1. No category

From SO/Sp instantons to W-algebra blocks
Christoph Keller
California Institute of Technology
based on 1101.4937 and 1107.xxxx
with Lotte Hollands and Jaewon Song
UNIFY Workshop
Frontiers in Theoretical Physics
Christoph Keller (Caltech)
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
1 / 17
Motivation and general idea
AGT correspondence: relate N = 2 gauge theories in 4d to 2d CFTs
[Alday,Gaiotto,Tachikawa]
Idea: wrap M5-branes on a Riemann surface of with punctures
R4 × C
duality frame of N = 2 gauge
theory
instanton partition function
⇔
pair of pants decomposition of C
⇔
conformal (chiral) block of CFT
Many checks have been performed for linear and cyclic quivers.
We want to check AGT for non-standard quivers (Sicilian gauge theories).
Christoph Keller (Caltech)
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
2 / 17
The AGT relation
N=2 SU(2) gauge theory
CFT
instanton partition function
conformal block
fundamental hyper of mass m
primary field of weight hm
Coulomb branch parameter a
internal channel of weight ha
Ω background deformation
central charge c
The building blocks of the quiver are fundamentals and bifundamentals
CFT: three point functions with
I
2 primaries, 1 descendant
I
1 primary, 2 descendants
To compare SU(2) to conformal block, compute U(2) instanton paritition
function, impose tracelessness and divide out ‘spurious U(1) factor’
Christoph Keller (Caltech)
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
3 / 17
Instanton counting for U(2)
Perform instanton counting for U(2) [Nekrasov; Nekrasov,Shadchin;. . . ]
Use localization to reduce to finite dimensional equivariant integral
In the end of the day, the kth instanton term reduces to a multiple integral of
a rational function
Z Y
k
(k )
)
~ ~
~
Zkinst =
dφi z(k
gauge (φi , a, 1 , 2 ) zmatter (φi , a, m, 1 , 2 ),
i=1
⇒ enumerate poles
⇒ For U(2), enumerated by sets of two Young diagrams (Y1 , Y2 ) with total
number of boxes k
Corresponds to the vector space
HVir ⊗ HU(1)
[Alday,Tachikawa; Alba,Fateev,Litvinov,Tarnopolsky;. . . ]
Christoph Keller (Caltech)
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
4 / 17
Trifundamental half-hyper
Try to repeat that for non-standard quivers, e.g. sphere with 6 punctures:
←→
For SU(2) quivers, this still has a weak coupling description
New ingredient: half-hypermultiplet in trifundamental
Problem: half-hypermultiplet only consistent if in pseudo-real representation
⇒ no U(2) trifundamental half-hyper
Christoph Keller (Caltech)
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
5 / 17
Instanton counting for Sp(1) and SO(4)
New strategy: use SU(2) ' Sp(1) (and SO(4) ' SU(2) × SU(2))
[Nekrasov,Shadchin] developed methods for general SO/Sp groups
Again reduce everything to integrals
Z Y
~ , 1 , 2 ),
Zkinst =
dφi zkgauge (φi , ~a, 1 , 2 ) zkmatter (φi , ~a, m
i
Problem: Structure of poles is much more complicated!
For Sp(1) one can express the poles as generalized Young diagrams with
signs, but it is much more complicated. (We managed up to order 6.)
SO(4): similar issues. In the unrefined case, there are useful methods
[Fucito, Morales, Poghossian; . . . ]
Christoph Keller (Caltech)
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
6 / 17
Comparing results
Let us compare the results of Sp(1) and (traceless) U(2):
Nf =0
Nf =0
(q) = ZSp(1)
(q)
ZU(2)
Nf =1
Nf =1
ZU(2)
(q) = ZSp(1)
(q)
Nf =2
Nf =2
Nf =0
ZU(2)
(q) = ZSp(1)
(q)[ZU(1)
(q)]1/2
q2
Nf =3
Nf =3
Nf =1
ZU(2)
(q) = ZSp(1)
(q)[ZU(1)
(q)]1/2 exp −
321 2
They agree in the asymptotically free cases,
but in the conformal case Nf = 4 they look completely different!
Nekrasov’s instanton counting introduces UV regulator ⇒ different
regularization schemes
⇒ results look different in the UV, but should agree in the IR
Christoph Keller (Caltech)
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
7 / 17
UV vs. IR
To check this, extract F0 from Z inst . Set 1 = −2 = ~ to get


X
Z inst = exp 
~2g−2 Fg 
g≥0
and use τIR = 21 ∂a2 F0 .
