✬
✩
An Exact prediction of N = 4 SUSYM theory
for string theory (II)
Nadav Drukker
Based on:
arXiv:1507.08659 - Jun Bourdier, N.D. and Jan Felix
arXiv:1510.07041 - Jun Bourdier, N.D. and Jan Felix
Quantum Field Theory, String Theory and Beyond
March 3, 2016
✫
1
✪
✬
✩
Outline
• During my PhD I wrote three papers with David.
✫
Nadav Drukker
2
Exact prediction (II)
✪
✬
✩
Outline
• During my PhD I wrote three papers with David.
• One was also with Hirosi, one with Arkady and one with Sunny.
✫
Nadav Drukker
2-a
Exact prediction (II)
✪
✬
✩
Outline
• During my PhD I wrote three papers with David.
• One was also with Hirosi, one with Arkady and one with Sunny.
• Immediately after graduating we published our fourth paper together computing the
exact VEV of the circular Wilson loop in N = 4 SYM.
• In the first half of this talk I will discuss that calculation, what I learned from it and
what progress has been made on that problem since.
• In the second half I will present my recent work, where we calculate exactly another
quantity in this theory — the Schur index, again by solving a matrix model.
✫
Nadav Drukker
2-b
Exact prediction (II)
✪
✬
✩
An Exact prediction of N = 4 SUSYM theory for string theory (I)
!
ND,Gross,2000
"
• The circular Wilson loop of N = 4 SYM is given by the Gaussian matrix model.
✫
Nadav Drukker
3
Exact prediction (II)
✪
✬
✩
An Exact prediction of N = 4 SUSYM theory for string theory (I)
!
ND,Gross,2000
"
• The circular Wilson loop of N = 4 SYM is given by the Gaussian matrix model.
!
"
• This was already observed by Erickson, Semenoff, Zarembo .
• We argued further for it, computed it at finite N and at strong coupling beyond the
leading large N and large λ limit.
✫
Nadav Drukker
3-a
Exact prediction (II)
✪
✬
✩
An Exact prediction of N = 4 SUSYM theory for string theory (I)
!
ND,Gross,2000
"
• The circular Wilson loop of N = 4 SYM is given by the Gaussian matrix model.
!
"
• This was already observed by Erickson, Semenoff, Zarembo .
• We argued further for it, computed it at finite N and at strong coupling beyond the
leading large N and large λ limit.
!
" # 2N
2
1
⟨W ⟩ =
dM Tr eM e− λ Tr M
Z
" λ # λ
= L1N −1 − 4N
e 8N
√
√
2
λ
I ( λ) + · · ·
= √ I1 ( λ) +
2 2
48N
λ
√ %
'
6p−3 &
∞
$
1
3(12p2 + 8p + 5)
1 e λ 2λ 4
√
+ O( )
1−
=
2p p!
p
N
π
96
λ
40
λ
p=0
✫
Nadav Drukker
3-b
Exact prediction (II)
✪
✬
✩
An Exact prediction of N = 4 SUSYM theory for string theory (I)
!
ND,Gross,2000
"
• The circular Wilson loop of N = 4 SYM is given by the Gaussian matrix model.
!
"
• This was already observed by Erickson, Semenoff, Zarembo .
• We argued further for it, computed it at finite N and at strong coupling beyond the
leading large N and large λ limit.
!
" # 2N
2
1
⟨W ⟩ =
dM Tr eM e− λ Tr M
Z
" λ # λ
= L1N −1 − 4N
e 8N
√
√
2
λ
I ( λ) + · · ·
= √ I1 ( λ) +
2 2
48N
λ
√ %
'
6p−3 &
∞
$
1
3(12p2 + 8p + 5)
1 e λ 2λ 4
√
+ O( )
1−
=
2p p!
p
N
π
96
λ
40
λ
p=0
• This last expression was given as a prediction for the 1/N and α′ expansion of the
semiclassical string in AdS.
✫
Nadav Drukker
3-c
Exact prediction (II)
✪
✬
✩
What I learned during my PhD (I):
• Calculate, calculate, calculate!
✫
Nadav Drukker
4
Exact prediction (II)
✪
✬
✩
What I learned during my PhD (I):
• Calculate, calculate, calculate!
• Relay on known results and push them as far as possible.
✫
Nadav Drukker
4-a
Exact prediction (II)
✪
✬
✩
What I learned during my PhD (I):
• Calculate, calculate, calculate!
• Relay on known results and push them as far as possible.
• email. . .
✫
Nadav Drukker
4-b
Exact prediction (II)
✪
✬
✩
Progress report
• The agreement of the leading large N result with classical string was already verified
earlier.
✫
Nadav Drukker
5
Exact prediction (II)
✪
✬
✩
Progress report
• The agreement of the leading large N result with classical string was already verified
earlier.
• The 1/N corrections with highest powers of λ can be captured by the Wilson loop in
!
the symmetric representation, given by a D3-brane.
• Likewise the antisymmetric representation is a D5-brane.
✫
Nadav Drukker
5-a
!
"!
"
Drukker Gomis
Fiol
Passerini
Hartnoll
Yamaguchi
Kumar
"!
"
Exact prediction (II)
✪
✬
✩
Progress report
• The agreement of the leading large N result with classical string was already verified
earlier.
• The 1/N corrections with highest powers of λ can be captured by the Wilson loop in
!
the symmetric representation, given by a D3-brane.
• Likewise the antisymmetric representation is a D5-brane.
• Leading large N all orders in λ given by
√2 I1 (
λ
Argued from limit of 1/4 BPS Wilson loop.
✫
Nadav Drukker
5-b
!
"!
"
Drukker Gomis
Fiol
Passerini
Hartnoll
Yamaguchi
Kumar
"!
"
!
"
√
λ):
Drukker
Exact prediction (II)
✪
✬
✩
Progress report
• The agreement of the leading large N result with classical string was already verified
earlier.
• The 1/N corrections with highest powers of λ can be captured by the Wilson loop in
!
the symmetric representation, given by a D3-brane.
!
• Likewise the antisymmetric representation is a D5-brane.
• Leading large N all orders in λ given by
√2 I1 (
λ
"!
"
Drukker Gomis
Fiol
Passerini
Hartnoll
Yamaguchi
Kumar
"!
"
!
Drukker
"
Faraggi,Pando Zayas
Silva,Trancanelli
"
√
λ):
Argued from limit of 1/4 BPS Wilson loop.
• Honest semiclassical string fluctuation calculation to reproduce
!
Still no agreement!
