T-DUALITY FOR MASSIVE STATES IN STRING THEORY Jnan Maharana March - PowerPoint PPT Presentation

• The antisymmetric tensor gets decomposed as � b µν ( X ) � B µα ( X ) B MN (ˆ ˆ X ) = B νβ ( X ) B αβ ( X ) • Note that there are d abelian gauge fields B µα ( X ) appearing due to compactification of ˆ B -field. • The worldsheet action for closed bosonic string in ˆ D -dimensions takes the following form in the presence of backgrounds ˆ G (ˆ X ) and ˆ B (ˆ X ) (depending in string coordinates) � � � S = 1 ν + ǫ ab ˆ ˆ ˆ ν ∂ a ˆ µ ∂ b ˆ ν ∂ a ˆ µ ∂ b ˆ X ˆ X ˆ X ˆ X ˆ ν d σ d τ G ˆ B ˆ µ ˆ µ ˆ 2 • REMARKS: 10 / 44

• The antisymmetric tensor gets decomposed as � b µν ( X ) � B µα ( X ) B MN (ˆ ˆ X ) = B νβ ( X ) B αβ ( X ) • Note that there are d abelian gauge fields B µα ( X ) appearing due to compactification of ˆ B -field. • The worldsheet action for closed bosonic string in ˆ D -dimensions takes the following form in the presence of backgrounds ˆ G (ˆ X ) and ˆ B (ˆ X ) (depending in string coordinates) � � � S = 1 ν + ǫ ab ˆ ˆ ˆ ν ∂ a ˆ µ ∂ b ˆ ν ∂ a ˆ µ ∂ b ˆ X ˆ X ˆ X ˆ X ˆ ν d σ d τ G ˆ B ˆ µ ˆ µ ˆ 2 • REMARKS: • When we compactify the on T d and backgrounds ˆ G and ˆ B are independent of compact coordinates Y α then the action can be expressed in terms of string coordinates X µ and Y α and these couple to backgrounds g , G , A α µ , b , B and B µα which depend on X µ ( σ, τ ) only. 10 / 44

• In this scenario, we introduce a set of dual coordinates ˜ Y corresponding to each compact coordinate Y and dual backgrounds B and define a dual action ˆ G and ˜ ˜ ˜ S . The two sets of equations of S and ˆ motion derived from ˆ ˜ S can be suitably combined to express O ( d , d ) covariant equations of motion (see JM and JHS for details). 11 / 44

• In this scenario, we introduce a set of dual coordinates ˜ Y corresponding to each compact coordinate Y and dual backgrounds B and define a dual action ˆ G and ˜ ˜ ˜ S . The two sets of equations of S and ˆ motion derived from ˆ ˜ S can be suitably combined to express O ( d , d ) covariant equations of motion (see JM and JHS for details). • We may ask whether it is possible to construct vertex operators for massive excited states compactified on T d in an O ( d , d ) invariant manner? 11 / 44

• In this scenario, we introduce a set of dual coordinates ˜ Y corresponding to each compact coordinate Y and dual backgrounds B and define a dual action ˆ G and ˜ ˜ ˜ S . The two sets of equations of S and ˆ motion derived from ˆ ˜ S can be suitably combined to express O ( d , d ) covariant equations of motion (see JM and JHS for details). • We may ask whether it is possible to construct vertex operators for massive excited states compactified on T d in an O ( d , d ) invariant manner? • Is is possible to generalize the results to superstring? 11 / 44

MASSIVE EXCITED STATES • Excited massive stringy states are interesting. • At very high energy (Planckian energy) scatterings stringy states are important (Gross, Mende, Amati, Ciafoloni, Veneziano..). It is conjectured that there might be symmetry enhancements in the limit α ′ → ∞ . • There are evidences for existence of gauge symmetry associated with massive states: Evans, Ovrut, Kubota, Veneziano... 12 / 44

MASSIVE EXCITED STATES • Excited massive stringy states are interesting. • At very high energy (Planckian energy) scatterings stringy states are important (Gross, Mende, Amati, Ciafoloni, Veneziano..). It is conjectured that there might be symmetry enhancements in the limit α ′ → ∞ . • There are evidences for existence of gauge symmetry associated with massive states: Evans, Ovrut, Kubota, Veneziano... • For closed strings in background of massless states when one computes β -function to higher loops and sums up it is necessary to include effects of massive excited states. (Das, Sathiapalan, Itoi, Watabiki). 12 / 44

MASSIVE EXCITED STATES • Excited massive stringy states are interesting. • At very high energy (Planckian energy) scatterings stringy states are important (Gross, Mende, Amati, Ciafoloni, Veneziano..). It is conjectured that there might be symmetry enhancements in the limit α ′ → ∞ . • There are evidences for existence of gauge symmetry associated with massive states: Evans, Ovrut, Kubota, Veneziano... • For closed strings in background of massless states when one computes β -function to higher loops and sums up it is necessary to include effects of massive excited states. (Das, Sathiapalan, Itoi, Watabiki). • Interactions in higher spin massless field theory gets insight from higher spin stringy state vertex operators in appropriate limit. Sagnotti, Tarona... 12 / 44

• T-duality Properties of Excited Massive States • We consider a simple scenario. Let us envisage evolution of the string in constant background G (0) , independent of coordinates. Under the interchange P ↔ X ′ the Hamiltonian remains invariant if G (0) ↔ G ( o ) − 1 . The vertex operator can be cast in a T-duality invariant form in this case. • Recall that this G (0) -tensor can carry coordinate dependence. • Let us recall some of the essential properties of the vertex operators. We work in the weak field approximation through out. • where T ++ = 1 ν ) , T −− = 1 µ ¯ G (0) G (0) 2(ˆ ν ∂ X ˆ µ ∂ X ˆ 2(ˆ ν ¯ ∂ X ˆ ∂ X ˆ ν ) µ ˆ ˆ µ ˆ ˆ G (0) • Here ˆ ν = (1 , − 1 , − 1 ... ). The stress energy momentum tensors are µ ˆ ˆ defined with flat target space metric. µ = ˙ µ + X ′ ˆ µ = ˙ µ − X ′ ˆ µ , ¯ • Moreover, ∂ X ˆ X ˆ ∂ X ˆ X ˆ µ 13 / 44

