CAN AXIAL U(1) ANOMALYDISAPPEAR AT HIGHTEMPERATURE?
µ
h@µ J5 i =

1
µ⌫ ⇢
✏
hF
F
i!
0?
µ⌫⇢
32⇡ 2

HIDENORI FUKAYA (OSAKA UNIV.)FOR JLQCD COLLABORATION
PRD96, NO.3, 034509(2017),
PRD93, NO.3, 034507 (2016)

DO YOU THINK AXIAL U(1) ANOMALYCAN DISAPPEAR (AT FINITE T) ?

YES

NO

U(1)A sym. may beat some T.

U(1)A is always
broken.

DO YOU THINK AXIAL U(1) ANOMALYCAN DISAPPEAR (AT FINITE T) ?
Naïve answer would be “NO!”with some reasonable reasons:Anomaly = symmetry breaking at cut-off.
Anomalous Ward-Takahashi identity
h@µ J5µ (x)O(x0 )if ermion =

1
µ⌫ ⇢
0
✏
F
F
(x)hO(x)if ermion
µ⌫⇢
2
32⇡
+ h A O(x)if ermion (x x0 )holds at any energy scale,and for any gluon background.

DO YOU KNOW ANY OTHERANOMALY WHICH CAN DISAPPEAR ?

YES

NO

DO YOU KNOW ANY OTHERANOMALY WHICH CAN DISAPPEAR ?
The answer should be “YES”.
Conformal anomaly in massless QCD
can disappear at IR
by tuning Nf(conformal window).
Which Nf?→ nontrivial.

Figure from
K-I. Ishikawa et al. 2013

DO YOU KNOW ANY OTHERANOMALY WHICH CAN DISAPPEAR ?
The answer should be “YES”.
Conformal anomaly in massless QCD
can disappear at IR

Figure from
K-I. Ishikawa et al. 2013by tuning Nf(conformal window).
Which Nf?→ nontrivial.

Why not axial U(1) (by tuning T ?)

WE ARE BIASED BYµh@µ J5 (x)O(x0 )if ermionh A O(x)if ermion (x

x0 )

1
µ⌫ ⇢
0
=
✏
F
F
(x)hO(x)if ermion
µ⌫⇢
2
32⇡

BUT THE REAL QUESTION IS
hh@µ J5µ (x)O(x0 )if ermion h A O(x)if ermion (x x0 )igluons
⌧
1
µ⌫ ⇢
0
=
✏
F
F
(x)hO(x)if ermion= 0???
µ⌫⇢
2
32⇡
gluonsMAIN MESSAGE OF THIS TALKIn high T QCD, whether
hh@µ J5µ (x)O(x0 )if ermion h A O(x)if ermion (x x0 )igluons
⌧
1
µ⌫ ⇢
0
=
✏
F
F
(x)hO(x)if ermion= 0???
µ⌫⇢
2
32⇡
gluonsor not is a non-trivial question, which can
only be answered by carefully integratingover gluons (by lattice QCD).
In particular, good control of chiralsymmetry (or continuum limit) is essential.

CAN U(1)ANOMALY DISAPPEAR
AT FINITE T? → MANY ANSWERS.

HotQCD 2012 (Domain-wall)
Before 2012
Aoki-F-Taniguchi 2012 (theory) After 2012
Cohen 1996, 1998 (theory)
Ishikawa et al2013, 2014,2017. (Wilson)
Bernard et al. 1996 (staggered) JLQCD 2013, 2016 (overlap)
Chandrasekharan et al. 1998 TWQCD 2013 (optimal DW)(staggered)
LLNL/RBC 2013 (Domain-wall) [may be at higher T]
HotQCD 2011 (staggered)
Pelisseto and Vicari 2013(theory)
Ohno et al. 2011 (staggered) Bonati et al. 2014, 2016(staggered)
Nakayama-Ohtsuki 2015, 2016(CFT)and many others
Sato-Yamada 2015(theory),
Kanazawa & Yamamoto 2015, 2016 (theory)
Dick et al. 2015 (OV in HISQ sea)
Red: YES
Sharma et al. 2015, 2016 (OV in DW sea)
Glozman 2015, 2016 (theory)
Blue: NO
Borasnyi et al. 2015 (staggered & OV)
Green: Not (directly)
Brandt et al. 2016 (Wilson)
answered but related Ejiri et al. 2016 (Wilson)
Azcoiti 2016,2017(theory)
Gomez-Nicola & Ruiz de Elvira 2017 (theory) ……