✫
Nadav Drukker
5-c
(
Forini,Giangreco,Puletti
Griguolo,Seminara,Vescovi
2 −3/4
:
πλ
"!
Exact prediction (II)
✪
✬
✩
Progress report
• The agreement of the leading large N result with classical string was already verified
earlier.
• The 1/N corrections with highest powers of λ can be captured by the Wilson loop in
!
the symmetric representation, given by a D3-brane.
!
• Likewise the antisymmetric representation is a D5-brane.
• Leading large N all orders in λ given by
√2 I1 (
λ
"!
"
Drukker Gomis
Fiol
Passerini
Hartnoll
Yamaguchi
Kumar
"!
"
!
Drukker
"
Faraggi,Pando Zayas
Silva,Trancanelli
"
√
λ):
Argued from limit of 1/4 BPS Wilson loop.
• Honest semiclassical string fluctuation calculation to reproduce
!
Still no agreement!
(
Forini,Giangreco,Puletti
Griguolo,Seminara,Vescovi
2 −3/4
:
πλ
"!
• Derivation of the matrix model from localization on S 4 and generalization to theories
!
"
with N = 2 SYM.
Pestun
• Explosion of calculations of exact results, AGT, indices, other dimension. . .
✫
Nadav Drukker
5-d
Exact prediction (II)
✪
✬
✩
An Exact prediction of N = 4 SUSYM theory for string theory (II)
• In the rest of the talk I will compute the exact Schur index of N = 4 SYM.
• I will define this quantity and review what was previously known about it. Namely it
was known as an elliptic matrix model.
✫
Nadav Drukker
6
Exact prediction (II)
✪
✬
✩
An Exact prediction of N = 4 SUSYM theory for string theory (II)
• In the rest of the talk I will compute the exact Schur index of N = 4 SYM.
• I will define this quantity and review what was previously known about it. Namely it
was known as an elliptic matrix model.
• Like the circular Wilson loop, this matrix model can be solved in closed form for fixed
N as well as in an exact (convergent) large N expansion.
• I will try to mention also some results on other 4d theories.
✫
Nadav Drukker
6-a
Exact prediction (II)
✪
✬
✩
Counting states
• An interesting question in a quantum mechanical system is the number of states.
• In supersymmetric theories we usually caonsider the Witten index is
I = Tr (−1)F
• It is the sum over all states in the Hilbert space weighted by their fermion number
✫
Nadav Drukker
7
Exact prediction (II)
✪
✬
✩
Counting states
• An interesting question in a quantum mechanical system is the number of states.
• In supersymmetric theories we usually caonsider the Witten index is
I = Tr (−1)F
• It is the sum over all states in the Hilbert space weighted by their fermion number
• In the presence of a global symmetry which commutes with the Hamiltonian and
superchange we can modify the sum to depend also on the charge Q
$
F Q
I(q) = Tr (−1) q =
(−1)Fn q Qn
n
✫
Nadav Drukker
7-a
Exact prediction (II)
✪
✬
The superconformal index of 4d theories
✩
• One can view a 4d theory on S 3 × R as a quantum mechanical system and apply the
same definition.
• Global symmetries include SO(4) rotations, R-symmetries and flavor symmetries.
!
• For an arbitrary theory with N = 1 SUSY we thus have
(a)
I(p, q, t; u
✫
Nadav Drukker
F E−2j1 −r E+2j1 −r r
) = Tr (−1) p
q
t
"!
"
Romels- Kinney,Maldacena
berger
Minwalla,Raju
)
e2iu
(a)
Q(a)
a
8
Exact prediction (II)
✪
✬
✩
The superconformal index of 4d theories
• One can view a 4d theory on S 3 × R as a quantum mechanical system and apply the
same definition.
• Global symmetries include SO(4) rotations, R-symmetries and flavor symmetries.
!
• For an arbitrary theory with N = 1 SUSY we thus have
(a)
I(p, q, t; u
F E−2j1 −r E+2j1 −r r
) = Tr (−1) p
q
t
"!
"
Romels- Kinney,Maldacena
berger
Minwalla,Raju
)
e2iu
(a)
Q(a)
a
• The index can be evaluated by enumerating all the “letters” within each
supermultiplet and can be expressed in terms of elliptic gamma functions.
!
Dolan
Osborn
"
• Gauge invariance is enforced by an integral over gauge charges.
✫
Nadav Drukker
8-a
Exact prediction (II)
✪
✬
✩
The superconformal index of 4d theories
• One can view a 4d theory on S 3 × R as a quantum mechanical system and apply the
same definition.
• Global symmetries include SO(4) rotations, R-symmetries and flavor symmetries.
!
• For an arbitrary theory with N = 1 SUSY we thus have
(a)
I(p, q, t; u
F E−2j1 −r E+2j1 −r r
) = Tr (−1) p
q
t
"!
"
Romels- Kinney,Maldacena
berger
Minwalla,Raju
)
e2iu
(a)
Q(a)
a
• The index can be evaluated by enumerating all the “letters” within each
supermultiplet and can be expressed in terms of elliptic gamma functions.
!
Dolan
Osborn
"
• Gauge invariance is enforced by an integral over gauge charges.
• The end result is a matrix model.
✫
Nadav Drukker
8-b
Exact prediction (II)
✪
✬
✩
The superconformal index of 4d theories
• One can view a 4d theory on S 3 × R as a quantum mechanical system and apply the
same definition.
• Global symmetries include SO(4) rotations, R-symmetries and flavor symmetries.
!
• For an arbitrary theory with N = 1 SUSY we thus have
(a)
I(p, q, t; u
F E−2j1 −r E+2j1 −r r
) = Tr (−1) p
q
t
"!
"
Romels- Kinney,Maldacena
berger
Minwalla,Raju
)
e2iu
(a)
Q(a)
a
• The index can be evaluated by enumerating all the “letters” within each
supermultiplet and can be expressed in terms of elliptic gamma functions.
!
Dolan
Osborn
"
• Gauge invariance is enforced by an integral over gauge charges.
• The end result is a matrix model.
• Being an index, I does not depend on continuous moduli of the theories
• Consequently, the index is the same for S-dual theories.
✫
Nadav Drukker
8-c
Exact prediction (II)
✪
✬
✩
The superconformal index of 4d theories
• One can view a 4d theory on S 3 × R as a quantum mechanical system and apply the
same definition.
• Global symmetries include SO(4) rotations, R-symmetries and flavor symmetries.
!
• For an arbitrary theory with N = 1 SUSY we thus have
(a)
I(p, q, t; u
F E−2j1 −r E+2j1 −r r
) = Tr (−1) p
q
t
"!
