Coulomb branches of 4d QFTs
PCA, Mario Martone,C. Long, M. Lotito, Y. Lü, ...
arXiv:1504.04814, 1601.00011, 1602.02764, 1609.04404, 1611.08602,
1704.05110, 1801.01122, 1801.06554, 1804.03152, & to appear...

Goal: systematically classify CB geometries
to constrain possible 4d N = 2 QFTs
Main problem: what kinds of singularities
in the CB geometry are allowed?

P. Argyres

4d QFT CBs

Strings2018

1 / 13Coulomb branch (CB) basics
CB is a component of the moduli space of vacua of a 4d N = 2
supersymmetric (non-gravitational) QFT with massless N = 2 U(1)gauge fields.
The kinetic terms of the effective action of the massless complex
scalars, Φ, in the U(1) gauge multiplets gives the CB a metricstructure. At its metrically smooth points N = 2 supersymmetry gives
it a special Kähler (SK) structure. At these points r := dimC (CB) is
its rank, and the low energy gauge group is U(1)r .
Each point is a distinct vacuum coordinatized by the vevs u = hΦi andnearby vacua are at finite distances: the CB is metrically complete andits metric topology coincides with its continuous topology.
The metric is non-analytic ("singular") at vacua where charged states
become massless. There need not be a singularity in the complex or
topological manifold structures on the CB at the metric singularities.
P. Argyres

4d QFT CBsStrings2018

2 / 13

Some assumptions
CB geometry at metric infinity (large vevs) reflect UV behavior of QFT.
Assume: large-distance asymptotics of CB are scale-invariant.
So these CBs can come from QFTs which flow from a UV f.p., which we
assume is a SCFT.
UV fpmIR fp⇒ Our classification of CB geometries will constrain N = 2 SCFTs.

P. Argyres

4d QFT CBsStrings2018

3 / 13

Some assumptions
CB geometry at metric infinity (large vevs) reflect UV behavior of QFT.
Assume: large-distance asymptotics of CB are scale-invariant.
So these CBs can come from QFTs which flow from a UV f.p., which we
assume is a SCFT.
HBconf. vac.
of UV fp

IR fp CFTs
R+m OCBm⇒ CB not lifted by local deformations, CB gives "map" of RG flow.
Unique SCFT vacuum ⇒ CB connected.
P. Argyres

4d QFT CBs

Strings2018

3 / 13

Some assumptions
Assume: large-distance asymptotics of CB are scale-invariant.
Metric singularities occur at vanishing central charge, Zq (u) = 0, but still...
Assume: singular locus is a complex-analytic set.
⇒ singular locus closed, not a natural boundary

P. Argyres

4d QFT CBs

Strings2018

3 / 13

Some assumptions
Assume: large-distance asymptotics of CB are scale-invariant.
Assume: singular locus is a complex-analytic set.
Still many possibilities...

P. Argyres

4d QFT CBsStrings2018

3 / 13

Some assumptions
Assume: large-distance asymptotics of CB are scale-invariant.
Assume: singular locus is a complex-analytic set.
More assumptions just to get started: (will try to lift them later)
Assume: rank(CB) = 1.
Assume: CB "planar": CB 'C C.
Assume: No interacting "rank-0" SCFTs.

⇒ finite number of singular points
P. Argyres

4d QFT CBs

Strings2018

3 / 13

Scaling forms of singularities
SK structure and unitarity (∆(u) ≥ 1) ⇒Possible singularities classified by conjugacy class of EM duality
SL(2, Z) monodromy [Mγ ].
Metric can have conical singularity with opening angle 2π/∆(u) plus
logarithmic corrections.
γγP. Argyres∆(u)

I0
1

II
6/5

III
4/3IV
3/2I0∗
2IV ∗3III ∗4II ∗6[Mγ ]

I(ST )−1

S −1(ST )−2−I(ST )2SST∆(u)

In1In∗2[Mγ ]Tn−T n(n > 0, not scale-invariant)

4d QFT CBs

Strings2018

4 / 13

SK deformations
SK defoComplex deformations of scale invariant geometries which preserve anSK strucure are characterized by the patterns of splitting of the UV
singularity which preserve the total EM monodromy ⇒ many possible
patterns.
Exists [SW’94,MN’96] maximal deformations X → {I1 n }, which arefamilies of CB geometries depending on complex parametersm ∈ Mmax ∼ Cf
All other deformations are restrictions of these maximal families tocertain subspaces M ⊂ Mmax [Caorsi Cecotti 1803.00531]
But: the IR physics of sub-maximal deformations can be qualitatively
different from that of the maximal deformation, so we should
distinguish them as inequivalent theories!
P. Argyres