CONTENTS
1. Is U(1)A symmetry theoretically possible ?
2. Lattice QCD at high T with chiral fermions
3. Result 1: U(1)A anomaly
4. Result 2: topological susceptibility
5. Summary

U(1)A AND SU(2)L XSU(2)R SHARE
DIM<=3 ORDER PARAMETER(S).
Among quark bi-linears hq̄ q(x)i
only

hq̄q(x)i

can have a VEV :No dim.<=3 operator breaks U(1)A without
breaking SU(2)LxSU(2)R .
How about higher dim. operators ?
-> our work [Aoki, F, Taniguchi 2012]

DIRAC SPECTRUM AND
SYMMETRIES [Aoki-F-Taniguchi 2012]q̄q = lim
m

0

d

2m
( ) 2
=
2
+m

(0)

[Banks-Casher 1980]

Our idea = generalization of BC relation
to higher dim operators (dim=6 operators
were done by T.Cohen 1996) :1
SU (2)VNN

dV q̄ i q

Constraints on=0lim ( )i1
U (1)AVN

NmdV q̄ i q
i0

= 0 ???

[Aoki-F-Taniguchi 2012]

OUR RESULT 1 : MANY ORDER
PARAMETERS ARE SHARED.
(under some “reasonable” assumptions)

Constraint we find

lim h⇢( )i = c| | (1 + O( )),

m!0

>2

is strong enough to show1
U (1)AVN

N

dV q̄ i q

= 0 for

i=

a

and

5

i

for any N (up to 1/V corrections):
these order parameters are shared by
SU(2)LxSU(2)R and U(1)A .

a

[Aoki-F-Taniguchi 2012]

OUR RESULT 2 :STRONG SUPPRESSION OF
TOPOLOGICAL SUSCEPTIBILITY
We also find (in the thermodynamical limit)Nm

Q2= 0 for any N ,Vwhich implies

1
Q=
32⇡ 2

Z

d4 x✏µ⌫⇢ tr[F µ⌫ F ⇢ ]Q2= 0 for m < mcr
V
Suggests 1st order chiral transition ?
(There’s no symmetry enhancement at finite quark mass.)

WHAT WE MEAN BY
U(1)A “SYMMETRY”
We call it “symmetry” if
1hany U (1)A breakingi = ↵ ,V↵>0

Cf. conformal symmetry at the IR
fixed point.

CONTENTS✔ 1.

Is U(1)A “symmetry” theoretically possible ?
Our answer = YES. SU(2)LxSU(2)R and U(1)A areconnected through Dirac spectrum.
2. Lattice QCD at high T with chiral fermions
3. Result 1: U(1)A anomaly
4. Result 2: topological susceptibility
5. Summary

JLQCD COLLABORATION
Machines at KEKHITACHI SR16000
and U of Tsukuba

Recently
shut down …

Oakforest-PACS

IBM BG/Q
Simulation code = IroIro++
https://github.com/coppolachan/IroIro

JLQCD FINITE T PROJECT (2014-)[JLQCD (Cossu et al.) 2015, JLQCD(Tomiya et al.) 2016]

We simulate 2-flavor lattice QCD.
1. good chirality :Mobius domain-wall & overlap fermion w/ OV/DW
reweighting (frequent topology tunnelings)

JLQCD FINITE T PROJECT (2014-)[JLQCD (Cossu et al.) 2015, JLQCD(Tomiya et al.) 2016]

We simulate 2-flavor QCD.
1. good chirality :Mobius domain-wall & overlap fermion w/ OV/DW
reweighting (frequent topology tunnelings)

2. different volumes : L=16,32,48 (2 fm-4 fm).

JLQCD FINITE T PROJECT (2014-)[JLQCD (Cossu et al.) 2015, JLQCD(Tomiya et al.) 2016]

We simulate 2-flavor QCD.
1. good chirality :Mobius domain-wall & overlap fermion w/ OV/DW
reweighting (frequent topology tunnelings)

2. different volumes : L=16,32,48 (2 fm-4 fm).
3. different lattice spacings : 0.07-0.1 fm.

