Bootstrapping with Supersymmetric
Bootstrapping
with
Supersymmetric Operator Algebras
Christopher Beem Oxford University
Simons Collaboration Simons
on
the Nonperturbative
Foundation
November 9
&
IO
,
Meeting 2017
Bootstrap
I
ofsignificant
CFTS
Conformal
bootstrap
( l )
amounts of
Theories
They
are
attractive for (
really
an
of such
space
theories
of independent interest /M theory
are
•
General
the
about
•
(3)
natural
are
.
Many opinions
(2)
Supersymmetry
string
-
mathematics
expectation that
another elaboration
Susy
specific on
tnathematical physics
reason
makes
reason
3)
things
.
easier
.
targets
For the
z
Extended
Super
Consequence of
theories
conformal
existence
of
are
algebraically
very
Supersymmetric operator
rich
( sub )
Super
OPE
conformal Algebra
.ek¥ / •¥aatne •
Finitekc
Commutative
Ring
Taft
.
Algebra
.
algebras
.
3
Study Aha racterize Klass ify (E) in
the
of
aorld
SCFTS
through
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their
"
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Shadows
"
.
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s
data
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for
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in
use
.
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,
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Rastelli
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•
•
o
van
Supersymmetric operator algebras
in
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Rees et al .
lot
of
.
Pufu et
j
.
alj Alday
progress
Some pure bootstrap problems structural questions
Many
Maybe
some
]
Seems to be
.
"
remain
be recycled
.
Bissi
,
can
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lot
more
to
say
.
.
.
conceptual inspiration regarding
more
general bootstrap program
.
4
Outline
•
•
[Review ] General Main
Main
•
example
I
example
I
:
Meta
•
-
algebraic
•
Conclusions
Operator Algebras
Quantization
Vertex Algebras from :
associating
constraints
:
Visions of representation
•
SUSY
of
Deformation
:
Algebraic constraints
•
Structure
&
4d
from
3d
N=
4
N=2
unitarily
Modularity -
theoretic
constraints
:
Defeats
-
General Structure
-
5
Basic well
story Known
of
Supersymmetric
.
Structurally
,
Follows
representation theory
Essentially
two flavours "
Genuinely
D
"
r
operator algebras
from
and
symmetry algebra
.
:
"
Simultaneous
cohomo
logical algebras "
co
homologies
(things like
( related Nil
chiral
to
ring
twists
)
.
)
.
is
Cohono a
°
logical operator algebras
single supercharge
"
€
acts
Supersymmetry r
{
{£
=
as
Q
,
as
of
cohomology
.
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s
constructed
are
a
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,
N
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derivation
Oixiozkz
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st
.
Ker (G)
Wirt
.
is
OPE
sub
algebra )
OPE
{ Q.0.rx.DO.kz )
+
Qrx
.
,
)
{Q
,
Qrx
.
)
]
Then
OPE
filters through
algebra acting
Q
cohomology
-
particular
,
( BPs/Short/semi Q
-
so
on
H•a(ops )= In
,
have consistent operator
such
each -
commentators
e.g.
short
)
algebra
operators
trivialized
Ppi { Q
w/Q Symmetries commuting
are
,
*
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r Q
)
defined
is
an
on
appropriate
setof
.
Q
on
Kerry
3
-
cohomology ⇒
Oµ[Ow]a=O
symmetries
of
cohomo
logical algebra
.
7
Another construction
is
{
usually
Q
;
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called
}eSCA(d,N )
8 .
ies
;
* mrainletr Q
Look
at
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intersection
,
?={
Qi
,Qj}=O "
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@
homologies
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(graded ) superalgebra
-
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( 20171
]
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Cohomology
>
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s
:{
QTNYSIA }
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representation theory
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N=2 s
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input
:
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free
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artistic
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of
this
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to
operators try don't I algorithmic
problem
→
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dime
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to
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failures
fie
think
of
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deformation
quantization
.
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s
subset
ivity
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In 3d , >
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physical
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theory
*
for
from
Lagrangian examples
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"
22
In
d=4 D
s
In
general
have
,
Situation
is
is
algebra b
the
In
we
of
cases
get •
•
bounds
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to
dyfs.ME
-12
,
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•
Q
:
Cudtfto ( Kud
How to
t
filtration
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a
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operators
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of
conditions
unitarily
ingenerdl on
.
vertex
the
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unitarily Cud
.
