Jose F. Morales (INFN, Tor Vergata, Roma)

1 Exact results

Napoli 19/11/2015

Motivations Non-perturbative effects in gauge theories:

•

QCD (Instantons): Confinement, gaugino condensation , chiral symmetry breaking, etc String Theory (D-branes):

• ✓

String dualities, AdS/CFT, AGT, etc.

Circular Wilson loop in N=4: loops

Localization:

2 I1 p W = YM

•

p

=1+ YM

YM

2 = gYM N

p

e = p 4 2⇡

YM

8

Seff =

d4 xd4 F( )

✓

+

2 YM

192

+ ...

Weak coupling

Exact results YM 5 4

YM

( 3+8

p

YM

+ . . .)

Strong coupling

!

holography

N=2 theories Z

Special theories and operators

F = F0 + F1

Prepotential

Seiberg Witten Theory, localization:

Jose F. Morales (INFN, Tor Vergata, Roma)

F( ) 2

Exact results

loop

+

⇥ k=1

Fk q k

Gauge instantons

Napoli 19/11/2015

Outline •

Chiral deformations of N=4: Wilson loops, gauge partition function :

✓

•

q-Exact formula for

✓

•

q-Exact formulas for

J

hW i

or

L

hW tr

tr

J2

. . .iL

cusp anomalous dimension

L=m

Recursion relations

F

J1

Z

d2 ✓(

2 2

+

2 3)

!

Self-duality of ADE & SO/Sp duality

SU(2)+funds SCFT at rational squashed sphe res AGT dual of Minimal Models :

✓

•

d4 ✓ ⌧J tr

Interacting Matrix Models

Mass deformations of N=4: Prepotentials for small mass:

L=

Z

Localization

q-Exact formulas for the gauge partition function

a0

Conclusions

Jose F. Morales (INFN, Tor Vergata, Roma)

ZS 4 =

X

3

Exact results

|Z(a0 )|2

Napoli 19/11/2015

Localization Localization formula:

•

Q⇠ ⌘ d + i⇠

i⇠ dxi ⌘

⇠x

i

Q2⇠ =

⇠

Q⇠ ↵ = 0

Z

•

M

⇠=✏

I=

Z

@ x @y

R2

X s

Example: ✓

•

↵ = ( 2⇡)

`

@ y @x

1

det 2 Q2⇠ (xs0 )

Gaussian integral: I =

◆

↵ = 2⇡

↵0 (xs0 )

✏e

2

Q =

✓

a(x20 +y02 )

2a✏

0 ✏

=

✏ 0

◆

Fixed Point

Z

R2

a(x2 +y 2 )

e

↵=e

dxdy a(x2 +y 2 )

dxdy

✏ e 2a

a(x2 +y 2 )

⇡ a

Supersymmetric theories:

Jose F. Morales (INFN, Tor Vergata, Roma)

Q⇠ 4

Supersymmetry charge and

Exact results

↵=e

Napoli 19/11/2015

S

⌧J -deformations •

The N=2 gauge action: Sclass =

"Z

n X i⌧J 4 4 d xd ✓ tr 2⇡J!

J

#

+ h.c. + . . . =

J=2

Z

d4 x Im⌧ (') F 2 +

Z

Re⌧ (') F ^ F + susy + . . .

scalar dependent coupling

•

The gauge partition function: Z(⌧J ) =

Z

D e

Zinst+tree (⌧ ) =

1 X

k=0

SYM ( ,⌧J )

Z

dMk e

= Zone Sinst (~ ⌧)

Localization

loop Zinst+tree (⌧J )

=

X Y

1 detY Q2

exp

fluctuations around instanton configuration

2⇡i ✏1 ✏2

p X

J=2

⌧J OJ,Y J!

Sum over Young tableaux (fixed points)

tr e

z' ˜0 Y

=

X z J OJ J

O2,Y = tra

2

J!

=

2 k ✏1 ✏2

Jose F. Morales (INFN, Tor Vergata, Roma)

X u

0

@e

zau

(1

e

z ✏1

)(1

e

z ✏2

!

⌧J -deformed YM action )

X

(i,j)2Yu

O3,Y = tra3

5

3 ✏1 ✏2

N X

X

(✏ + 2

(i,j) )

ez

(i,j)

1 A

u=1 (i,j)2Yu

Exact results

Napoli 19/11/2015

Instanton partition function :

ZY =

QN

u,v ZYu ,Yv (auv + m) QN u,v ZYu ,Yv (auv )

ZYu ,Yv (x) =

Y

(x

✏1 (kvj

i) + ✏2 (1 + k˜ui

j))

✏2 (k˜vi

j))

(i,j)2Yu

⇥

Y

(x + ✏1 (1 + kuj

i)

(i,j)2Yv

One-loop partitionn function :

Zone

loop

=

QN

m) 2 (auv 2 (auv QN u