���
Page limit
�� B a metric on S�e�Sd�� for each dim� xI
metrics on EK�ne�Sm�� �Lu�Page�Pope�����
generalization of B
Kerr�A dS with n J �s �M�J�� ��� Jn���Page limit
�� metrics on S�e�Sd�� and S��Sd��
�Gibbons�Lu�Page�Pope������ generalization of A
�� �Cvetic�Lu�Page�Pope������
������� �
Kerr��A�dS with n J �s Sasaki�Einstein
Ji � J
twist
�� Sasaki�Einstein metrics on circle bundles
�Hashimoto�Yasui�MS���� over C Pne�S�� Yp�q
xII �Gauntlett�Martelli�Sparks�Waldram��� �
Ji �� Jj
twist
�� Lp�q�r���� �Cvetic�Lu�Page�Pope�����
de�nitions of Sasaki�Einstein X�k��
� C�X�k��� is d � �k Calabi�Yau ds�C�X� � dr� � r�ds�X
� circle bundle over d � �k � � K�ahler�Einstein
ds� � �d� � ��� � ds�KE where d��� is the K�ahler form on the KE
� � a Killing vector b of unit length �called Reeb vector�
such that rirjbk � �ki bj � gijbk and Einstein Rij � �gij�
� Wb��� � � and W���� � ��
������W����
where b � ���� �Reeb vector� and b �� f� � �g
D3
X5
C(X )5
������� �
D��brane sitting at the apex of C�X��
ds� � H
�����r�ds�R��� �H
����r�ds�C�X��
��
AdS� �X� as r � �
R��� � C�X�� as r ��
where H�r� � � � a�r�is a harmonic func� on C�X��
AdS�CFT correspondence �Maldacena����
IIB string on AdS� �X�
m
��dim� N � � gauge theory on D��brane at the apex of C�X��
D3
X5
C(X )5
������� �
toric Sasaki�Einstein
� Killing �elds of T��action on the C�X�� �� toric data
� �Bertolini et at�Benvenuti et at����
IIB strings on AdS��X�AdS�CFT
�� ��dim� N � � SCFT �quiver gauge theory
Yp�q �Gauntlet�Martelli�Sparks�Waldram����� �� Yp�q quiver �Benvenuti et al����
Lp�q�r �Cvetic�Lu�Page�Pope�� �� �� Lp�q�r quiver �Franco et al�� �
�Bevenuti�Kruczenski�� �
�Butti et al�� �
Xp�q Xp�q quiver �Hanany et al�� ��
� spectrum on Yp�q �Kihara�Yasui�MS�� � xIII
� program to construct a metric from a toric data �Martelli�Sparks�Yau�� �� xIV
������� �
I� Page Limit
������� �
��dim� Kerr�AdS black hole
��dim� Kerr�AdS black hole �M�a� b�� � ����� �Hawking et al �����
gBH� � �
�r��
�dt�a sin� �
�a
d��b cos� �
�b
d���
���
�rdr� ���
��d��
��� sin� �
��
�adt�r� � a�
�a
d��� ��� cos� �
��
�bdt�r� � b�
�b
d���
� � r���
r���
�abdt�b�r� � a�� sin� �
�a
d��a�r� � b�� cos� �
�b
d���
where�� � r� � a� cos� � � b� sin� �� �r �
r��r� � a���r� � b��� � r����� M�
�� � � a��� cos� � � b��� sin� �� �a � � a���� �b � � b���
� Killing horizons of Killing vector elds �i �
��t� a�a
r�i�a��
��� b�b
r�i�b��
��
at real roots
r � ri of �r � �
� R�U����U��� isometry generated by � ��t� ���� ����
������� ��
Page limit of ��dim� Kerr�AdS black hole
Wick rotation �Hashimoto�Yaui�MS�����
t� �i� as well as �a� b�� ��i���i�� for the reality of the metric
� sign ip
AdS with � � ���� to dS with � � ���� so that �� �i�
Euclidean Kerr�dS ��� r� �� �� � R�U�� �U�� isometry
g� �
�r�
�d� �� sin� �
��
d�� cos� �
��
d���
��
�rdr� ��
��d���
��� sin� �
�
��d� �r� � ��
��
d��
��� cos� �
�
��d� �r� � ��
��
d���
�
� r���
r��
���d� ���r� � ��� sin� �
��
d���r� � ��� cos� �
��
d���
where� � r� � �� cos� � � �� sin� �� �r �
r��r� � ����r� � ����� r����� M�
�� � � ���� cos� � � ���� sin� �� �� � � ����� �� � � ����
������� ��
Page limit
� � �horizons� sit at � roots of �r � ��
Consider an �extremal black hole� in which two horizons degenerate� �r � ��r �
r���e��r�� and then the number of the parameters reduces� We choose parameter M
such that M � M����� ��� �� where �� � ��r�� �� � ��r� �
� Let us consider the narrow region between two horizons at r � r� � � of the nearly
extreme black hole� For this� we introduce a new coordinate as r � r� � � cos
�� � � �� It follows that �r � ��e��r�� sin� � O����� and we scale � as � � ��
for non�singular metric under the limit �� ��
� Take the limit �� �� then the dimensionful parameter � disappears�
As a result� g��M�J�� J�� �� �� g�����
S3
2S (η, χ)
(φ, ψ, θ)
������� ��
g���� � h�� �d � �
�Xi�j��
aij� ��i� �j � b�� �gS��
��i � d�i � ki cosd�
gS� � d� � sin�d��
where ��� ��� ���� ���
h� �
�� ���
cos� � ���
sin�
�� ���cos� � ���sin� � b� ���� ���
� ������� ���
cos� � ���
sin� �
�� � ������
� ������
� �������
�
a�� �
�
������ ������� ���
� ����
� � ������
� ������
� ������
����� ���cos� � ���
sin� � sin�
�� ���
cos� � ���
sin�
�
a�� �
�
������ ������� ���
� ����
� � ������
� ������
� ������
����� ���
cos� � ���sin� � cos�
�� ���
cos� � ���
sin�
�
a�� � ���������� �������� �������� ���
� ����
�� � ������
� ������
� ��������
sin� cos�
�� ���
cos� � ���
sin� �
���
�
������ �������
�� ���
� ���
� ���
������� �������
�� ���
� ���
�
k� �
����� ������� ���
� �������
� � ������
� ������
� ������
� k� ������ ������� ���
� �������
� � ������
� ������
� ������
�
������� ��
� detaij��� � � at � � �� ��� �� � � � � ���
resolve orbifold singularities at � � � and ���
� � � � g � const��
d�� � ����d���
�� gLk�
� � ��
� g � const��
d���
� ��� � �����
� ���d���
�� gLk�
where Lki is the lens space� Lki � L�ki� � � S��Zki
�� � � �i � � and ki � Z
� � � �� � ��� connection on
the principal T��bundle over S�
� g������ metric on associated S��
bundle of the principal T��bundle
over S�
0
� S��bundles over S� are classi�ed by ��� � ���SO��� � Z�� where � � S� � SO���
In the present case� ��� � k� � k� mod �
�� �non�trivial S��bundle when k� � k� �even�odd��
������� ��
Theorem �� �HSY������ Let �� and �� be real numbers in the region ���� ��
� � � and
��� �� ��� together with the integral conditions
k� ������ ������� ��� � �����
��
� � �����
� � �����
� � ������
�
� k� ������ ������� ��� � �����
��
� � �����
� � �����
� � ������
�
�
where �k�� k�� � Z � Z� Then fg����g gives an in�nite series of inhomogeneous
Einstein metrics with positive scalar curvature ���� � �����
����� � ��� � ���� on S��
bundles over S�� If the integer k� � k� is even �odd�� then the bundle is trivial
�non�trivial��
� unique pair ���� ��� for each �k�� k�
� ������ � standard metric on S�
� these are not the metrics proved to exist by Bohm�
� These metrics further examined in �Gibbons�Hartnoll�Yausi���� �
SU��� � U��� isometry� Laplacian separates in variables� Schwarzschild with M�
instead of S� is stable only for T ����k� � k� � ��� elliptic curve is of genus � �� for
�inhomogeneous case�
� generalized to metrics on S�e�Sn and S��Sn �Gibbons et al������
������� ��
Theorem �� �HSY������ Let �k be real numbers satisfying k � �k���
k � �������k � ��
� Z� Then� fg�k
�
����kg�k�kg gives an in�nite series of homogeneous Einstein
metrics with positive scalar curvature ���� � ��k���� � ��k� on S� � S��
� represents the homogeneous Einstein metric on M���k�� � circle bundles over C P��C P�
studied by Wang and Ziller�
������� ��
Page limit of d�dim� Kerr�AdS black hole
d�dim� Kerr�AdS black hole �M�a�� � ����� �Hawking et al �����
gBHd � �
�r
���dt�a
�sin� �d��� ���
�rdr� ���
��d��
��� sin� �
��
�adt�r� � a�
�
d��� � r� cos� �gSd��
where �r � �r� � a��� � ��r��� Mr��d� �� � � a��� cos� ��
� � � a���� �� � r� � a� cos� �
Euclidean Kerr dS � Wick rotation t� i� together with �a� ��� ��i���i��
gd �
�r
���d� ��
�sin� �d������
�rdr� ���
��d��
��� sin� �
��
��d� �r� � ��
�
d��� � r� cos� �gSd��
where �r � �r� � ����� ��r��� Mr��d� �� � � ���� cos� ��
� � � ����� �� � r� � �� cos� �
������� ��
g� � h����d�� �
�Xi��
ai����i� �i � b����gSd��
�
�����
�� � cos�d�� sin� sin�d�
�� � � sin�d�� cos� sin�d�
�� � d� � k cos�d�
where �i are Maurer�Cartan one�forms of SU���� and
h� ��� �� cos� �
�� � cos� �� a� � a� ��� �� cos� �e� �
a� ��
��
�� ��
�� ���
�� � cos� �
�� �� cos� �sin� � � b� � cos� � �
with e� � �d����d��������d����d�������d����d���
d����d�����
and � � �d�������d�����
d����d�����
�
Theorem �� �HSY��� Let � be a real number satisfying �� � and
���d� �� �d� �����
�d� ���d� �� � �d� ���d� ���� � �d� ���d� �� �� �
Then g� gives an inhomogeneous Einstein metric with positive scalar curvature on
S�e�Sd��� which is the non�trivial Sd���bundle over S��
������� ��
� The metric is of cohomogeneity one with principal orbits S��Sd�� if d � �� In the
case of d � �� it reproduces the Page metric on S�e�S� � C P��C P� with principal
orbits S��
� generalized to metrics on EK�ne�Sm�� �Lu�Page�Pope����
������� ��
II� Sasaki�Einstein Twist
������� ��
��dim� Kerr�AdS black hole
��dim� Kerr�AdS black hole �m� a� b�� � ����� �Hawking et al �����
gBH� � �
�r��
�dt�a sin� �
�a
d��b cos� �
�b
d���
���
�rdr� ���
��d��
��� sin� �
��
�adt�r� � a�
�a
d��� ��� cos� �
��
�bdt�r� � b�
�b
d���
� � r���
r���
�abdt�b�r� � a�� sin� �
�a
d��a�r� � b�� cos� �
�b
d���
where�� � r� � a� cos� � � b� sin� �� �r �
r��r� � a���r� � b��� � r����� m�
�� � � a��� cos� � � b��� sin� �� �a � � a���� �b � � b���
� Killing horizons of Killing vector elds �i �
��t� a�a
r�i�a��
��� b�b
r�i�b��
��
at real roots
r � ri of �r � �
� R�U����U��� isometry generated by � ��t� ���� ����
������� ��
Wick rotation
t� �i as well as �a� b�� ��ia��ib� for reality of the metric
� sign ip
AdS with � � ���� to dS with � � ���� so that �� �i�
Euclidean Kerr�dS �� r� �� �� �� R�U����U��� isometry
g� �
�r��
�d �a sin� �
�a
d��b cos� �
�b
d���
���
�rdr� ���
��d���
��� sin� �
��
�ad �r� � a�
�a
d���
��� cos� �
��
�bd �r� � b�
�b
d���
�
� r���
r���
�abd �b�r� � a�� sin� �
�a
d��a�r� � b�� cos� �
�b
d���
where
�� � r� � a� cos� � � b� sin� �� �r �
r��r� � a���r� � b���� r����� m�
�� � � a��� cos� � � b��� sin� �� �a � � a���� �b � � b���
������� ��
Sasaki�Einstein twist
Sasaki limit �Hashimoto�Yaui�MS����� CLPP�����
set a� � ���
��� � b� � ���
��� � m�� ��
��� and �� �
gBH�
�m�a� b� ��
���
�� ���g���� �� � g���� �� � �d� � ��� � gKE
where d�� is the K�ahler form on the locally K�ahler Einstein gKE
� BPS limit M � Z�M � m���a � ��b � �a�b���a�b��
Z � ��Ja � Jb� � Ja � ma���a�b � Jb � Ja�a� b�
� eigenvalues of Weyl curvature W���� �i �i � �� � � � � ��� �Yasui�MS������
f���� ���� ���� ���� ��� ��g � f��� ���� ��g
� b � ��� is a Killing vector of unit length Reeb vector�
�� g���� �� is locally Sasaki�Einstein with orbifold singularities�
������� ��
twist resolves orbifold singularities
isometry R � T � � isometry T �
��
����
����
���
twist
��
��
�����
�����
����
� � � �i � ��
�� � � �BH with equal angular momenta� �Hashimoto�Yasui�MS���
�� � �� � � � � � �� � ��� � �� � �� � ��p� q�� ��
��p� q �
����
where p� q�p � q� � Z and �� � ��q� � �p� � pp
p� � �q���q�
g��� � � reproduces the Sasaki�Einstein Yp�q constructed by �Gauntlett�Martelli�
Sparks�Waldram���
������� ��
inhomogeneous metric on Y p�q ���� ��� ���� ��� �� ��
ds� �
�� y�
�d�� � sin� �d���
�
dy�
w�y�q�y��q�y�
��d� � cos �d���
�w�y� �d�� f�y��d� � cos �d���
� yy1 2
xx ψ
with w�y� � ��b�y��
��y
� q�y� � b��y���y�
b�y�
� f�y� � b��y�y�
��b�y��
� b � �� �
p���q�
�p�
pp� � �q� �
Singularities at y � y�� y� �yi are roots of q�y�� are resolved by y� � y � y� �
y� � � � � � � � � � � � � � � � � � � � � � � � � where
� q
�q���p��pp
�p���q��
�� � �� �Cvetic�Lu�Page�Pope���
g�� �� � is Sasaki�Einstein Lp�q�r �p� q� r � Z�
Y p�q � Lp�q�p�q�p
higher dimensional Y p�q and Lp�q�r���� are obtained by the Sasaki�Einstein twist�
������� ��
III� Scalar Laplacian on Yp�q
������� ��
gravity�gauge theory correspondence
IIB string on AdS� �X� � low energy limit of world�volume theory
on D��brane at the apex of C�X��
examples of homogeneous X�
X� isometry dual gauge theory
S� SO��� N � � SU�N�� adj Xi �i � � � ��
T��� SU����U��� N � � SU�N�� with SU��� �avor�
bifund Ai� Bj �i� j � �� �� quartic superpot
D3
X5
C(X )5
Toric Sasaki�Einstein manifolds Y p�q provide new examples
� isometry SU���U����
� T� action on C�Y p�q� �T� action on ��dim Einstein�K�ahler base�
toric data of Y p�q � quiver diagram
� �Bertolini et al Benvenuti et al�������
IIB on AdS� � Y p�q � N � � SU�N��p quiver gauge theory with SU���U��� �avor
bifund U� V � Y and Z
cubic � quartic superpotential
������� ��
Yp�q quiver gauge theory
(p,p)
(p-q-1,p-q)
(0,0) (1,0)
V
U
V
V
U
U U
Y
Y
Y
Y
Y
Y
Y
Z
q
Y ���
toric data � quiver diagram
node � gauge group
�node��p � SU�N��p
arrow � bifundamental �elds
� �U�� V �� Y� Z� ��p� q� p�q� p�q� � � �� �
triangles and quadrangles
� cubic and quartic superpotentials
Field J QR U���F
Y � ��q����p� � �q� � �pq � ��p � q�p