Formulate
cannot
:
problem
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impose
operator ambiguity problem
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worse
This
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to
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unitarily
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VOA
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[ lens
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ingeneral
(filtration
or
otherwise
)
.
-
Meta algebraic -
constraints
-
23
Supersymmetric algebras interrelations
Already gr
>
trivial
a
in
As
r
has
.
non
=
to
interpret
3d
in
Q[ M ] ,
by
-
by
this
N=4
R
-
charge
a
.
.
of deformation
given generally
as
case
.
construction
commutative
,
wish
sense
filtration F by
a
As
satisfied
r
we
constrained
trivially
non
.
used this
More
should be
their
assoc
3d
.
algebra
N=4
,
quantization
this
SCFT
is
.
a
.
constraint
if
In
4d
,
the
situation
richer
is
24 .
X Associated
✓ Been
,
]
Rastelli
For
:
✓
Poisson
VOA
Conjecture [
→
Variety
V the
associated to
voa
XUEM ( there Constraint
is
:
a
&
large
µµ
growing not
is
D
In this
s
This turns
case
,
just
V
is
out to
collection of
Poisson
called
have
,
it
Variety
is
Quasi
4dN=2
a
.
this
examples supporting
hdomorphicsympleetic -
Lisse
profound
SCFT
.
consequences
.
.
,
25
Xo%|)
In
9%4
futqli
=
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,
/
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character
Theorem [
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Supercar
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For ✓
:
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quasi
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Corollary
Suggests
:
XY (g)
transforms
novel
a
as
a
vector
organizing principle "
Simple
"
theories
linear
-
-
Lisse
VOA
modular
valued
-
those of low
modular
order
a
of
index
finite
equation ( LMDFD
differential
form
:
are
formal
XY (g) satisfies
,
quasi
limit
Schur
LMDES
•
order
,
manic
,
26
List of simple theories Free
s
(A
o
D
, ,
Az )
"
s
"
Wsu (2) F-
one
(A extra
"
Fu
,
deg
(deg
&
Q classify
)
2
=
LMDE
theories (
Gz symmetry ruled
out
,
?
) )
"
Deligne (
=
(
2) LMDE
deg
(deg
theory singularities ,
D.C.
LMDE
)
I
=
gaugegroup
Aa , Du , Eo Ez Eg ,
Possibly
:
LMDE
Argyres Douglas
SVM
rank
(
multiplet
vector
N=4
°
( deg
hyper multiplet
Free
7
:
-
deg
LMDE
Cvitanovic LMDE
hypothetically
=
LMDEEZ
2) =
2)
theories
2)
.
by anomaly arguments
theories cldeg
=
?
Is
[ Shimizu
?
is
,
Tachikaua
this
,
Zafrir
reasonable
]
?
-
Representations Categorical Constraints
-
This characterization
of
suggests considering representations
"
algebras
our
.
.me?*fEz.*EEyHe .
¥
.
Edionnoean As ) '
-
It
Bimod
As
Q
:
c.
(As ,Bs )
mod
-
Bimod
? Whatsortsofmodulesan.se?*Whatareconstraintsof3dunitarity Could
[
.
H,
module
.
level
f. Baltimore,Dimofte,GaioHo , Hilburn ( 2016 )
]
constraints
Fix
quantization
?
28
This characterization
suggests considering representations
of
our
|4I| at th
algebras
.
surface
BPS
defeat
( 2,2
)
.gg#iH9*toa.n.. 1
Quasi
-
Unitarily
Lisse
condition
imposes
⇒
restrictions
Aaa
s
s
Q
:
What
is
Cg
the
"
Finitely many
representations
on
=
I
nice
hmiznn
hmm
category
of
-
I
2
,Ijw=w=o]
representations ; characters
"
transform
in
v.
:
594 0
[
,
(
(
representations
Cardy
Hofman
&
how
like
behavior
Maldacena
.
is
it
)
constrained
by unitarily
.
v.
mf
.
