Relating Berkovits and A ∞ superstring field theories; large Hilbert space perspective

Journal of High Energy Physics, Feb 2016

We lift the dynamical field of the A ∞ superstring field theory to the large Hilbert space by introducing a gauge invariance associated with the eta zero mode. We then provide a field redefinition which relates the lifted field to the dynamical field of Berkovits’ superstring field theory in the large Hilbert space. This generalizes the field redefinition in the small Hilbert space described in earlier works, and gives some understanding of the relation between the gauge symmetries of the theories. It also provides a new perspective on the algebraic structure underlying gauge invariance of the Wess-Zumino-Witten-like action.

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:

https://link.springer.com/content/pdf/10.1007%2FJHEP02%282016%29121.pdf

Relating Berkovits and A ∞ superstring field theories; large Hilbert space perspective

HJE superstring eld theories; Theodore Erler 0 1 2 0 Theresienstrasse 37 , 80333 Munich , Germany 1 Arnold Sommerfeld Center, Ludwig-Maximilians University 2 Therefore, the eld strength F We lift the dynamical eld of the A1 superstring eld theory to the large Hilbert space by introducing a gauge invariance associated with the eta zero mode. We then provide a eld rede nition which relates the lifted eld to the dynamical eld of Berkovits' superstring eld theory in the large Hilbert space. This generalizes the eld rede nition in the small Hilbert space described in earlier works, and gives some understanding of the relation between the gauge symmetries of the theories. It also provides a new perspective on the algebraic structure underlying gauge invariance of the Wess-Zumino-Witten-like action. String Field Theory; Superstrings and Heterotic Strings - Relating Berkovits and A 1 Introduction Recap 3.1 3.2 3.3 3.4 4.1 4.2 4.3 A1 action in the large Hilbert space Potentials and eld strengths Variation of the action and gauge invariance Higher potentials and the Maurer-Cartan equation Little potentials 4 Field rede nition From A1 to Berkovits From Berkovits to A1 Okawa's approach 5 Mapping gauge invariances A Some computations involving cyclicity A.1 Proof of (3.13) A.2 Proof of (3.78) A.3 Proof of (3.126) like open superstring eld theory of Berkovits [3, 4] and a new form of open superstring eld theory based on A1 algebras [ 5 ].1 The rst issue one encounters in this regard is that the A1 theory uses a string eld in the small Hilbert space, while the Berkovits theory uses a string eld in the large Hilbert space. The large Hilbert space comes with an additional gauge symmetry associated with the eta zero mode, on top of the usual gauge symmetry associated with the BRST operator. In earlier works, this discrepancy was resolved by xing the eta part of the gauge invariance [10], so that the remaining degrees of freedom of the Berkovits theory could be described by a single string eld in the small Hilbert space. Then one can compare the gauge- xed eld of the Berkovits theory to the string eld of the A1 theory. Here we take a complementary approach. Instead of partially gauge xing the Berkovits theory, we lift the eld of the A1 theory to the large Hilbert space, producing a new gauge 1For corresponding investigations in heterotic string eld theory [6, 7], see [8, 9]. { 1 { symmetry associated with the eta zero mode. This can be done as follows. One substitutes the original string eld Hilbert space according to A in the A1 action with a new dynamical eld A in the large where = 0 is the eta zero mode. The lifted A1 theory automatically possesses an additional gauge symmetry A = A; 0A = A + A; (1.1) (1.2) since the action only depends on A in the combination A. Therefore we can search for a eld rede nition relating the lifted eld A to the string eld B of the Berkovits theory. No gauge xing is required. The eld rede nition we propose comes in the form of a \Wilson line" which relates a path through eld space in the Berkovits theory to a path through eld space in the lifted A1 theory. When the actions are expressed in Wess-Zumino-Witten-like form, this reduces their equivalence to an identity. One advantage of this approach is that it gives a clearer understanding of the relation between the gauge symmetries of the two theories. This question is more di cult to study in the small Hilbert space since the partially gauge- xed Berkovits theory does not exhibit a cyclic A1 structure.2 We show that the gauge transformations of the two theories map into each other, while the