• T-duality Properties of Excited Massive States • We consider a simple scenario. Let us envisage evolution of the string in constant background G (0) , independent of coordinates. Under the interchange P ↔ X ′ the Hamiltonian remains invariant if G (0) ↔ G ( o ) − 1 . The vertex operator can be cast in a T-duality invariant form in this case. • Recall that this G (0) -tensor can carry coordinate dependence. • Let us recall some of the essential properties of the vertex operators. We work in the weak field approximation through out. • The vertex operators ˆ Φ n ( X ˆ µ ) , n , referring to the level of excited state are required to be (1 , 0) and (0 , 1) primaries with respect to T ++ and T −− respectively. • where T ++ = 1 ν ) , T −− = 1 µ ¯ G (0) G (0) 2(ˆ ν ∂ X ˆ µ ∂ X ˆ 2(ˆ ν ¯ ∂ X ˆ ∂ X ˆ ν ) µ ˆ ˆ µ ˆ ˆ G (0) • Here ˆ ν = (1 , − 1 , − 1 ... ). The stress energy momentum tensors are µ ˆ ˆ defined with flat target space metric. µ = ˙ µ + X ′ ˆ µ = ˙ µ − X ′ ˆ µ , ¯ • Moreover, ∂ X ˆ X ˆ ∂ X ˆ X ˆ µ 13 / 44

• These vertex operators satisfy ’Equations of Motion’ and certain ’Transversality (gauge) Conditions’ due to above requirements. • Example: Consider the graviton background in the weak field approximation: µ ) = G (0) ν ( X ˆ ν ( X ˆ µ ). The corresponding constrains are: • G ˆ ν + h ˆ µ ˆ µ ˆ µ ˆ ˆ ∇ 2 h ˆ ν = 0 , and ∂ ˆ µ h ˆ ν = 0 µ ˆ µ ˆ 14 / 44

• These vertex operators satisfy ’Equations of Motion’ and certain ’Transversality (gauge) Conditions’ due to above requirements. • Example: Consider the graviton background in the weak field approximation: µ ) = G (0) ν ( X ˆ ν ( X ˆ µ ). The corresponding constrains are: • G ˆ ν + h ˆ µ ˆ µ ˆ µ ˆ ˆ ∇ 2 h ˆ ν = 0 , and ∂ ˆ µ h ˆ ν = 0 µ ˆ µ ˆ • First Excited Massive Level: The vertex operator is V ( 1 ) V ( 2 ) V ( 3 ) V ( 4 ) Φ 1 = ˆ ˆ + ˆ + ˆ + ˆ 1 1 1 1 V ( i ) • We define ˆ 1 , i = 1 , 2 , 3 , 4 as vertex functions. Thus a vertex operators is sum of vertex functions as above. 14 / 44

• The four vertex functions are µ ′ ¯ ν ¯ V (1) = A (1) ˆ ν ′ ( X ) ∂ X ˆ µ ∂ X ˆ ∂ X ˆ ∂ X ˆ ν ′ 1 ˆ µ ˆ ν, ˆ µ ′ ˆ ν ¯ V (2) = A (2) ˆ µ ′ ( X ) ∂ X ˆ µ ∂ X ˆ ∂ 2 X ˆ µ ′ 1 ˆ µ ˆ ν, ˆ µ ′ ¯ µ ¯ V (3) = A (3) ˆ ν ′ ( X ) ∂ 2 X ˆ ∂ X ˆ ∂ X ˆ ν ′ 1 ˆ µ, ˆ µ ′ ˆ V (4) = A (4) µ ¯ ˆ µ ′ ( X ) ∂ 2 X ˆ ∂ 2 X ˆ µ ′ 1 µ, ˆ ˆ 15 / 44

• The four vertex functions are µ ′ ¯ ν ¯ V (1) = A (1) ˆ ν ′ ( X ) ∂ X ˆ µ ∂ X ˆ ∂ X ˆ ∂ X ˆ ν ′ 1 µ ˆ ˆ ν, ˆ µ ′ ˆ ν ¯ V (2) = A (2) ˆ µ ′ ( X ) ∂ X ˆ µ ∂ X ˆ ∂ 2 X ˆ µ ′ 1 ˆ µ ˆ ν, ˆ µ ′ ¯ µ ¯ V (3) = A (3) ˆ ν ′ ( X ) ∂ 2 X ˆ ∂ X ˆ ∂ X ˆ ν ′ 1 ˆ µ, ˆ µ ′ ˆ V (4) = A (4) µ ¯ ˆ µ ′ ( X ) ∂ 2 X ˆ ∂ 2 X ˆ µ ′ 1 ˆ µ, ˆ • Unprimed and primed indices correspond to right moving and left µ and ∂ X ˆ µ ′ ). When we demand ˆ moving sectors respectively ( ∂ X ˆ Φ 1 to be (1 , 1) with respect to T ±± , then ˆ V ( i ) 1 ( i . e . A ( i ) ) are constrained - they are not all independent. V (1) (i) Only ˆ is (1 , 1) on its own; however the other three vertex 1 V (2) V (4) V (1) functions: ˆ − ˆ are related to ˆ . 1 1 1 15 / 44

• The (1 , 1) constraint on ˆ Φ 1 implies (a) Each A ( i ) satisfies a mass shell condition. ∇ 2 − 2) A (1) ∇ 2 − 2) A (2) ( ˆ ( ˆ ν ′ ( X ) = 0 , µ ′ ( X ) = 0 µ ˆ ˆ ν, ˆ µ ′ ˆ µ ˆ ˆ ν, ˆ ∇ 2 − 2) A (3) ∇ 2 − 2) A (4) ( ˆ ( ˆ ν ′ ( X ) = 0 , µ ′ ( X ) = 0 µ, ˆ ˆ µ ′ ˆ µ, ˆ ˆ ∇ 2 is ˆ where ˆ D -dimensional Laplacian defined in flat space. (b) Following relations show how A ( i ) are related: Transversality (gauge) conditions are A (2) ν ′ A (1) ν ′ , A (3) ν A (1) ν ′ , A (4) ν A (1) µ ′ = ∂ ˆ ν ′ = ∂ ˆ µ ′ = ∂ ˆ ν ′ ∂ ˆ µ ˆ ˆ ν, ˆ µ ˆ ˆ ν, ˆ µ ′ ˆ µ, ˆ ˆ µ ′ ˆ µ ˆ ˆ ν, ˆ µ ′ ˆ µ, ˆ ˆ µ ˆ ˆ ν, ˆ µ ′ ˆ ν ′ The other set i.e. Transversality (gauge) conditions are A (1)ˆ µ ν A (1) ν ′ +2 ∂ ˆ µ ∂ ˆ , ˆ ν ′ = 0 µ ′ ˆ ˆ µ ˆ ˆ ν, ˆ µ ′ ˆ µ and µ ′ ˆ A (1) ν ′ A (1) + 2 ∂ ˆ µ ′ ∂ ˆ ν ′ = 0 ˆ µ ˆ ν, ˆ µ ′ µ ˆ ˆ ν, ˆ µ ′ ˆ 16 / 44

