Construction of action for heterotic string field theory including the Ramond sector

Journal of High Energy Physics, Dec 2016

Extending the formulation for open superstring field theory given in arXiv:​1508.​00366, we attempt to construct a complete action for heterotic string field theory. The action is non-polynomial in the Ramond string field Ψ, and we construct it order by order in Ψ. Using a dual formulation in which the role of η and Q is exchanged, the action is explicitly obtained at the quadratic and quartic order in Ψ with the gauge transformations.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://link.springer.com/content/pdf/10.1007%2FJHEP12%282016%29157.pdf

Construction of action for heterotic string field theory including the Ramond sector

Received: July Construction of action for heterotic string eld theory including the Ramond sector Keiyu Goto 0 1 3 Hiroshi Kunitomo 0 1 2 Open Access, c The Authors. 0 Kitashirakawa Oiwakecho , Sakyo-ku, Kyoto 606-8502 , Japan 1 Komaba , Meguro-ku, Tokyo 153-8902 , Japan 2 Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University 3 Institute of Physics, The University of Tokyo Extending the formulation for open superstring arXiv:1508.00366, we attempt to construct a complete action for heterotic string theory. The action is non-polynomial in the Ramond string . Using a dual formulation in which the role of changed, the action is explicitly obtained at the quadratic and quartic order in gauge transformations. including; the; String Field Theory; Superstrings and Heterotic Strings - Complete action for open superstring eld theory The NS sector of heterotic string eld theory Contents 1 Introduction 2 3 3.1 Basic ingredients WZW-like action Dual formulation 4 Inclusion of the Ramond sector Ramond string eld and restricted Hilbert space Perturbative construction 4.3 Fermion expansion Cubic interaction in SR Quartic interaction in SR Quadratic in fermion Quartic in fermion Summary and discussion A Construction of the dual gauge product B Four-point amplitudes with external fermions B.1 Propagators and vertices B.2 Four-fermion amplitude B.3 Two-fermion-two-boson amplitude Introduction the A1=L theory are interrelated by a partial gauge xing [8]. In spite of this success, it had been for a long time. of the heterotic string eld theory. introducing R string from the constructed action. Complete action for open superstring eld theory eld theory. Sen [12, 13]. 2See also [20, 21]. action for the NS sector. The original expression given in [3] is S = dt hA~t(t); A~Q(t)i; is the zero mode of (z) and A~t and A~Q are the left-invariant forms A~t(t) = g 1(t)@tg(t); A~Q(t) = g 1(t)Qg(t); and its one-parameter extension (t) are related (0) = 0. the role of and Q is exchanged: S = dt hAt(t); QA (t)i; They satisfy the relations implying the operator XY is a projector: XY X = X; Y XY = Y; (XY )2 = XY: + ( 0 + c0G) : The former constraint imposes that is in the small Hilbert space, and the latter restricts the form of expanded in the ghost zero-modes as where At(t) and A (t) are the right-invariant forms At(t) = (@tg(t))g 1(t); A (t) = ( g(t))g 1(t): which the A eld theory in which two operators and Q do not appear symmetrically but act di erently on the closed string products. string eld by the conditions3 = 0; space at picture number 3=2 and X = ( 0) G0 + 0( 0) b0; Y = string eld theory [23{25]. 