BRST gauge transformation in one theory maps into a combination of BRST, , and trivial gauge transformations in the other. Another consequence of our analysis is a new perspective on the algebraic structure underlying the Wess-Zumino-Witten-like (WZW-like) action. We show that the \potentials" which appear in the WZW-like action are generally part of a hierarchy of higher-form potentials which together provide a solution to a certain Maurer-Cartan equation. Maurer-Cartan gauge transformations implement eld rede nitions and relate equivalent realizations of the WZW-like action. In this sense, the Maurer-Cartan equation plays a role in the large Hilbert space somewhat analogous to the role of cyclic A1 algebras in the small Hilbert space. These results may be a useful step towards a better understanding of the role of the large Hilbert space in superstring eld theory, and in particular the problem of quantization [11{15]. 2 Recap whose conventions we follow. In this paper, string This section contains a repository of de nitions and formulae that we will need in our calculations. See earlier works for a more extended introduction to the formalism, especially [2], elds are always elements of the Neveu-Schwarz state space H of an open superstring quantized in the RNS formalism, including bc and bosonized superconformal ghosts ; ; e [16]. A string eld A is in the small Hilbert space if A = 0, otherwise it is in the large Hilbert space. The degree of a string eld A, denoted deg(A), 2The full nonlinear gauge invariance of the partially gauge- xed Berkovits theory has recently been derived in [15], and it would be interesting to investigate the relation to the A1 gauge invariance. { 2 { is de ned to be its Grassmann parity (A) plus one: When using the degree grading, it is natural to work with a 2-product m2 and a symplectic form !L related, respectively, to Witten's open string star product and the BPZ inner product by a sign: deg(A) = (A) + 1 mod Z2: m2(A; B) = ( 1)deg(A)A B; !L(A; B) = ( 1)deg(A)hA; BiL: We will often drop the star when writing the star product, that is AB A B. The subscript L denotes the BPZ inner product and symplectic form computed in the large We will also encounter the symplectic form in the small Hilbert space denoted !S(A; B). We will omit the subscript S or L for equations that hold for symplectic forms in both the large and small Hilbert space. The de nition of the A1 theory requires an operator built from the ghost I dz where f (z) is a function which is holomorphic in the neighborhood of the unit circle, is BPZ even, and [ ; ] = 1.3 Using this operator, the symplectic form in the small Hilbert space can be related to the symplectic form in the large Hilbert space by where A and B are string elds in the small Hilbert space. An n-string product cn(A1; : : : ; An) can be viewed as a linear map from n copies of the state space H into one copy: We write cn : H n ! H: 3In contrast to [2], in this paper we set the open string coupling constant to 1. { 3 { cn(A1; : : : ; An) = cn A1 : : : An; where the right hand side is interpreted as the linear map cn acting on the tensor product of states A1 : : : An. The degree of cn is de ned to be the degree of its output minus the sum of the degrees of its inputs. We consider the tensor algebra T H generated by taking sums of tensor products of states: T H = H 0 H H 2 : : : : 1T H A = A 1T H = A; Here H satisfying 0 consists of scalar multiples of the identity element of the tensor algebra 1T H, (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) n denotes the projection onto the n-string component of the tensor algebra and I n is the identity operator on H n products of states as follows. If bk;m is a linear map H ` n, then their tensor product is de ned to acts as . The tensor product of linear maps acts on tensor m ! H k and c`;n is a linear map : : : Now suppose we have a list of degree even products H0; H1; H2; : : :. With this data we can de ne an operator on the tensor algebra called a cohomomorphism H^ = 0 + X 1 1 X `=1 k1;:::;k`=0 (Hk1 : : : Hk` ) k1:::+k` : A typical example of a cohomomorphism is the identity operator on the tensor algebra IT H, which is de ned by taking H1 = I and the remaining Hk to vanish. Given a degree even string eld A, we can de ne an element of the tensor algebra called a group-like element : 1 1 A 1T H + A + (A A) + (A A A) + : : : : In our calculations we will often need to act coderivations and cohomomorphisms on grouplike elements. We note the formulas product D1, a 2-string product D2 and so on so that all products are either degree even or degree odd. From this data we de ne an operator on the tensor algebra called a coderivation D = 1 X `;m;n=0 (I ` copies of the state space into complex numbers: Likewise, a cohomomorphism H^ is cyclic if it satis es which is graded antisymmetric upon interchange of its arguments: !