"
Romels- Kinney,Maldacena
berger
Minwalla,Raju
)
e2iu
(a)
Q(a)
a
• The index can be evaluated by enumerating all the “letters” within each
supermultiplet and can be expressed in terms of elliptic gamma functions.
!
Dolan
Osborn
"
• Gauge invariance is enforced by an integral over gauge charges.
• The end result is a matrix model.
• Being an index, I does not depend on continuous moduli of the theories
• Consequently, the index is the same for S-dual theories.
• A lot of the work on the superconformal index in recent years went into studying the
!
index for non-Lagrangian theories.
✫
Nadav Drukker
8-d
"
"!
Gadde,Rastelli,
Rastelli
Razamat Razamat,W. Yan
Exact prediction (II)
✪
✬
✩
The superconformal index of 4d theories
• One can view a 4d theory on S 3 × R as a quantum mechanical system and apply the
same definition.
• Global symmetries include SO(4) rotations, R-symmetries and flavor symmetries.
!
• For an arbitrary theory with N = 1 SUSY we thus have
(a)
I(p, q, t; u
F E−2j1 −r E+2j1 −r r
) = Tr (−1) p
q
t
"!
"
Romels- Kinney,Maldacena
berger
Minwalla,Raju
)
e2iu
(a)
Q(a)
a
• The index can be evaluated by enumerating all the “letters” within each
supermultiplet and can be expressed in terms of elliptic gamma functions.
!
Dolan
Osborn
"
• Gauge invariance is enforced by an integral over gauge charges.
• The end result is a matrix model.
• Being an index, I does not depend on continuous moduli of the theories
• Consequently, the index is the same for S-dual theories.
• A lot of the work on the superconformal index in recent years went into studying the
!
index for non-Lagrangian theories.
✫
Nadav Drukker
"
"!
Gadde,Rastelli,
Rastelli
Razamat Razamat,W. Yan
I will be old fashioned: Will study regular Lagrangian theories!
8-e
Exact prediction (II)
✪
✬
!
The Schur limit
Gadde,Rastelli
Razamat,Yan
"
✩
• We shall consider theories with N = 2 SUSY.
• Those have extra supercharges and we can “unrefine” the index by counting only
states invariant under a pair of them
• Under this restriction we set t = q and are left with a single charge E − R with
fugacity q.
✫
Nadav Drukker
9
Exact prediction (II)
✪
✬
!
The Schur limit
Gadde,Rastelli
Razamat,Yan
"
✩
• We shall consider theories with N = 2 SUSY.
• Those have extra supercharges and we can “unrefine” the index by counting only
states invariant under a pair of them
• Under this restriction we set t = q and are left with a single charge E − R with
fugacity q.
✟
☛
Pairs of elliptic gamma functions
combine to Jacobi theta functions ✠
✡
✫
Nadav Drukker
9-a
Exact prediction (II)
✪
✬
!
The Schur limit
Gadde,Rastelli
Razamat,Yan
"
✩
• We shall consider theories with N = 2 SUSY.
• Those have extra supercharges and we can “unrefine” the index by counting only
states invariant under a pair of them
• Under this restriction we set t = q and are left with a single charge E − R with
fugacity q.
✟
☛
Pairs of elliptic gamma functions
combine to Jacobi theta functions ✠
✡
• The resulting expression for a hypermultiplet in the fundamental representation is
) q −1/12 η(τ )
"
#
=
ϑ4 α + u
i
Ifund
• For a vector multiplet we have
Ivec =
q
−
rG
6
η(τ )
|W|π N
2rG
!
π
0
dN α
#
"
) ϑ21 αi − αj
1
i<j
q 3 η(τ )2
• Finally for a hypermultiplet in a bifundamental representation
✫
Nadav Drukker
Ibi-fund =
)
i,j
q −1/12 η(τ )
" (1)
#
(2)
ϑ4 α i − α j + u
9-b
Exact prediction (II)
✪
✬
✩
The matrix model
• I will mostly focus on N = 4 SYM.
• At this stage I can consider an arbitrary conformal circular quiver:
– L vector multiplets with parameters α(a) .
– L bi-fundamental hypermultiplets without flavor fugacity (u = 0).
– No fundamental fields.
✫
Nadav Drukker
10
Exact prediction (II)
✪
✬
✩
The matrix model
• I will mostly focus on N = 4 SYM.
• At this stage I can consider an arbitrary conformal circular quiver:
– L vector multiplets with parameters α(a) .
– L bi-fundamental hypermultiplets without flavor fugacity (u = 0).
– No fundamental fields.
• The index of this theory (with U (N )L gauge symmetry) is given by the matrix model
" (a)
*
2
(a) #
L
2
3N L ! π )
− LN
ϑ
α
−
α
4
q
η(τ )
i
j
i<j 1
N (a)
I=
d
α
"
*
(a)
(a+1) #
N !L π N L
0 a=1
ϑ4 α − α
i,j
✫
Nadav Drukker
10-a
i
j
Exact prediction (II)
✪
✬
✩
The matrix model
• I will mostly focus on N = 4 SYM.
• At this stage I can consider an arbitrary conformal circular quiver:
– L vector multiplets with parameters α(a) .
– L bi-fundamental hypermultiplets without flavor fugacity (u = 0).
– No fundamental fields.
• The index of this theory (with U (N )L gauge symmetry) is given by the matrix model
" (a)
*
2
(a) #
L
2
3N L ! π )
− LN
ϑ
α
−
α
4
q
η(τ )
i
j
i<j 1
N (a)
I=
d
α
"
*
(a)
(a+1) #
N !L π N L
0 a=1
ϑ4 α − α
i,j
i
j
• This expression has been known for many years.
• There have been no attempts in the literature to solve this matrix model.
✫
Nadav Drukker
10-b
Exact prediction (II)
✪
✬
✩
The matrix model
• I will mostly focus on N = 4 SYM.
• At this stage I can consider an arbitrary conformal circular quiver:
– L vector multiplets with parameters α(a) .
– L bi-fundamental hypermultiplets without flavor fugacity (u = 0).
– No fundamental fields.
• The index of this theory (with U (N )L gauge symmetry) is given by the matrix model
" (a)
*
2
(a) #
L
2
3N L ! π )
− LN
ϑ
α
−
α
4
q
η(τ )
i
j
i<j 1
N (a)
I=
d
α
"
*
(a)
(a+1) #
N !L π N L
0 a=1
ϑ4 α − α
i,j
i
j
• This expression has been known for many years.
