Boundaries, Interfaces, & Duality in 3d SCFTs
Natalie M. Paquette
California Institute of Technology
Based on 1712.07654 with Tudor Dimofte and Davide Gaiotto & WIP

Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Outline of this talkIntroduction
Boundary Conditions in 3d N = 2 theories
Dual pairs of boundary conditions
Duality interfaces(0, 2) theories and the 4-simplex
Other Directions

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Overview
Introduction
Boundary Conditions in 3d N = 2 theories
Dual pairs of boundary conditions
Duality interfaces(0, 2) theories and the 4-simplex
Other Directions

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Motivation:
I There is more to QFT than just local operators. Extendedoperators can distinguish QFTs, phases thereof...
I Organizing dualities: bulk duality y lower dimensional

QFTs supported on extended operators
I SUSY + twisting: equalities among local ops/correlation

fns in dual theories can admit precise mathematical
reformulations leading to interesting (conjectural)equivalences. What about extended operators?

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Goals:I Can we enrich and extend bulk (supersymmetric) dualities

by adding (half-BPS) defects?
Today: 3d N = 2 IR (Seiberg-like) dualities, codim-1 objects
I Given such dual pairs, can we understand the resulting

equivalences of underlying mathematical objects?

5/30Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Boundaries and interfaces
I Interface: domain wall between T , T 0

I Boundary: “domain wall” between T , trivial theory

These objects compose in interesting ways.
T_I

Tcollide

B

flow

TIR

B _= collision ofI, B

flow

TIR
id

T_

BIR

collide=TIR
BIR
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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Overview
Introduction
Boundary Conditions in 3d N = 2 theories
Dual pairs of boundary conditions
Duality interfaces(0, 2) theories and the 4-simplex
Other Directions

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Distinct UV Lagrangian theories that flow to common
superconformal fixed point
Simple examples to keep in mind:
I SQED/XYZ: U(1)0 , Φ, Φ̃ ↔ W = XYZ[Aharony-Hanany-Intriligator-Seiberg-Strassler], [de Boer-Hori-Oz]

I SUSY level-rank: U(N)k+N,k ↔ SU(k)−k−N [Hsin-Seiberg]

Identify UV boundary conditions B, B ∨ that flow to the same
superconformal BIR
For simplicity, focus on theories with unitary gauge groups and fundamental/antifundamental matter

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Semiclassical bc’s in UV YM-CS + matter theories
Study (dual pairs of) 3d N = 2 theories on a half-space(expands on [Gadde-Gukov-Putrov])

R1,1 × R≤0Half-BPS boundary conditions preserving:
I 2d N = (0, 2) SUSYI 3d U(1)R (inherited by 2d superalgebra)Q+ , Q̄+ = −2P+
Q− , Q̄− = 2P−

Q+ , Q̄− = −2i(P⊥ − iZ)

x.tt#g.?EIo

Q− , Q̄+ = 2i(P⊥ + iZ)

NB: Our UV computations will fail if emergent IR U(1)R !

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Building a boundary
Step One:
Write 3d N = 2 multiplets/Lagrangians in 2d N = (0, 2) language→i.e. the 3d theory is a 2d theory on R1,1 whose fields are valued inmaps f : R → 3d targetI Φ3d → Φ = φ + θ + ψ+ + . . . , Ψ = ψ̄− + θ + f + . . .

I V3d → A = θ + θ̄ + A+ , V− = A− + θ + λ− + . . .
S = σ + iA⊥ − θ+ λ̄+ . . . (WZ gauge)

Kinetic terms in x⊥ repackaged into (0, 2) interactions
Fermionic
superpotential,“J-term”:
R +Rdθ ΨJ(Φ) = dθ+ Ψ∂⊥ Φ → f ∼ ∂⊥ φ̄
NB: Also E-terms, D̄+ Ψ = E(Φ), E.J = 0 for SUSY
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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Step Two:
Identify simple reference boundary conditions, then deform:∂S∂ (u)|∂ = 0v|∂ = 0 → v +∂u
Shortcut: boundary conditions from boundary action e.o.m’s:
ZZv∂⊥ u →v∂⊥ u + S∂ (u)x⊥ ≤0x⊥ ≤0

Reference b.c’s:
Neumann: Ψ|∂ = 0 ⇒ ∂⊥ φ|∂ = 0, ψ− |∂ = 0Dirichlet: Φ|∂ = 0 ⇒ φ|∂ = 0, ψ+ |∂ = 0
See e.g. [Witten] for general discussion of elliptic boundary conditions
Rmk: SUSYc bdy action → ⊥ comp. of supercurrent total deriv. on ∂11/30

Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Step One, cont’d:Reference b.c’s for gauge fields:
Neumann: F⊥± |∂ = 0, σ|∂ = 0, λ+ |∂ = 0
preserves G gauge symmetry on boundary(bdy FI term: σ|∂ = t∂ )Dirichlet: A, V− |∂ = 0 ⇒ A± |∂ = 0, D3d − ∂⊥ σ|∂ = 0, λ− |∂ = 0
trivializes gauge field at bdy, preserves only G global symmetry
Neumann b.c’s require projection onto gauge invariants
Dirichlet b.c’s admit boundary monopole operators!
Rmk: Can define bdy conditions defined by singular behavior of bulk fields → ∂; simple examples in paper

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Step Three:
Couple to extra boundary matter.
∂Deforms boundary conditions, e.g. Neumann → F⊥± |∂ = J±I Neumann b.c. on chirals in presence of W3d (Φ):
RPa⊥
a Ea J = W(Φ)|∂ , add 2d Fermis Γ s.t.
x⊥ ≤0 dx

Pii EΓ Ji,Γ = −W(Φ)|∂

choice of matrix factorization

I Neumann b.c. on gauge fields

gauge anomaly cancellation
Dual pairs? Match ‘t Hooft anomalies and flavor symmetries!

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Anomaly Interlude
Claim: A 3d fermion with b.c. ψ± |∂ = 0 contributes ± 12 f2 , i.e. half thecontribution of a purely 2d fermion, to the anomaly polynomial.
I Add a real mass term imαβ ψαβI Flow to IR
I Match UV-IR anomalies
Two effects:
1. Integrating out massive fermion generates bckgd CS term at level 12 sign(m)2. Normalizable edge modes (2d fermions) may survive at boundary

Net result:
ψ+ ∂ = 0 has total IR anomaly
10sign(m) +12if m > 0if m < 0

ψ− ∂ = 0 has total IR anomaly1−1sign(m) +02if m > 0if m < 0

=12=−12

NB: detailed (but analogous) computation also for gauginos in paper
Rmk: see also [Moore-Freed] for a geometric avatar
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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Half-Index
IIT ,B (x; q) = TrOpsB (−1)R qJ+R/2 xe
Dual pair of b.c’s, same half-index (N gauge b.c: [Gadde-Gukov-Putrov]):
I Path integral on HS2 × S1 localization for N gauge b.c.: [Yoshida-Sugiyama]
I Operatorial count of Q̄+ -closed boundary local operators,vacuum character of bdy chiral algebra
IIN (x; q) =Yn≥0IID (x; q) =Yn≥011=:
1 − qn x(x; q)∞(1 − qn+1 x−1 ) = (qx−1 ; q)∞

Augment with lines to compute characters of other modules:

LB
ML

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Overview
Introduction
Boundary Conditions in 3d N = 2 theories
Dual pairs of boundary conditions
Duality interfaces(0, 2) theories and the 4-simplex
Other Directions

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Example: Level-rank duality

Proposal (G simple, k > 0):
1. N = 2 Gk+h∨ YMCS with D b.c. flows to chiral Gk WZWmodel coupled to TFT[Gk ].
2. N = 2 G−k−h∨ YMCS with N b.c. MUST be coupled to T2d ,flows to T2d /Gk coupled to TFT[G−k ].
U(N)−k−N,−k N + k fund. Fermis ↔ SU(k)k+N D

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Half-index of SU(2)1+2 with Dirichlet boundary conditions ongauge field
IIT1 ,D = χ0 (SU(2)1 )(1)

Half-index of U(1)−2 with Neumann bc’s + 2 fund. Fermis
IIT2 ,N = χ0 (SU(2)1 )(2)Each equality is nontrivial.
Here, equality is a manifestation of level-rank dual bdy CFTs!
2 fermions' SU(2)1U(1)2
More generally: k fermions' SU(k)1U(1)k18/30Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

More Examples
3d N = 2 dual pairs:
I U(N), (Nf , N̄f ) ↔ U(Nf − N), (Nf , N̄f ) + singlets, superpotential [Aharony]
I U(N)k+N ↔ U(k)−k−N (more SUSY level-rank)
I free chiral ↔ U(1)1/2 + chiral ↔ U(1)−1/2 + chiral (SUSYparticle-vortex)