• Simple Example: Graviton Vertex Consider the Graviton vertex in ˆ D -dimension and compactify on T d . Look at the part involving compact coordinates Y α . In weak field approximation: G αβ ( X ) = δ αβ + h αβ ( X ) , α, β = 1 , 2 .. d V h = h αβ ′ ∂ Y α ¯ ∂ Y β ′ Y α to express the vertex operator as use P α = δ αβ ˙ V h = h αβ ′ P α P β − h αβ ′ Y ′ α Y ′ β ′ − h α β ′ P α Y ′ β ′ + h α ′ β P α ′ Y ′ β We have retained the memory where P and Y ′ came from through appearance of unprimed and primed indices. 17 / 44

• Simple Example: Graviton Vertex Consider the Graviton vertex in ˆ D -dimension and compactify on T d . Look at the part involving compact coordinates Y α . In weak field approximation: G αβ ( X ) = δ αβ + h αβ ( X ) , α, β = 1 , 2 .. d V h = h αβ ′ ∂ Y α ¯ ∂ Y β ′ Y α to express the vertex operator as use P α = δ αβ ˙ V h = h αβ ′ P α P β − h αβ ′ Y ′ α Y ′ β ′ − h α β ′ P α Y ′ β ′ + h α ′ β P α ′ Y ′ β We have retained the memory where P and Y ′ came from through appearance of unprimed and primed indices. • The vertex operator can be expressed as V h = H mn W m W n − K n m W m W n Note that W is the O ( d , d ) vector defined earlier. V h be O ( d , d ) invariant if n H m ′ n ′ , W m → Ω m H mn → Ω m ′ n Ω n ′ m ′ W m ′ , K n m → Ω m ′ m Ω n n ′ K n ′ m ′ 17 / 44

• The vertex operator along noncompact directions (setting A α µ = 0 for simplicity) is V h = h µν ′ ( X ) ∂ X µ ¯ ∂ X ν ′ in the weak field approximation. • The spacetime coordinates and the tensors are inert under O ( d , d ) transformations. Therefore, the full vertex operator is T-duality invariant. • For excited massive levels, we follow a similar approach. However, we shall encounter complications since number of vertex functions keep increasing as we go to higher and higher excited level. Therefore, we have to adopt a suitable procedure. 18 / 44

• The vertex operator along noncompact directions (setting A α µ = 0 for simplicity) is V h = h µν ′ ( X ) ∂ X µ ¯ ∂ X ν ′ in the weak field approximation. • The spacetime coordinates and the tensors are inert under O ( d , d ) transformations. Therefore, the full vertex operator is T-duality invariant. • For excited massive levels, we follow a similar approach. However, we shall encounter complications since number of vertex functions keep increasing as we go to higher and higher excited level. Therefore, we have to adopt a suitable procedure. • As we noticed, when graviton and antisymmetric tensor backgrounds are considered in lower dimensions, we see appearance of scalars and gauge fields in addition to graviton and antisymmetric tensors, g ( X ) and b ( X ). We consider the case analogous to Hassan-Sen prescription in what follows. 18 / 44

T-DUALITY FOR EXCITED STATES V ( i ) • Let us consider the scenario where tensors ˆ ( i . e . A ( i ) ) are 1 coordinate independent - this is analog of constant G and B . Focus V (1) on ˆ 1 µ ′ ¯ ν ¯ V (1) = A (1) ˆ ν ′ ∂ X ˆ µ ∂ X ˆ ∂ X ˆ ∂ X ˆ ν ′ 1 µ ˆ ˆ ν, ˆ µ ′ ˆ µ = P ˆ µ + X ′ ˆ ∂ X ˆ µ . We can express the above equation in terms of µ and X ′ ˆ P ˆ µ . It has following form in the expanded version. ( I ). G (1) ρ P ˆ λ - Products of momenta only. λ P ˆ µ P ˆ ν P ˆ ρ ˆ µ ˆ ˆ ν, ˆ ( II ). G (2) ρ X ′ ˆ λ - Product of X ′ only. λ X ′ ˆ µ X ′ ˆ ν X ′ ˆ ρ ˆ µ ˆ ˆ ν, ˆ λ - Product of three momenta and one X ′ . ( III ). G (3) ρ X ′ ˆ λ P ˆ µ P ˆ ν P ˆ ρ ˆ µ ˆ ˆ ν, ˆ There are four such terms altogether. ( IV ). G (4) ρ P ˆ λ - Product of three X ′ and one P . There λ X ′ ˆ µ X ′ ˆ ν X ′ ˆ ρ ˆ µ ˆ ˆ ν, ˆ are also four such terms. 19 / 44

• ( V ). G (5) ρ X ′ ˆ λ - There are six terms of this type: Product λ P ˆ µ P ˆ ν X ′ ˆ ρ ˆ µ ˆ ˆ ν, ˆ of pair of P ’s and a pair of X ′ ’s. 20 / 44

• ( V ). G (5) ρ X ′ ˆ λ - There are six terms of this type: Product λ P ˆ µ P ˆ ν X ′ ˆ ρ ˆ µ ˆ ˆ ν, ˆ of pair of P ’s and a pair of X ′ ’s. • There are total 16 terms in ( I ) − ( V ). A careful inspection shows that the vertex function A (1) remains invariant under P ↔ X ′ if following relations hold: G (1) ↔ G (2) , G (3) ↔ G (4) 20 / 44

• ( V ). G (5) ρ X ′ ˆ λ - There are six terms of this type: Product λ P ˆ µ P ˆ ν X ′ ˆ ρ ˆ µ ˆ ˆ ν, ˆ of pair of P ’s and a pair of X ′ ’s. • There are total 16 terms in ( I ) − ( V ). A careful inspection shows that the vertex function A (1) remains invariant under P ↔ X ′ if following relations hold: G (1) ↔ G (2) , G (3) ↔ G (4) • The six terms in ( V ) rearrange themselves appropriately. 20 / 44

• ( V ). G (5) ρ X ′ ˆ λ - There are six terms of this type: Product λ P ˆ µ P ˆ ν X ′ ˆ ρ ˆ µ ˆ ˆ ν, ˆ of pair of P ’s and a pair of X ′ ’s. • There are total 16 terms in ( I ) − ( V ). A careful inspection shows that the vertex function A (1) remains invariant under P ↔ X ′ if following relations hold: G (1) ↔ G (2) , G (3) ↔ G (4) • The six terms in ( V ) rearrange themselves appropriately. • This duality in analog of G ↔ G − 1 under τ ↔ σ - A (1) are constant as was G . 20 / 44