• There have been no attempts in the literature to solve this matrix model.
In the next 15 minutes I will evaluate this integral (for L = 1)
at finite N and to all orders in the large N expansion!
✫
Nadav Drukker
10-c
Exact prediction (II)
✪
✬
✩
What I learned during my PhD (II):
• Calculate, calculate, calculate!
✫
Nadav Drukker
11
Exact prediction (II)
✪
✬
✩
What I learned during my PhD (II):
• Calculate, calculate, calculate!
• Rely on known results and push them as far as possible.
✫
Nadav Drukker
11-a
Exact prediction (II)
✪
✬
✩
What I learned during my PhD (II):
• Calculate, calculate, calculate!
• Rely on known results and push them as far as possible.
• Spam filter. . .
✫
Nadav Drukker
11-b
Exact prediction (II)
✪
✬
✩
What I learned during my PhD (II):
• Calculate, calculate, calculate!
• Rely on known results and push them as far as possible.
• Spam filter. . .
• And more. . .
✫
Nadav Drukker
11-c
Exact prediction (II)
✪
✬
An Exact prediction of N = 6 SCSM theory for M-theory
!
Kapustin
Willett,Yaakov
"!
✩
"!
"! "
Marino
Drukker
...
Putrov Marino,Putrov
• The ABJM matrix model without a Wilson loop is
*
! )
2
2
N
!
(2
sinh
π(λ
−
λ
))
(2
sinh
π(ν
−
ν
))
2
2
1
i
j
i
j
i<j
Z=
dλi dνi eπik i (λi −νi )
*
2
2
(N !)
(2
cosh
π(λ
−
ν
))
i
j
i,j
i=1
• Most effective solution: Use the Cauchy determinant identity with x = e2πλ , y = e2πν
*
N
$
)
1
i<j (xi − xj )(yi − yj )
*
=
(−1)τ
(xi − yτ (i) )
i,j (xi − yj )
i=1
τ ∈SN
✫
Nadav Drukker
12
Exact prediction (II)
✪
✬
An Exact prediction of N = 6 SCSM theory for M-theory
!
Kapustin
Willett,Yaakov
"!
✩
"!
"! "
Marino
Drukker
...
Putrov Marino,Putrov
• The ABJM matrix model without a Wilson loop is
*
! )
2
2
N
!
(2
sinh
π(λ
−
λ
))
(2
sinh
π(ν
−
ν
))
2
2
1
i
j
i
j
i<j
Z=
dλi dνi eπik i (λi −νi )
*
2
2
(N !)
(2
cosh
π(λ
−
ν
))
i
j
i,j
i=1
• Most effective solution: Use the Cauchy determinant identity with x = e2πλ , y = e2πν
*
N
$
)
1
i<j (xi − xj )(yi − yj )
*
=
(−1)τ
(xi − yτ (i) )
i,j (xi − yj )
i=1
τ ∈SN
• The resulting model has 1d fermions with a very complicated Hamiltonian, but no
interactions.
• From the asymptotic density of states one can extract the full 1/N perturbative
answer
+
,
"
#
2
k
1
−1/3 A(k)
−1/3
Z(N ) = C
e
Ai C
(N − B) + O e−N ,
C= 2 , B=
−
π k
24 3k
• More work allows to recover nonperturbative large N terms.
✫
Nadav Drukker
12-a
Exact prediction (II)
✪
✬
An Exact prediction of N = 6 SCSM theory for M-theory
!
Kapustin
Willett,Yaakov
"!
✩
"!
"! "
Marino
Drukker
...
Putrov Marino,Putrov
• The ABJM matrix model without a Wilson loop is
*
! )
2
2
N
!
(2
sinh
π(λ
−
λ
))
(2
sinh
π(ν
−
ν
))
2
2
1
i
j
i
j
i<j
Z=
dλi dνi eπik i (λi −νi )
*
2
2
(N !)
(2
cosh
π(λ
−
ν
))
i
j
i,j
i=1
• Most effective solution: Use the Cauchy determinant identity with x = e2πλ , y = e2πν
*
N
$
)
1
i<j (xi − xj )(yi − yj )
*
=
(−1)τ
(xi − yτ (i) )
i,j (xi − yj )
i=1
τ ∈SN
• The resulting model has 1d fermions with a very complicated Hamiltonian, but no
interactions.
• From the asymptotic density of states one can extract the full 1/N perturbative
answer
+
,
"
#
2
k
1
−1/3 A(k)
−1/3
Z(N ) = C
e
Ai C
(N − B) + O e−N ,
C= 2 , B=
−
π k
24 3k
• More work allows to recover nonperturbative large N terms.
• All terms have a natural M-theory interpretation.
✫
Nadav Drukker
12-b
Exact prediction (II)
✪
✬
An Exact prediction of N = 6 SCSM theory for M-theory
!
Kapustin
Willett,Yaakov
"!
✩
"!
"! "
Marino
Drukker
...
Putrov Marino,Putrov
• The ABJM matrix model without a Wilson loop is
*
! )
2
2
N
!
(2
sinh
π(λ
−
λ
))
(2
sinh
π(ν
−
ν
))
2
2
1
i
j
i
j
i<j
Z=
dλi dνi eπik i (λi −νi )
*
2
2
(N !)
(2
cosh
π(λ
−
ν
))
i
j
i,j
i=1
• Most effective solution: Use the Cauchy determinant identity with x = e2πλ , y = e2πν
*
N
$
)
1
i<j (xi − xj )(yi − yj )
*
=
(−1)τ
(xi − yτ (i) )
i,j (xi − yj )
i=1
τ ∈SN
• The resulting model has 1d fermions with a very complicated Hamiltonian, but no
interactions.
• From the asymptotic density of states one can extract the full 1/N perturbative
answer
+
,
"
#
2
k
1
−1/3 A(k)
−1/3
Z(N ) = C
e
Ai C
(N − B) + O e−N ,
C= 2 , B=
−
π k
24 3k
• More work allows to recover nonperturbative large N terms.
• All terms have a natural M-theory interpretation.
• A lot more work required to recover them all from honest M-theory calculations.
✫
Nadav Drukker
12-c
Exact prediction (II)
✪
✬
✩
Back to the Schur index
• The theta functions in the Schur index formula also satisfy a determinant identity
!