4d QFT CBs

Strings2018

5 / 13

"Frozen" IR fixed pointsExample:
In singularity arises from an IR free U(1) gauge theory with n q = ±1
massless hypermultiplets.
This has n−1 relative masses = deformations splitting In → {I1 n }.
√But In singularity also arises from U(1) with a single q = ± n hyper.
This doesn’t have a relative mass deformation ⇒ "frozen singularity".

This is familiar already in the simplest lagrangian example:
SU(2) gauge theory with Nf = 4 ⇒ I0∗ → {I1 6 }SU(2) with Nadj = 1 ⇒ I0∗ → {I1 2 , I4 }, found by restricting parameters√Constrained by Dirac quantization: the set { ni } commensurate ⇒ ...
28 possible rank-1 planar CB geometries for SCFTs

P. Argyres

4d QFT CBs

Strings2018

6 / 13

CB geometry and flavor symmetry
The global flavor symmetry, F , of a submaximal deformation 6⊂ Fmax ,the flavor symmetry of the maximal deformation.

P. Argyres

4d QFT CBs

Strings2018

7 / 13

CB geometry and flavor symmetry
The global flavor symmetry, F , of a submaximal deformation 6⊂ Fmax ,the flavor symmetry of the maximal deformation.
This is because F is not unambiguously determined by a deformation
family of CB geometries: gives only a complex reflection group Γ anda flavor charge lattice Λ.

P. Argyres

4d QFT CBsStrings2018

7 / 13

CB geometry and flavor symmetry
The global flavor symmetry, F , of a submaximal deformation 6⊂ Fmax ,the flavor symmetry of the maximal deformation.
This is because F is not unambiguously determined by a deformation
family of CB geometries: gives only a complex reflection group Γ anda flavor charge lattice Λ.
This allows a finite number of possible F st Weyl(F ) ⊂ Γ and rootlattice ΛF ⊂ Λ with [Λ : ΛF ] < ∞.

P. Argyres

4d QFT CBs

Strings2018

7 / 13

CB geometry and flavor symmetry
The global flavor symmetry, F , of a submaximal deformation 6⊂ Fmax ,the flavor symmetry of the maximal deformation.
This is because F is not unambiguously determined by a deformation
family of CB geometries: gives only a complex reflection group Γ anda flavor charge lattice Λ.
This allows a finite number of possible F st Weyl(F ) ⊂ Γ and rootlattice ΛF ⊂ Λ with [Λ : ΛF ] < ∞.
NB: in all cases in our classification, Γ is actually a real
crystallographic reflection group, i.e., a Weyl group.

P. Argyres

4d QFT CBs

Strings2018

7 / 13

CB geometry and flavor symmetry
The global flavor symmetry, F , of a submaximal deformation 6⊂ Fmax ,the flavor symmetry of the maximal deformation.
This is because F is not unambiguously determined by a deformation
family of CB geometries: gives only a complex reflection group Γ anda flavor charge lattice Λ.
This allows a finite number of possible F st Weyl(F ) ⊂ Γ and rootlattice ΛF ⊂ Λ with [Λ : ΛF ] < ∞.
Some of this F ambiguity lifted by demanding RG flow consistency:
m00

F0m0

F2 m
2P. Argyres

F1m1

do assumed assignments of Fi to f.p.s
match breakings due to mi ∈ adj(Fi )?

F34d QFT CBsStrings2018

7 / 13

CB geometry and flavor symmetry
The global flavor symmetry, F , of a submaximal deformation 6⊂ Fmax ,the flavor symmetry of the maximal deformation.
This is because F is not unambiguously determined by a deformation
family of CB geometries: gives only a complex reflection group Γ anda flavor charge lattice Λ.
This allows a finite number of possible F st Weyl(F ) ⊂ Γ and rootlattice ΛF ⊂ Λ with [Λ : ΛF ] < ∞.
Some of this F ambiguity lifted by demanding RG flow consistency:
m00

F0m0

F2 m2F1m1

do assumed assignments of Fi to f.p.s
match breakings due to mi ∈ adj(Fi )?