(A; B) = ( 1)deg(A)deg(B)!(B; A): We write h j ! A symplectic form ! if its coderivation cn satis es B = !(A; B). An n-string product cn is cyclic with respect to the This summarizes most of what we need from the coalgebra formalism. At the margins, a few computations are helped by introducing the coproduct. We will review this in appendix A. Now let's review the A1 superstring eld theory. The dynamical string eld A is in the Neveu-Schwarz (NS) sector,4 is degree even, lives in the small Hilbert space, and carries ghost number 1 and picture number 1. The action is 1 1 1 multi-string products in the small Hilbert space which satisfy the relations of a cyclic A1 algebra. The products Q; M2; M3; : : : de ne a coderivation M = Q + M2 + M3 + : : : : The statement that the products are in the small Hilbert space can be expressed by the equation5 where is the coderivation corresponding to the eta zero mode. The statement that the products form an A1 algebra is expressed by the equation In addition, the products form a cyclic A1 algebra because 4In this paper we only discuss the NS sector. The generalization to the Ramond sector [17{21] will be considered in [22]. 5Commutators of products and coderivations are always graded with respect to degree. Commutators of string elds, with the multiplication de ned by Witten's open string star product, are always graded with respect to Grassmann parity. h!j : H 2 ! C: h!j 2cn = 0: { 5 { eld theory is that the coderivation M can be related to Q using a similarity transformation in the large Hilbert space. The similarity transformation is provided by an invertible, cyclic cohomomorphism G^ satisfying [ 1, 5 ] h!Lj 2G^ = h!Lj 2 M = G^ 1QG^ ; m2 = G^ G^ 1 ; (2.26) (2.27) (3.1) (3.2) (3.3) (3.5) (3.6) A1 action in the large Hilbert space In this section we reformulate the A1 superstring eld theory by replacing A in the small Hilbert space with a new dynamical eld A in the large Hilbert space via the substitution derivation follow [1], but we will make some re nements. The n-string vertex in the A1 action takes the form where m2 is the coderivation corresponding to the open string star product m2. The construction of G^ requires the operator in (2.4), and is described in [ 5 ]. dt 2 !S n !S( A; Mn 1( A; : : : ; A)): A(0) = 0; as an integral of a total derivative with respect to t: A(t); Q A(t) + A(t); M2( A(t); A(t)) + : : : : (3.4) Acting the t-derivative on the n-string vertex produces n terms containing a factor of _ A(t) = d A(t)=dt. Since the vertices are cyclic, each of these terms are equal, canceling the factor of 1=n. Using cyclicity to place _ A(t) in the rst entry of the symplectic form, we can therefore write the action which can be written more compactly as 1 3 !S { 6 { _ A(t); 1M 1 1 A(t) : Since this form of the action was obtained from the integral of a total derivative, by construction it only depends on the value of A(t) at t = 1. The next step is to lift to the large Hilbert space by making the substitution A(t) is an interpolating 1-parameter family of string elds subject to the boundary and A is the new dynamical string eld in the large Hilbert space. The new eld degree odd (but Grassmann even) and carries ghost and picture number zero. The action HJEP02(16) where conditions becomes (3.7) (3.8) A is (3.9) (3.10) (3.11) (3.12) (3.13) where the gauge parameter A is degree even, ghost number 1 and picture number 1. The in nitesimal BRST gauge transformation is Q A = 1M 1 1 A A 1 1 A ; where the gauge parameter A is degree even, ghost number 1 and picture 0. Acting on both sides of this equation produces the standard A1 gauge transformation of symplectic form is in the small Hilbert space. Therefore in the rst entry of the symplectic form we can replace with + = 1 at no cost. We can therefore write dt !L _ A(t); 1M 1 1 A(t) ; where we replaced the small Hilbert space symplectic form by the large Hilbert space symplectic form. The action only depends on the value of A(t) at t = 1 (in fact, it only depends on the value of A(t) at t = 1), and has a new gauge invariance related to the eta zero mode A ! 