*
(a)
i<j ϑ1 αi
"
(a) # " (a+1)
− α j ϑ1 α i
−
" (a)
*
(a+1) #
− αj
i,j ϑ4 αi
=
Here
-
ϑ2
ϑ4 ϑ3
.N
Frebenius
"
(a+1) #
αj
2
(a)
(a+1)
N2
/ ""
0
)
ϑ3
q − 4 eiN (A −A
(a)
(a+1) # 2 #
"
# det cn αi − αj
ϑ3
ij
ϑ3 N (A(a) − A(a+1) + πτ
)
2
– cn is a Jacobi elliptic function with modulus k = ϑ22 /ϑ23 .
1N
(a)
1
(a)
– A = N i=1 αi are the centers of mass.
✫
Nadav Drukker
13
Exact prediction (II)
✪
✬
✩
Back to the Schur index
• The theta functions in the Schur index formula also satisfy a determinant identity
!
*
(a)
i<j ϑ1 αi
"
(a) # " (a+1)
− α j ϑ1 α i
−
" (a)
*
(a+1) #
− αj
i,j ϑ4 αi
=
Here
-
ϑ2
ϑ4 ϑ3
.N
Frebenius
"
(a+1) #
αj
2
(a)
(a+1)
N2
/ ""
0
)
ϑ3
q − 4 eiN (A −A
(a)
(a+1) # 2 #
"
# det cn αi − αj
ϑ3
ij
ϑ3 N (A(a) − A(a+1) + πτ
)
2
– cn is a Jacobi elliptic function with modulus k = ϑ22 /ϑ23 .
1N
(a)
1
(a)
– A = N i=1 αi are the centers of mass.
• If we can eliminate the dependence on A(a) , then this becomes a system of free
particles on the circle.
(a)
• Eigenvalue αi
✫
Nadav Drukker
(a−1)
interacts only with αi′
13-a
(a+1)
and αi′′
.
Exact prediction (II)
✪
✬
(3)
ϑ−1
4 (αi
(3)
ϑ21 (αi
(2)
ϑ21 (αi
−
(4)
− αj )
(4)
(3)
ϑ21 (αi
− αj )
(3)
(2)
ϑ−1
4 (αi
αi
(3)
αj )
✩
(4)
− αj )
(4)
αi
(2)
(2)
− αj ) αi
(1)
ϑ−1
4 (αi
(2)
− αj )
(1)
αi
(1)
ϑ21 (αi
(1)
− αj )
(3)
(4)
cn(αi′′ − αi′′′ )
(3)
(2)
cn(αi′
−
αi
(3)
αi′′ )
(4)
αi
(2)
αi
(1)
cn(αi
(2)
− αi′ )
(1)
αi
✫
Nadav Drukker
14
Exact prediction (II)
✪
✬
✩
• Three different ways to remove the A(a) dependence in
-
ϑ2
ϑ4 ϑ3
.N
2
− N4
2
(a)
(a+1)
ϑ3 q
eiN (A −A
"
ϑ3 N (A(a) − A(a+1) +
)
πτ
2 )
/
# det cn
– L = 1 i.e., N = 4 SYM — no complications.
ij
""
(a)
αi
−
(a+1) # 2 #
αj
ϑ3
0
– SU (N ) — delta function on A(a) , expanded in Fourier modes.
– U (N ) — expand 1/ϑ3 in Fourier modes.
✫
Nadav Drukker
15
Exact prediction (II)
✪
✬
✩
• Three different ways to remove the A(a) dependence in
-
ϑ2
ϑ4 ϑ3
.N
2
− N4
2
(a)
(a+1)
ϑ3 q
eiN (A −A
"
ϑ3 N (A(a) − A(a+1) +
)
πτ
2 )
/
# det cn
– L = 1 i.e., N = 4 SYM — no complications.
ij
""
(a)
αi
−
(a+1) # 2 #
αj
ϑ3
0
– SU (N ) — delta function on A(a) , expanded in Fourier modes.
– U (N ) — expand 1/ϑ3 in Fourier modes.
• In the two latter cases we have a Fourier expansion of a function of A(a) or
A(a) − A(a+1) .
• The Fourier coefficients of the delta function are clearly simpler.
• I will outline what we get for SU (N ) and L > 1 later.
✫
Nadav Drukker
15-a
Exact prediction (II)
✪
✬
✩
N = 4 SYM
• Using the determinant identity and extracting a prefactor we have
ϑ
" 3
# Z(N ) ,
ϑ3 πτ N/2
! π
N
)
"
#
1 $
ϑ22
σ
N
Z(N ) =
(−1)
d α
cn (αi − ασ(i) )ϑ23
N!
2π
0
i=1
I(N ) = q −N
2
/2
σ∈SN
• Z(N ) is the partition function of N free fermions on a circle.
✫
Nadav Drukker
16
Exact prediction (II)
✪
✬
✩
N = 4 SYM
• Using the determinant identity and extracting a prefactor we have
ϑ
" 3
# Z(N ) ,
ϑ3 πτ N/2
! π
N
)
"
#
1 $
ϑ22
σ
N
Z(N ) =
(−1)
d α
cn (αi − ασ(i) )ϑ23
N!
2π
0
i=1
I(N ) = q −N
2
/2
σ∈SN
• Z(N ) is the partition function of N free fermions on a circle.
• The Fermi gas partition function is completely determined by the spectral traces
! π
Zℓ = Tr (ρℓ ) =
dα1 . . . dαℓ ρ (α1 , α2 ) . . . ρ (αℓ , α1 )
0
• Where the density can be written in position space or in a Fourier expansion as
2
$ ei(2p−1)(α−α′ )
"
#
"
#
1
ϑ
ρ α, α′ = 2 cn (α − α′ )ϑ23 =
2π
π
q p−1/2 + q −p+1/2
p∈Z
✫
Nadav Drukker
16-a
Exact prediction (II)
✪
✬
✩
N = 4 SYM
• Using the determinant identity and extracting a prefactor we have
ϑ
" 3
# Z(N ) ,
ϑ3 πτ N/2
! π
N
)
"
#
1 $
ϑ22
σ
N
Z(N ) =
(−1)
d α
cn (αi − ασ(i) )ϑ23
N!
2π
0
i=1
I(N ) = q −N
2
/2
σ∈SN
• Z(N ) is the partition function of N free fermions on a circle.