3In this paper we use the same symbol to denote the string eld in the Ramond sector both for the open superstring and for the heterotic string eld. We will not confuse them since two cases never appear simultaneously. X = fQ; ( 0)g; (x) is the Heaviside step function satisfying introduce the following operator which is more suitable for use in the large Hilbert space [10]: = 0 + ( ( 0) 0 0)P 3=2 + ( 0 0)P 1=2 ; Hilbert space at picture number operator F (t) as F (t) = 1 + (D (t) = 1 + X( (D (t) D (t)A A (t)A + ( 1)AAA (t); D (t)F (t) = F (t) ; and thus the dressed Ramond string eld F (t) with the Ramond string eld by the constraints (2.5) is annihilated by D (t). Now a complete gauge invariant action is given by S = dthAt(t); QA (t) + (F (t) )2i; invariant under the gauge transformations [9]: A = Q + fF ; F (fF ; g = Q + X F are gauge parameters in the NS sector and is a gauge parameter in the Ramond sector satisfying = 0; = 0: The NS sector of heterotic string eld theory to the nilpotency 2 = 0. (z) = @ (z)e (z) = e (z) (z): b0Vi = 0; L0 Vi = 0; (i = 1; 2); hV1; V2i = hV1jc0 jV2i; It satis es in [21] hQV1; V2i = ( 1)V1 hV1; QV2i; h V1; V2i = ( 1)V1 hV1; V2i: The n-string product carries ghost number 2n + 3 (and picture number 0). The string and cyclic with respect to the inner product: De ning [V ] the L1-relations: 0 = [ [V (1); : : : ; V (m)]; V (m+1); : : : ; V (n)] : The operator acts as a derivation on the string products: V1; : : : ; Vn = It is useful to introduce new string products [ [Bm; V1; If B satis es the Maurer-Cartan equation QB + X [Bn] = 0; 0 = [ [V (1); : : : ; V (m)]B; V (m+1); : : : ; V (n)]B : the shifted string products (3.14) satisfy the identical L 1 relation to (3.9): charge, (QB)2 = 0, de ned by [V ]B = QV + [Bm; V ] : WZW-like action also satis es the closed string constraints b0 Ve = 0 ; L0 Ve = 0 : operators @t and as well as act as derivations on the string products: X[Ve1(t); : : : ; Ven(t)] = X( 1)X(1+Ve1+ +Vek 1)[Ve1(t); : : : ; XVek(t); : : : ; Ven(t)]; QG(Ve ) + X [G(Ve )n] = 0 : X(Ve ), which we call an associated eld, satisfying QG(XG) = 0 : @ G( Ve ) = [G( Ve )m; Ve ] = QG( Ve )Ve ; = 0, and set = 1. on (3.20), we have We denote t(Ve ) for @t (Ve ) for simplicity. The associated eld (Ve ) is Grassmanneven and carries ghost number 2 and picture number 1. The associated elds and X( Ve ) = XVe + Ve ; X( Ve ) G( Ve ); with the initial condition, X = 0 at = 0, and set = 1. at order 2 [22]. Utilizing these functionals G and X, a gauge-invariant action can be written in the WZW-like form: SWZW = dth t(t); G(t)i; t(t); G(t) = @t (t); G(t) ; SWZW = (Ve ); G(Ve )i; G(Ve ) = 0; = QG e + X(Ve (t)) and G(t) G(Ve (t)). One can show that the variation of the integrand becomes a total derivative in t and thus the variation of the action is given by since Ve (0) = 0, and is given by Dual formulation the role of appendix A. In the dual formulation, an L1-structure starting with plays a central role. Note that, in the case of the open string, a set of products f ; g satisfy the A1-relations: is nilpotent, As a natural extension of f ; g, we introduce a set of products satisfying L1-relations, which we call the dual sting products : eld, and cyclic: hV1; [V2; 5Note that is invertible as a function of Ve . See also [3] and [28]. They satisfy the L and picture number n on the dual string products: ( 1) (fV g) [V (1); : : : ; V (k)] ; V (k+1); : : : ; V (n) = 0; Q V1; : : : ; Vn + X( 1)V1+ +Vk 1 V1; : : : ; QVk; : : : ; Vn = 0: In the dual formulation, we denote the NS string eld as V , which is a Grassmann-odd closed string constraint: b0 V = 0 ; L0 V = 0 : with ghost number 2 and picture number 1 satisfying the Maurer-Cartan equation dual to (3.20): acts as a derivation on the dual string pro X[V1; : : : ; Vn] = X( 1)X(1+V1+ +Vk 1)[V1; : : : ; XVk; : : : ; Vn] : [V1; V2; [(G )m; V1; V2; 0 = @ G ( V ) = products as (D )2V1 = 0; = V + [(G )m; V ] 1 relation: The shifted dual string products satisfy the L dual shifted two string products: = 0 : shifted -operator D as the shifted one-string product [ ]G : iteratively with BX = 0 at = 0, and then set = 0 ; ( 1)XYYX D [V1; V2]G [D V1; V2]G ( 1)V1 [V1; D V2]G : The operator X = Q, @t, or acts on the shifted dual products as In particular, Acting X on the Maurer-Cartan equation (3.37) we have X[V1; : : : ; Vn]G + ( 1)X [XG ; V1; : : : ; Vn]G : D XG (V ) = 0 : XG (V ) = ( 1)XD BX(V ); carries ghost number 2 and picture number 0, and Bt ( B@t ) and B are Grassmanndi erential equations @ BX( V ) = XV + with Bt(t) lated as motion is given by Bt(V (t)) and G (t) G (V (t)). The variation of the action can be calcuand the action is invariant under the gauge transformation from the de nition (3.40). using these functionals G (V ) and Bt(V ) by S = dt hBt(t); QG (t)i ; S = hB (V ); QG (V )i ; QG (V ) = 0 ; B = Q The gauge parameters having ghost number 0 carry picture number 0 and 1, 2 = Q2 = 0, and QG = D BQ. dual n-string products for n 3 themselves written as a BRST variation of some products ) which we call the dual gauge products: ; Vn] = Q(V1; X( 1)V1+ +Vk 1 (V1; ; QVk; and is commutative and cyclic: hV1; (V2; ; V (n)) = ( 1) (fV g)(V1; act as a derivation also on this product, fV1; : : : ; Vng to ; Vn) = X( 1)(V1+ +Vi 1)(V1; It is useful again to de ne the shifted dual gauge products ( ; Vn)G = X ((G )m; V1; hV1; (V2; ; Vn)G ; Vn+1i: ; Vn]G = X [(G )m; = X = Q(V1; Q((G )m; V1; m((G ) X( 1)V1+:::+Vk 1((G )m; V1; X( 1)V1+ +Vk 1(V1; ; QVk; ; QVk; (QG ; V1; gauge product but satis es the relation ; Vn)G = X(V1; ; XVk; ; Vn)G + (XG ; V1; Inclusion of the Ramond sector Ramond string eld and restricted Hilbert space constrained in the restricted Hilbert space, = 0; = ; { 12 { which make the operator XY a projector: XY X = X; Y XY = Y ; (XY )2 = XY: 1 = XY Q 1 = XY QXY 1 = XY XQY 1 = XQY 1 = QXY 1 = Q 1 eld has the form where G = G0 + 2b0 0. = b0 = 0 = 0; = b0 = 0 = 0; picture number The picture changing operators X and Y are de ned by = L = 0: X = ( 0)G0 + 0( 0)b0; Y = 3=2 and action for the Ramond sector to be which is invariant under the gauge transformation hhA; Bii = hhAjc0 jBii: S0 = = Q : { 13 { the picture number p. The gauge parameter also satis es the same constraints as : = L = 0; = : X is BRST trivial in the large Hilbert space [26]: X = fQ; ( 0)g; with the Heaviside step function (x). More general operator suitable in the large Hilbert space is de ned by [10] )P 3=2 + ( )P 1=2 ; Then we generalize the operator X to the one given by X = fQ; g ; Hilbert space: hhXV1; V2ii = hhV1; XV2ii: Perturbative construction can be expanded in powers of fermion: For the NS sector, SNS remaining part, SR S(0), we adopt the dual WZW-like action de ned in (3.51). The S = { 14 { and interaction terms between two sectors. We can further expand the action in the coupling constant : SNS = S(0) + S(0) + 2S(0) + O 0 1 2 The gauge transformations can also be expanded in V = V = V (0) = Q ; V (0) = (1) = 0; (1) = 0; V = V (2) = 0; (1) = Q ; is a gauge parameter in the NS sector, is another gauge parameter in the NS sector, and S(0) = S(2) = order by order in by requiring the gauge invariance. action (3.51): included. Starting from the kinetic terms S(0) = S(0) = 4! hV; Q[V; ( V )2] i + 4! hV; Q[V; [V; V ] ] i: { 15 { V (0) = V (0) = [V; [V; Q ] ; V (0) = V (0) = [V; V; Q ] + [V; [V; Q ]]; [V; V; which keep the action (4.33) invariant at each order in : ( )(0)S(0) + ( )(0)S(0) = 0; ( )(00)S(0) + ( )(10)S(0) = 0; 1 0 candidate of cubic interaction term given by S(2) = (11)(= 