-
Conclusions
-
To
conclude >
°
°
29
:*
SCFTS In
some
cast
cases
a
the
with
of
Shadow
of
,
Basic questions interplay
variety
about
unitarily
algebraic crossing associative
these remain
to
be
Shadows
is
still
a
algebras
answered
.
.
powerful &
their
constraint
.
To
conclude D
30
: .
>
s
VOA
in
>
Higgs
Could there be
With these
>
constraints
of
Source
°
different
Relationships amongst
Here
algebras
Branch
is
to
the
say
main
example
about
relations
to modularity developments starting
algebras also
relating
there to
is
CFT
a
potent
of
basic
data
their
.
bn
see
,
terms
be
.
something
in
could
modules
.
work .
to be done
extracting /
dimensions
?
To
conclude >
D
:*
There of
>
31
This
If
is
many is
an
a
ideas
we
invitation need
rich
lot of
a
dreamed
here
realizing toy
,
from come
double talk
to
"
' '
,
about
for
general
!
break
you Convex optimization
have
structure
discontinuities
Mets
.
&
versions
CFTS
.
-
Thanks
-
with
Supersymmetric Operator Algebras
Christopher Beem Oxford University
Simons Collaboration Simons
on
the Nonperturbative
Foundation
November 9
&
IO
,
Meeting 2017
Bootstrap
I
ofsignificant
CFTS
Conformal
bootstrap
( l )
amounts of
Theories
They
are
attractive for (
really
an
of such
space
theories
of independent interest /M theory
are
•
General
the
about
•
(3)
natural
are
.
Many opinions
(2)
Supersymmetry
string
-
mathematics
expectation that
another elaboration
Susy
specific on
tnathematical physics
reason
makes
reason
3)
things
.
easier
.
targets
For the
z
Extended
Super
Consequence of
theories
conformal
existence
of
are
algebraically
very
Supersymmetric operator
rich
( sub )
Super
OPE
conformal Algebra
.ek¥ / •¥aatne •
Finitekc
Commutative
Ring
Taft
.
Algebra
.
algebras
.
3
Study Aha racterize Klass ify (E) in
the
of
aorld
SCFTS
through
simpler operator algebras
their
"
various
Shadows
"
.
Theprogramir Messages Sidbeneit :
s
data
OPE
for
[ of
in
use
.
CB
,
encoded
numerical
Rastelli
,
made
Have
a
"
•
•
o
van
Supersymmetric operator algebras
in
bootstrap investigations
Rees et al .
lot
of
.
Pufu et
j
.
alj Alday
progress
Some pure bootstrap problems structural questions
Many
Maybe
some
]
Seems to be
.
"
remain
be recycled
.
Bissi
,
can
a
lot
more
to
say
.
.
.
conceptual inspiration regarding
more
general bootstrap program
.
4
Outline
•
•
[Review ] General Main
Main
•
example
I
example
I
:
Meta
•
-
algebraic
•
Conclusions
Operator Algebras
Quantization
Vertex Algebras from :
associating
constraints
:
Visions of representation
•
SUSY
of
Deformation
:
Algebraic constraints
•
Structure
&
4d
from
3d
N=
4
N=2
unitarily
Modularity -
theoretic
constraints
:
Defeats
-
General Structure
-
5
Basic well
story Known
of
Supersymmetric
.
Structurally
,
Follows
representation theory
Essentially
two flavours "
Genuinely
D
"
r
operator algebras
from
and
symmetry algebra
.
:
"
Simultaneous
cohomo
logical algebras "
co
homologies
(things like
( related Nil
chiral
to
ring
twists
)
.
)
.
is
Cohono a
°
logical operator algebras
single supercharge
"
€
acts
Supersymmetry r
{
{£
=
as
Q
,
as
of
cohomology
.
QESCA
s
constructed
are
a
(d
,
N
)
( symmetry global
derivation
Oixiozkz
) }=
st
.
Ker (G)
Wirt
.
is
OPE
sub
algebra )
OPE
{ Q.0.rx.DO.kz )
+
Qrx
.
,
)
{Q
,
Qrx
.
)
]
Then
OPE
filters through
algebra acting
Q
cohomology
-
particular
,
( BPs/Short/semi Q
-
so
on
H•a(ops )= In
,
have consistent operator
such
each -
commentators
e.g.
short
)
algebra
operators
trivialized
Ppi { Q
w/Q Symmetries commuting
are
,
*
In
r Q
)
defined
is
an
on
appropriate
setof
.