• The Fermi gas partition function is completely determined by the spectral traces
! π
Zℓ = Tr (ρℓ ) =
dα1 . . . dαℓ ρ (α1 , α2 ) . . . ρ (αℓ , α1 )
0
• Where the density can be written in position space or in a Fourier expansion as
2
$ ei(2p−1)(α−α′ )
"
#
"
#
1
ϑ
ρ α, α′ = 2 cn (α − α′ )ϑ23 =
2π
π
q p−1/2 + q −p+1/2
p∈Z
• The integrals over αi identify the Fourier modes giving
.ℓ
$1
Zℓ =
p−1/2 + q −p+1/2
q
p∈Z
✫
Nadav Drukker
16-b
Exact prediction (II)
✪
✬
✩
Finite N expressions
• There is an algorithm to express such sums as elliptic integrals K and E.
!
Zucker
"
• For example
kK
,
Z1 =
π
KE − (1 − k 2 )K 2
,
Z2 =
π2
✫
Nadav Drukker
17
Z1
(1 − k 2 )kK 3
Z3 =
−
,
8
2π 3
Z2
(1 − k 2 )k 2 K 4
Z4 =
−
.
6
3π 4
Exact prediction (II)
✪
✬
✩
Finite N expressions
• There is an algorithm to express such sums as elliptic integrals K and E.
!
Zucker
"
• For example
Z1
(1 − k 2 )kK 3
kK
,
Z3 =
−
,
Z1 =
π
8
2π 3
Z2
(1 − k 2 )k 2 K 4
KE − (1 − k 2 )K 2
,
Z4 =
−
.
Z2 =
π2
6
3π 4
• In terms of the spectral traces the partition function is
Z(N ) =
$ ′ ) (−1)(ℓ−1)mℓ
ℓ
{mℓ }
where the sum is over sets satisfying
• This gives
✫
Nadav Drukker
1
ℓ
mℓ ! l m ℓ
Zℓmℓ
ℓmℓ = N .
q −1/4 √
q −1
K(K − E) ,
I(1) =
kK ,
I(2) =
π
2π 2
#
q −9/4 √ "
2
2
2
I(3) =
kK
12K(K
−
E)
−
4(1
+
k
)K
+
π
,
24π 3
#
q −4 "
2
2 3
2
K 3K(K − E) − 2k K + π (K − E) .
I(4) =
24π 4
17-a
Exact prediction (II)
✪
✬
✩
The grand index
• The grand canonical partition function of the free fermions is
2 ∞
3
.
∞
ℓ
$
$ (−κ)
)κ
Ξ(κ) = 1 +
Z(N )κN = exp −
Zℓ =
1 + p−1/2
−p+1/2
ℓ
q
+
q
N =1
ℓ=1
✫
Nadav Drukker
18
p∈Z
Exact prediction (II)
✪
✬
✩
The grand index
• The grand canonical partition function of the free fermions is
2 ∞
3
.
∞
ℓ
$
$ (−κ)
)κ
Ξ(κ) = 1 +
Z(N )κN = exp −
Zℓ =
1 + p−1/2
−p+1/2
ℓ
q
+
q
N =1
ℓ=1
p∈Z
• This is in fact a theta function
.
∞ /
2p−1
p−1/2 2
)
1+q
+ κq
1
κ 1/2 0
2 1
Ξ(κ) =
=
ϑ3
arccos , q
2p−1
1
+
q
ϑ
ϑ
2
2
3
4
p=1
• We can also use Watson’s identity to write it as a sum
- /
.
/
κ 0 ϑ2
κ 0
1
ϑ3 arccos , q +
ϑ2 arccos , q
Ξ(κ) =
ϑ4
2
ϑ3
2
✫
Nadav Drukker
18-a
Exact prediction (II)
✪
✬
✩
The grand index
• The grand canonical partition function of the free fermions is
2 ∞
3
.
∞
ℓ
$
$ (−κ)
)κ
Ξ(κ) = 1 +
Z(N )κN = exp −
Zℓ =
1 + p−1/2
−p+1/2
ℓ
q
+
q
N =1
ℓ=1
p∈Z
• This is in fact a theta function
.
∞ /
2p−1
p−1/2 2
)
1+q
+ κq
1
κ 1/2 0
2 1
Ξ(κ) =
=
ϑ3
arccos , q
2p−1
1
+
q
ϑ
ϑ
2
2
3
4
p=1
• We can also use Watson’s identity to write it as a sum
- /
.
/
κ 0 ϑ2
κ 0
1
ϑ3 arccos , q +
ϑ2 arccos , q
Ξ(κ) =
ϑ4
2
ϑ3
2
• Recall the rescaling relating I(N ) and Z(N ) which from quasiperiodicity distinguishes
odd and even N
⎧
⎨q −N 2 /4 ,
N even
2
ϑ
3
#=
q −N /2 "
⎩q −N 2 /4 ϑ3 , N odd
ϑ3 πτ N/2
ϑ2
✫
Nadav Drukker
18-b
Exact prediction (II)
✪
✬
✩
The grand index
• The grand canonical partition function of the free fermions is
2 ∞
3
.
∞
ℓ
$
$ (−κ)
)κ
Ξ(κ) = 1 +
Z(N )κN = exp −
Zℓ =
1 + p−1/2
−p+1/2
ℓ
q
+
q
N =1
ℓ=1
p∈Z
• This is in fact a theta function
.
∞ /
2p−1
p−1/2 2
)
1+q
+ κq
1
κ 1/2 0
2 1
Ξ(κ) =
=
ϑ3
arccos , q
2p−1
1
+
q
ϑ
ϑ
2
2
3
4
p=1
• We can also use Watson’s identity to write it as a sum
- /
.
/
κ 0 ϑ2
κ 0
1
ϑ3 arccos , q +
ϑ2 arccos , q
Ξ(κ) =
ϑ4
2
ϑ3
2
• Recall the rescaling relating I(N ) and Z(N ) which from quasiperiodicity distinguishes
odd and even N
⎧
⎨q −N 2 /4 ,
N even
2
ϑ
3
#=
q −N /2 "
⎩q −N 2 /4 ϑ3 , N odd
ϑ3 πτ N/2
ϑ2
• It is then natural to to define the grand index Ξ̂
✫
Nadav Drukker
Ξ̂(κ) ≡ 1 +
∞
$
N =1
18-c
I(N )q N
2
/4 N
κ
Exact prediction (II)
✪
✬
✩
• It is easy to evaluate Ξ̂ by splitting Ξ the to contributions from even and odd N
0
1/
Ξ± ≡
Ξ(κ) ± Ξ(−κ)
2
• Ξ± are just the two summands in Ξ
/
κ 0
1
ϑ3 arccos , q ,
Ξ+ (κ) =
ϑ4
2
/
ϑ2
κ 0
Ξ− (κ) =
ϑ2 arccos , q
ϑ4 ϑ3
2
• So the expression for the grand index Ξ̂ is very elegant
/
ϑ3
1 + /
κ 0
κ 0,
Ξ̂(κ) = Ξ+ +
Ξ− =
ϑ3 arccos , q + ϑ2 arccos , q
ϑ2
ϑ4
2
2
✫
Nadav Drukker
19
Exact prediction (II)
✪
✬
✩
Large N expansion
• We can get the index from the grand index by the integral transform
!