• ( V ). G (5) ρ X ′ ˆ λ - There are six terms of this type: Product λ P ˆ µ P ˆ ν X ′ ˆ ρ ˆ µ ˆ ˆ ν, ˆ of pair of P ’s and a pair of X ′ ’s. • There are total 16 terms in ( I ) − ( V ). A careful inspection shows that the vertex function A (1) remains invariant under P ↔ X ′ if following relations hold: G (1) ↔ G (2) , G (3) ↔ G (4) • The six terms in ( V ) rearrange themselves appropriately. • This duality in analog of G ↔ G − 1 under τ ↔ σ - A (1) are constant as was G . µ ′ depends on ν ¯ • Consider the case where A (1) ν ′ ( X ) ∂ X ˆ µ ∂ X ˆ ∂ X ˆ µ ˆ ˆ ν, ˆ µ ′ ˆ X µ , µ = 0 , 1 , .. D − 1 and is independent of internal coordinates Y α . 20 / 44

• The vertex function will be decomposed into following classes. 21 / 44

• The vertex function will be decomposed into following classes. • (i) A tensor A (1) µν,µ ′ ν ′ which has spacetime Lorentz indices. • (ii) Another: Three Lorentz indices and one internal index. • (iii) A tensor with two Lorentz indices and two internal indices. • (iv) A tensor with one Lorentz index and three internal indices. • (v) A tensor with all internal indices: A (1) αβ,α ′ β ′ which contracts with ∂ Y α ′ ¯ ∂ Y α ∂ Y β ¯ ∂ Y β ′ . If we express this vertex function in terms of P α and Y ′ α etc. we note that there will be 16 terms - for discrete T-duality transformations along internal directions the tensor A (1) αβ,α ′ β ′ behaves like a constant tensor. 21 / 44

• The vertex function will be decomposed into following classes. • (i) A tensor A (1) µν,µ ′ ν ′ which has spacetime Lorentz indices. • (ii) Another: Three Lorentz indices and one internal index. • (iii) A tensor with two Lorentz indices and two internal indices. • (iv) A tensor with one Lorentz index and three internal indices. • (v) A tensor with all internal indices: A (1) αβ,α ′ β ′ which contracts with ∂ Y α ′ ¯ ∂ Y α ∂ Y β ¯ ∂ Y β ′ . If we express this vertex function in terms of P α and Y ′ α etc. we note that there will be 16 terms - for discrete T-duality transformations along internal directions the tensor A (1) αβ,α ′ β ′ behaves like a constant tensor. • our goal in to construct O ( d , d ) invariant vertex operators and we take clue from construction of graviton vertex operator. Recall � P � W = Y ′ 21 / 44

• Let us closely look at the types of terms we have (I) P α P β P α ′ P β ′ (II) Y ′ α Y ′ β Y ′ α ′ Y ′ β ′ . (III) P α P β P α ′ Y ′ β ′ and three more terms. (IV) Y ′ α Y ′ β P α ′ Y ′ β ′ Also three more terms. (V) P α P β Y ′ α ′ Y ′ β ′ . Altogether six terms. • These 16 terms originate from products of ∂ Y α ¯ ∂ Y α ′ ... . There are terms with positive and negative signs in ( III ) , ( IV ) and ( V ). When we inspect ( I ) and ( II ) we note that they can be combined to form product of W -vectors which will contract with a suitable O ( d , d ) tensor. 22 / 44

• Let us closely look at the types of terms we have (I) P α P β P α ′ P β ′ (II) Y ′ α Y ′ β Y ′ α ′ Y ′ β ′ . (III) P α P β P α ′ Y ′ β ′ and three more terms. (IV) Y ′ α Y ′ β P α ′ Y ′ β ′ Also three more terms. (V) P α P β Y ′ α ′ Y ′ β ′ . Altogether six terms. • These 16 terms originate from products of ∂ Y α ¯ ∂ Y α ′ ... . There are terms with positive and negative signs in ( III ) , ( IV ) and ( V ). When we inspect ( I ) and ( II ) we note that they can be combined to form product of W -vectors which will contract with a suitable O ( d , d ) tensor. • The terms appearing in ( III ) and ( IV ) can be appropriately rearranged to obtain products of W vectors. • The six terms in ( V ) have pair of momenta and Y ′ . They nicely arrange to be products of W -vectors. 22 / 44

• Let us closely look at the types of terms we have (I) P α P β P α ′ P β ′ (II) Y ′ α Y ′ β Y ′ α ′ Y ′ β ′ . (III) P α P β P α ′ Y ′ β ′ and three more terms. (IV) Y ′ α Y ′ β P α ′ Y ′ β ′ Also three more terms. (V) P α P β Y ′ α ′ Y ′ β ′ . Altogether six terms. • These 16 terms originate from products of ∂ Y α ¯ ∂ Y α ′ ... . There are terms with positive and negative signs in ( III ) , ( IV ) and ( V ). When we inspect ( I ) and ( II ) we note that they can be combined to form product of W -vectors which will contract with a suitable O ( d , d ) tensor. • The terms appearing in ( III ) and ( IV ) can be appropriately rearranged to obtain products of W vectors. • The six terms in ( V ) have pair of momenta and Y ′ . They nicely arrange to be products of W -vectors. • Therefore, we have products W ’s which we contract with appropriate O ( d , d ) tensors such that resulting vertex functions are O ( d , d ) invariant. 22 / 44

• If we have a doublet � Y ′ � P we can express it as η W which is also an O ( d , d ) vector. Therefore, there are products of W and η W . 23 / 44

• If we have a doublet � Y ′ � P we can express it as η W which is also an O ( d , d ) vector. Therefore, there are products of W and η W . • The other vertex functions (look at only internal directions) αβ,α ′ ( X ) ∂ Y α ∂ Y β ¯ V (2) = A (2) ∂ 2 Y α ′ 1 V (3) = A (3) α,α ′ β ′ ( X ) ∂ 2 Y α ∂ Y α ′ ∂ Y β ′ 1 α,α ′ ( X ) ∂ 2 Y α ¯ V (4) = A (4) ∂ 2 Y α ′ 1 • Explicit calculations show that these can be also cast in duality invariant forms; however it is not so straight forward since higher derivatives of σ and τ appear. Moreover, as we consider higher and higher excited levels more vertex functions appear with higher derivatives. 23 / 44

• We encounter, generically, terms of two different types: (a) Products like: ∂ Y ∂ Y ... ¯ ∂ Y ¯ ∂ Y .. which has only ∂ or ¯ ∂ acting on string coordinates. (b) Higher derivatives and their products, typically of the form: ∂ m Y ∂ Y ... ¯ ∂ n Y ¯ ∂ Y .. The structure of a vertex function, at a given level, is constrained by level matching condition and by requirement that it should be (1 , 1). 24 / 44