JLQCD FINITE T PROJECT (2014-)[JLQCD (Cossu et al.) 2015, JLQCD(Tomiya et al.) 2016]

We simulate 2-flavor QCD.
1. good chirality :Mobius domain-wall & overlap fermion w/ OV/DW
reweighting (frequent topology tunnelings)

2. different volumes : L=16,32,48 (2 fm-4 fm).
3. different lattice spacings : 0.07-0.1 fm.
Other comments
T= 190-330MeV (Tc~180MeV) with Lt=8,10,12.
3-10 different quark masses (w/ reweighting).
long MD time 20000-30000 for reweighting.

OVERLAP VS DOMAIN-WALL �
Overlap Dirac operator has exact chiral symmetry

1+m 1 mDov (m) =+5 sgn(HM )22

Measure for how
much chiral sym.
is violated

mres = 0.

(Monius) domain-wall operator is an approximation of overlap.

1+m 1
4D
DDW
(m) =
+
2

m
2

1 (T (HM ))Ls
5
1 + (T (HM ))Ls

mres ⇠ 1MeV
HM =

2DW
5
2 + DW

We thought domain-wall fermion
was good enough. But…

VIOLATION OF CHIRAL SYMMETRY
ENHANCED AT FINITE TEMPERATURE[JLQCD (Cossu et al.) 2015, JLQCD(Tomiya et al.) 2016]

Examine chiral symmetry for each eigen-mode of
Mobius domain-wall Dirac operator:gi =† D 5+vi ,5D

aRD 5 Divi

→ very bad modes appear above Tc (〜180MeV).
Domain-wall, L3xLt=323x8, T= 217MeV (β=4.10)
Domain-wall (am=0.01)
Domain-wall (am=0.005)
Domain-wall (am=0.001)

Cf.) residual mass 1
is (weighted)
0.8
average of them.
For T=0, gi areconsistent withresidual mass.

|gi|

0.6
0.4
0.2
0
0

20

40

60

80
λ (MeV)

100

120

140

U(1)A ANOMALY IS SENSITIVE TO
THE BAD MODES.
GW violation part in U(1)Asusceptibility (definition will begiven later.)Mobius domain-wall
fermion is not good13233x8 β=4.07, T=203MeV
32 x8 β=4.10, T=217MeV
3233x12 β=4.23, T=191MeV
32 x12 β=4.24, T=195MeV
1633x8 β=4.07, T=203MeV
16 x8 β=4.10, T=217MeV

enough (at high T) ! 0.8
0.6

is 20%-100% .

0.4(10 times of mres)

⁄
∆GW
/∆

GW violation effect

0.2
0
0

5

10
m (MeV)

15

20

25

[JLQCD (Cossu et al.) 2015, JLQCD(Tomiya et al.) 2016]

OVERLAP/DOMAIN-WALL REWEIGHTING(fermion action can be changed AFTER simulations)�
30
20

0.3
0.2

R

det[Dov (m)]2R⌘
4D (m)]2
det[DDW

0.4

Rov∆ xR/<R>

10
0

0.1
0

2000

4000

6000
MD timeRdAO[det Dov (m)]2 e SG
hOioverlap = R
dA[det Dov (m)]2 e SGR
4D
dAOR[det DDW(m)]2 e SG
= R
4D (m)]2 e SGdAR[det DDW
hORidomain wall=
hRidomain wall0
8000 10000 12000∆ovxR/<R>

a~0.08fm, T=1.4Tc m=6MeVOVERLAP/DOMAIN-WALL REWEIGHTINGALLOWS TOPOLOGY TUNNELINGS�Q (L=48)3

m=0.001m=0.0025
m=0.00375
m=0.0052

L=48 (3.6fm)
a = 0.07fm
T = 220 MeV

1
0
-1
-2
-3
0

2000

4000

6000

8000

10000

12000

14000

trj

Auto-correlation time of topology is O(100),
small enough compared to our long
trajectory length, 20000-30000 MD time.

CONTENTS✔ 1.

