Symmetries in supersymmetric gauge theory on the graph Kazutoshi - PowerPoint PPT Presentation

Symmetries in supersymmetric gauge theory on the graph Kazutoshi Ohta (Meiji Gakuin University) Based on N. Sakai and KO, PTEP 2019 043B01, and work in progress with S. Kamata, S. Matsuura and T. Misumi “Discrete Approaches to the Dynamics of Fields and Space-Time 2019”, Shimane, 9/10/2019

Introduction 2d supersymmetric (topological) gauge theory can be well formulated on generic graphs (discretized Riemann surface or polyhedra) ⇒ a generalization of the supersymmetric lattice gauge theory (the so- called Sugino model) Simplicial complexes (graph) with the same Euler characteristics � χ Γ = 2 � χ h = 2 S 2

Introduction We would like to consider properties (symmetries) of the discretized gauge theory on the 2d graph Question: How much can we discuss symmetries on the graph in parallel with the continuous field theory? ✤ Supersymmetries ✤ Global symmetries ✤ Index theorem, heat kernel, zero modes ✤ BRST symmetries, etc.

SUSY on curved Riemann surface 4d N =1 (4 supercharges) · Riemann surface α � A M , D ; ψ α , ¯ ψ with genus h dimensional reduction on � Σ h × T 2 · � A μ , Φ = A 3 + iA 4 , ¯ α Φ = A 3 − iA 4 , D ; ψ α , ¯ ψ turn on a background R -gauge field ∇ R μ ξ ≡ ∇ μ ξ + i 𝒝 R μ ξ = 0 Preserves 2 supercharges at least Killing eq. μ ¯ ξ ≡ ∇ μ ¯ μ ¯ ∇ R ξ − i 𝒝 R ξ = 0

SUSY on curved Riemann surface original helicity R -charge redefined fields fields Φ , ¯ Φ , ¯ 0 0 0-form Φ Φ A = A μ dx μ A μ ±1 0 1-form volume form Y ≡ D ω − F 0 0 2-form D field strength λ = λ μ dx μ ±1/2 ±1/2 1-form ψ 1 , ¯ ψ · 1 0-form η ψ 2 , ¯ ψ · ∓ 1/2 ±1/2 χ = 1 2 2 χ μν dx μ ∧ dx ν 2-form as the same as the topological twist


 Isometries and supercharges ✤ 4 supercharges are decomposed into: 
 Q μν ( ˜ � on generic curved Riemann surface 
 Q , Q μ , Q ) 0-form 1-form 2-form ✤ 2 supercharges are nilpotent up to gauge transformation: 
 Q 2 = ˜ Q 2 = δ g � ✤ If there exist isometries, associated supercharges are preserved: 
 Lie derivative Q 2 � I = δ g + ℒ I e.g. (squashed) sphere ⇒ 1 isometry ⇒ 3 supercharges 
 torus ⇒ 2 isometries ⇒ 4 supercharges (2d N =(2,2) SUSY)

� � � � SUSY transformation ✤ We consider Abelian gauge theory only in this talk ✤ We can define SUSY transformations for one of the supercharges Q Q ϕ = 0, Q ¯ ϕ = 2 η , Q η = 0 Q λ = − d ϕ QA = λ , QY = 0, Q χ = Y Note that � Q 2 = δ ϕ ✤ The action can be written in the Q -exact form � S = − 1 2 g 2 Q ∫ [ d ¯ ϕ ∧ * λ + χ ∧ * ( Y − 2 F ) ]

� � � 
 
 SUSY action ✤ Bosonic part of the SUSY action: 
 2 g 2 ∫ [ d ¯ 1 ϕ ∧ * d ϕ − Y ∧ * ( Y − 2 F ) ] S b = 2 g 2 ∫ [ d ¯ 1 ϕ ∧ * d ϕ + F ∧ * F ] ⇒ � ✤ Fermionic part of the SUSY action: 
 2 g 2 ∫ Ψ T ∧ * i / 1 1 S f = D Ψ ≡ 2 g 2 ( Ψ , i / D Ψ ) where 
 Ψ = ( η 0 − d † χ ) , 0 adjoint exterior derivative d † ≡ − * d * i / D = , λ d † d 0 (co-differential) − d 0 0