• We encounter, generically, terms of two different types: (a) Products like: ∂ Y ∂ Y ... ¯ ∂ Y ¯ ∂ Y .. which has only ∂ or ¯ ∂ acting on string coordinates. (b) Higher derivatives and their products, typically of the form: ∂ m Y ∂ Y ... ¯ ∂ n Y ¯ ∂ Y .. The structure of a vertex function, at a given level, is constrained by level matching condition and by requirement that it should be (1 , 1). • Consider a vertex function (leading trajectory) of second massive level: ∂ Y α ′ ¯ ∂ Y β ′ ¯ αβγ,α ′ β ′ γ ′ ( X ) ∂ Y α ∂ Y β ∂ Y γ ¯ V (1) = C (1) ∂ Y γ ′ 2 • We can express V (1) in terms of P α , Y ′ α as before. Rearrange the 2 terms ( here we 64 terms!) and cast in O ( d , d ) invariant form. • For second level the vertex operator has nine vertex functions and up to ∂ 3 Y and ¯ ∂ 3 Y . Thus present method becomes unmanageable soon. 24 / 44

HIGHER EXCITED LEVELS • Observations: (a) The basic building blocks of any vertex functions are: ∂ Y α = P α + Y ′ α and ¯ ∂ Y α = P α − Y ′ α (b) Each vertex function at every mass level is EITHER string of products of these basic blocks OR these blocks are acted upon by ∂ and ¯ ∂ respectively so that the vertex function has desired dimensions. Note terms like ∂ ¯ ∂ Y do not appear since they vanish by worldsheet equations of motion. (c) It is not convenient to deal with P ± Y ′ for our purpose - we project them out from O ( d , d ) vector W . 25 / 44

HIGHER EXCITED LEVELS • Observations: (a) The basic building blocks of any vertex functions are: ∂ Y α = P α + Y ′ α and ¯ ∂ Y α = P α − Y ′ α (b) Each vertex function at every mass level is EITHER string of products of these basic blocks OR these blocks are acted upon by ∂ and ¯ ∂ respectively so that the vertex function has desired dimensions. Note terms like ∂ ¯ ∂ Y do not appear since they vanish by worldsheet equations of motion. (c) It is not convenient to deal with P ± Y ′ for our purpose - we project them out from O ( d , d ) vector W . • Introduce projection operators � 1 � P ± = 1 0 2( 1 ± ˜ σ 3 ) , σ 3 = ˜ 0 − 1 Here 1 is 2 d × 2 d matrix and the diagonal elements of ˜ σ 3 are d × d unit matrix. 25 / 44

• The canonical momentum P α and Y ′ α are projected out as follows. Y ′ = P − W P = P + W , Consequently, � � � � P + Y ′ = 1 , P − Y ′ = 1 P + W + η P − W P + W − η P − W 2 2 26 / 44

• The canonical momentum P α and Y ′ α are projected out as follows. Y ′ = P − W P = P + W , Consequently, � � � � P + Y ′ = 1 , P − Y ′ = 1 P + W + η P − W P + W − η P − W 2 2 • Note: η flips lower component Y ′ of W to upper component. 26 / 44

• The canonical momentum P α and Y ′ α are projected out as follows. Y ′ = P − W P = P + W , Consequently, � � � � P + Y ′ = 1 , P − Y ′ = 1 P + W + η P − W P + W − η P − W 2 2 • Note: η flips lower component Y ′ of W to upper component. • When we have only products of P + Y ′ and P − Y ′ ,we first express them as products of O(d,d) vectors and contract these vector indices with suitably constructed O ( d , d ) tensors. 26 / 44

• The canonical momentum P α and Y ′ α are projected out as follows. Y ′ = P − W P = P + W , Consequently, � � � � P + Y ′ = 1 , P − Y ′ = 1 P + W + η P − W P + W − η P − W 2 2 • Note: η flips lower component Y ′ of W to upper component. • When we have only products of P + Y ′ and P − Y ′ ,we first express them as products of O(d,d) vectors and contract these vector indices with suitably constructed O ( d , d ) tensors. • Some of the vertex functions have structures where ∂ and ¯ ∂ act on P + Y ′ and P − Y ′ respectively. 26 / 44

• In order to cast these vertex functions in desired form, we define ∆ ± ( τ, σ ) = 1 ∆ τ = P + ∂ τ , ∆ σ = P + ∂ σ , 2(∆ τ ± ∆ σ ) 27 / 44

• In order to cast these vertex functions in desired form, we define ∆ ± ( τ, σ ) = 1 ∆ τ = P + ∂ τ , ∆ σ = P + ∂ σ , 2(∆ τ ± ∆ σ ) • There are two useful relations � � ∂ ( P + Y ′ ) = ∆ + ( τ, σ ) P + W + η P − W � � ¯ ∂ ( P − Y ′ ) = ∆ − ( τ, σ ) P + W − η P − W 27 / 44

• In order to cast these vertex functions in desired form, we define ∆ ± ( τ, σ ) = 1 ∆ τ = P + ∂ τ , ∆ σ = P + ∂ σ , 2(∆ τ ± ∆ σ ) • There are two useful relations � � ∂ ( P + Y ′ ) = ∆ + ( τ, σ ) P + W + η P − W � � ¯ ∂ ( P − Y ′ ) = ∆ − ( τ, σ ) P + W − η P − W • We utilize above two relations to express the basic buliding blocks and their worldsheet derivatives appearing in any vertex function as products of O ( d , d ) vectors 27 / 44

• In order to cast these vertex functions in desired form, we define ∆ ± ( τ, σ ) = 1 ∆ τ = P + ∂ τ , ∆ σ = P + ∂ σ , 2(∆ τ ± ∆ σ ) • There are two useful relations � � ∂ ( P + Y ′ ) = ∆ + ( τ, σ ) P + W + η P − W � � ¯ ∂ ( P − Y ′ ) = ∆ − ( τ, σ ) P + W − η P − W • We utilize above two relations to express the basic buliding blocks and their worldsheet derivatives appearing in any vertex function as products of O ( d , d ) vectors • The product of such vectors are to be contracted with appropriate O ( d , d ) tensors to construct T-duality invariant vertex functions. Recall that when we construct O ( d , d ) invariant vertex operators for G αβ and B αβ we introduce M -matrix and contract W -vector with it. 27 / 44

• With these prescriptions, we shall construct duality invariant vertex functions for excited massive levels. Let us consider n th excited massive level. 28 / 44

• With these prescriptions, we shall construct duality invariant vertex functions for excited massive levels. Let us consider n th excited massive level. • The dimension of all right movers constructed from ∂ Y and powers of ∂ acting on ∂ Y should be n + 1. Same holds for left moving sector. 28 / 44

• With these prescriptions, we shall construct duality invariant vertex functions for excited massive levels. Let us consider n th excited massive level. • The dimension of all right movers constructed from ∂ Y and powers of ∂ acting on ∂ Y should be n + 1. Same holds for left moving sector. ∂ Y α i and left moving • Consider right moving sector of the type Π n +1 1 sector of the same type i.e. Π n +1 ¯ ∂ Y α i 1 28 / 44