1D1 ; 1D2 1 1 A = ! To simplify further, recall from [1] that a cyclic cohomomorphism H^ satis es the identity { 7 { where A is a string eld and D1 and D2 are arbitrary coderivations. We provide another proof of this identity in appendix A. Since the cohomomorphism G^ is cyclic with respect to !L, this identity allows us to write !L string product. We can further write This is the Wess-Zumino-Witten-like action for the lifted A1 theory. Note that with this de nition of At and A we will not be able to express the action in the standard WZW-like form 1 2 SA 6= A similar obstruction prevents the action of heterotic string eld theory from being expressed in standard WZW-like form [6]. The action (3.20) turns out to be more general, and will be the basis of our computations. The above embedding of the A1 theory into the large Hilbert space is trivial | the action only depends on the large Hilbert space eld in the combination A. One might ask whether there is a more interesting extension of the A1 theory to the large Hilbert space. In this regard it is useful compare to the partially gauge- xed Berkovits theory [2, 10], whose extension to the large Hilbert space should apparently be Berkovits' open superstring eld theory. If B is the eld of the partially gauge- xed Berkovits theory, we can attempt to replace it with a eld B in the large Hilbert space in the same way as we did for the A1 theory: B = B ? (3.22) However, the resulting theory in the large Hilbert space is not Berkovits' superstring eld theory. To get the Berkovits theory we must do something more re ned. First we start with the potentials expressed in terms of the star product, , and B. Then, for every instance where operates on B, we should substitute B, and for every other case we 1 A(t) ; Q 1G ^ 1 A(t) The action (3.12) therefore becomes Now we de ne \potentials" ^ 1G 1 At A 1 A(t) ^ 1G ^ 1G 1 1 The potential At is degree odd but Grassmann even. The potential A is degree even but Grassmann odd. Switching to the Grassmann grading the action is therefore expressed 0 1 1 should substitute B. The resulting potentials de ne the WZW-like action for Berkovits' superstring eld theory. In principle, we can follow the same procedure for the A1 theory, producing a string eld theory in the large Hilbert space with a nonlinear gauge invariance. However, for the Berkovits theory this procedure has a special property: it allows one to eliminate all instances of from the partially gauge- xed action, so that the theory in the large Hilbert space can be expressed solely in terms of Q; and the star product. In the A1 theory the same procedure does not eliminate all insertions of , since in some cases acts on products of elds rather than the eld itself. The upshot is that we do not know of a nontrivial embedding of the A1 theory in the large Hilbert space which is more \natural" than the trivial one. The trivial embedding is su cient for our purposes, and will be the focus for the remainder of the paper. 3.1 Potentials and eld strengths The structure of the WZW-like action can be understood in terms of the properties of potentials and eld strengths. Suppose that we have a collection of graded derivations of the open string star product which commute. We write the derivations @I with I ranging over an index set, and they satisfy where the commutator [; ] is graded with respect to Grassmann parity. In particular, if @I eld AI with the same Grassmann parity, ghost and picture number. We refer to AI as the potential corresponding to the derivation @I . The eld strength is de ned FIJ where (I) denotes the Grassmann parity of @I , and the commutator of string elds is computed with the star product and is graded with respect to Grassmann parity. The eld strength is graded antisymmetric: FIJ = ( 1) (I) (J)FJI : Note that diagonal elements of the eld strength do not necessarily vanish. If the derivation AI AI ); (I) = 1 mod Z2: It is useful to de ne a gauge covariant derivative We have the identities and rI [AI ; ]: rJ AI + FIJ [rI ; rJ ] = [FIJ ; ]: { 9 { In particular, covariant derivatives commute if the associated eld strength vanishes. (3.23) (3.24) (3.25) (3.26) (3.27) (3.28) (3.29) When computing the variation of the WZW-like action we need four derivations of the star product, with associated potentials: The variational derivative denotes an arbitrary variation of the interpolating eld In the lifted A1 theory, the potentials are naturally de ned by where in the last equation the string eld aQ(t) is de ned aQ(t) A comment about notation: the potentials, eld strengths, and related objects are functions of the interpolating variable t. To avoid clutter, we will often leave this implicit. When the dependence is not explicitly indicated, we will always assume that