� � � � Another supercharge ✤ If we exchange a role between 0-forms ( � ) and 1-forms ( � ), we can find η χ ˜ another SUSY transformation � Q ˜ Q ( ϕω ) = 0, Q ( ¯ ˜ ˜ ϕω ) = 2 χ , Q χ = 0 ˜ Q λ = − d † ( ϕω ) QA = * λ , ˜ ˜ QY = 0, Q η = − * Y Q 2 = δ ϕ Again � ˜ ✤ The same � -exact action also can be written in the � -exact form ˜ Q Q 1 Q ∫ [ d ¯ 2 g 2 ˜ ϕ ∧ λ + η ( Y − 2 F ) ] � S =

� 
 � � � vs � ˜ Q Q ✤ The action is invariant under � and � (both � and � exact) since the ˜ ˜ Q Q Q Q action can be written simply by 
 1 Q ] ∫ [ ¯ 4 g 2 [ Q , ˜ ϕ F + ηχ ] S = and 
 { Q , ˜ Q } = 0 ˜ ✤ Thus 2 supercharges � and � are preserved on the Riemann surface Q Q Σ h

� 
 � current U (1) A ✤ The action is invariant under the � rotation 
 U (1) A ϕ → e − 2 i θ A ¯ ¯ ϕ → e 2 i θ A ϕ , η → e − i θ A η , λ → e i θ A λ , χ → e − i θ A χ ϕ , ✤ Associated � current is given by 
 U (1) A J A = ( ϕ d ¯ ϕ − d ϕ ¯ ϕ + ηλ + * χ * λ ) / g 2 � ✤ � current has an anomaly 
 U (1) A d † J A = 1 scalar curvature on � Σ h � 4 π ℛ ∫ d † J A ω = 2 − 2 h = χ h In particular, � Euler characteristic of � Σ h

� 
 
 � 
 
 current U (1) V ✤ We call another global symmetry � U (1) V � δ V Ψ = θ V γ V Ψ where 
 γ V = ( η ↔ * χ λ ↔ * λ Q ↔ ˜ , � , � , etc. Q − * 0 0 0 ) � − * 0 0 ω 0 ✤ Associated � current is given by 
 U (1) V J V = (* χλ − η * λ )/ g 2 � ✤ � current associates with supercurrents � and � U (1) V J Q J ˜ Q ˜ � QJ V = − J Q QJ V = J ˜ Q , d † J V = 0 ⇒ d † J Q = d † J ˜ So we find that � Q = 0

Graph � v 1 � e 1 � e 2 ✤ A (connected and directed) graph � Γ � v 2 consists of vertices � and edges � V E � v 5 � f 1 � e 3 � e 4 ✤ We also consider faces � , which are F � e 8 � e 5 � v 4 � f 2 � f 3 surrounded by closed edges � e 6 � e 7 ✤ A dual graph � � f 5 � v 6 is defined by Γ * � f 4 � e 10 � v 3 exchanging � and � (also � and � ) 
 E * V F E � e 9 � e 11 � v 7 graph � Γ

Graph � ¯ f 1 ✤ A (connected and directed) graph � Γ � ¯ e 1 � ¯ v 1 consists of vertices � and edges � � ¯ e 2 V E � ¯ � ¯ f 5 f 2 e 4 � ¯ e 3 � ¯ ✤ We also consider faces � , which are F e 5 � ¯ � ¯ f 4 � ¯ e 8 v 3 � ¯ surrounded by closed edges v 2 � ¯ � ¯ e 7 e 6 � ¯ � ¯ e 10 ✤ A dual graph � � ¯ is defined by Γ * f 6 � ¯ e 11 � ¯ f 3 v 4 � ¯ v 5 � ¯ exchanging � and � (also � and � ) 
 E * V F E � ¯ e 9 � ¯ f 7 dual graph � Γ *

Differential forms and graph ✤ There is a good correspondence between the differential forms (fields) on the Riemann surface � and the objects on the graph � Σ h Γ Differential Fields Variables Graph objects forms ϕ v , ¯ ϕ , ¯ ϕ v 0-form Vertex ϕ U e ≡ e iA e Bosons 1-form Edge A Y f 2-form Face Y η η v 0-form Vertex Λ e ≡ e i λ e Fermions 1-form Edge λ χ χ f 2-form Face

� 
 
 
 
 
 