• With these prescriptions, we shall construct duality invariant vertex functions for excited massive levels. Let us consider n th excited massive level. • The dimension of all right movers constructed from ∂ Y and powers of ∂ acting on ∂ Y should be n + 1. Same holds for left moving sector. ∂ Y α i and left moving • Consider right moving sector of the type Π n +1 1 sector of the same type i.e. Π n +1 ¯ ∂ Y α i 1 • A vertex function at this level takes the form n +1 ( X ) Π n + 1 ∂ Y α i Π n + 1 ∂ Y α ′ ¯ V α 1 ,α 2 ...α n +1 ,α ′ i 1 α ′ 2 ...α ′ 1 1 28 / 44

• With these prescriptions, we shall construct duality invariant vertex functions for excited massive levels. Let us consider n th excited massive level. • The dimension of all right movers constructed from ∂ Y and powers of ∂ acting on ∂ Y should be n + 1. Same holds for left moving sector. ∂ Y α i and left moving • Consider right moving sector of the type Π n +1 1 sector of the same type i.e. Π n +1 ¯ ∂ Y α i 1 • A vertex function at this level takes the form n +1 ( X ) Π n + 1 ∂ Y α i Π n + 1 ∂ Y α ′ ¯ V α 1 ,α 2 ...α n +1 ,α ′ i 1 α ′ 2 ...α ′ 1 1 ¯ • We know how to covert Π n +1 ∂ Y α ′ i and Π n +1 ∂ Y α ′ i to products 1 1 projected O ( d , d ) vectors W . 28 / 44

• With these prescriptions, we shall construct duality invariant vertex functions for excited massive levels. Let us consider n th excited massive level. • The dimension of all right movers constructed from ∂ Y and powers of ∂ acting on ∂ Y should be n + 1. Same holds for left moving sector. ∂ Y α i and left moving • Consider right moving sector of the type Π n +1 1 sector of the same type i.e. Π n +1 ¯ ∂ Y α i 1 • A vertex function at this level takes the form n +1 ( X ) Π n + 1 ∂ Y α i Π n + 1 ∂ Y α ′ ¯ V α 1 ,α 2 ...α n +1 ,α ′ i 1 α ′ 2 ...α ′ 1 1 ¯ • We know how to covert Π n +1 ∂ Y α ′ i and Π n +1 ∂ Y α ′ i to products 1 1 projected O ( d , d ) vectors W . • A generic vertex function is of the form ∂ p Y α i ∂ q Y α j ∂ r Y α k ... ¯ ∂ p ′ Y α ′ i ¯ ∂ q ′ Y α ′ j ¯ ∂ r ′ Y α ′ k ..., With constraints: p + q + r = n + 1 , p ′ + q ′ + r ′ = n + 1 28 / 44

• APPENDIX • Look at second excited level: Φ 2 = V (1) + V (2) + V (3) + V (4) + V (5) + V (6) + V (7) + V (6) + V (8) 2 2 2 2 2 2 2 2 2 where ∂ Y α ′ ¯ ∂ Y β ′ ¯ αβ,α ′ β ′ γ ′ ∂ 2 Y α ∂ Y β ¯ V (2) = C (2) ∂ Y γ ′ 2 ∂ 2 Y α ′ ¯ αβγ,α ′ β ′ ∂ Y α ∂ Y β ∂ Y γ ¯ V (3) = C (3) ∂ Y β ′ 2 α,α ′ β ′ γ ′ ∂ 3 Y α Y α ′ ¯ ∂ Y β ′ ¯ V (4) = C (4) ∂ Y γ ′ 2 αβγ,α ′ ∂ Y α ∂ Y β ∂ Y γ ¯ V (5) = C (5) ∂ 3 Y α ′ 2 ∂ 2 Y α ′ ¯ αβ,α ′ β ′ ∂ 2 Y α ∂ Y β ¯ V (6) = C (6) ∂ Y β ′ 2 ∂ 2 Y α ′ ¯ ∂ Y β ′ + C (8) α α ′ β ′ ∂ 3 Y α ¯ αβ,α ′ ∂ 2 Y α ∂ Y β ¯ V (7) = C (7) ∂ 3 Y α ′ 2 α,α ′ ∂ 3 Y α ¯ V (8) = C (9) ∂ 3 Y α ′ 2 This illustrates how the number of vertex functions grow with levels. 29 / 44

• REMARKS: • To achieve this objective, we go through following steps: STEP I: Rewrite ∂ p Y = ∂ p − 1 ( P + Y ′ ) , ∂ p ′ ( P − Y ′ ) = ¯ ¯ ∂ p ′ − 1 ( P − Y ′ ) STEP II: Using the projection operators ∂ p − 1 ( P + Y ′ ) = ∆ + p − 1 ( P + Y ′ ) , ∂ p ′ − 1 ( P − Y ′ ) = ∆ − p ′ − 1 ( P − Y ′ ) ¯ 30 / 44

• REMARKS: • The above expression has to be converted into products of W ’s and derivatives to be an O ( d , d ) tensor. The constraints satisfied by p , q , r .. determine the rank of the tensor. This tensor is contracted with a tensor with appropriate O ( d , d ) transformation properties in order to obtain the duality invariant vertex operator. • To achieve this objective, we go through following steps: STEP I: Rewrite ∂ p Y = ∂ p − 1 ( P + Y ′ ) , ∂ p ′ ( P − Y ′ ) = ¯ ¯ ∂ p ′ − 1 ( P − Y ′ ) STEP II: Using the projection operators ∂ p − 1 ( P + Y ′ ) = ∆ + p − 1 ( P + Y ′ ) , ∂ p ′ − 1 ( P − Y ′ ) = ∆ − p ′ − 1 ( P − Y ′ ) ¯ 30 / 44

• STEP III: � � ∆ + p − 1 ( P + Y ′ ) = ∆ + p − 1 P + W + η P − W , � � ∆ p ′ − 1 ( P − Y ′ ) = ∆ − p ′ − 1 P + W − η P − W With these relations a generic vertex function an expressed as product of W vectors its ∆ ± derivatives which are contracted with suitable tensors. + ∆ q − 1 A klm .., k ′ l ′ m ′ .. ( X )∆ + p − 1 W k W l + ∆ + r − 1 W m V n +1 = + .. + ∆ − p ′ − 1 W k ′ − ∆ − p ′ − 1 W l ′ − ∆ − p ′ − 1 W m ′ (1) − where W ± = ( P + W ± η P − W ). p + q + r = n + 1 and p ′ + q ′ + r ′ = n + 1 . Indices { k , l , m ; k ′ , l ′ , m ′ } of W ± correspond to O ( d , d ) components. A klm .., k ′ l ′ m ′ .. ( X ) is O ( d , d ) tensor. 31 / 44