1(0) ) requiring the gauge invariances in this order (# of R elds before transformation). Cubic interaction in SR Under the gauge transformation (01) in (4.30), the variation of S1(2) is given by (1) = 0 : { 16 { ( )(00)S(2) = 2 1hQ ; [V; ] i 1 = 2 1h ; [QV; ] i = 2 1h = 2 1h[ ; ; [Q V; ] i + 2 1h ] ; Q V i + 2 1h[ V; 0(0)S(2) + 1(0)S(2) + 1(2)S(0) = 0 : 1 0 0 gauge transformation V (0) in (4.24) is calculated as 1hQ ; [ 2] i = 2 1h ; [ ; Q ] i = This can be cancelled by ( )(10)S0(2) if we take (1) = V (0) in (4.27) is given by ; [ 2] i = ] i = 0; because of = 0, and so we have where we used the fact that a relation, h ; Bi = h ; Bi = h holds for general string eld B since the parameter is in the small Hilbert space. This variation (4.44) can be canceled by ( )(12)S(0) + ( )(10)S(2) with 0 0 V (2) = Quartic interaction in SR (11), which is calculated as 4 12hX [ ; ] ; [V; ] i 4 12h[ ; ] ; X[ V; ] i : h [ ; ] ; [Q V; ] i h[ [ V; ] ; ] ; Q i h[ ; [ ; ] ] ; Q V i h [ ; ] ; [ V; Q ] i hQ ; [ ; [ V; ] ] i h[ V; [ ; ] ] ; Q i Then we nd terms with two Ramond strings as S(2) = ] i + 2h Q ; [V; 2 = 2 2hQ ; [ V; 2 + 2 2hQ ; [ ; [V; ] ] i 4 2h[ V; ; ] ; Q i The variation of the second term in S(2) can similarly be calculated as 3hQ ; [ ; [ V; ] ] i + 3h ; [V; [ Q ; ] ] i; 3hQ ; [ ; [V; ] ] i + 2 3hQ ; [ ; [ V; ] ] i: (4.52) Therefore, in total, we have 2hQ ; [V; [ 2] ] i + (2 2 + 3)hQ ; [ ; [V; ] ] i + 2 3hQ ; [ ; [ V; ] ] i From (4.49) and (4.53), we nd that the constants 1 3 should be chosen to be 1 = 2 = 3 = Then we have i + h[ V; [ ; ] ] ; Q + h[ [ V; ] ; ] ; Q These terms can be cancelled by ( ) and ( ) if we choose (2) = (1) = X [ V; ; ] + X [ V; [ ; ] ] + X [ [ V; ] ; ] : Note that = ( ) = ( ) transformation with the parameter in this order holds: . Thus the gauge invariance under at this order can easily be calculated as ] i = ; [V; [ 2] ] i = and hence The correction at this order is not necessary: (2) = 0 ; (1) = 0 : h[QV; V ] ; ( )(12)V i = ] i = h respectively, where we used a relation The variation ( )(00)S2(2) is given by Substituting the relation Q = Q , this can further be calculated as hQ ; [V; [ V; ] ] i hQ ; [ V; [V; ] ] i : (4.66) hA; X Bi = h A; X Bi = hh A; X Bii = hhX A; Bii = h X A; Bi = ( 1)AhX A; Bi : = h[ V; ; In total we have ( )(12)S1(0)+( )(10)S(2) + ( )(00)S(2) 1 2 V (2) = (1) = X [[V; V ] ; holds at quadratic order in both coupling constant and the Ramond elds: ( )(12)S(0) + ( )(10)S(2) + ( )(00)S(2) + ( )(22)S(0) + ( )(20)S(2) = 0 : 1 1 2 0 0 { 19 { V (2) = 0 ; V (2) = V (2) = (1) = Q ; (1) = X [ V; (1) = (3) = Fermion expansion V (0) = V (0) = V (0) = (1) = 0 ; (1) = 0 ; (1) = 0 : [V; V; The gauge transformation with the gauge parameter in the Ramond sector is given by V = ] ; (4.110) X [[V; V ] ; The equations of motion are therefore given by S(2n) = E(0) + E(2) + E(4) + E(1) + E(3) + E(5) + = 0 ; = 0 ; for the NS and the Ramond string elds, respectively. We can also expand the gauge transformation in powers of the Ramond string eld as B = B(0) + B(2) + B(4) + requiring the gauge-invariance at each order: 0 = (2n 2k+1); Y E(2k 1) 0 = hB(0); E(0)i: S(2) = ; Y (E(1))ii + B ; E(2) ; Y (Q eld under the following gauge transformations at this order B(2) = (1) = F 1(t) = 1 + (D (t) ) = D (t) : (1) so that the 0 = (1); Y E(1) ii + hB(2); E(0)i + hB(0); E(2)i: by the same form of that for the open superstring (2.15): S(2) = where F (t) is the linear operator de ned by F (t) = 1 + (D (t) = 1 + X (D (t) We should note that this has the same form as (2.12) but D de ned in (3.40) contains variation of the action S(2) becomes as summarized in the previous section. Quadratic in fermion