Nonlinear gauge invariance and WZW-like action for NS-NS superstring field theory

Journal of High Energy Physics, Sep 2015

We complete the construction of a gauge-invariant action for NS-NS superstring field theory in the large Hilbert space begun in arXiv:1305.3893 by giving a closedform expression for the action and nonlinear gauge transformations. The action has the WZW-like form and vertices are given by a pure-gauge solution of NS heterotic string field theory in the small Hilbert space of right movers.

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Nonlinear gauge invariance and WZW-like action for NS-NS superstring field theory

JHE Open Access 0 1 c The Authors. 0 1 0 Komaba , Meguro-ku, Tokyo 153-8902 , Japan 1 Institute of Physics, University of Tokyo We complete the construction of a gauge-invariant action for NS-NS superstring field theory in the large Hilbert space begun in arXiv:1305.3893 by giving a closedform expression for the action and nonlinear gauge transformations. The action has the WZW-like form and vertices are given by a pure-gauge solution of NS heterotic string field theory in the small Hilbert space of right movers. String Field Theory; Superstrings and Heterotic Strings; Gauge Symmetry - Nonlinear gauge invariance and WZW-like action for NS-NS superstring field theory 1 Introduction Nonlinear gauge invariance Quartic vertex WZW-like expression Coalgebraic description of vertices Gauge-invariant insertions NS string products Wess-Zumino-Witten-like action Nonlinear gauge invariance Gauge-invariant insertions of picture-changing operators B Some identities A Heterotic theory in the small Hilbert space Introduction While bosonic string field theories have been well-understood [1–8], superstring field theories remain mysterious. A formulation of supersymmetric theories in the early days [9], which is a natural extension of bosonic theory, has some disadvantages caused by picturechanging operators inserted into string products: singularities and broken gauge invariances [10]. To remedy these, various approaches have been proposed within the same There exists an alternative formulation of superstring field theory: large space theory [18–26]. Large space theories are formulated by utilizing the extended Hilbert space of operators. One can check the variation of the action, the equation of motion, and gauge invariance without taking account of these operators. Of course, the action implicitly includes picture-changing operators, which appear when we concretely compute scattering amplitudes after gauge fixing. The singular behaivor of them is, however, completely regulated and there is no divergence [28, 29]. The cancellation of singularities can also occur in the small Hilbert space. Recently, by the brilliant works of [30, 31], it is revealed how to obtain gauge-invariant insertions of picture-changing operators into (super-) string products in the small Hilbert space: the NS and NS-NS sectors of superstring field theories in the small Hilbert space is completely formulated. In this paper, we find that using the elegant technique of [31], one can construct the WZW-like action for NS-NS superstring field theory in the large Hilbert space. A pure-gauge solution of small-space theory is the key concept of WZW-like formulation of NS superstring field theory in the large Hilbert space, which determines the vertices of theory, and we expect that it goes in the case of the NS-NS sector. There is an attempt to construct non-vanishing interaction terms of NS-NS string fields utilizing a pure-gauge solution GB of bosonic closed string field theory [22]. However, the construction is not complete: the nonlinear gauge invariance is not clear and the defining equation of GB is ambiguous. To obtain nonlinear gauge invariances, we have to add appropriate terms to these interaction terms defined by GB at each order. Then, the ambiguities of vertices are removed and we obtain the defining equation of a suitable pure-gauge solution GL, which we explain in the following sections. In this paper, we complete this construction begun in [22] by determining these additional terms which are necessitated for the nonlinear gauge invariance and by giving closed-form expressions for the action and nonlinear gauge