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 Boundaries GLSM Low energy NLSM or LG model D-branes Target space = vacuum manifold of GLSM

What we did • Construction of N=(2,2) GLSMs on 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 on hemispheres • Hemisphere partition functions and their properties

N=(2,2) GLSMs on hemispheres

Bulk data of 2d N=(2,2) GLSM Gauge group G ( A µ , σ 1 , σ 2 , λ , ¯ Vector multiplet λ , D) a-th chiral multiplet in irreducible rep. ( φ a , ψ a , F a ) R a Complexified FI parameter t = r − i θ FI parameter / theta angle (Defined for each abelian factor) Superpotential : holomorphic function of φ = ( φ a ) W Complexified twisted masses m = ( m a ) R-charge and real twisted masses for flavor symmetry

Supersymmetry We choose the supercharge constructed by Gomis-Lee (2013). Deformation of the sphere does not change the partition functions. semi-infinite cylinder with a cap at infinity R-symmetry flux B-type supersymmetry A-type twist Periodic around circle → Ramond-Ramond sector Long propagation through cylinder → Zero energy state boundary state state without any insertion h B| 1 i Setting of Cecotti-Vafa (1991)

Boundary conditions For vector multiplet: Gauge symmetry preserving condition ✏� = ✏ ¯ � 1 = D 1 � 2 = A 1 = F 12 = ¯ � = · · · = 0 (Gauge symmetry broken condition? ) For chiral multiplets: Neumann condition D 1 � = D 1 ¯ ✏� 3 = ✏� 3 ¯ � = ¯ = · · · = 0 Dirichlet condition � = ¯ ✏ = ✏ ¯ � = ¯ = · · · = 0 These conditions determine the submanifolds on which D-branes are wrapped.

Boundary interactions V = V e ⊕ V o -graded Chan-Paton space Z 2 brane / anti-brane Inclusion of the Wilson loop at the boundary  ✓ ◆� I d ϕ A ϕ Str V P exp i ϕ + i σ 2 ) + ρ ∗ ( m ) − i 2 {Q , ¯ A ˆ ϕ = ρ ∗ ( A ˆ Q} + . . . Q (¯ Tachyon profile Q ( φ ) , ¯ odd operators on φ ) V Matrix factorization Q 2 = W · 1 V , ¯ Q 2 = ¯ W · 1 V → supersymmetry preserved

Low energy behavior is not changed by (1) boundary D-term deformation (deformation of fibre metric) (2) brane anti-brane annihilation Brane / anti-brane bound state Tachyon condensation 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.

Hemisphere partition functions and their properties

Hemisphere partition functions d rk( G ) σ 1 Z Z hem ( B ; t ; m ) = (2 π i ) rk( G ) | W ( G ) | σ ∈ i t × Str V [ e − 2 π i ( σ + m ) ] e t · σ Z 1-loop ( B ; σ ; m ) Boundary interaction Classical action Chiral multiplets (Neumann) Vector multiplet α · σ sin( πα · σ ) Y Y Y Z 1-loop ( B ; σ ; m ) = Γ ( w · σ + m a ) − π α > 0 a ∈ Neu w ∈ R a − 2 π ie π i ( w · σ + m a ) Chiral multiplets Y Y × (Dirichlet) Γ (1 − w · σ − m a ) a ∈ Dir w ∈ R a

Hemisphere partition function = B-brane central charge 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) | 1 i h B| Comparison with the large volume formula obtained by anomaly inflow argument Minasian-Moore (1997) Aspinwall (hep-th/0403166) Z q ˆ ch( E ) e B + i ω A ( TM ) Re t → ∞ Z hem ( B , t, m = 0) ' M up to overall factor, higher derivative corrections and (worldsheet) instanton corrections.

Example: Quintic Calabi-Yau A hypersurface in determined by a degree 5 polynomial P 4 Z d σ 2 π ie − 2 π in σ ( e − 5 π i σ − e 5 π i σ ) e t σ Γ ( σ ) 5 Γ (1 − 5 σ ) Z hem [ O M ( n )] = i R ✓ t ✓ t ◆ ! ◆ 2 = − 20 3 π 4 − 400 π i ζ (3) + O ( e − t ) 2 + 5 2 π i − n 2 π i − n higher derivative corrections instanton corrections ✓ t ✓ t ◆ ! ◆ 2 A ( TM ) = − 5 Z q + O ( e e − t ) ˆ ch( O M ( n )) e B + i ω 2 + 5 2 π i − n 2 π i − n 12 M Identification of Kähler parameter B + i ω = it 2 π + O ( e − t ) in large volume limit

Hilbert space interpretation g ( − σ 1 − i σ 2 ) f ( σ 1 − i σ 2 ) h g |B a i χ ab = h B a |B b i h B b | f i h g | f i = h g |B a i χ ab h B b | f i g ( − σ 1 − i σ 2 ) f ( σ 1 − i σ 2 ) 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 fundamentals gauge group U ( N ) N F Duality map: ( N, N F , t, m ) → ( N F − N, N F , t, − m ) Gr( N, N F ) ' Gr( N F � N, N F ) Z hem [Gr( N, N F ); B ; t ; m ] = Z hem [Gr( N F − N, N F ); B ∨ ; t ; − m ] appropriate “dual” brane X Note that m f = 0 wrapped on the same submanifold f N F fundamental / anti-fundamental matters, 1 adjoint matter T ∗ Gr( N, N F ) ' T ∗ Gr( N F � N, N F ) cf: Kapustin-Willett-Yaakov (1012.4021) Nontrivial duality relation Kim-Kim-Kim-Lee (1204.3895) Ito-Maruyoshi-Okuda (2013)

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 Stong tests • Hilbert space interpretation of our results! • Seiberg-like Dualities

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