Is U(1)A symmetry theoretically possible ?
Our answer = YES. SU(2)LxSU(2)R and U(1)A areconnected through Dirac spectrum.
✔ 2. Lattice QCD at high T with chiral fermions
U(1)A at high T is sensitive to lattice artifact.
We need good chiral sym (or careful cont. limit.).
3. Result 1: U(1)A anomaly
4. Result 2: topological susceptibility
5. Summary

WHAT WE OBSERVE
Axial U(1) susceptibility
4

=
=

d x[
d

0

a

(x)

a

a

(0)

2m2
( ) 2
( + m2 ) 2

2.5x108

We compute

⌘

ov
⇡

N0 : # of zero modes(

4

N0/V (MeV )¯ ov⇡

1/ V )(x) (0) ] ,β=4.07, =203MeV, m=0.001
β=4.10, T=217MeV, m=0.001
β=4.10, T=217MeV, m=0.01

2x108

2N0
.
2
Vm

a

1.5x108
1x108
5x107
0

0

0.1

0.2

0.3
1/L

3/2

0.4
-3/2

(fm

)

0.5

0.6

U(1)A ANOMALY VANISHESIN THE CHIRAL LIMIT
Coarse (a>0.08fm) lattice [JLQCD(Tomiya et al.) 2016]
0.7

3233x12 β=4.23, T=191MeV
32 x12β=4.24, T=195MeV
3
323x8 β=4.07, T=203MeV
323x8 β=4.10, T=217MeV
163x8 β=4.07, T=203MeV
16 x8 β=4.10, T=217MeV

0.6∆ov(GeV2)

0.5
0.4

T = 1.1 – 1.2 Tc

L=16 (1.8fm)
and 32 (3.6fm )results are
consistent.

0.3
0.2
0.1

(Mscreen L > 5.)005Physical point1015
m (MeV)

20

25

30

MESON CORRELATOR ITSELFSHOWS U(1) ANOMALY VANISHING[C. Rohrhofer et al. 2017 ]C(nz) / C(nz=1)

SU(2)xSU(2)[blue] and U(1)A(red) partners
are degenerate.

10[similar results
reported by Brandt et al.
2016]

10

10

10

10

10

218 MeV

256 MeV

330 MeV

380 MeV

-2

-4

-6

10

C(nz) / C(nz=1)

Furtherenhancement toSU(4) ? [Glozman 2015]

10

0

0

-2

PS
S
Vx
Ax
Tt
Xt

-4

-6

0

5

nz

10

15

0

5

nz

10

15

CONTENTS✔ 1.

Is U(1)A symmetry theoretically possible ?
Our answer = YES. SU(2)LxSU(2)R and U(1)A areconnected through Dirac spectrum.
✔ 2. Lattice QCD at high T with chiral fermions
U(1)A at high T is sensitive to lattice artifact.
We need good chiral sym (or careful cont. limit.).
3. Result 1: U(1)A anomaly✔
U(1)A anomaly at T~1.1-1.4Tc (Tc~180MeV)
in the chiral limit is consistent with zero.
4. Result 2: topological susceptibility
5. Summary

TOPOLOGICAL SUSCEPTIBILITY

2

Q
t =
V
1
Q=
32 2

d4 xTr µFµ F

another direct probe for U(1)A anomaly.

TOPOLOGICAL SUSCEPTIBILITY
Above Tc, it is sensitive to lattice artifact.
We need (reweighted) overlap fermion fora>0.08fm.
1x108

m=10MeV Gluonic Q on DW
m=10MeV OVindex on OV

χt [MeV4]8x107 T=220MeV
6x107

Gluonic Q on
DW sea4x107

OV index(reweighted)

7

2x10

0
0

0.02

0.04

0.06

0.08
a (fm)

0.1

0.12

0.14

index ofoverlap Dirac
operator isstable against
lattice cut-off.

TOPOLOGICAL SUSCEPTIBILITYVANISHES BEFORE THE CHIRAL LIMIT[JLQCD preliminary]

1.6x108

1.4x108

T=220MeV
T=260MeV
T=330MeV
T = 220 gluonic
MeV w/o reweighting

χt [MeV4]

1.2x108

L=48 (3.6fm)& L=32 (2.4fm) results
are consistent.

1x108
8x10

7

T = 260 MeV

6x107

On our fine lattices(a~0.07fm) OV index
and gluonic def. after
Wilson flow(
8t
0.47 fm )are also consistent.