F3

For one of the 28 geometries, this constraint has no solutions:
it is ruled out as inconsistent
P. Argyres

4d QFT CBsStrings2018

7 / 13

Rank-0 SCFTs
This classification depended on the interpretation of the In and In∗
singularities as IR free gauge theories in order to use the Diracquantization condition.

P. Argyres

4d QFT CBs

Strings2018

8 / 13

Rank-0 SCFTs
This classification depended on the interpretation of the In and In∗
singularities as IR free gauge theories in order to use the Diracquantization condition.
If there existed interacting rank-0 SCFTs with global flavor symmetries
F , then by weakly gauging a U(1) or SU(2) ⊂ F gives In , In∗singularities which are non-free fixed points

P. Argyres

4d QFT CBs

Strings2018

8 / 13

Rank-0 SCFTs
This classification depended on the interpretation of the In and In∗
singularities as IR free gauge theories in order to use the Diracquantization condition.
If there existed interacting rank-0 SCFTs with global flavor symmetries
F , then by weakly gauging a U(1) or SU(2) ⊂ F gives In , In∗singularities which are non-free fixed points⇒ our classification collapses (hundreds of possible distinct rank-1
planar CB geometries).

P. Argyres

4d QFT CBs

Strings2018

8 / 13

Rank-0 SCFTs
This classification depended on the interpretation of the In and In∗
singularities as IR free gauge theories in order to use the Diracquantization condition.
If there existed interacting rank-0 SCFTs with global flavor symmetries
F , then by weakly gauging a U(1) or SU(2) ⊂ F gives In , In∗singularities which are non-free fixed points⇒ our classification collapses (hundreds of possible distinct rank-1
planar CB geometries).
This was why we assumed no rank-0 SCFTs.

P. Argyres

4d QFT CBsStrings2018

8 / 13

Rank-0 SCFTs
This classification depended on the interpretation of the In and In∗
singularities as IR free gauge theories in order to use the Diracquantization condition.
If there existed interacting rank-0 SCFTs with global flavor symmetries
F , then by weakly gauging a U(1) or SU(2) ⊂ F gives In , In∗singularities which are non-free fixed points⇒ our classification collapses (hundreds of possible distinct rank-1
planar CB geometries).
This was why we assumed no rank-0 SCFTs.
Conversely: the success of our classification could be evidence for their
non-existence.

P. Argyres

4d QFT CBsStrings2018

8 / 13

Comparison to rank-1 SCFTs
Evaluate success by comparing to known rank-1 SCFTs.

P. Argyres

4d QFT CBs

Strings2018

9 / 13

Comparison to rank-1 SCFTs
Evaluate success by comparing to known rank-1 SCFTs.
"Known" = there is evidence for their existence from:lagrangian theories,
S-class constructions,S-fold constructions,
RG flows from the above, andgauging of certain discrete symmetries of above.

P. Argyres

4d QFT CBs

Strings2018

9 / 13

Comparison to rank-1 SCFTs
Evaluate success by comparing to known rank-1 SCFTs.
"Known" = there is evidence for their existence from:lagrangian theories,
S-class constructions,S-fold constructions,
RG flows from the above, andgauging of certain discrete symmetries of above.
We find:
Of the 27 consistent CB geometries, 25 are associated to known CFTs.
There are four cases where the same CB geometry corresponds to 2
different SCFTs.

P. Argyres

4d QFT CBs

Strings2018

9 / 13

Comparison to rank-1 SCFTs
Evaluate success by comparing to known rank-1 SCFTs.
"Known" = there is evidence for their existence from:lagrangian theories,
S-class constructions,S-fold constructions,
RG flows from the above, andgauging of certain discrete symmetries of above.
We find:
Of the 27 consistent CB geometries, 25 are associated to known CFTs.
There are four cases where the same CB geometry corresponds to 2
different SCFTs.
So, AFAIK:SCFT → CB geometry map is neither 1-to-1 nor onto.
P. Argyres

4d QFT CBs

Strings2018

9 / 13

CB chiral rings
The assumption that CB 'C C is equivalent to the chiral ring, OCB , of theCB operators in the UV SCFT being freely generated: OCB = C[u].