2
q −N /4 iπ
dµ Ξ̂(eµ )e−µN
I(N ) =
2πi
−iπ
✫
Nadav Drukker
20
Exact prediction (II)
✪
✬
✩
Large N expansion
• We can get the index from the grand index by the integral transform
!
2
q −N /4 iπ
dµ Ξ̂(eµ )e−µN
I(N ) =
2πi
−iπ
• Given the large κ expansion of arccos
√
e
1 + 1 − 4e−2µ
e
= i arccosh
= iµ + i log
arccos
2
2
2
µ
µ
• we find
e−µN log 1+
−µN
µ
e
Ξ̂(e ) =
:e
ϑ4
• Expanding in powers of e−2µ gives
e−µN :elog
1+
√
1−4e−2µ
2
∂µ
: = e−µN
√
1−4e−2µ
2
∂µ
:
"
ϑ3 (iµ) + ϑ2 (iµ)
#
n−1
∞
$
(−1)n −(N +2n)µ )
e
∂µ
(∂µ − n − k)
+
n!
n=1
k=1
✫
Nadav Drukker
20-a
Exact prediction (II)
✪
✬
• In the integral
✩
2
q −N /4
I(N ) =
2πi
!
iπ
dµ Ξ̂(eµ )e−µN
−iπ
• the derivatives are removed by integration by parts
! iπ
l ! iπ
1
(N
+
2n)
dµ e−µ(N +2n) ∂µl ϑ3 (iµ) =
dµ e−µ(N +2n) ϑ3 (iµ)
2πi −iπ
2πi
−iπ
• Then use the Fourier expansion
! iπ
"
#
(N +2n)2
1
−µ(N +2n)
dµ e
ϑ3 (iµ) + ϑ2 (iµ) = q 4
2πi −iπ
✫
Nadav Drukker
21
Exact prediction (II)
✪
✬
✩
• In the integral
2
q −N /4
I(N ) =
2πi
!
iπ
dµ Ξ̂(eµ )e−µN
−iπ
• the derivatives are removed by integration by parts
! iπ
l ! iπ
1
(N
+
2n)
dµ e−µ(N +2n) ∂µl ϑ3 (iµ) =
dµ e−µ(N +2n) ϑ3 (iµ)
2πi −iπ
2πi
−iπ
• Then use the Fourier expansion
! iπ
"
#
(N +2n)2
1
−µ(N +2n)
dµ e
ϑ3 (iµ) + ϑ2 (iµ) = q 4
2πi −iπ
• we finally obtain
&. .'
∞
$
2
N +n
N +n−1
1
(−1)n
+
q nN +n
I(N ) =
ϑ4 n=0
N
N
✫
Nadav Drukker
21-a
Exact prediction (II)
✪
✬
✩
• In the integral
2
q −N /4
I(N ) =
2πi
!
iπ
dµ Ξ̂(eµ )e−µN
−iπ
• the derivatives are removed by integration by parts
! iπ
l ! iπ
1
(N
+
2n)
dµ e−µ(N +2n) ∂µl ϑ3 (iµ) =
dµ e−µ(N +2n) ϑ3 (iµ)
2πi −iπ
2πi
−iπ
• Then use the Fourier expansion
! iπ
"
#
(N +2n)2
1
−µ(N +2n)
dµ e
ϑ3 (iµ) + ϑ2 (iµ) = q 4
2πi −iπ
• we finally obtain
&. .'
∞
$
2
N +n
N +n−1
1
(−1)n
+
q nN +n
I(N ) =
ϑ4 n=0
N
N
This full large N expansion converges to
the expressions we found for finite N !
✫
Nadav Drukker
21-b
Exact prediction (II)
✪
✬
✩
Prediction for string theory
&. .'
∞
2
1 $
N
+
n
N
+
n
−
1
I(N ) =
+
q nN +n
(−1)n
N
N
ϑ4 n=0
• Ignoring exponential corrections
I(N ) =
✫
Nadav Drukker
22
1
+ O(q N )
ϑ4
Exact prediction (II)
✪
✬
✩
Prediction for string theory
&. .'
∞
2
1 $
N
+
n
N
+
n
−
1
I(N ) =
+
q nN +n
(−1)n
N
N
ϑ4 n=0
• Ignoring exponential corrections
I(N ) =
1
+ O(q N )
ϑ4
• Matches state counting in AdS5 × S 5 .
• To date there is no match to SUGRA action.
• Also not when including the “Casimir energy”.
✫
Nadav Drukker
22-a
Exact prediction (II)
✪
✬
✩
Prediction for string theory
&. .'
∞
2
1 $
N
+
n
N
+
n
−
1
I(N ) =
+
q nN +n
(−1)n
N
N
ϑ4 n=0
• Ignoring exponential corrections
I(N ) =
1
+ O(q N )
ϑ4
• Matches state counting in AdS5 × S 5 .
• To date there is no match to SUGRA action.
• Also not when including the “Casimir energy”.
• If we identify q = e−β with β the radius of the compact direction, the corrections
should match some objects in SUGRA wrapping the compact direction.
• For wrapping number n there is also an n2 correction. . .
✫
Nadav Drukker
22-b
Exact prediction (II)
✪
✬
✩
L-node circular quiver
• As mentioned we have L Fourier series.
• One gets removed by integration, so we are left with a sum over ⃗n ∈ ZL−1 .
✫
Nadav Drukker
23
Exact prediction (II)
✪
✬
✩
L-node circular quiver
• As mentioned we have L Fourier series.
• One gets removed by integration, so we are left with a sum over ⃗n ∈ ZL−1 .
• For fixed ⃗n the grand canonical partition function is still a product of theta functions
"
#"
#
!L
(a) 2 2L
∞ *2L
−2idj p−1/2
2idj p−1/2
) )
)
1
+
e
q
1
+
e
q
a=1 (n
"
#
q
j=1
1/2
Ξ⃗n =
ϑ3 d j , q
#=
#"
*L "
L ϑL
(a) 2p−1
(a) 2p−1
2n
−2n
ϑ
q
1+q
q
4 3
p=1
a=1 1 + q
j=1
• The arguments are related to the roots of a polynomial of degree 2L.