Q
on
Kerry
3
-
cohomology ⇒
Oµ[Ow]a=O
symmetries
of
cohomo
logical algebra
.
7
Another construction
is
{
usually
Q
;
Simultaneous cohomology
called
}eSCA(d,N )
8 .
ies
;
* mrainletr Q
Look
at
"
intersection
,
?={
Qi
,Qj}=O "
Ilskerrailt
of
@
homologies
H{•ai}( ops )=
its
Standard
Q
things
:
"
key
,
)
liketflchiralringinldareofthistype
Isthereanythingmoretosayabrutthis
.
construction ?
-
Main Examples
-
q
4dN±2 Nz
We've
the
most
for
Gd
work
For
and
3d
4
Maybe ( Also
done
some
these
Q
the most
are
Are there
:
Ni C 2,0 ) ;
similar
interesting
to
cases
4dN=2
)
.
.
operator interesting supersymmetric
algebras
in
{ ÷¥a§ } :o) ,
First
"
a
,
map
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of
algebras
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.
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.
10
|| ...
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symplectic duality
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[
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Higgs Algebra Associative Algebra ]
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Algebra ]
-
→
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V
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chiral
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c.
f.
Baltimore , Dimofte , Gaiatto
,
I
Hilburn
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Algebra
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i
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commutative derivation
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sN=2 Semi
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ring d.es#ai&d:gmebkrtan9oh "
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o
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chiral / /
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I
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,
Algebra]
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c.
f.
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Fredrickson
,
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Poisson
;
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*
i
Pei
,Yan,Ye(
2017 )
][
Neitzke
,
,
reduced
algebra
F. Yan
(
)
2017
topological
)
e
Hall
Poisson
embed
)][ Neitzke
,
elaaotient
.li#leuoodChiralRing
(graded ) superalgebra
-
twist
Fredrickson
( 20171
]
3dN=4
Cohomology
>
Trivialisation
s
:{
QTNYSIA }
Supercharges
H
:
S
:
-
Eta !I= '
:
,•#* (ops ,
Q 't
)±{
dependence
is
Ors
SSE
+
;⇐e=o
LIKE
)
st
.
exact
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'
12
1,2
as :#
.
Nilpotent Supercharge
s
A=
( E=R
)
Sed
(R
)
'=l=o
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branch
operators
o.rs ) .
L
a
[
C. B
.
,
W
.
C
Algebra
Peeters
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L
.
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:
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.
cR3
commutative
Chester
,
J
.
Lee ,
,
Spafu
associative
,
R
.
Yacoby ]
algebra
.
•
qrs CR
.
'
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)
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Nilpotent Supercharge
>
-
Cohomology Trivialisation
>
>
Symmetry
:
:{ QE.Q~E.SI.SI }
Supercharges
It
:
E
dependence
-
SIC 2) zis
:
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He:( ops
:
,oa=Q'
,
I Zz
1
CB
,
Lemos
Algebra ,
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,
vertex
:
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Rastelli
Rees
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.
.
'=¥=R} in
Schur
operators
.
.
=f(
[
z
z
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,zz
...
,
,Z
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operator algebra ( van
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exact
Os 's )
QIZD
s
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.
.
13
1,2
Si
+
.
asymmetry
42
=
Oh .#w=o=o)st
Etna
Is
I
voa
)
(
aka
Chiral
algebra
)
.
-
Associativity
Constraints
-
:
14
Higgs 3dN=4
take
branch
as
input µµ
D
7
1
[M
hyper
a
a
,
]
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cone
Poisson
algebra
(
grading
by
sub )
# Ap =p
A
•
Kahler
.
Ap A• qetptq
•
{ Ap Ag ) ,
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associative
algebra
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.
is
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of
A
.
,z
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In
fact
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,
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isacxequivariant
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even
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quantization
off
' 5 .
Lp÷9j
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setVMµ •
•
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>
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,
...
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ftp.q.zk
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d.
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even
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Remarkably .
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16
-
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*
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Arises
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three times
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+
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hyper 11=0
(
a
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branch
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these of
are
interesting
hs A)
values from perspective
representation theory
.
