Geometric approach for 3d interfaces at strong coupling Based on: - - PowerPoint PPT Presentation

Geometric approach for 3d interfaces at strong coupling

Based on: 2005.05983 with J.J. Heckman, T. B. Rochais, E. Torres String Pheno Seminar - July 7, 2020 Markus Dierigl

Outline

• Non-dynamical example: Topological insulator
• The fundamental domain of Maxwell theory and its T-invariant subspace
• Generalization to other duality groups
• A six-dimensional interpretation
• Other construction of interfaces in four dimensions
• Conclusions and Outlook

Γ ⊂ SL(2, Z)

Topological insulator

System with unbroken global U(1) and time-reversal symmetry. Couple U(1) to background gauge field with local term Breaks time-reversal invariance unless A

θ ∈ {0, π} mod 2π

trivial phase topological insulator phase

[Kane, Mele ’05,’06], [Bernevig, Hughes, Zhang ’06], [Moore, Balents, ’06], …

θ 8π2 F ∧ F

Constructing an interface

x3

θ = 0

θ = π

Induces a Chern-Simons term: 1 8π A ∧ F

breaks time-reversal invariance (half-integer level) To restore it, one needs to add fields on the interface

• charged 3d Dirac fermion
• topological field theory (gapped phases) e.g. [Seiberg, Witten ‘16]

What if U(1) is dynamical?

L = 1 2g2 F ∧ ∗F + θ 8π2 F ∧ F

Has SL(2,Z) duality [Montonen, Olive ’77], [Witten ’95] τ → aτ + b cτ + d , ✓ qe qm ◆ → ✓a b c d ◆ ✓ qe qm ◆

with complexified coupling: τ = 4πi g2 + θ 2π

Time-reversal acts as: θ → −θ , τ → −τ

anti-holomorphic involution

Time-reversal revisited

Why is time-reversal invariant? θ = π

θ = π → −π ∼ −π + 2π = π

using T generator of SL(2,Z) What if one uses S generator instead? τ → −1 τ = − τ |τ|2

acts as time-reversal for |τ| = 1

Potential for time-reversal invariant phases at strong coupling

Time-reversal in pure Maxwell

Re(τ)

Im(τ)

1 2

− 1

2

i

trivial

strongly coupled

topological insulator

F-theory toolkit:

Parametrize the physically inequivalent values for by the complex structure

• f an auxiliary torus

τ

Weierstrass form: y2 = x3 + fx + g

J(τ) = j(τ) 1728 = 4f 3 4f 3 + 27g2 = 4f 3 ∆

time-reversal J ∈ R

Building interfaces

Let vary with respect to one coordinate in flat space τ

x3

For time-reversal invariant system J(τ) ∈ R

(can be realized by realm of real elliptic curves) f, g ∈ R

J = 1

J → ±∞

J = 0

τ → i∞

τ = i

τ = eπi/3

Passing through:

topological insulator J = ∞

(one expects localized dynamics and indeed ) ∆ = 0

J = 1

also leads to ∆ = 0

(strongly coupled localized dynamics?)

Other SL(2,Z) subgroups

Assume that the actual duality group is a subgroup of SL(2,Z) Γ0(N) , Γ1(N) , Γ(N) ⊂ SL(2, Z)

Γ(N) = n γ ∈ SL(2, Z) : γ = ✓1 1 ◆ mod N

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e.g. Changes fundamental domain or modular curve We are interested in the time-reversal invariant subspace: X(Γ)R = {τ ∈ X(Γ) : −τ = τ} = {τ ∈ H : −τ = γτ with γ ∈ Γ}

compactification in the presence of charged fields

Example:

Γ = Γ(2)

generated by ✓1 2 1 ◆ , ✓3 −2 2 −1 ◆ , ✓−1 −1 ◆

Re(τ)

Im(τ)

1

−1

equivalent of the -function, the Hauptmodul, given by elliptic -function J

λ

Single closed component with three cusps:

τ → i∞

topological insulator interface τ = 0

related via

S = ✓ 0 1 −1 ◆

γ = ✓1 1 1 ◆

τ = 1

related via

electric states magnetic states dyonic states

Seiberg-Witten theory

[Seiberg, Witten ’94]

SYM in 4d with gauge algebra N = 2

gauge theory in IR su2

U(1)

y2 = (x − u)(x − Λ2)(x + Λ2) , u = 1

2trφ2

duality group given by: Γ = Γ(2)

• interface crossing localized electric fields (W-bosons)
• interface crossing localized magnetic fields (monopole point)
• interface crossing localized dyonic fields (dyon point)

τ = i∞

τ = 0

τ = 1

Similar analysis for other theories (more exotic AD theories) N = 2

Mathematical classification

[Snowden ’11]

• time-reversal invariant configuration sticks to one component
• only certain combinations of interfaces realizable
• localized states from coset representatives (interesting statistics

between surfaces)

Real components of modular curves classified: Splits into disconnected sets of topological circles with “special points”

(1, 1) (1,0) (0,1)

(1, 0) i

SL(2, Z)

Γ(2)

(a, 0) (0, b) (N/2, b) (a, N/2)

Γ(N) = Γ(4n)

several components

6d interpretation

Maxwell theory as anti-symmetric 6d tensor on a torus B

A = Z

C

B , AD = − Z

C

B

charged fields form strings coupling to B

literally the complex structure of torus τ

∆ = 0

1-cycle pinches massless states charges by type of 1-cycle C = qeC − qmC

Congruence subgroups

Demand invariance of certain line operators in 4d exp ⇣ i Z

L×(rC+sC)

B ⌘ = exp ⇣ ir Z

L

A − is Z

L

AD ⌘

restricts the identifications: SL(2, Z) → Γ

(example if and are measured mod and the line operators are invariant) Γ(2)

r

s

2

Alternatively: as torsion points on Jacobian J (E) = H1(E, R)/H1(E, Z) ' e E

Nσ ∈ H1(E, Z) : B ∼ ˜ A ∧ σ

with decomposition: direct relation to congruence subgroups (example: preserves torsion)

Γ(2)

Z2 × Z2

natural generalization to higher-genus

Other geometrical interfaces

Why stop there?

So far: varied the shape of a torus along one direction (+ subtleties involving time-reversal)

Vary instead: genus, fluxes, … Some control over chiral degrees of freedom via anomaly Again use interplay between real geometry and time-reversal to secure protection of localized states

Conclusions

• Use duality to explore time-reversal invariant regions in moduli space
• Uncovers connection to real elliptic curves for Maxwell theory
• Deduce localized degrees of freedom from time-reversal invariance of

interfaces (associated to special points in moduli space, e.g. cusps)

• Passes tests where we have control ( supersymmetric theories)
• Embedding into higher-dimensional theories makes the geometric

perspective more apparent

• Explore further (more exotic) possibilities for geometrically engineered

interfaces

N = 2

Outlook

• Anomalies in duality groups [Tachikawa, Yonekura ’17], [Cordova, Freed, Lam, Seiberg

‘19], [Hsieh, Tachikawa, Yonekura ’19]

• Non-Abelian groups with 1-form center symmetries [Aharony, Seiberg,

Tachikawa ’13],… (relation to MW-torsion, e.g. [Hajouji, Oehlmann ’19])

• Realization within higher-dimensional internal space (Spin-7

manifolds) e.g. recent progress in [Cvetic, Heckman, Rochais, Torres, Zoccarato ’20]

• Tests beyond SUSY