Structure of Anomalies in
4d SCFTs from M5-branes
Emily Nardoni
University of California, San Diego
1803.00136 w/ Ibrahima Bah
1807.xxxxx w/ Ibrahima Bah, Federico Bonetti, Ruben Minasian

Strings 2018 | Okinawa Institute of Science and Technology | June 25, 2018

Arena:N M5 branes

Eﬀective worldvolume theory is

6d (2,0) SCFTs of type AN-1
Partial topological twist over∑g,n (genus g, n punctures)to preserve susy

4d theory of “Class S”[Witten] - early constructions[Gaiotto; Gaiotto, Moore, Neitzke] - N=2[Bah, Beem Bobev, Wecht] - N=1Arena:

N M5 branes

Eﬀective worldvolume theory is

6d (2,0) SCFTs of type AN-1
Partial topological twist over∑g,n (genus g, n punctures)to preserve susy

The 4d SCFTs are labeled by:• Euler characteristic 𝜒(∑ g,n) = -(2g-2+n)• Local data at the punctures.

4d theory of “Class S”

punctures→ boundary conditions for the branes→ flavor symmetries in 4d,
1-1 with Young tableaux[Chacaltana, Distler, Tachikawa]• This setup allows us to organize a large space of(generically strongly coupled!) QFTs geometrically.
• Much of the richness of Class S comes from the punctures!

Goal: First principles way of computing ’t Hooft anomalies
when compactifying M5-branes on a Riemann surface, forboth N=1 and N=2 in 4d.
Anomalies here
are related
to central
charges, and are topologicalFor•simplicity,
I’ll restrict
to N=2.

—so can be computed even for these strongly coupled QFTs!

4d anomalies from 6d• In general, one can reduce anomalies to lower dimensions.
This will capture the contribution due to the compactification(although might fail with accidental symmetries!).
• For class S, when n=0 this method has been powerful in computing theanomalies of the 4d theories. [Bah, Beem Bobev, Wecht]✴ With punctures, there are additionalcontributions. These are known indirectlyfrom QFT dualities + Higgsing.
[Chacaltana, Distler, Tachikawa]

6d (2,0) SCFTs of type AN-1

4d theory of Class S

I8 [AN1]

I6S = I6 (⌃g,n ) +︸depends on/ (⌃g,n )our work: compute
these directlynXi=1

I6 (Pi )︸depends on Young
Tableaux data

Method: Anomaly Inflow• In the presence of M5-branes, bulk diffeomorphisms localized on
the brane are anomalous. The anomalies are canceled byterms in the bulk 11d sugra effective action:
Z(1)Se↵ = 2⇡I6
W6inflowI6(1) related to anomaly polynomial I8 of 6d theory
via the descent procedure,(0)I8 = dI7 ,(0)(1)
I7 = dI6

W6[Witten; Freed, Harvey, Minasian, Moore;
Harvey, Minasian, Moore]

Roadmap: Inflow for M5-branes on M1,3 x ∑g,n[Bah, EN]

1. M5-branes magnetically source G4:

dG4 = N (5) (r)dr ^ d⌦4(*)

angular form on the 5 transverse
directions = SO(5)-bundle*(**) our workN d⇢(r) ^ e4bump form gauge-invariant, closed,globally defined
angular form( ) subtle! see [Freed, Harvey, Minasian, Moore](**)

2. Generalizing to the Class S geometries, the⌦Nd⇢(r)^enormal bundle reduces SO(5) → SO(3) x SO(2):2 ^ e2 (F⌃ )3. The RHS is a function of connection forms for background field strengths.
F ∑ over ∑g,n has sources at the punctures.
4. 𝛿Seff has new contributions from boundary terms in the integration by parts.

Punctures imply sources[Bah, EN; Bah, Bonetti, Minasian, EN]• Motivated by the holographic description, [Gaiotto, Maldacena; Bah]the curvature form F ∑ gets explicit sources localized at the punctures
as well as along a transverse direction 𝜇 to the branes:X
𝜇 E.g. 1 puncture: dF⌃ =2⇡ (~x)dx1 ^ dx2 ^ (µ µa )dµ...
..

a𝜇2✴ Each source is a KK monopole in
M-theory that some of the N M5branes can end on.

𝜇1✴ Monopole charge ka = # of M5’s
ending on the monopole.

x2

k2 M5’s ✴

x1

k1 M5’sU(ka) induced on each set of branes,→ 4d global symmetry.

✴ Sources for F ∑ imply sources for G4,which must be smoothed.

New components for inflow with punctures:[Bah, EN; Bah, Bonetti, Minasian, EN]• There are now two types of terms in the variation:✓◆
Z
3gives 6d (2,0) I8SNSe↵inf(1)(1)=ee4 ^ N Ie6
ee4 ^ ee2
2 ⇥S 1 ⇥[µ]⇥⌃2⇡
62⇡
1,3S⌦
g,n ⇥M✓◆Z
3
localizes on the braneN(0)inf(1)(1)e+ee3 ^ N I6ee4 ^ ee22 ⇥S 1 ⇥@([µ]⇥⌃91,3S⌦g,n )⇥M✓1p2 (T W6 ) + p2 (N W6 )48!
12
[p1 (T W6 ) p1 (N W6 )]4
pk = Pontryagin classes

I8inf =I6S = I6 (⌃g,n ) +nX
i=1

I6 (Pi )(0)

ee4 = dee3 ,(0)(1)

ee3 = dee2

✴ This has the structure of the usual bulk integrand, but now e4 ! ee4 depends on
the improved connection forms. Reducing over ∑g,n gives the known answer.

New components for inflow with punctures:[Bah, EN; Bah, Bonetti, Minasian, EN]• There are now two types of terms in the variation:✓◆Z
3SNSe↵
inf(1)(1)
e=ee4 ^ N I6
ee4 ^ ee2
2 ⇥S 1 ⇥[µ]⇥⌃
2⇡
62⇡
1,3S⌦
g,n ⇥M✓Z(0)inf(1)e+ee3 ^ N I6
2 ⇥S 1 ⇥@([µ]⇥⌃1,3S⌦g,n )⇥M

localizes on the boundaries

I6S = I6 (⌃g,n ) +nX

3N(1)ee4 ^ ee2
9I6 (Pi )i=1✴ Reducing over the boundaries gives the additional puncture anomalies.
The topological structure of the 4d anomalies follows directlyfrom inflow arguments.

◆Thanks!
ありがとうございます