T = 330 MeV

4x107
2x107
0
0

Physical point

20

40

60
m [MeV]80

agrees with our prediction[Aoki, F, Taniguchi 2012]

100

120

Q2
= 0 for m < mcr
VFINITE VOLUME DEPENDENCE[JLQCD preliminary]
2x108

T=220MeV L=32 (2.4fm)
T=220MeV L=24 (1.8fm)

1x108
4x107
5x10

T=330MeV L=48 (3.6fm)
T=330MeV L=32 (2.4fm)

7

7

3.5x103x107
2.5x107

0
0

5

10

15
m [MeV]20

25

χt [MeV4]

χt [MeV4]

1.5x10830

2x107
1.5x107
1x107
5x106
0
-5x106

0

20

40

60
m [MeV]

80

100

120

FINITE LATTICE SPACING
DEPENDENCE[JLQCD preliminary]
2x108

T=220MeV a=0.07fm
T=220MeV a=0.09fm

8

χt [MeV4]

1.5x10

1x108

5x107

0
0

5

10

15
m [MeV]

20

25

30

STRONG SUPPRESSION OF
TOPOLOGICAL SUSCEPTIBILITY
If our data indicatesQ2= 0 for m < mcr
V
Chiral phase transition is likely to be

1st order.
(There’s no symmetry enhancement at finite quark mass.)

If mu , md < mcr , there may begravitational waves from QCD bubble
collision in the early universe.

CAN AXION BE A DARK MATTER?
If our result really indicatest =0

and 1st order phase transition,
Figure from
Kitano-san’s
seminar slide(2015)

Axion cannot be a dark matter since toomuch DM created (to expand our universe).

TEMPERATURE DEPENDENCE
Shows a sharp drop!
1.2x107m=13.2 MeV
m=26.4 MeV
ChPT (T<Tc)χt/m [MeV3]

1x107
8x106
6x106
4x106
2x106
0
0

50

100

150

200
T [MeV]

250

300

350

400

TEMPERATURE DEPENDENCE
But so does instanton model (1/T8).
1.2x107

χt/m [MeV3]

1x10

m=13.2 MeV
m=26.4 MeV
ChPT (T<Tc)
8
instanton (c/T )

7

8x106
6x106
4x106
2x106
0
0

50

100

150

200 250
T [MeV]

300

350

400

THE DROP IS STILL IMPRESSIVE.
2x108

T=220MeV a=0.07fm
T=220MeV a=0.09fm

8

χt [MeV4]

1.5x10

1x108

5x107

0
0

Dilute instanton gas model
does not predict this“discontinuity”.

5

10

15
m [MeV]

20

25

30

[JLQCD preliminary]

CONTENTS✔ 1.

Is U(1)A symmetry theoretically possible ?
Our answer = YES. SU(2)LxSU(2)R and U(1)A areconnected through Dirac spectrum.
✔ 2. Lattice QCD at high T with chiral fermions
U(1)A at high T is sensitive to lattice artifact.
We need good chiral sym (or careful cont. limit.).
3. Result 1: U(1)A anomaly✔
U(1)A anomaly at T~1.1-1.4Tc (Tc~180MeV)
in the chiral limit is consistent with zero.
✔ 4. Result 2: topological susceptibility
Topological susceptibility drops before the chiral limit.
5. Summary

SUMMARY
1. U(1)A anomaly at high T is a non-trivial problem.
2. U(1)A and SU(2)LxSU(2)R order prms. connected.
3. U(1)A is sensitive to lattice artifact at high T-> We need overlap fermion for a>0.08 fm.
For a=0.07 fm, Mobius domain-wall is O.K.
4. In our simulation with chiral fermions at 3
volumes and 3-10 quark masses at T=1.1-1.8Tc(Tc~180MeV), U(1)A anomaly disappears[ before the chiral limit ](suggesting 1st order transition ?).

MAIN MESSAGE OF THIS TALKIn high T QCD, whether
hh@µ J5µ (x)O(x0 )if ermion h A O(x)if ermion (x x0 )igluons
⌧
1
µ⌫ ⇢
0
=
✏
F
F
(x)hO(x)if ermion= 0???
µ⌫⇢
2
32⇡
gluonsor not is a non-trivial question, which can
only be answered by carefully integratingover gluons (by lattice QCD).