 
 � � � � 
 � � � Differential forms and graph ✤ We can define the SUSY on the graph as well as the cont. field theory 
 Q ϕ v = 0, Q ϕ = 0, ϕ v = 2 η v , Q η v = 0 Q ¯ Q ¯ ϕ = 2 η , Q η = 0 QA e = i λ e , Q λ e = − L ev ϕ v Q λ = − d ϕ QA = λ , QY f = 0, Q χ f = Y f QY = 0, Q χ = Y L ev where � is an incidence matrix on the graph � e 1 � e 4 ✤ The action also can be written in the � -exact form: 
 Q Q [ ¯ S = − 1 e λ e + χ f ( Y f − 2 Ω f ) ] ϕ v ( L T ) v � � f � e 2 2 g 2 0 Ω f � e 3 where � is a function of the plaquette (face) variable, which goes to 
 Ω f ≡ − i 2 ( U f − U f † ) → F � � U f = U 1 U 2 U 3 U − 1 in the continuum limit 4

Incidence matrix ✤ Incidence matrix L : V ( Γ ) → E ( Γ ) ( n e × n v matrix) if s ( e ) = v s ( e ) = v 1 t ( e ) = v 2 +1 L ev = e if t ( e ) = v − 1 v 1 v 2 others 0 v 1 + - e.g. v 1 v 2 v 3 e 1 e 3   +1 − 1 0 e 1 - + L ( Γ ) = 0 +1 − 1   e 2   v 3 v 2 + - e 2 − 1 0 +1 e 3 Known as charge matrix (toric data) for the bi-fundamental matters in quiver gauge theory


 
 SUSY action on the graph ✤ Bosonic part of the SUSY action: 
 1 0 [ ¯ ϕ v L Tve L ev ′ � ϕ v ′ � − Y f ( Y f − 2 Ω f ) ] S b = � 2 g 2 1 0 [ ¯ v ′ � ϕ v ′ � + Ω f Ω f ] ϕ v ( Δ V ) v Δ V ≡ L T L ⇒ � where � is the graph Laplacian 2 g 2 ✤ Fermionic part of the SUSY action: 
 1 Ψ T i / � D Ψ S f = incidence matrix on 2 g 2 0 the dual graph � Γ * where 
 η v − L T 0 0 e ≡ δ Ω f δ A e ∝ (ˇ λ e ( D T ) f L T ) f � Ψ = , i / D = , L 0 D e χ f − D T 0 0


 
 
 
 
 Properties of the “Dirac operator” ✤ We can see the correspondence between the (co)differentials and incidence matrices 
 0 − d † − L T 0 0 0 D ( Σ h ) = D ( Γ ) = � i / � i / d † L 0 D 0 d − D T − d 0 0 0 0 ✤ � is a square root of the graph Laplacians 
 D / Δ V 0 0 L T L 0 0 D 2 = LL T + DD T � ≡ 0 Δ E 0 / 0 0 � e e � ¯ D T D Δ F 0 0 0 0 where we have used the orthogonality between � and � : 
 L D d 2 = d †2 = 0 L T D = D T L = 0 � (corresponds to � )

� � 
 
 
 
 
 
 
 � � � � � 
 � SUSY on the dual graph ✤ We can dualize the SUSY on � to one on the dual graph � Γ Γ * Q ϕ v = 0, Q ϕ f = 0, ˜ ϕ v = 2 η v , Q η v = 0 ϕ f = 2 χ f , Q χ f = 0 Q ¯ Q ¯ ˜ ˜ QA e = λ e , Q λ e = − L ev ϕ v QA e = λ e , Q λ e = − ˇ ˜ ˜ L ef ϕ f QY f = 0, Q χ f = Y f QY v = 0, Q η v = Y v ˜ ˜ e = e ¯ where we have used the relation � , � , � v = f ¯ ¯ f = v ˜ ✤ The dual action is defined as the � -exact form: 
 Q Q [ λ ¯ S = − 1 ϕ f + η v ( Y v − 2 Ω v ) ] ef ¯ ˜ ˜ e ˇ L ¯ � ˜ Q , Q I , Q 2 g 2 0 Ω v ≡ M vf Ω f where � , etc., but we find 
 S ≠ ˜ S { Q , ˜ Q } ≠ 0 ˜ on the graph � , � , � QS ≠ 0