Multiplying it by from the left or by D from the right, we have F 1(t) = F 1(t)D (t) = D (t) ; = D (t)F (t) ; fD (t); F (t) g = 1 : We further have left and right and using It is also shown on the dual string products is given by [X; F (t)]V1 = F (t)[X; F 1(t)]F (t)V1 F (t)(X X)(D (t) )F (t)V1 F (t) [XG (t); F (t)V1]G (t) : We also summarize the properties of F (t) for later use. Since F (t) = D (t)F (t) and = 0, F (t) is D (t)-exact: D (t)F (t) + F (t) D (t) = F (t) + F (t) D (t) = F (t) 1 + (D (t) ) = 1 : = F (t)f ; g = D (t)F (t) Acting with QF (t) on , (4.133) leads to = F (t) Q + X F (t) F (t) [QG (t); F (t) ] = D (t)F (Q + X F (t) ) F (t) [QG (t); F (t) ] , XF (t) can be transformed into the following form: = F (t)X = F (t)X + ( 1)X F (t) D (t)[BX(t); F (t) ] + ( 1)X [BX(t); F (t) ] D (t)F (t) [BX(t); F (t) ] Now let us consider the variation of S(2): S(2) = dt hBt(t); [F (t) ; F (t) ] term, and obtain For the second term, utilizing (4.137), we nd Then the total variation is given by = @t ; Y X F = 0, we eventually nd the integrand of the interaction term, can be calculated as follows. Since [F ; F ] G is D -exact, we can use (3.49) for the rst Bt; [ G ; F ; F ] B ; [@tG ; F ; F ] where we assumed that @tB + [B ; Bt]G ; [F ; F ] B ; [Bt; [F ; F ]G ]G : [Bt; F ]G ; [B ; F ] 2 [Bt; F ]G ; [B ; F ] 2 B ; [F ; [Bt; F ]G ]G D F [B ; F ] Bt; [ G ; F ; F ] Bt; [D B ; F ; F ] D [B ; F ; F ] G + [B ; [F ; F ]G ]G 2[F ; [B ; F ]G ]G B ; [@tG ; F ; F ] B ; [Bt; [F ; F ]G ]G + 2 B ; [F ; [Bt; F ]G ]G S(2) = ; Y (Q { 25 { E(1) = Q + X F E(2) = By requiring (4.122), let us determine (1) and B(2) for each of gauge transformations with the parameters , and . Let us rst consider the invariance under the transformation with the parameter : 0 = = D F E(1) it can be calculated as B(0); E(2) = = h B(2) = (1) = G + 2[F ; F [F ; ]G ]G D [F ; ]G ; E(1)i : X F 2[F ; F [F ; ]G ]G ; The invariance under the transformation with the parameter Since the second term is again known and calculated as 0 = hB(0); E(2)i = hD ; E(2)i = h ; D E(2)i = 0 ; we conclude that B(2) = 0; (1) = 0 : Finally, for the invariance under the transformation with : where we decomposed 0 = (1); Y E(1)ii + B(2); E(0) (1); Y E(1)ii + B(2); E(0) ; (1) into the free part (4.30) and remaining: (1) = ii = hhQ ; Y (Q + X F )ii = = h[F ; F The invariance (4.154) holds if we take B(2) = (1) = X F D = X F : Thus, in total, the gauge transformation at this order becomes (4.127). which has the same form as that of the open superstring eld theory, and thus is its 0 = (1); Y E(3) (3); Y E(1) + B(0); E(4) + B(2); E(2) + B(4); E(0) ; in which, in particular, we nd hB(2); E(2)i 6= 0 : (3) so that the equation (4.158) is satis ed. we have = Q(F ; F ; F ) 3(F ; F ; QF ) hB(2); E(2)i = ; D [F ; F ; F ]G i F ; [F ; F ; F ]G i (QG ; F ; F ; F ) { 27 { and thus 6 hQF ; (F ; F ; F ) 2 h(F ; F ; F )G ; QF i 6 h(F ; F ; F ; F )G ; QG i 6 hQF ; (F ; F ; F ) G i = + X F ); Y X F D (F ; F ; F ) hB(2); E(2)i = F ; Q(F ; F ; F ) F ; (F ; F ; QF ) F ; (QG ; F ; F ; F ) D (F ; F ; F )G ; Y (Q under an arbitrary variation of , where we used satis es the constraint (4.1) and therefore D F = F can integrate it, and obtain = F . Since the shifted dual gauge products are cyclic, we S(4) = F ; (F ; F ; F ) variation of the NS string eld, we have S(4) = F [ G ; F ]G ; (F ; F ; F ) B ; D [F ; F (F ; F ; F )G ]G F ; ( G ; F ; F ; F ) B ; D (F ; F ; F ; F ) respectively, and we eventually have hB(2); E(2)i = D (F ; F ; F ) From this form of E(3), the action S(4) has to satisfy ; Y X F D (F ; F ; F ) E(3) = B(4) = (3) = X F D (F ; F ; F ) [F ; F (F ; F ; F )G ]G ; X F D (F ; F ; F ) D (F ; F ; F )G ; Y E(1) (F ; F ; F ; F ) [F ; F (F ; F ; F )G ]G [F (F ; F ; F )G ; F ] (F ; F ; F ; F ) [F (F ; F ; F )G ; F ] S(4) = = D B . Thus we obtain E(4) = { 28 { D (F ; F ; F ; F ) D [F ; F (F ; F ; F )G ]G : Let us consider the invariance under the parameter rst. The action is invariant if we can determine B(4) and (3) so that they satisfy 0 = (3); Y (Q ; E(4) + B(4); QG : However, since the second term vanishes, we can consistently take ; E(4)i = h ; D E(4)i = 0 ; B(4) = 0 ; (3) = 0 : that one can determine (3) and B(4) so that the condition (4.120) at quartic order, 0 = (1); Y E(3) (3); Y E(1) + B(0); E(4) + B(2); E(2) + X F and E(0) = QG , which can be compensated by appropriately determining (3) and B(4), respectively: 0 = B(0); E(4) + B(2); E(2) useful to note that = 0 ; fQ; D g = 0 ; Q(F ; : : : ; F ) = [F ; : : : ; F ]G : Utilizing them, we have Q[B1; : : : ; Bn]G (1); Y E(3) ii = D (F ; F ; F ) [F ; ]G ; D F QF D (F ; F ; F ) [F ; ]G ; D F [F ; F ; F ] [F ; ]G ; QD F (F ; F ; F ) ; [F ; [F ; F ; F ]G ]G F [F ; ]G ; D [F ; F ; F ] ; [F ; QD F (F ; F ; F )G ]G { 29 { where we used D F QD F F D )QD F one can show that the remaining two terms become hB(0); E(4)i = hB(2); E(2)i = ; [F ; QD F (F ; F ; F )G ]G ; D [F ; F ; F ; F ] ; [F ; F ; [F ; F ]G ]G F [F ; ]G ; [F ; [F ; F ]G ]G F [F ; ]G ; D [F ; F ; F ]G +3[F ; [F ; F ]G ]G = 0 : By picking up the terms with E(1) and E(0), the transformations (3) and B(4) can be explicitly determined as B(4) = [(F ; F ; F ; F )G ; ] [[F ; F (F ; F ; F )G ]G ; ] (F ; F ; F ; F ; D (F ; F ; F ; F [F ; D [F ; F (F ; F ; F ; D [F (F ; F ; F )G ; F ; D [F ; F (F ; F ; F [F ; D [F ; F [F (F ; F ; F )G ; D [F (F ; F ; F )G ; F [F ; D X F D [F (F ; F ; F )G ; D (3) = X F D (F ; F ; F ; D X F D (F [F ; D Summary and discussion Using the expansion in the number of the Ramond string eld, we have constructed in quartic order: S = dt Bt(t); QG (t) + [F (t) ; F (t) ]G (t) F ; (F ; F ; F )G i + O( 6): [(F ; F ; F ; F )G ; ] [[F ; F (F ; F ; F )G ]G ; ] (F ; F ; F ; F ; D (F ; F ; F ; F [F ; D [F ; F (F ; F ; F ; D [F (F ; F ; F )G ; F ; D [F ; F (F ; F ; F [F ; D [F ; F [F (F ; F ; F )G ; D [F (F ; F ; F )G ; F [F ; D X F D (F ; F ; F ; D ) X F D (F ; F ; F [F ; D X F D [F (F ; F ; F )G ; D X F D [F ; ] with the parameter , and with the parameter , = D = O( 5); = Q 2[F ; F [F ; ]G ]G = Q + X F X F D (F ; F ; F ) from this action are ENS = QG + (F ; F ; F ; F ) 4[F ; F (F ; F ; F )G ]G + O( 6); (5.8) ER = Q + X F X F D (F ; F ; F ) at each gauge transformation with the parameter does not subject to change any more. One can nd that the gauge transformations are obtained by replacing elds in the equations of motion with gauge parameters:7 E(2k+1) = E(2k) = D B(2k); E(2k+1) = for k = 0; 1; 2 ; for k = 0; 1 ; for k = 1; 2 ; for k = 0; 1 : to complete an action to all orders. form of the rst-order equations of motion obtained in [15]: where Be = P1 ( + Q)Be + m! [Bem] = 0; [Bem1] = 0 ; D Be 1=2 = 0 ; Be 1 = G ; Be 1=2 = F can be solved as The next two with p = QG + QBe 1=2 + [Be0; Be 1=2]G + { 32 { 0). In the original formulation simply become which can be solved as QBen=2 + Ben(n=+24) = = 0 ; QG + QF = 0 ; = 0 ; (5.20) Be 1=2 = F D F (F ; F ; F ) rede nition of the Ramond string eld , we have to nd a way to reproduce the higher for constructing a complete gauge invariant action. Acknowledgments Construction of the dual gauge product a multilinear map dn : H^n ! H, where ^ is the symmetrized tensor product satisfying { 33 { algebra S(H) = H^0 to S(H) itself called a coderivation. A coderivation N ) = (dn ^ IN n)( 1 ^ (n+1) ^ (N) ; (A.1) S(H), and it