BACK UP SLIDESSUM OF NON-ZERO
QUANTITY NONZERO ?
NOT always.
Example: chiral condensate

hq̄qi =

R

dATrD 1 det De SGZ

can be zero and non-zero. How about
hh@µ J5µ (x)O(x0 )if ermion h A O(x)if ermion (x x0 )igluons
⌧
1
µ⌫ ⇢
0
=
✏
F
F
(x)hO(x)if ermion= 0???
µ⌫⇢
2
32⇡
gluonsSHARP DROP IS NOT DUE TO
TOPOLOGY FREEZING
8[JLQCD preliminary, parallel talk by Y. Aoki]

1x10

χt [MeV4]

8x107

OVindex 48x8
gluonic w/o reweighting

T = 220 MeV
Filled symbols =mass reweightingfrom heavier mass,
where topology isfluctuating well.

6x107

T = 260 MeV
T = 330 MeV

4x107
2x107
0

0
Physical point

20

40

60
m [MeV]

80

100

120

WHY BAD MODES ONLY ABOVE TC?
Suppose bad modes are always there but
sparse.
At T=0, they mix with MANY good lowlying modes, then relative lattice artifact iscomparable to residual mass.
At T>Tc, good modes are also SPARSE,
the lattice artifacts remain large.

PHASE DIAGRAM?
larger than physical point?

WHY DIFFERENT ?
Overlap and non-chiral fermions may be indifferent phases:
Lattice artifact= another axis

staggeredWHEN WE USE DOMAIN-WALL
FERMIONS WE MUST
1.4x109χt(MeV4)

1.2x101. Check massdependence at T=0. 1x10

2. Check massdependence at T>
Tc : if m=0 limit isconsistent withzero.
Otherwise, yourresults could becontaminated bylattice artifacts.

8x10

slab β=4.17, ms=0.04
slab β=4.17, ms=0.03
slab β=4.17, ms=0.04 (L=48)
global Q β=4.17, ms=0.04
global Q β=4.17, ms=0.03
global Q β=4.17, ms=0.04 (L=48)
SU(2) LO ChPT (Σ1/3=270 MeV)

9
9
8

6x108
4x108
2x108
0

Latticeartifact

0

10

20
30
mud (MeV)

40

50

RESOLUTION OF <Q2>
Topological charge is discrete.
→ resolution of topological

T=220 MeV

susceptibility islimitted by

β=4.3 GluonicQ on DW
β=4.3 OVindex on OV

8

1.5×104

χt [MeV ]# of confs.

8

2.0×108

1.0×107

5.0×10

0.0
0

5

10

15
m [MeV]We put max(standard err, 1/N)
to estimate the statistical error.

20

25

30

ASSUMPTIONS INAOKI-FUKAYA-TANIGUCHI 2012
1. SU(2) x SU(2) fully recovered at Tc.
2.

,

3.

finite4.
(4 can be removed.)OUR OVERLAP DIRAC OPERATOR
X



1+m 1 mM
M
M
Dov (m) =+5 sgn( i ) | i ih i |
2
2
M
| M
|<
i
th
2
3
X
4D
M 5
+ DDW (m) 41| M
i ih i | ,M <| M |ithMi : eigenvalue of HM .

“EFFICIENCY” OF OV/DW REWEIGHTING :
Nef f
hRi
=
N
N max(R)On our 2-4 fm lattices at T=1.1-1.8Tc (Tc~180MeV),
a ~ 0.1 fm : O.K. for L=2 fm, Nef f /N ⇠ 1/20
but does not work for 4 fm. Nef f /N < 1/1000.
(→ we approximate it by O(10) low-modes.)a ~ 0.08 fm : works well (3 fm). Nef f /N ⇠ 1/10
a ~ 0.07 fm : domain-wall & overlap are
consistent (2.4, 3.6 fm).
Nef f /N > 1/10

“EFFICIENCY” OF OV/DW REWEIGHTING :
Nef f
hRi
=
N
N max(R)On our 2-4m lattices at T=1.1-1.8Tc (Tc~180MeV),
a ~ 0.1 fm : O.K. for L=2 fm : Nef f /N ⇠ 1/20
but does not work for 4 fm.Nef f /N < 1/1000.
(→ we approximate it by O(10) low-modes.)a ~ 0.08 fm : works well (3 fm). Nef f /N ⇠ 1/10
a ~ 0.07 fm : domain-wall & overlap are
consistent (2.4, 3.6 fm).
Nef f /N > 1/10
Our focus in this talk