4d
N=2 s
•
°
input
:
fix
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many
finitely
Q
:
0in
free
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17
artistic
more
(
stronggenerators
finitely
if
is
.
)
Ojru )
coefficients
Q ( z ,)
%fi¥gtjI¥h¥aFtIg
constrained ,
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have
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generators
opinions
by =
associatively
(0
to solve
about
;
rzi Ojftz
)
crossing
(
)
or
) Okfzs ) rule
out
)
strong generators
.
?
4d
2
N=
input
:
efix
a
bit
more
associate
delicate
:
Not
.
of
this
would then be
to
operators try don't I algorithmic
problem
→
saasneabloisut
add
dime
-
to
)
failures
fie
think
of
.
( ?)
deformation
quantization
.
Alternatively -
x
an
(
strong generators
✓ input hdomorphi topological algebra •
s
subset
ivity
18
artistic
more
guaranteed
•
e
is
,
Many
focus universal
This has been numerical
main
on
"
canonical
sub
algebras
"
constraints
for
avenue
bootstrap
.
Feeding back
into
ftp.tfifff?alaggnyEogIIIIYfdgYy?tgt
)
-
Unitarily
Constraints
-
19
In 3d , >
Given
Ag •
•
•
Problem
D
This
In
"
is
,
how to
Of
Compute
TO
§
•
→
F
.sOg
§
suffers
"
>
basis
Ct
~
~
( f*g )
.
at
.
gauge
,
,+5fu+
.
.
equivalence
=
0
if
(ftxg )
positive
.
.
.
"
must
conditions
extra ,
Ck ( Ap Ag ) CT
H*g*I ) "
,
•
.
?
data
ordering ambiguities
"
"
beginning
Ag
c.
fiftsf
-
the
CFT
Keegan >
aforementioned
physical
just
is
recover
Identify
fo
:
•
>
associating
solving
K
>
hold
.
minlpq )
definite
' '
( positivity "
.
( truncation ) "
"
)
Unitary
\
parameter for space truncated
bases
Deformation
dim 'l
space
2°
( M ,+=e¥D
of
.
a
Finite
Quantization
of
"
truncated
bases
"
j
quantization parameter
impose
positivity laboriously
.
Unitary
Quantization
Deformation
(M#=eYEy) :*
[
truncated
bases
riauantiptaaatin
yiparamelvfan
Q
*
:
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[ Dedushenko
,
Pufu
way
,
to find "
one
Yacoby (
.
2016
quantization parameter
? Input unitary quantization
dimensional
)]
"
theory
*
for
from
Lagrangian examples
?
"
22
In
d=4 D
s
In
general
have
,
Situation
is
is
algebra b
the
In
we
of
cases
get •
•
bounds
"
[
to
dyfs.ME
-12
,
•
•
Q
:
Cudtfto ( Kud
How to
t
filtration
certain
a
finding
operators
some
I
of
conditions
unitarily
ingenerdl on
.
vertex
the
.
unitarily Cud
.
Formulate
cannot
:
problem
a
impose
operator ambiguity problem
same
worse
This
•
to
harder
unitarily
,
Kid
"
1
+
-
,
Ramirez
BPLLRVR
forget
unitarily
charge ambiguity ,
,
Sea
is
absent
and
]
]
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(g)
impose
R
.
Liendo
[
allow
on
sucz
)
VOA
[
[ lens
BPLLRVR
,
Liendo
]
]
ingeneral
(filtration
or
otherwise
)
.
-
Meta algebraic -
constraints
-
23
Supersymmetric algebras interrelations
Already gr
>
trivial
a
in
As
r
has
.
non
=
to
interpret
3d
in
Q[ M ] ,
by
-
by
this
N=4
R
-
charge
a
.
.
of deformation
given generally
as
case
.
construction
commutative
,
wish
sense
filtration F by
a
As
satisfied
r
we
constrained
trivially
non
.
used this
More
should be
their
assoc
3d
.
algebra
N=4
,
quantization
this
SCFT
is
.
a
.
constraint
if
In
4d
,
the
situation
richer
is
24 .