�p� � �q�� ��
Z � ��q����p� � �q� � �pq � ��p� q�p
�p� � �q�� ��
U� ��
�p�q���p�p
�p� � �q�� �
V � ��
��q��q � �p�p
�p� � �q�� ��
closed paths � mesonic operators �chiral primary�
S � tr�ZUYqU� � tr�UV Y �
L� � tr�ZUV UV UV U�
L�
� tr�YqY Y Y U� � tr�YqY Y UY �
� tr�YqY UY Y � � tr�YqUY Y Y �
Meson J QR N�
S � � �
L�
p�q�
p� q � ���
�
L�
p�q�
p� q � ���
��
������� ��
Scalar Laplacian on Yp�q
inhomogeneous metric on Y p�q ���� ��� ���� ��� �� ��
ds� �
�� y�
�d�� � sin� �d���
�
dy�
w�y�q�y��q�y�
��d� � cos �d���
�w�y� �d�� f�y��d� � cos �d���
� yy1 2
xx ψ
with w�y� � ��b�y��
��y � q�y� � b��y���y�
b�y�
� f�y� � b��y�y�
��b�y��� b � ��� p���q�
�p�
pp� � �q� �
Singularities at y � y�� y� �yi are roots of q�y�� are resolved by y� � y � y� � y� � � �
� � � � � � � � � � � � � � � � � � � � � where � q
�q���p��pp
�p���q��
scalar Laplacian
� �y��� y�w�y�q�y�y
�� y
��
� �QR��
��
� � �y �QR��
w�y�q�y�
�
��� y
��K � �����
where �QR � � � ��� is the Reeb�Killing vector and �K is the second Casimir of SU����
������� ��
Let � � exp�
i�N���N�� �N�
����
R�y����� with N�� N�� N� � Z� then
�� � �E� reduces to two di�erential equations� �K� � �J�J � ��� � whose regular
solution is given by Jacobi polynomials� and
Heun�s equation �four regular singularities are at fy�� y�� y���g�
d�dy�R�
��X
i��
�y � yi
�d
dyR� v�y�R � � v�y� �
�H�y�
���y
E �
�Xi��
��iH��yi�
y � yi�
where H�y� �Q�
i���y � yi� � � � E�
� ��J�J � �� � ���
��
�N�
�
�QR��
and
�� � ��
N��
p� q �
���
��QR
� �� � ��
N��
p� q �
���
��QR
�
�� � ��
�
N��
��p� � q� � pp
p� � �q�
q
�
���
��QR
�� QR � �N� �
���N�
The exponents at the singularities are �i at yi and �� � � at � where we put
E � �� ���
Scalar Laplacian on Lpqr reduces to two Heun�s equations OotaYasui��� ���
������� ��
It is convenient to transform singularities fy�� y�� y���g to f�� �� a � y��y�
y��y���g by
x � y�y�
y��y�and R � xj��j��� x�j��j�a� x�j��jh�x�� so that
d�dx�h�x� ��
�x�
�x� ��
�x� a
�d
dxh�x� �
��x� k
x�x� ���x� a�h�x� � �
which is the standard form of Heun�s equation� Heun�s parameters are
� � ����X
i��j�ij � � � � � ��
�Xi��j�ij � � � � � �j��j � � � � � �j��j � � � � � �j��j �
and the �accessory� parameter k is
k � �j��j� j��j��j��j� j��j� ��� j��j� � a�
�j��j� j��j��j��j� j��j� ��� j��j��� �
with � � � �
y��y��� y����� ��� and a � ��
�� �p
�p���q�
q
�
For regularity� h�x� should be a Heun function� h�x� has zero exponents at the boundaries
x � �� �� This condition is non�trivial even for a polynomial solution which is the simplest
example�
������� ��
constant solution as BPS
We examine constant solutions� � � � �� � �� and k � �� Let j�ij be given by
j��j � ���
�N��
p� q � ���
��QR�
� j��j � ��
�N��
p� q �
���
��QR�
�
j��j � ���
�N�
���p� � q� � pp
�p� � �q�
q
� ���
��QR
�
which is true for all mesonic operators� One �nds
that the condition � � �� �P
i j�ij � � implies
� � ��QR� Because the conformal dimension of the
dual operator is given by the formula
� � �� �p� �E � �� � ��� � � ���
Meson J QR N�
S � � �
L� p�q�
p� q � ���
�
L�
p�q�
p� q � ���
��
the states satisfy the BPS condition � � ��QR� In addition� we impose the condition
k � �� This is solved by QR � �N��q���p��pp
�p���q�
�q
� �J � which is true for mesonic
operators�
Summarizing� we �nd that constant solutions �ground states� are dual to mesonic
operators and their composites in the dual quiver gauge theory�
������� ��
polynomial solution as near�BPS
We consider n�th excited states on the ground states of N mesons� Charges of N
mesons are given by those of S�L� multiplied by N � fNJ�NQR� NN�g� � is given by
� � ��NQR � n�
�rst exited states are speci�ed by � � ��NQR � �� We �nd that there exists the
corresponding Heun function h��x� given by the polynomial solution of degree ��
h��x� ��������
�������� �p��q�p
�p���q�
q�p
�p���q�
x for S �
� ��Np���Npq��q��p�p
�p���q�
��p�q����Np� �x� �� for L� �
�� �q��p�Npq����Np�p
�p���q�
���Np��q�p
�p���q��
x for L� �
n�th excited states Numerical analysis strongly indicates that there exist corresponding
Heun functions� In the large N limit� we obtain polynomial solutions of degree n
corresponding to near�BPS operators hn�x�����
��h��x�n for S
xn for L�
��� x�n for L��
������� ��
IV� toric Sasaki�Einstein metric
from toric data
������� ��
toric data from Sasaki�Einstein metric
toric Sasaki�Einstein Yp�q � T��action on C�Y p�q� �Martelli�Sparks�������
moment map � is de�ned by �ei � � d�i where � is the K�ahler form on C�Y p�q�
�symplectic form� and ei �i � �� �� �� is basis for e�ectively acting T��
The image of the moment map � � C�Y p�q� � R� is the subset C � f� � R�j��� va� �
�� a � �� � � � � g with
v� � ����� ���
v� � �������p� q�� �
v� � ������p� �
v� � �� �� �� �
C�Y p�q� as T ���bration over the
polyhedral cone
u3(x,v2)=0
(x,v1)=0
(x,v3)=0
(x,v4)=0
u4
u2
u1
������� ��
� K�ahler quotient �symplectic quotient�
U�� charges Qa s�t�P�
a��Qa � � andP
Qava � �� �� ��
� X � C ���U��� with �Q�� � � � � Q�� �constraintP
Qajzaj� � ��
C�X� is the vacuum of the U�� gauged linear sigma model �without FI term�
������� ��
Sasaki�Einstein metric from toric data
� Reeb vector b minimizes Z�minimization �Martelli�Sparks�Yau����������
Z�b� �b� � �
b�
Xa
�va��� va� va���
�b� va��� va��b� va� va���
� T ��invariant CY metric on C�Y p�q�
gC �
�Xi�j��
Gij���d�id�j �
�Xi�j��
Gij���d�id�j � Gij �
��G
��i��j
G is the symplectic potential
G � Gcan �Gb � g �
�Gcan � ��P
a��� va� log��� va�
Gb ��
�P
a��� b� log��� b���
����P
a va� log���P
a va�
and g��� is a homogeneous degree one function� g is determined by the Monge�Ampere
equation �Ricci �at condition� metric induced by the K�ahler quotient is not Ricci �at�
detGij � exp�
��i�G
��i�
� � � ���� � � � �� R�� � �
������� ��
Open Questions
Lorentzian Einstein Sasaki�Einstein gauge theory
AdS�Kerr BH �� Yp�q
�� Yp�q quiver
Lp�q�r Lp�q�r quiver
�� �� Xp�q �� �� Xp�q quiver
AdS Black Ring ���� � � �
Black Ring �S��S� horizon�
S��rotating �Emparan�Reall���
S��rotating �Mishima�Iguchi� ��� Figueras� � �
S�� and S��rotating ��