✫
Nadav Drukker
23-a
Exact prediction (II)
✪
✬
✩
L-node circular quiver
• As mentioned we have L Fourier series.
• One gets removed by integration, so we are left with a sum over ⃗n ∈ ZL−1 .
• For fixed ⃗n the grand canonical partition function is still a product of theta functions
"
#"
#
!L
(a) 2 2L
∞ *2L
−2idj p−1/2
2idj p−1/2
) )
)
1
+
e
q
1
+
e
q
a=1 (n
"
#
q
j=1
1/2
Ξ⃗n =
ϑ3 d j , q
#=
#"
*L "
L ϑL
(a) 2p−1
(a) 2p−1
2n
−2n
ϑ
q
1+q
q
4 3
p=1
a=1 1 + q
j=1
• The arguments are related to the roots of a polynomial of degree 2L.
• For even L the degree of the polynomial can be reduced to L.
• Expressions with square roots happen only for L = 1 and L = 2.
• Found closed form expressions also for L = 2.
• For other L can approximate di at large κ as
idj,± = ±
• this gives
Ξ⃗n =
✫
Nadav Drukker
q
!L
(L + 1 − 2j)iπ
niπτ
µ
+
+
+ O(κ−2/L ) ,
2L
2L
2L
a=1 (n
(a) 2
L
ϑL
4 ϑ3
)
L
))
± j=1
ϑ3
j = 1, . . . , L .
(L + 1 − 2j)π
nπτ 1/2 0
+
±
,q
+ O(κ−2/L )
2L
2L
2L
/ iµ
23-b
Exact prediction (II)
✪
✬
✩
• This product over elliptic functions is
Ξ⃗n = q
✫
!L
Nadav Drukker
(a) 2
)
a=1 (n
"
#
η L (τ )
L
L
L
L
ϑ3 (iµ, q )ϑ3 (nπτ, q ) + ϑ2 (iµ, q )ϑ2 (nπτ, q )
L ) η(Lτ )
ϑL
ϑ
(0,
q
4
3
24
Exact prediction (II)
✪
✬
✩
• This product over elliptic functions is
Ξ⃗n = q
!L
(a) 2
)
a=1 (n
"
#
η L (τ )
L
L
L
L
ϑ3 (iµ, q )ϑ3 (nπτ, q ) + ϑ2 (iµ, q )ϑ2 (nπτ, q )
L ) η(Lτ )
ϑL
ϑ
(0,
q
4
3
• Integrating over µ we obtain Z⃗n
!L
η(τ )L
L
2ϑL
3 ϑ4 (0, q ) η(Lτ )
"
##
"
"
#
L
L
N
N
+ ...
× (1 + (−1) )ϑ3 πτ n, q + (1 − (−1) )ϑ2 πτ n, q
2
(a)
2
Z⃗n (N ) = q a=1 (n ) q LN /4
• Summing over ⃗n finally gives at leading order at large N
L-nodes
ISU
(N )
✫
Nadav Drukker
q L/6
+ ...
= L
η (τ )η 2 (τ L/2)
24-a
Exact prediction (II)
✪
✬
✩
L=2
• Works just like for N = 4 SYM.
• The exact answer for all N and q is
1
I
✫
L=2
Nadav Drukker
∞
∞
(N + k − 1)!(N + l − 1)! N (k+l)+2kl
q3 $$
q
.
(N + k + l)
(N ) = 4
η (τ )
N !(N − 1)!k!l!
k=0 l=0
25
Exact prediction (II)
✪
✬
✩
L=2
• Works just like for N = 4 SYM.
• The exact answer for all N and q is
1
I
L=2
∞
∞
(N + k − 1)!(N + l − 1)! N (k+l)+2kl
q3 $$
q
.
(N + k + l)
(N ) = 4
η (τ )
N !(N − 1)!k!l!
• At finite N :
k=0 l=0
1
q− 6 k K 2
L=2
,
I
(1) = 4
η (τ ) π 2
5
q − 3 −3E 2 K 2 + 2(2 − k 2 )EK 3 − (1 − k 2 )K 4
L=2
I
(2) = 4
,
η (τ )
6π 4
- 2 4
.
− 25
2
5
2 2 6
3
2 4
2
6 k
q
6E
K
−
6(1
−
k
)EK
+
(1
−
k
)
K
EK
+
k
K
K
I L=2 (3) = 4
−
+
,
η (τ )
12π 6
24π 4
192π 2
− 23
3
−3E 4 K 4 + 4(2 − k 2 )E 3 K 5 − 6(1 − k 2 )E 2 K 6 + (1 − k 2 )2 K 8
q
L=2
I
(4) = 4
η (τ )
72π 8
✫
15E 3 K 3 − 15(2 − k 2 )E 2 K 4 + (11 − 11k 2 − 4k 4 )EK 5 + 2(1 − k 2 )(2 − k 2 )K 6
+
1080π 6
3E 2 K 2 − 2(2 − k 2 )EK 3 + (1 − k 2 )K 4 0
−
.
432π 4
Nadav Drukker
25-a
Exact prediction (II)
✪
✬
✩
Summary
• Completely explicit results for the Schur index of N = 4 SYM:
– For all N , an exact q series.
– Generating function is a Jacobi theta function.
– For fixed N polynomials of elliptic integrals.
• Same is true for the two-node quiver.
✫
Nadav Drukker
26
Exact prediction (II)
✪
✬
✩
Summary
• Completely explicit results for the Schur index of N = 4 SYM:
– For all N , an exact q series.
– Generating function is a Jacobi theta function.
– For fixed N polynomials of elliptic integrals.
• Same is true for the two-node quiver.
• Is useful to do the large N expansion by first summing over all N .
• Large N expressions for all other conformal circular quivers.
✫
Nadav Drukker
26-a
Exact prediction (II)
✪
✬
✩
Summary
• Completely explicit results for the Schur index of N = 4 SYM:
– For all N , an exact q series.
– Generating function is a Jacobi theta function.
– For fixed N polynomials of elliptic integrals.
• Same is true for the two-node quiver.
• Is useful to do the large N expansion by first summing over all N .
• Large N expressions for all other conformal circular quivers.
• Many predictions which aren’t understood yet for AdS.
• Can this be generalized to non-Lagrangian theories?
• Relation to other representations of the index?
• Many other extensions possible.
✫
Nadav Drukker
26-b
Exact prediction (II)
✪
✬
✩
Happy birthday David
✫
Nadav Drukker
27
Exact prediction (II)
✪