• Vertex function V n + 1 will be O ( d , d ) invariant if the tensor A klm .., k ′ l ′ m ′ .. ( X ) transforms as m ... Ω p ′ k ′ Ω q ′ A klm .., k ′ l ′ m ′ .. → Ω p k Ω q l ′ Ω r ′ l Ω r m ′ A pqr .., p ′ q ′ r ′ .. 32 / 44

• Vertex function V n + 1 will be O ( d , d ) invariant if the tensor A klm .., k ′ l ′ m ′ .. ( X ) transforms as m ... Ω p ′ k ′ Ω q ′ A klm .., k ′ l ′ m ′ .. → Ω p k Ω q l ′ Ω r ′ l Ω r m ′ A pqr .., p ′ q ′ r ′ .. • So far we have discussed only those vertex functions which have internal indices contracting with derivatives of compact coordinates. There are two other possibilities: (a) Set of vertex functions which have only have spacetime coordinates (like ∂ X , ¯ ∂ X and higher derivatives) contracting with spacetime tensors. Such vertex functions are O ( d , d ) invariant since T-duality transformations do not affect them. (b) There are set of vertex functions with mixed indices (Lorentz and internal). It the terms like ∂ Y , ¯ ∂ Y and higher derivatives which need to be converted to W ’s and the derivatives to obtain T-duality invariant vertex functions. 32 / 44

• Vertex function V n + 1 will be O ( d , d ) invariant if the tensor A klm .., k ′ l ′ m ′ .. ( X ) transforms as m ... Ω p ′ k ′ Ω q ′ A klm .., k ′ l ′ m ′ .. → Ω p k Ω q l ′ Ω r ′ l Ω r m ′ A pqr .., p ′ q ′ r ′ .. • So far we have discussed only those vertex functions which have internal indices contracting with derivatives of compact coordinates. There are two other possibilities: (a) Set of vertex functions which have only have spacetime coordinates (like ∂ X , ¯ ∂ X and higher derivatives) contracting with spacetime tensors. Such vertex functions are O ( d , d ) invariant since T-duality transformations do not affect them. (b) There are set of vertex functions with mixed indices (Lorentz and internal). It the terms like ∂ Y , ¯ ∂ Y and higher derivatives which need to be converted to W ’s and the derivatives to obtain T-duality invariant vertex functions. • A vertex function of the form ∂ m ′ X µ ′ ¯ j ∂ m X µ ∂ p Y α i ∂ q Y α j ... ¯ ∂ p ′ Y α ′ i ¯ ∂ q ′ Y α ′ T µα i α j µ ′ α ′ j .. i α ′ can expressed in T-duality invariant manner (note that spacetime vectors.. are T-duality-inert). 32 / 44

T-DUALITY FOR NSR STRING • We encounter some difficulties when we investigate T-duality for NSR string. Consider free NSR string. The σ ↔ τ duality (analog of P ↔ X ′ for closed bosonic string) is not exhibited since the worldsheet spinor part is not endowed with this symmetry. 33 / 44

T-DUALITY FOR NSR STRING • We encounter some difficulties when we investigate T-duality for NSR string. Consider free NSR string. The σ ↔ τ duality (analog of P ↔ X ′ for closed bosonic string) is not exhibited since the worldsheet spinor part is not endowed with this symmetry. • Issues related to T-duality for NSR strings have been addressed in the past: (Das+JM, Siegel, E. Alvarez, L. Alvarez-Gaume, Lozano, Hassan, Curtright, Uematsu, Zachos, ....). 33 / 44

T-DUALITY FOR NSR STRING • We encounter some difficulties when we investigate T-duality for NSR string. Consider free NSR string. The σ ↔ τ duality (analog of P ↔ X ′ for closed bosonic string) is not exhibited since the worldsheet spinor part is not endowed with this symmetry. • Issues related to T-duality for NSR strings have been addressed in the past: (Das+JM, Siegel, E. Alvarez, L. Alvarez-Gaume, Lozano, Hassan, Curtright, Uematsu, Zachos, ....). • Basically the gaol is two fold: (i) Consider NSR string in background B , compactify on T d and explore the T-duality symmetry. (ii) G and ˆ ˆ Whether vertex functions of excited massive levels can be cast in T-duality invariant form. 33 / 44

T-DUALITY FOR NSR STRING • We encounter some difficulties when we investigate T-duality for NSR string. Consider free NSR string. The σ ↔ τ duality (analog of P ↔ X ′ for closed bosonic string) is not exhibited since the worldsheet spinor part is not endowed with this symmetry. • Issues related to T-duality for NSR strings have been addressed in the past: (Das+JM, Siegel, E. Alvarez, L. Alvarez-Gaume, Lozano, Hassan, Curtright, Uematsu, Zachos, ....). • Basically the gaol is two fold: (i) Consider NSR string in background B , compactify on T d and explore the T-duality symmetry. (ii) G and ˆ ˆ Whether vertex functions of excited massive levels can be cast in T-duality invariant form. • The convenient starting point is to consider two dimensional worldsheet action in superspace in the presence of backgrounds and then compactify on T d . 33 / 44

• The action in the NS-NS massless background is � � � S = − 1 d σ d τ d 2 θ D ˆ Φ ˆ µ ˆ ν (ˆ Φ) − γ 5 ˆ ν (ˆ D ˆ Φ ˆ ν Φ) G ˆ B ˆ µ ˆ µ ˆ 2 Here ˆ µν (ˆ Φ) and ˆ µν (ˆ G ˆ B ˆ Φ) are the graviton and and 2-form backgrounds which depend on the superfield ˆ Φ. It has expansion in component fields as µ θ + 1 µ = X ˆ µ + ¯ µ + ¯ Φ ˆ ˆ θψ ˆ ψ ˆ θθ F ˆ ¯ µ 2 µ and F ˆ µ are the bosonic, fermionic and auxiliary fields where X ˆ µ , ψ ˆ respectively. 34 / 44

• The action in the NS-NS massless background is � � � S = − 1 d σ d τ d 2 θ D ˆ Φ ˆ µ ˆ ν (ˆ Φ) − γ 5 ˆ ν (ˆ D ˆ Φ ˆ ν Φ) G ˆ B ˆ µ ˆ µ ˆ 2 Here ˆ µν (ˆ Φ) and ˆ µν (ˆ G ˆ B ˆ Φ) are the graviton and and 2-form backgrounds which depend on the superfield ˆ Φ. It has expansion in component fields as µ θ + 1 µ = X ˆ µ + ¯ µ + ¯ Φ ˆ ˆ θψ ˆ ψ ˆ θθ F ˆ ¯ µ 2 µ and F ˆ µ are the bosonic, fermionic and auxiliary fields where X ˆ µ , ψ ˆ respectively. • The covariant derivatives in superspace are defined to be ∂ D α = − ∂ + i (¯ − i ( γ a θ ) α ∂ a , θγ a ) α ∂ a D α = ∂ ¯ θ α ∂θ α where ∂ a stands for worldsheet derivatives ( σ and τ ) and the convention for γ matrices are � 0 � � 0 � � 1 � 1 − 1 0 γ 0 = , γ 1 = , γ 5 = γ 0 γ 1 = 1 0 1 0 0 − 1 34 / 44

