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Page 1: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� �

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�

Page 2: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� �

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����������

Page 3: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� �

Introduction and Summary

‘‘BPS’’

R�� � �g��

Page 4: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� �

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������

Page 5: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� �

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

Page 6: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� �

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

Page 7: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� �

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

Page 8: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� �

I� Page Limit

Page 9: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� �

��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 10: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i 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 11: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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)

Page 12: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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� ������ ������� ���

� �������

� � ������

� ������

� ������

Page 13: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

� 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��

Page 14: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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������

Page 15: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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 16: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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� �

Page 17: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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��

Page 18: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

� 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����

Page 19: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

II� Sasaki�Einstein Twist

Page 20: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

��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 21: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i 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���

Page 22: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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�

Page 23: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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���

Page 24: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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�

Page 25: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

III� Scalar Laplacian on Yp�q

Page 26: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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

Page 27: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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 � ���

��

Page 28: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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����

Page 29: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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��� ���

Page 30: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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�

Page 31: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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�

Page 32: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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��

Page 33: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

IV� toric Sasaki�Einstein metric

from toric data

Page 34: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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

Page 35: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

� 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�

Page 36: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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�� � �

Page 37: Manifolds SasakiEinstein and Holes Blackresearch.kek.jp/group/riron/workshop/theory2006/slides/...C X examples of homogeneous X X isometry dual gauge theo ry S i SO N SUN adj X i T

������� ��

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 ��