Obtain two different UV-IR relations:
qU(2)
=
θ2 (qIR )4
θ3 (qIR )4
2
qSp(1)
=
16
[Grimm,Klemm,Marino,Weiss]
2 4
θ2 (qIR
)
2 )4
θ3 (qIR
This leads to a very simple UV-UV relation:
qU(2) =
Christoph Keller (Caltech)
qSp(1)
2
1 + qSp(1) /4
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
8 / 17
AGT for Sp/SO
We want to find an AGT-type correspondence that gives Sp(1)/SO(4)
instanton partition function
New ingredients:
I
the W-algebra is W(2, 2): two copies of the Virasoro tensor T
Z2 twist lines (branch cuts) lead to twisted representations
I
tubes with twist lines: Sp(1)
I
tubes without twist lines: SO(4)
I
Christoph Keller (Caltech)
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
9 / 17
AGT for Sp/SO
N=2 gauge theory
CFT
SO(4)/Sp(1) quiver
SO(4)/Sp(1) Gaiotto curve
Sp(1) fund. hyper (µ1 , µ2 )
untwisted W(2, 2) representation (hµ1 , hµ2 )
SO(4) fund. hyper µ
twisted W(2, 2) representation hµ
Sp(1) − SO(4) bifund. hyper
twist vacuum σ
Sp(1) Coulomb par. a
weight of twisted int. channel ha
SO(4) Coulomb pars. (a1 , a2 )
weights of untwisted int. channel (ha1 , ha2 )
Christoph Keller (Caltech)
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
10 / 17
Twisted representations
Two copies of the Virasoro tensor T A (z), T B (z)
exhange T B , T A when crossing a twist line
I T + (z) = T A (z) + T B (z): always integer modes
I T − (z) = T A (z) − T B (z): half-integer modes around twist fields
To compute chiral block, map C to its double cover ⇒ T A , T B combine into
single copy T (z)
The double cover of C is exactly the Gaiotto curve of the SU(2)
configuration!
Christoph Keller (Caltech)
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
11 / 17
Geometric interpretation
I
I
I
ˆ H σ(q 2 )V A,B (0)i
Base: FSp(1) (q) = hV1A,B (∞)σ(1)P
a
2
P
i hi B
2
A
ˆ H V A (q)V B (−q)i
Cover: FSp(1) (q) = 1 − q
hV1 (−1)V1 (1)P
a
2
2
B
B
A
A
ˆ
˜ )V (0)i
˜ ) = hV (∞)V (1)PH V (q
SU(2): FSU(2) (q
1
˜=
Cross ratios must agree ⇒ q
Christoph Keller (Caltech)
1
a
2
2
2q
(1+q)2
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
12 / 17
Sp-SO bifundamental as trifundamental
We can now compute linear and cyclic Sp(1) − SO(4) quivers:
The double cover computations of these are effectively non-standard SU(2)
quivers!
Here we use the fact that the the Sp(1) − SO(4) bifundamental is equivalent
to the (specialized) SU(2) trifundamental.
Christoph Keller (Caltech)
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
13 / 17
Asymptotically free theories
From this it is clear that the trifundamental should correspond to a
three-point function
hV (φ1I1 , z1 )V (φ2I2 , z2 )V (φ3I3 , z3 )i
φiIi : arbitrary Virasoro descendant of φi , φiIi = L−n1 L−n2 · · · φi
Because we are dealing with descendants, the exact expression depends
on choice of z1 , z2 , z3 !
I
different choices will lead to non-trivial UV-UV relations, as in Sp(1) vs.
U(2)
I
but different choices will still agree in the IR
We can circumvent this problem by considering asymptotically free theories.
Christoph Keller (Caltech)
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
14 / 17
Gaiotto states
There are two ways to compute conformal blocks of asymptotically free
theories:
I
take the infinite mass limit of the conformal case
I
use Gaiotto states |h, Λi [Gaiotto]:
|h, Λi =
∞
X
Λn |vn i L1 |h, Λi = Λ|h, Λi
Ln |h, Λi = 0
n≥2
n=0
⇒ |h, Λi eigenstate of L1
Christoph Keller (Caltech)
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
15 / 17
Trifundamental
The trifundamental is the three point function of three Gaiotto states:
hh1 , Λ1 |V (|h2 , Λ2 i, 1)|h3 , Λ3 i
Compare to instanton counting:
I
agrees with Sp(1) − SO(4)
I
reduces to SU(2) bifundamentals
I
to do: compare to Sp(1) trifundamental
Note that the correlator is essentially independent of choice of insertions.
Under Möbius transformations:
γ 00
Λ
|h, Λi 7→ e 2γ 0 (γ 0 )h |h, γ 0 Λi
Christoph Keller (Caltech)
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
16 / 17
Thank You!
Christoph Keller (Caltech)
From SO/Sp instantons to W-algebra blocks
UNIFY Workshop, Porto 2011
17 / 17