transformations in the NS-NS sector of closed superstring field theory. We propose the action S = the NS heterotic string equation of motion in the small Hilbert space of right movers. The action has the WZW-like form and the almost same algebraic properties as the large-space action for NS open and NS closed (heterotic) string field theory [20, 21]. This paper is organized as follows. In section 2 we show that cubic and quartic actions can be determined by adding appropriate terms and imposing gauge invariance. In section 3, we briefly review the method of gauge-invariant insertions of picture-changing operaIn section 4, first, we give the defining equation of GL and associated fields which are necessitated to construct the NS-NS action. Then, we derive the WZW-like expression for the string equation of motion just as other large-space theories. We end with some conclusions. Nonlinear gauge invariance string field theory in powers of κ: S = α2′ P number 0, and right-mover picture number 0 state. The free action S2 is given by S2 = use the symbol (G|p, p˜) which denotes that the total ghost number is G, the left-mover picture number is p, and the right-mover picture number is p˜. Then, ghost-and-picture that the inner product hA, Bi gives a nonzero value if and only if the sum of A’s and B’s S2 is invariant under the gauge transformation We would like to construct nonzero and nonlinear gauge-invariant interacting terms invariance remains to be linear in our interacting theory. Consequently, with two nonlinear and one trivial gauge invariances, the theory is Wess-Zumino-Witten-likely formulated. In section 2, starting with the action proposed in [22] and adding appropriate terms at Cubic vertex zero modes of left- and right-moving picture-changing operators respectively. The (n + 2)to the result of first quantization, where [A1, . . . , An] is the bosonic string n-product and gauge invariance not clear. In this subsection, as the simplest example, we show that a gauge-invariant cubic term S3 can be obtained by adding appropriate terms to Cyclicity. It would be helpful to consider the cyclicity of vertices. A (n + 1)-point vertex Vn is called a (BPZ-) cyclic vertex if Vn satisfies 1 Sn+1 is given by Sn+1 = (n+1)! hΨ, Vn(Ψn)i using a cyclic vertex Vn, its variation becomes For example, bosonic string field theories and superstring field theories in the small Hilbert Next, we consider the following case: a vertex Vn′ is not cyclic but has the following Nonzero Wn implies that the cyclicity of Vn is broken. For instance, this relation holds this case, in general, a zero divisor of Vn′ + Wn gives the generator of gauge transformations. However, when Wn consists of lower Vk<n, there exists a special case that the zero divisor of Vn gives the generator of gauge transformations just as WZW-like formulations of superstring field theories in the large Hilbert space, which is the subject of this paper. Adding terms. We know that naive insertions of operators which do not work as derivaclear. We show that a cubic vertex satisfying the special case of (2.6) can be constructed by adding appropriate terms and by imposing gauge invariances. Computing the variation of (2.3), we obtain this term is given by = hδΨ, ηn[QQeΨ, η˜Ψ] + Qe[QΨ, η˜Ψ] + [Qη˜Ψ, QeΨ] Averaging these three terms, we obtain the cubic action satisfying (2.6) S3 = The variation of this cubic action is given by 3 Qe[QΨ, η˜Ψ] + [QQeΨ, η˜Ψ] + [Qη˜Ψ, QeΨ] i 3 Qe[Ψ, ηQη˜Ψ] − [QeΨ, ηQη˜Ψ] − [η˜Ψ, ηQQeΨ] i Q-gauge transformation of S3 is given by Note that the zero mode Xe of the right-mover picture-changing operator is inserted cyclicly. We find that under the following second order gauge transformation define the following new string product which includes the zero mode Xe of the right-mover picture-changing operator [A, B]L := Xe[A, B] + [XeA, B] + [A, XeB] . This new product [A, B]L satisfies the same properties as original product [A, B], namely, that when we use this new product, the cubic term S3 of the action is given by S3 = under the following Q-gauge transformations Quartic vertex We can construct the quartic term S4 and, in principle, higher interaction terms Sn>4 by for example, as follows Therefore, we have to consider the following terms [A, B]L, C L = Xe [XeA, B] + [A, XeB], C + [XeA, B] + [A, XeB], XeC Xe[XeA, B] + Xe[A, XeB], C + Xe[A, B], XeC + Xe Xe[A, B], C + Xe2[A, B], C , quartic term S4 has to include the following types of terms Of course, we can repeat similar computations of above terms as we did in subsection 2.1. However, there exists a reasonable shortcut. Note that, for example, the following relation + Xe ξ˜[QA, B], C + (−)AXe ξ˜[A, QB], C + (−)A+BXe ξ˜[A, B], QC + Xe Xe[A, B], C = 0. Hence, the three product L2+2(A, B, C) defined by L2+2(A, B, C) := 9 · 2 1 n2 ξ˜Xe[A, B], C − ξ˜ Xe[A, B] + [XeA, B] + [A, XeB], C − Xe [ξ˜A, B], C − Xe[ξ˜A, B], C − [ξ˜A, B], XeC − (−)A − (−)A+B + (B, C, A)-terms + (C, A, B)-terms This new three product L2+2(A, B, C) possesses the symmetric property and the deriva 1 n 3 · 2 Xe [A, B], C + ξ˜ [XeA, B] + [A, XeB] , C + [A, B], XeC o L1+3) = 0. Note that the following types of products have the Q-derivation property N (A, B, C) = Xe ξ˜[A, B], C + ξ˜ Xe[A, B], C , and N -types of products, whose coefficients are fixed by the cancellation of the second line of (2.29) and the sum of N -type products: L1+3(A, B, C) := + (−)A [A, ξ˜XeB], C + (−)A+B [A, B], ξ˜XeC o 8 Xe XeA, B, C + Xe A, XeB, C + Xe A, B, XeC + XeA, XeB, C + XeA, B, XeC + A, XeB, XeC o + Xe [ξ˜A, B], C − [ξ˜A, XeB], C − [ξ˜A, B], XeC + (−)A Xe [A, ξ˜B], C − [XeA, ξ˜B], C − [A, ξ˜B], XeC + (B, C, A)-terms + (C, A, B)-terms . define the following new three string product [A, B, C]L := L2+2(A, B, C) + L1+3(A, B, C), which satisfies the same properties as the original three product [A, B, C], namely, the Quartic vertex S4. Let us now consider the quartic vertex having the property (2.6) and is necessitated. The variation of this term becomes Hence, the quartic term S4 defined by S4 = has the property (2.6) and its variation is given by we obtain the third order Q-gauge transformation In principle, we can construct higher vertices S5, S6, . . . by repeating these steps at each because higher order vertices consist of a lot of terms and each term has complicated operator insertions. To obtain a closed form expression of all vertices in an elegant way, we necessitate another point of view, which we explain in section 3. Gauge-invariant insertions of picture-changing operators In this section, we briefly review the coalgebraic description of string vertices [32–34] and Coalgebraic description of vertices Let T (H) be a tensor algebra of the graded vector space H. As the quotient algebra of T (H) S(H) is graded commutative and associative as follows A1A2 = (−1)deg(A1)deg(A2)A2A1, A1(A2A3) = (A1A2)A3, Fock space of superstrings, and the Z2 grading, called degree, is just Grassmann parity. The product of two multilinear maps L : Hn → Hl, M : Hm → Hk also becomes a map L · M : Hn+m → Hk+l which acts as L · M (A1A2 · · · An+m) = X(−)σ(n,m)L(Aσ(1) · · · Aσ(n)) · M (Aσ(n+1) · · · Aσ(n+m)). (3.2) Note that the n-product of the identity map I : H → H on symmetric algebras is different from the n-tensor product or the identity In on Hn: In ≡ 1 Iz ·}·|· {I = Iz ⊗ ·}·|· ⊗ I . In other words, In is the sum of all permutations, In gives the sum of equivalent permutations, and Ik · Il is equivalent to the sum of different (k, l)-partitions of k + l. Hn → H. We can naturally define a coderivation Ln : S(H) → S(H) from the map LnA = (Ln · IN−n)A, LnB = 0, or, more simply, Gauge-invariant insertions consider a series of these inserted products [[L(s), L(s)]] = 0, L(s) = L[m](t) := [[L1, Ln]] + [[L2, Ln−1]] + · · · + [[Ln, L1]] = 0, where s is a real parameter and L(s) is the generating function for the bosonic string N+1 be a (N + 1)-product including n-insertions of picture-changing operators. We where t is a real parameter and L(mn+)n+1 acts on S(H) as (3.4). Note that the upper index [m] on L[m](t) indicates the deficit in picture number of the products relative to what is needed for superstrings: L[0](t) is the sum of all superstring products and L[m](0) is the (m + 1)-product of bosonic strings. To associate the generating functions L[0](t) of the NS superstring products with (3.8), we define the following generating function L(s, t) := X smL[m](t) = where s is a real parameter. Note that powers of t count the picture number, and powers of s count the deficit in picture number. These two parameters