vanishes when acting on H^N<n. The graded map [[bn; cm]] : H^n+m 1 ! H which is de ned by [[bn; cm]] = bn(cm ^ In 1) ( 1)deg(bn)deg(cm)cm(bn ^ Im 1) : Then the L1-relation can be written as [[L; L]] = 0 ; where L = L1 + L2 + L3 + and Lk is a coderivation derived from the k-string product. de ne a cohomomorphism bf : S(H) ! S(H0), which acts on S(H) as n) = i n k1< <ki=n ^ fki ki 1 ( ki 1+1; : : : ; n) : space, S(H) ! H, as 3 ^ 4 ^ 5 + : : : = pLp+1, had ghost number 1 by a similarity transformation of the BRST operator Q as LNS( ) = Gb 1( )Q Gb( ) ; Gb( ) = P exp where [0]( ) = P1 p [p0+] 2, called gauge products, can be determined iteratively. The (p + 2)-gauge product [p0+] 2 carries ghost number cohomomorphisms Gb( ) and Gb 1( ) satisfy The L1-relations are followed from the nilpotency of Q as [[LNS( ); LNS( )]] = 2 LNS( ) @ Gb( ) = Gb( ) [0]( ) ; @ Gb 1( ) = = 2 Gb 1( )Q2 Gb( ) = 0 : initial gauge product [0], whose explicit example is given in [7]. pLp+1 which provides a set of the dual string products by This n-th dual product Ln carries ghost number 3 2n and picture number n By construction, they satisfy the L1-relation L ( ) = Gb( ) Gb 1( ) : [[L ( ); L ( )]] = 0 : for example, [V1; V2] = [V1; V2]; [V1; V2; V3] = X0[V1; V2; V3]+[X0V1; V2; V3]+[V1; X0V2; V3]+[V1; V2; X0V3] + ( 1)V1 0[V1; [V2; V3]] ( 1)V1 [ 0V1; [V2; V3]] + [V1; [ 0V2; V3]]+( 1)V2 [V1; [V2; 0V3]] considering Vi is either the NS string eld or the Ramond string eld, which preserves the dual string products. respect to can be written as the commutator of Q and a product : @2L ( ) = [[Q; ( )]] = The dual gauge products can be read from (V1; V2; : : : ; Vn) = is related to the gauge products [0]( ) as [[ ; [0]( )]] = L[1]( ) ; and satis es L[1]( = 0) = LB, where L2B is a coderivation derived from the simple two2 introduce a coderivation [1]( ) = P1 n+3 derived from a set of intermediate gauge products with de cit picture 1 [7]. It satis es the relation, we can rewrite L as The integrand in the second term becomes { 36 { Then eventually the dual string products L can be written as L ( ) = 00)[[Q; Gb( 00) [1]( 00) Gb 1( 00)]] : 00 d 0 and carried out 0-integral. In this expression, we obtain @ L ( ) = @2L ( ) = d 00[[Q; Gb( 00) [1]( 00) Gb 1( 00)]] ; [[Q; Gb( ) [1]( ) Gb 1( )]] : Therefore we can de ne the product by two gauge products [0] and [1] as ( ) = G( ) [1]( )G 1( ) : The cyclicity of obtain the following expressions for the rst few orders: 3 = 4 = 5 = Four-point amplitudes with external fermions in this paper. We follow the notations and conventions in [29]. { 37 { S0 = V = Q = Q ; b0+V = 0V = 0 ; = 0 : d2T ( 0b0 b0+) e T L0+ i L0 d2T (b0 b0+X ) e T L0+ i L0 R = R = = 0 ; R( 0Y X ) = are obtained from (3.51) and (4.123) as j ih j = XY This is invariant under the gauge transformations which we x here by gauge conditions be found as expanding them in the coupling constant : S(0) = S(0) = S(2) = S(2) = S(4) = amplitudes including external fermions. { 38 { three-string interaction (B.10) as (ABjCD) = ( = 2 = 2 + d2Tsh A B( 0b0 b0 ) C DiWs + d2Tshh A B(b0 b0 ) C DiiWs ; coordinate on the NS propagator.9 The symbols A; ; D represent the wave functions and the on-shell condition Q if necessary. (ACjBD) = 2 (ADjBC) = 2 d2Tthh A C (b0 b0+) B DiiWt ; + d2Tuhh A D(b0 b0 ) B C iiWu : which lls the gap in the moduli integration [18]: (ABCD) = 2 d 1d 2hh(bC1bC2) A B C DiiW4 ; ghost insertions, the details of which are given in [20]. AF 4 = AF 4 9These are denoted c and bc in [14]. rst quantized formulation. Two-fermion-two-boson amplitude vertices (B.8) and (B.10): AF 2B2 (ABjCD) = d2Ts h A B( 0b0 b0+) (QVC ) ( VD) + ( VC ) (QVD) iWs where we denoted H d tor (B.5) and the vertex (B.10): AF 2B2 (ACjBD) = d2Tt h AVC (b0 b0+X ) BVDiWt : as to be the correlation function of the same external vertices, A, B, ((X0 VC ) ( VD) + AF 2B2 (ACjBD) = d2Tt hh A (X0 VC ) (b0 b0+) B ( VD)iiWt + + hh A ( VC ) (b0 b0 ) B (X0 VD)iiWt tices (B.11): AF 2B2 (ABCD) = + d2Tu hh A (X0 VD)(b0 b0 ) B ( VC )iiWu + + hh A ( VD)(b0 b0 ) B (X0 VC )iiWu d 1d 2h( 0bC1bC2) A B (QVC ) ( VD) + ( VC ) (QVD) iW4 B VDiWth b0 A VDiWt Tt=0 A VC iWu Tu=0 d 1d 2hh(bC1bC2) A B (X0 VC ) ( VD) + ( VC ) (X0 VD) iiW4 AF 2B2 (ADjBC) = d2Tu h AVD(b0 b0+X ) BVC iWu Summing up all these contributions, one can AF 2B2 = AF 2B2 + hh A ( VC ) (b0 b0 ) B (X0 VD)iiWt + d2Tu hh A (X0 VD)(b0 b0 ) B ( VC )iiWu + hh A ( VD)(b0 b0 ) B (X0 VC )iiWu Open Access. JHEP 11 (2004) 038 [hep-th/0409018] [INSPIRE]. superstring eld theory, arXiv:1505.01659 [INSPIRE]. 2014 (2014) 093B07 [arXiv:1407.0801] [INSPIRE]. Theory, JHEP 11 (2015) 199 [arXiv:1506.05774] [INSPIRE]. superstring eld theory, arXiv:1512.03379 [INSPIRE]. Theories, Nucl. Phys. B 276 (1986) 366 [INSPIRE]. Covariant Superstring, Nucl. Phys. B 278 (1986) 833 [INSPIRE]. Superstring Field Theory, Phys. Lett. B 173 (1986) 134 [INSPIRE]. [1] N. Berkovits , SuperPoincare invariant superstring eld theory, Nucl . Phys . B 450 ( 1995 ) 90 [2] Y. Okawa and B. Zwiebach , Heterotic string eld theory , JHEP 07 ( 2004 ) 042 [3] N. Berkovits , Y. Okawa and B. Zwiebach , WZW-like action for heterotic string eld theory , [4] H. Matsunaga , Construction of a Gauge-Invariant Action for Type II Superstring Field [9] H. Kunitomo and Y. Okawa , Complete action for open superstring eld theory , PTEP 2016 [11] S. Konopka and I. Sachs , Open Superstring Field Theory on the Restricted Hilbert Space , JHEP 04 ( 2016 ) 164 [arXiv:1602.02583] [INSPIRE]. Sector , JHEP 08 ( 2015 ) 025 [arXiv:1501.00988] [INSPIRE]. [12] A. Sen , Gauge Invariant 1PI E ective Superstring Field Theory: Inclusion of the Ramond [13] A. Sen , BV Master Action for Heterotic and Type II String Field Theories , JHEP 02 ( 2016 ) [14] H. Kunitomo , The Ramond Sector of Heterotic String Field Theory , PTEP 2014 ( 2014 ) [15] H. Kunitomo , First-Order Equations of Motion for Heterotic String Field Theory , PTEP [16] T. Erler , S. Konopka and I. Sachs , Ramond Equations of Motion in Superstring Field [17] N. Berkovits , The Ramond sector of open superstring eld theory , JHEP 11 ( 2001 ) 047 [18] M. Saadi and B. Zwiebach , Closed String Field Theory from Polyhedra , Annals Phys . 192 [19] T. Kugo , H. Kunitomo and K. Suehiro , Nonpolynomial Closed String Field Theory, Phys. [20] T. Kugo and K. Suehiro , Nonpolynomial Closed String Field Theory: Action and Its Gauge [22] K. Goto and H. Matsunaga , A1=L1 structure and alternative action for WZW-like [23] Y. Kazama , A. Neveu , H. Nicolai and P.C. West , Symmetry Structures of Superstring Field [24] Y. Kazama , A. Neveu , H. Nicolai and P.C. West , Space-time Supersymmetry of the [25] H. Terao and S. Uehara , Gauge Invariant Actions and Gauge Fixed Actions of Free [26] C.R. Preitschopf , C.B. Thorn and S.A. Yost , Superstring Field Theory, Nucl. Phys . B 337 [27] D. Friedan , E.J. Martinec and S.H. Shenker , Conformal Invariance , Supersymmetry and [28] T. Erler , Relating Berkovits and A1 superstring eld theories; large Hilbert space perspective,


This is a preview of a remote PDF: http://link.springer.com/content/pdf/10.1007%2FJHEP12%282016%29157.pdf

Keiyu Goto, Hiroshi Kunitomo. Construction of action for heterotic string field theory including the Ramond sector, Journal of High Energy Physics, 2016, DOI: 10.1007/JHEP12(2016)157