X Associated
✓ Been
,
]
Rastelli
For
:
✓
Poisson
VOA
Conjecture [
→
Variety
V the
associated to
voa
XUEM ( there Constraint
is
:
a
&
large
µµ
growing not
is
D
In this
s
This turns
case
,
just
V
is
out to
collection of
Poisson
called
have
,
it
Variety
is
Quasi
4dN=2
a
.
this
examples supporting
hdomorphicsympleetic -
Lisse
profound
SCFT
.
consequences
.
.
,
25
Xo%|)
In
9%4
futqli
=
Tras
)
,HFq*t¥=Iwq ✓
,
/
acuff
=
,
(
character
Theorem [
Arakawa
Supercar
,
Kauasetsif
For ✓
:
a
quasi
hdomophic ,
Corollary
Suggests
:
XY (g)
transforms
novel
a
as
a
vector
organizing principle "
Simple
"
theories
linear
-
-
Lisse
VOA
modular
valued
-
those of low
modular
order
a
of
index
finite
equation ( LMDFD
differential
form
:
are
formal
XY (g) satisfies
,
quasi
limit
Schur
LMDES
•
order
,
manic
,
26
List of simple theories Free
s
(A
o
D
, ,
Az )
"
s
"
Wsu (2) F-
one
(A extra
"
Fu
,
deg
(deg
&
Q classify
)
2
=
LMDE
theories (
Gz symmetry ruled
out
,
?
) )
"
Deligne (
=
(
2) LMDE
deg
(deg
theory singularities ,
D.C.
LMDE
)
I
=
gaugegroup
Aa , Du , Eo Ez Eg ,
Possibly
:
LMDE
Argyres Douglas
SVM
rank
(
multiplet
vector
N=4
°
( deg
hyper multiplet
Free
7
:
-
deg
LMDE
Cvitanovic LMDE
hypothetically
=
LMDEEZ
2) =
2)
theories
2)
.
by anomaly arguments
theories cldeg
=
?
Is
[ Shimizu
?
is
,
Tachikaua
this
,
Zafrir
reasonable
]
?
-
Representations Categorical Constraints
-
This characterization
of
suggests considering representations
"
algebras
our
.
.me?*fEz.*EEyHe .
¥
.
Edionnoean As ) '
-
It
Bimod
As
Q
:
c.
(As ,Bs )
mod
-
Bimod
? Whatsortsofmodulesan.se?*Whatareconstraintsof3dunitarity Could
[
.
H,
module
.
level
f. Baltimore,Dimofte,GaioHo , Hilburn ( 2016 )
]
constraints
Fix
quantization
?
28
This characterization
suggests considering representations
of
our
|4I| at th
algebras
.
surface
BPS
defeat
( 2,2
)
.gg#iH9*toa.n.. 1
Quasi
-
Unitarily
Lisse
condition
imposes
⇒
restrictions
Aaa
s
s
Q
:
What
is
Cg
the
"
Finitely many
representations
on
=
I
nice
hmiznn
hmm
category
of
-
I
2
,Ijw=w=o]
representations ; characters
"
transform
in
v.
:
594 0
[
,
(
(
representations
Cardy
Hofman
&
how
like
behavior
Maldacena
.
is
it
)
constrained
by unitarily
.
v.
mf
.
-
Conclusions
-
To
conclude >
°
°
29
:*
SCFTS In
some
cast
cases
a
the
with
of
Shadow
of
,
Basic questions interplay
variety
about
unitarily
algebraic crossing associative
these remain
to
be
Shadows
is
still
a
algebras
answered
.
.
powerful &
their
constraint
.
To
conclude D
30
: .
>
s
VOA
in
>
Higgs
Could there be
With these
>
constraints
of
Source
°
different
Relationships amongst
Here
algebras
Branch
is
to
the
say
main
example
about
relations
to modularity developments starting
algebras also
relating
there to
is
CFT
a
potent
of
basic
data
their
.
bn
see
,
terms
be
.
something
in
could
modules
.
work .
to be done
extracting /
dimensions
?
To
conclude >
D
:*
There of
>
31
This
If
is
many is
an
a
ideas
we
invitation need
rich
lot of
a
dreamed
here
realizing toy
,
from come
double talk
to
"
' '
,
about
for
general
!
break
you Convex optimization
have
structure
discontinuities
Mets
.
&
versions
CFTS
.
-
Thanks
-