t and s connect the generating L(s, 0) for the bosonic string products. Starting with these relations, we can construct the NS superstring products L(0, t) satisderivation conditions [[L(s, 0), L(s, 0)]] = 0, [[L(0, t), L(0, t)]] = 0, we have to solve the following pair of differential equations The solution of this pair of differential equations generates all products including appropriate insertions of picture-changing operators. Further, the equation (3.15) leads to the equation Therefore, when L(s, t) satisfies the equation (3.16), we obtain [[L(s, t), L(s, t)]] 1 ∂ 2 ∂s string products. As a result, the pair of equations (3.15) and (3.16) generates inserted Expanding (3.15) and (3.16) in powers of (s, t), we obtain the following formulae L(mn++n1+)2 = k=0 l=0 [[η, Ξm+n+2]] = (m + 1)L(mn+)n+2, (n+1) L(kk+)l+a, Ξm+n+2−k−l , (n−k+1) m + n + 3 ξL(mn+)n+2 − L(mn+)n+2 ξ · Im+n+1 Therefore, we can always derive explicit forms of these inserted products as follows: L(n0+)1 = given, L(n2+)1 = L(nn+)1 = n − 1 L(n1+)1 = [[Q, Ξ(n1+)1]] + [[L(20), Ξ(n1)]] + · · · + [[L(n0), Ξ(21)]], + [[L(n0−)1, Ξ(32)]] + [[L(n1−)1, Ξ(31)]] + [[L(n1), Ξ(21)]] , [[Q, Ξ(nn+)1]] + [[L(21), Ξ(nn−1)]] + · · · + [[L(nn−1), Ξ(21)]] . X2 A, B, C + [X2A, B, C, ] + [A, X2B, C, ] + [A, B, X2C] For example, we find that the lowest inserted product L(1) is given by is given by X[A, B] + [XA, B] + [A, XB] , L(32)(A, B, C) = where L(30)(A, B, C) = [A, B, C] and + (−)A[[A, ξXB], C] + (−)A+B[[A, B], ξXC]o X[XA, B, C] + X[A, XB, C] + X[A, B, XC] + [XA, XB, C] + [XA, B, XC] + [A, XB, XC]o + (−)A+B X[[A, B], ξC]−[[XA, B], ξC]−[[A, XB], ξC] o + (B, C, A)-terms + (C, A, B)-terms . − (−)A − (−)A+B + (B, C, A)-terms + (C, A, B)-terms NS string products The generating function L(0, t) of the superstring products, as well as that of bosonic ones L(s, 0), has nice properties, which we explain in this subsection. Note that the above this sense, we write L(nn+,01) for this L(nn+)1, an NS superstring product with insertions of left -moving picture-changing operators. L(nn+,01) gives the (n + 1)-product of NS (heterotic) superstring field theory in the small Hilbert space of left movers [31]. By construction, we can also obtain an NS superstring product L(n0+,n1) with insertions of [A0, A1, . . . , An]L := L(n0+,n1)(A0, A1, . . . , An). X(−1)|σ(A)| [Aσ(1), . . . , Aσ(k)]L, Aσ(k+1), . . . , Aσ(n) L = 0. Let G be a ghost-and-picture BRST operator QG and shifted right-mover NS string products QG A ≡ [A ]GL := QA + [A1, . . . , An]GL := F (G) ≡ QG + n=1 (n + 1)! Gn, G L = 0, in the same manner as shifted bosonic string products. Provided that the state G shifting theory in the small Hilbert space of right movers X(−1)|σ| [Aσ(1), . . . , Aσ(k)]GL, Aσ(k+1), . . . , Aσ(n) GL = 0. Then, QG becomes a nilpotent operator. WZW-like expression In this section, first, we gives the defining equations of a formal pure-gauge GL and associents of our construction. Then, we present a closed form expression of WZW-like action for NS-NS string field theory, the equation of motion, and the gauge invariance of the action. A pure-gauge GL of right-mover NS theory. We can build a formal pure-gauge solution GL of NS heterotic string field theory in the small Hilbert space of right-movers ψX[τ ] = X η˜Ψ + κ η˜Ψ, ψX[τ ] GLL[τ] so-called an associated field, satisfying ΨX[τ ] = (−1)XXΨ + κ η˜Ψ, ΨX[τ ] GLL[τ], are given by + . . . . (4.10) which appears in the action for NS-NS string fields with general t-parametrization. Note Wess-Zumino-Witten-like action Let GL = P∞ n=0 κnG(n) be the expansion of the pure-gauge GL in powers of κ. Here, L we propose a large-space WZW-like action utilizing the pure-gauge GL(t) and the large gauge-invariant cubic vertex V2 of S3 = 31! hΨ, V2(Ψ2)i Recall that the kinetic term S2 = 21 hΨ, V1(Ψ)i V4(Ψ4) = η (Qη˜Ψ)3, η˜Ψ L + [(Qη˜Ψ)2, η˜Ψ]L, η˜Ψ L (n + 1)-point vertex Vn. Therefore, the pure-gauge solution GL + . . . , (4.15) Note that all coefficients of Vn+1 and G(n) match by the t-integral. L n=1 (n + 1)! explain in the rest. Since the relation Thus, we propose the following WZW-like action for NS-NS string field theory 2 Z 1 S = which reduces to (4.16) or the familiar WZW form (see appendix B) S|Ψ(t)=tΨ = − α′ hΨη, GLi + κ The equation of motion is given by which is