P. Argyres

4d QFT CBs

Strings2018

10 / 13

CB chiral ringsThe assumption that CB 'C C is equivalent to the chiral ring, OCB , of theCB operators in the UV SCFT being freely generated: OCB = C[u].
A singularity in the CB complex structure then gives a non-freely-generated
CB chiral ring. For example,
OCB = C[a, b]/ha3 − b 2 i.
In terms of the normalizing coordinate,u = a1/2 = b 1/3 ,the CB looks like the complex u-plane, but with a complex singularity
hidden at the origin.

P. Argyres

4d QFT CBsStrings2018

10 / 13

CB chiral rings
The assumption that CB 'C C is equivalent to the chiral ring, OCB , of theCB operators in the UV SCFT being freely generated: OCB = C[u].
A singularity in the CB complex structure then gives a non-freely-generated
CB chiral ring. For example,
OCB = C[a, b]/ha3 − b 2 i.
In terms of the normalizing coordinate,u = a1/2 = b 1/3 ,the CB looks like the complex u-plane, but with a complex singularity
hidden at the origin.
Now a = hΦa i, b = hΦb i are vevs of SCFT local operators, but u is not:Can have ∆(u) < 1 without violating the SCFT unitarity bound.
P. Argyres

4d QFT CBs

Strings2018

10 / 13Non-freely-generated CB chiral rings?
Without the ∆(u) ≥ 1 constraint, there are now an infinite number of SK
singular geometries for each EM monodromy conjugacy class, with negative
curvatures localized at u = 0.

P. Argyres

4d QFT CBs

Strings2018

11 / 13Non-freely-generated CB chiral rings?
Without the ∆(u) ≥ 1 constraint, there are now an infinite number of SK
singular geometries for each EM monodromy conjugacy class, with negative
curvatures localized at u = 0.
⇒ Infinitely many possible consistent rank-1 CB geometries.

P. Argyres

4d QFT CBsStrings2018

11 / 13Non-freely-generated CB chiral rings?
Without the ∆(u) ≥ 1 constraint, there are now an infinite number of SK
singular geometries for each EM monodromy conjugacy class, with negative
curvatures localized at u = 0.
⇒ Infinitely many possible consistent rank-1 CB geometries.
There are no known examples of rank-1 SCFTs with a non-freely-generated
CB chiral ring.

P. Argyres

4d QFT CBs

Strings2018

11 / 13Non-freely-generated CB chiral rings?
Without the ∆(u) ≥ 1 constraint, there are now an infinite number of SK
singular geometries for each EM monodromy conjugacy class, with negative
curvatures localized at u = 0.
⇒ Infinitely many possible consistent rank-1 CB geometries.
There are no known examples of rank-1 SCFTs with a non-freely-generated
CB chiral ring.
Can show [PCA,Lü,Martone 1704.05110] unitarity & SK ⇒ if SCFT CB
has a complex singularity, then under any relevant deformation a complex
singularity persists.

P. Argyres

4d QFT CBsStrings2018

11 / 13Non-freely-generated CB chiral rings?
Without the ∆(u) ≥ 1 constraint, there are now an infinite number of SK
singular geometries for each EM monodromy conjugacy class, with negative
curvatures localized at u = 0.
⇒ Infinitely many possible consistent rank-1 CB geometries.
There are no known examples of rank-1 SCFTs with a non-freely-generated
CB chiral ring.
Can show [PCA,Lü,Martone 1704.05110] unitarity & SK ⇒ if SCFT CB
has a complex singularity, then under any relevant deformation a complex
singularity persists.
Ff such theories exist, they form a separate set under RG flows.

P. Argyres

4d QFT CBs

Strings2018

11 / 13

Rank > 1?
Much less is known. 2 notable results:[PCA,Martone 1801.06554, 1804.03152]
Can construct (lagrangian, N = 4!) SCFTs with CB complex
singularities by discrete gauging.
CB chiral ring is not freely-generated.
In these examples, CBs are normal varieties (unlike rank-1 singularvarieties), and no negative curvatures.
Also: [Bourget, Pini, Rodriguez-Gómez 1804.00118].

Can compute the (finite, rational) set of allowed CB scaling
dimensions for any rank r ≥ 2.
Assumes CB 'C Cr .
Also: [Caorsi, Cecotti 1801.04542].

In progress: exploring the tight structure tying stratified topology of thelocus of metric singularities to the set of EM duality monodromies aroundthese singularities.
P. Argyres

4d QFT CBs

Strings2018

12 / 13

EndThanks!

P. Argyres

4d QFT CBs

Strings2018

13 / 13