• The equations of motion are � � ν = 0 ˆ ν (ˆ Φ) − γ 5 ˆ ν (ˆ D ˆ Φ ˆ D G ˆ G ˆ Φ) µ ˆ µ ˆ Let us adopt Hassan-Sen compactification scheme � g µν ( φ ) � � B µν ( φ ) � 0 0 ˆ , ˆ ν = ν = G ˆ B ˆ µ ˆ µ ˆ 0 G ij ( φ ) 0 B ij ( φ ) 35 / 44

• The equations of motion are � � ν = 0 ˆ ν (ˆ Φ) − γ 5 ˆ ν (ˆ D ˆ Φ ˆ D G ˆ G ˆ Φ) µ ˆ µ ˆ Let us adopt Hassan-Sen compactification scheme � g µν ( φ ) � � B µν ( φ ) � 0 0 ˆ , ˆ ν = ν = G ˆ B ˆ µ ˆ µ ˆ 0 G ij ( φ ) 0 B ij ( φ ) • Note: backgrounds depend only on spacetime superfields, φ µ . We µ = ( φ µ , W i ) where decompose the superfields: ˆ Φ ˆ µ, ν = 0 , 1 , 2 .. D − 1 and i , j = 1 , 2 , .. d with ˆ D = D + d . The two superfields can be expanded as ψ µ θ + 1 φ µ = X µ + ¯ θψ µ + ¯ θθ F µ ¯ 2 and χ i θ + 1 W i = Y i + ¯ θχ i + ¯ ¯ θθ F i 2 35 / 44

• Consider equation of motion of superfields along compact directions. The action � � � S = − 1 d σ d τ d 2 θ DW i DW j G ij ( φ ) − γ 5 B ij ( φ ) 2 The equations of motion are �� � � DW j D G ij ( φ ) − γ 5 B ij ( φ ) = 0 This is just a conservation law since backgrounds depend on φ µ . We may introduce a free dual superfield, � W i , satisfying following equation locally � � DW j = D � G ij ( φ ) − γ 5 B ij ( φ ) W i satisfying the constraints: DD � W i = 0. 36 / 44

• Consider equation of motion of superfields along compact directions. The action � � � S = − 1 d σ d τ d 2 θ DW i DW j G ij ( φ ) − γ 5 B ij ( φ ) 2 The equations of motion are �� � � DW j D G ij ( φ ) − γ 5 B ij ( φ ) = 0 This is just a conservation law since backgrounds depend on φ µ . We may introduce a free dual superfield, � W i , satisfying following equation locally � � DW j = D � G ij ( φ ) − γ 5 B ij ( φ ) W i satisfying the constraints: DD � W i = 0. • This is analog of dual coordinate introduced in case of closed compactified bosonic string. Introduce a Lagrangian density in first order formalism � � L = 1 Σ j − ¯ � ¯ Σ i D � Σ i G ij ( φ ) − γ 5 B ij ( φ ) W i 2 36 / 44

Σ i variation leads to • The ¯ � � Σ j = D � G ij ( φ ) − γ 5 B ij ( φ ) W i Σ i = 0. Therefore, when Σ i = DW i we and � W i variation implies D ¯ recover original equation. 37 / 44

Σ i variation leads to • The ¯ � � Σ j = D � G ij ( φ ) − γ 5 B ij ( φ ) W i Σ i = 0. Therefore, when Σ i = DW i we and � W i variation implies D ¯ recover original equation. • Introduce a dual Lagrangian density in terms of the dual superfields, � W i and a set of dual backgrounds G ij ( φ ) and B ij ( φ ); whereas the former of the two backgrounds is symmetric in its indices the latter is antisymmetric. � � W = − 1 2 D � D � G ij ( φ ) − γ 5 B ij ( φ ) L f W i W j 37 / 44

Σ i variation leads to • The ¯ � � Σ j = D � G ij ( φ ) − γ 5 B ij ( φ ) W i Σ i = 0. Therefore, when Σ i = DW i we and � W i variation implies D ¯ recover original equation. • Introduce a dual Lagrangian density in terms of the dual superfields, � W i and a set of dual backgrounds G ij ( φ ) and B ij ( φ ); whereas the former of the two backgrounds is symmetric in its indices the latter is antisymmetric. � � W = − 1 2 D � D � G ij ( φ ) − γ 5 B ij ( φ ) L f W i W j • The two dual backgrounds, ( G , B ), are related to the original background fields, ( G , B ) as follows: � � − 1 � � − 1 G − BG − 1 B G − BG − 1 B BG − 1 G = and B = − 37 / 44

• Note that G is symmetric and B is antisymmetric (the combination ( G − BG − 1 B ) is symmetric). The equation of motion resulting from dual Lagrangian is �� � � D � G ( φ ) − γ 5 B ( φ ) = 0 D W Note: We may identify W as the dual superfield of � W from here. 38 / 44

• Note that G is symmetric and B is antisymmetric (the combination ( G − BG − 1 B ) is symmetric). The equation of motion resulting from dual Lagrangian is �� � � D � G ( φ ) − γ 5 B ( φ ) = 0 D W Note: We may identify W as the dual superfield of � W from here. • Strategy: combine the equations of motion from original Lagrangian and the dual Lagrangian in such a way that combined equation is in O ( d , d ) covariant form (both the equations motion are conservation laws). The two equations are DW i = γ 5 ( G − 1 B ) i j DW j + G − 1 ij D � W j D � W i = γ 5 ( G − 1 B ) j i D � W j + G − 1 ij DW j 38 / 44

• Note that G is symmetric and B is antisymmetric (the combination ( G − BG − 1 B ) is symmetric). The equation of motion resulting from dual Lagrangian is �� � � D � G ( φ ) − γ 5 B ( φ ) = 0 D W Note: We may identify W as the dual superfield of � W from here. • Strategy: combine the equations of motion from original Lagrangian and the dual Lagrangian in such a way that combined equation is in O ( d , d ) covariant form (both the equations motion are conservation laws). The two equations are DW i = γ 5 ( G − 1 B ) i j DW j + G − 1 ij D � W j D � W i = γ 5 ( G − 1 B ) j i D � W j + G − 1 ij DW j • Define an O ( d , d ) vector � W i � U = � W i 38 / 44