Exact Result for boundaries (and domain walls) in 2d supersymmetric - - PowerPoint PPT Presentation

Exact Result for boundaries (and domain walls) in 2d supersymmetric theory

Daigo Honda（本田） University of Tokyo, Komaba Based on the paper arXiv:1308.2217 Collaboration with Takuya Okuda @ YITP workshop on “Field Theory and String Theory” 2013

3 papers about the supersymmetric localization with boundary appeared from Japan!!!

Sugishita-Terashima 1308.1973 Honda-Okuda 1308.2217 Hori-Romo 1308.2438

The three groups coordinated the submission to the arXiv.

2d N=(2,2) GLSMs on hemispheres

GLSM Low energy NLSM or LG model Boundaries D-branes

Target space = vacuum manifold of GLSM

What we did

• Construction of N=(2,2) GLSMs
• n hemispheres
• Supersymmetric localization
• Derivation of hemisphere partition functions

and their various properties

• Domain walls (non-dynamical)

Motivations

• String theoretic (boundaries)
• D-branes in Calabi-Yau manifold
• Mirror symmetry
• Gauge theoretic (domain walls)
• Line + Surface operators in 4d theory
• Integrable structure

Plan of this talk

• Construction of N=(2,2) GLSMs
• n hemispheres
• Hemisphere partition functions and

their properties

N=(2,2) GLSMs

• n hemispheres

Gauge group Vector multiplet a-th chiral multiplet in irreducible rep. Complexified FI parameter Superpotential : holomorphic function of Complexified twisted masses

Bulk data of 2d N=(2,2) GLSM

G

(Aµ, σ1, σ2, λ, ¯ λ, D)

(φa, ψa, Fa)

t = r − iθ

W

φ = (φa)

m = (ma)

FI parameter / theta angle

(Defined for each abelian factor)

R-charge and real twisted masses for flavor symmetry

Ra

Supersymmetry

We choose the supercharge constructed by Gomis-Lee (2013). Deformation of the sphere does not change the partition functions. B-type supersymmetry semi-infinite cylinder with a cap at infinity A-type twist Setting of Cecotti-Vafa (1991) Periodic around circle → Ramond-Ramond sector Long propagation through cylinder → Zero energy state

hB|1i

boundary state state without any insertion R-symmetry flux

For vector multiplet: Gauge symmetry preserving condition (Gauge symmetry broken condition? ) For chiral multiplets: Neumann condition Dirichlet condition These conditions determine the submanifolds on which D-branes are wrapped.

Boundary conditions

1 = D12 = A1 = F12 = ¯ ✏ = ✏¯ = · · · = 0 = ¯ = ¯ ✏ = ✏ ¯ = · · · = 0 D1 = D1 ¯ = ¯ ✏3 = ✏3 ¯ = · · · = 0

Boundary interactions

• graded Chan-Paton space

Z2

StrV  P exp ✓ i I dϕAϕ ◆

V = Ve ⊕ Vo

Inclusion of the Wilson loop at the boundary brane / anti-brane Tachyon profile Q(φ), ¯

Q(¯ φ)

Matrix factorization Q2 = W · 1V, ¯ Q2 = ¯ W · 1V

• dd operators on

A ˆ

ϕ = ρ∗(A ˆ ϕ + iσ2) + ρ∗(m) − i

2{Q, ¯ Q} + . . .

V

→ supersymmetry preserved

Low energy behavior is not changed by (1) boundary D-term deformation (deformation of fibre metric) (2) brane anti-brane annihilation D-brane wrapped on the zero locus of U = {Q, ¯

Q}

IR equivalence of UV descriptions = Quasi-isomorphism in the derived category of the coherent sheaves

Herbst-Hori-Page (0803.2045)

Any B-brane is obtained as (quasi-isomorphic to) the bound state (complex) of space filling branes.

Brane / anti-brane bound state Tachyon condensation

Hemisphere partition functions and their properties

Hemisphere partition functions

Z1-loop(B; σ; m) = Y

α>0

α · σ sin(πα · σ) −π Y

a∈Neu

Y

w∈Ra

Γ(w · σ + ma) × Y

a∈Dir

Y

w∈Ra

−2πieπi(w·σ+ma) Γ(1 − w · σ − ma)

Zhem(B; t; m) = 1 |W(G)| Z

σ∈it

drk(G)σ (2πi)rk(G) ×StrV[e−2πi(σ+m)]et·σZ1-loop(B; σ; m) Boundary interaction Classical action Vector multiplet Chiral multiplets (Neumann) Chiral multiplets (Dirichlet)

D-brane central charge = central charge of the SUSY algebra for non-compact dimensions in Calabi-Yau compactification = central charge of the D-brane Ooguri-Oz-Yin (1996) Comparison with the large volume formula obtained by anomaly inflow argument Minasian-Moore (1997) Aspinwall (hep-th/0403166) up to overall factor, higher derivative corrections and (worldsheet) instanton corrections.

Hemisphere partition function = B-brane central charge

hB|

|1i

Zhem(B, t, m = 0) ' Z

M

ch(E)eB+iω q ˆ A(TM) Re t → ∞

Example: Quintic Calabi-Yau

A hypersurface in determined by a degree 5 polynomial

P4

Zhem[OM(n)] = Z

iR

dσ 2πie−2πinσ(e−5πiσ − e5πiσ)etσΓ(σ)5Γ(1 − 5σ) = −20 3 π4 ✓ t 2πi − n ◆ 2 ✓ t 2πi − n ◆2 + 5 ! − 400πiζ(3) + O(e−t)

Z

M

ch(OM(n))eB+iω q ˆ A(TM) = − 5 12 ✓ t 2πi − n ◆ 2 ✓ t 2πi − n ◆2 + 5 ! + O(ee−t)

higher derivative corrections instanton corrections Identification of Kähler parameter in large volume limit B + iω = it 2π + O(e−t)

Hilbert space interpretation

χab = hBa|Bbi

f(σ1 − iσ2) g(−σ1 − iσ2)

hg|Bai hBb|fi

f(σ1 − iσ2) g(−σ1 − iσ2)

hg|fi = hg|Bai χab hBb|fi Comparison with the large volume formula → Fixing overall factor of the hemisphere partition function Cylinder partition function = index → No ambiguity We can fix the ambiguity of the sphere partition function!

Seiberg-like dualities

Gr(N, NF) ' Gr(NF N, NF)

Note that X

f

mf = 0

appropriate “dual” brane wrapped on the same submanifold Duality map:

Zhem[Gr(N, NF); B; t; m] = Zhem[Gr(NF − N, NF); B∨; t; −m]

(N, NF, t, m) → (NF − N, NF, t, −m) T ∗Gr(N, NF) ' T ∗Gr(NF N, NF)

NF fundamental / anti-fundamental matters, 1 adjoint matter

Nontrivial duality relation

Kapustin-Willett-Yaakov (1012.4021) Kim-Kim-Kim-Lee (1204.3895) Ito-Maruyoshi-Okuda (2013) cf:

gauge group U(N)

fundamentals

NF

Conclusion

• Constructed 2d N=(2,2) GLSMs on

hemispheres with general B-type boundary conditions and boundary interactions.

• Determined properties of hemisphere

partition functions.

• D-brane central charge
• Hilbert space interpretation
• Seiberg-like Dualities

Stong tests

• f our results!

Comments on domain walls

• Domain walls are boundaries in folded theories.
• Line operators on surface operators and AGT

correspondence (open Verlinde operators)

• Affine Hecke algebra from monodromy domain

wall algebra (Integrability suggests the presence of quantum group symmetry.)

• Relations between domain walls and geometric

representation theory