derived in subsection 4.3. Although the action includes the integral over a real Nonlinear gauge invariance Here, we derive the equation of motion and the closed form expression of nonlinear gauge L = −h∂tΨδ(t) − κ[Ψδ(t), ψt]GL(t), QGL(t)ψη(t)i L = h∂tΨδ(t), η GL(t)i − κhη GL(t), [Ψδ(t), ψt]GL(t)i. L = hΨδ(t), η(QGL(t)ψt) + κ[η GL(t), ψt]GL(t)i L = hΨδ(t), ∂t η GL(t) i + κhη GL(t), [Ψδ(t), ψt]GL(t)i. which does not include t-parametrized fields. The equation of motion is, therefore, given by (4.24) and it is independent of t-parametrization of fields. As a result, although the action has three generators of gauge transformations, since one of these gauge invariances reduces to trivial, the resulting theory is Wess-Zumino-Wittenlikely formulated with two nonlinear gauge invariances. In this paper, we proposed WZW-like expressions for the action and nonlinear gauge transformations in the NS-NS sector of superstring field theory in the large Hilbert space. Alit does not depend on t-parametrization. Vertices are determined by a pure-gauge solution of NS (heterotic) string field theory in the small Hilbert space of right movers, which is constructed by NS closed superstring products (except for the BRST operator) including insertions of right-moving picture-changing operators [31]. Gauge equivalent vertices. edge points at the diamonds of products in figure 5.1 of [31]. It would be possible to write the large-space NS-NS action utilizing another but gauge-equivalent products in [31] instead of the (−, NS) string products. Ramond sectors. We have not analyzed how to incorporate the R sector(s). Our largespace NS-NS action has the almost same algebraic properties as the large-space action for NS closed string field theory. Thus, we can expect that the method proposed in [25, 26] also goes in the NS-NS case. It is very important to obtain clear understandings of the geometrical meaning of theory, gauge fixing [35–37], the relation between two formulations: large- and small-space formulations. However, our large-space formulation is purely algebraic and these aspects remain mysterious. Acknowledgments The author would like to express his gratitude to the members of Komaba particle theory group, in particular, Keiyu Goto and my supervisors, Mitsuhiro Kato and Yuji Okawa. The author is also grateful to Shingo Torii. This work was supported in part by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists Heterotic theory in the small Hilbert space The action for heterotic string field theory in the small Hilbert space of right movers is S = n=1 (n + 2)! small Hilbert space of right movers and right-moving picture-changing operators Xe inserted action is invariant under the following gauge transformation [6, 7] Just as bosonic theory [5, 7], the equation of motion is given by n=1 (n + 1)! and a pure-gauge GL is constructed by infinitisimal gauge transformations [6, 20, 21]. differential equation Some identities or heterotic string products, and a derivation operator X satisfy hA, Bi = (−)(A+1)(B+1)hB, Ai, hXA, Bi = (−)AXhA, XBi, where c0− = 12 (c0 − c˜0) and X = Q, η, η˜. The Maurer-Cartan element. A pure-gauge solution GL satisfies the equation of moUsing the defining equation of GL, we find that F (GL) = n=1 (n + 1)! The standard WZW form. Recall that when there exist higher sting products [A1, . . . , An] > 2), a field-strength-like object fXY FXY ≡ XΨY + (−)(X+1)(Y +1)Y ΨX + κ[ΨX , ψY ]GL L S = dt hFηt − ∂tΨη − κ[Ψt, ψη]GLL , GLi − hΨt, QGL ψηi dth h∂tΨη, GLi + hΨη, ∂tGLi + κhΨt, [ψη, GL]GLL ii. (B.6) = 0. Hence, provided that S|Ψ(t)=tΨ = − α′ hΨη, GLi + κ Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Phys. Rev. D 34 (1986) 2360 [INSPIRE]. field theory, Phys. Rev. D 35 (1987) 1356 [INSPIRE]. Nucl. Phys. B 303 (1988) 455 [INSPIRE]. Lett. Math. Phys. 26 (1992) 259 [INSPIRE]. Nucl. Phys. B 390 (1993) 33 [hep-th/9206084] [INSPIRE]. foundations, Nucl. Phys. B 505 (1997) 569 [hep-th/9705038] [INSPIRE]. Nucl. Phys. B 314 (1989) 209 [INSPIRE]. Nucl. Phys. B 337 (1990) 363 [INSPIRE]. [17] B. Jurˇco and K. 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Hiroaki Matsunaga. Nonlinear gauge invariance and WZW-like action for NS-NS superstring field theory, Journal of High Energy Physics, 2015, 11, DOI: 10.1007/JHEP09(2015)011