I U(N)k+N−

Nf +Na2[Benini-Closset-Cremonesi], (Nf , Na ) ↔ U(k)−k−N+

Nf +Na, (Na , Nf )2I SU(2)1 + adjoint ↔ free chiral [Jafferis-Yin]

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

In other examples, many dual pairs we find are the simplest
possible choices consistent with:
1. matching boundary flavor symmetries
2. matching boundary ’t Hooft anomalies
3. gauge anomaly cancellation
4. superpotential factorization

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Boundary chiral algebras[Costello-Dimofte-Gaiotto]

Dual bdy conditions support ' chiral algebras in Q̄+ -cohI ∂⊥ , ∂z̄ exact, but ∂z is not

I “Holomorphic/topological” twist, see [Aganagic-Costello-McNamara-Vafa]
I Bulk ops ↔ trivial OPE (except superpotentials → quantum corrections)

SQED: (N NN) + Γ dual to XYZ: (NDD)
Gauge-invt, Q̄+ -closed operators with nontrivial OPE?
X(z)φφ̃(z)(ψY )(z)(ψZ )(w) ∼ (z−w)(φγ)(z)(φ̃γ̄)(w) ∼ (z−w)

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Overview
Introduction
Boundary Conditions in 3d N = 2 theories
Dual pairs of boundary conditions
Duality interfaces(0, 2) theories and the 4-simplex
Other Directions

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Build a duality interface
TB B!

Tdualize

T_T
B_ B !

coupleT

B B!

couple

T_T

flow

TIR

TIR
id

T
I := coupling of B _, B !
flow

=TIR

TIR
id23/30

Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

ExampleU(N)−k−N,−k |Γ|SU(k)k+N
Neumann b.c’s on either side
minimally coupled to Nk free Fermi multiplets Γ(flavor symmetry from Fermis gauged by coupling to RHS)⇒Γ form a bifundamental representation of the gauge groups
Can check that total anomaly cancels for k, N > 024/30

Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Overview
Introduction
Boundary Conditions in 3d N = 2 theories
Dual pairs of boundary conditions
Duality interfaces(0, 2) theories and the 4-simplex
Other Directions

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Geometric interpretation?
[Dimofte-NMP], to appear

3d-3d correspondence[Dimofte-Gaiotto-Gukov]

6d (2, 0) thy on 3-mfld M3 ↔ 3d N = 2 theory T [M3 ]I IR duality of SQED/XYZ ↔ change of triangulation

I2-3 move

I Geometry: Cobordism between two different

triangulations = 4-simplex
I Field theory: Change of triangulation implemented by(0, 2) duality interface
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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories“Duality” of duality interfaces
I Continue this logic:

Different ways of gluing 4-simplex together ↔ Different sequence of duality interfaces

I Different gluings realize homeomorphic 4-simplices ∴ IR equivalence of sequences/collisions of dualityinterfaces
Associate bdy chiral algebras to simple 4-manifold operation (+ functional identities from half-indices)
see also [Cecotti-Cordova-Vafa, Nakajima, Vafa-Witten, Gadde-Gukov-Putrov, Dedushenko-Gukov-Putrov,
Nidaiev-Moore, Feigin-Gukov, ...]
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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Overview
Introduction
Boundary Conditions in 3d N = 2 theories
Dual pairs of boundary conditions
Duality interfaces(0, 2) theories and the 4-simplex
Other Directions

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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Many interesting adjacent areas!
1. Similar techniques employed for non-SUSY dual pairs[Gaiotto’s Nati-Fest talk, Aitken-Baumgartner-Karch-Robinson, Aitken-Karch-Robinson]

2. Consistency of our proposal ← 2d dualities, e.g. (0, 2)
dualities of Gadde-Gukov-Putrov. Generate new (0, 2)
dualities?
3. Lift dual b.c’s to Seiberg-dual surface operators in 4d
N = 1 theories?
4. Techniques apply to 3d theories with more SUSY,
half-index useful to study mathematical counterparts[Costello-Gaiotto, Aganagic-Okounkov, . . . ]5. String/M-theory construction of 3d N = 4 boundaryconditions + applications to geometry, rep theory [WIP w/Dimofte, Hilburn]

..
.
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Introduction Boundary Conditions in 3d N = 2 theories Dual pairs of boundary conditions Duality interfaces (0, 2) theories

Thank you for your attention!

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