the interpolating variable has been set equal to t. So, for example, A should be interpreted as A (t), and r should be interpreted as r (t). The exception to this rule will be the dynamical string eld and gauge parameters, where the dependence on t is always indicated except when t = 1. The key property of the potentials (3.34){(3.37) is that the associated eld strengths vanish along the As we will review in a moment, this property implies the expected formula for the variation of the WZW-like action. Therefore, the vanishing of these eld strengths is the basis for the claim that the lifted A1 action can be written in WZW-like form. It is not di cult to show that these eld strengths vanish by direct substitution of the provided de nitions [1], but it will be helpful to give a slightly more general argument. Suppose the list derivations @I includes and other derivations which we denote collectively as @i (with a lower case index): F F = 0; = 0; Ft = 0; FQ = 0: For example, @i should include Q; d=dt and . We assume that the @is commute among The string eld ai(t) will be called the little potential, and is de ned where the coderivation Di is the image of @i under mapping with G^ : ai(t) = ( 1)deg(i) Z t = ( 1)deg(i) Z t 1Di 1 ds 1Di d ds ds 1Di 1 1 1 A(t) 1 : A(s) 1 A(s) ; Now let us demonstrate that the eld strength Fi vanishes. Compute and @i is the coderivation corresponding to @i. This de nition agrees with aQ(t) given in (3.38), while in the t and directions it simpli es to since d=dt and happen to commute with G^ . The little potential satis es the identity and are derivations of the open string star product. By a similar argument it follows that [Di; Dj ] = 0. Now act on the little potential ai and 1 A(t) 1 where deg(i) = (i) is the degree of @i. To see that this identity holds, note that commutes r Ai = 1( = 1G ^ = 1G ^ 1 1 m2)G^ 1 1 1 A(t) A(t) 1 A(t) ai(t) ai(t) ai(t) 1 1 1 1 1 A(t) A(t) ; : (3.44) (3.45) (3.46) (3.47) (3.48) (3.49) (3.50) (3.51) (3.52) (3.53) (3.54) (3.55) r Ai = ( 1)deg(i) ^ 1G 1 1G^ Di 1 1 1 A(t) 1 1 A(t) hF t; QA iL = hF t; r AQiL; that F vanishes is a di erent but straightforward computation, given in [1]. ( 1) (i) r Ai vanishes as claimed. The proof large Hilbert space with the same quantum numbers as A. The action is de ned by the potentials B (t) B (t) ( e B(t))e ( e B(t))e B(t); B(t); Bt(t) BQ(t) d dt e B(t) e B(t); (Qe B(t))e B(t): Bi(t) B(t): All eld strengths in the Berkovits theory vanish. By contrast, except along the direction, eld strengths in the lifted A1 theory do not vanish. However, they satisfy the Bianchi identity: rI FJK + ( 1) (I)( (J)+ (K)) rJ FKI + ( 1) (K)( (I)+ (J)) rK FIJ = 0: Suppose we choose @I to be . Then the second two terms in this equation vanish, and we nd that the nonvanishing eld strengths are covariantly constant in the gauge invariance. 3.2 Variation of the action and gauge invariance Consider the variation of the integrand of the WZW-like action: hAt; QA iL = h At; QA iL + hAt; Q A iL; = hrtA ; QA iL + hF t; QA iL + hQr A ; AtiL; where in the second step we applied (3.28) to express the variation in terms of A . The term with the eld strength actually vanishes because (3.56) (3.57) (3.58) (3.59) (3.60) (3.61) (3.62) (3.63) (3.64) (3.65) and the eld strength is annihilated by r . Pulling Q and the r o the A in the last term of (3.64), we therefore have hAt; QA iL = hrtA ; QA iL hA ; r QAtiL: Now apply (3.28) again in the second term to replace Q by a t derivative: hAt; QA iL = hrtA ; QA iL hA ; r (rtAQ + FQt)iL: Again the eld strength does not contribute since it is annihilated by r . Moreover, we HJEP02(16) can commute r and rt since Ft = 0. Therefore hAt; QA iL = hrtA ; QA iL hA ; rtr AQiL; = hrtA ; QA iL + hA ; rtQA iL; d = dt hA ; QA iL: SA = hA ; QA iLjt=1 : Integrating t from 0 to 1, we obtain This is the expected variation of a WZW-like action. We would now like to use this formula to prove that the lifted A1 theory has the expected gauge invariances. First consider the gauge invariance (3.10). For this transformation, the potential A takes the form (3.66) (3.67) (3.68) (3.69) (3.70) (3.71) (3.72) where for later reference we de ne Plugging in we nd A jt=1 = 1G ^ = 1G ^ = 1( = r 1 1 1 1 A m2)G^ A 1 Bjt=1 ; B ^ 1G 1 1 A 1 A A A A 1 1 1 1 A A A ; ; 1 1 1 A : SA = hr B; QA iLjt=1 = 0; 1 A ; since QA is annihilated by r . This proves that the action is invariant under this gauge symmetry. Next consider the BRST gauge invariance (3.11). For this transformation, the potential A takes the form A Q jt=1 = 1G ^ where we substituted h!Lj ^ 1G ^ 1G 1 1 ^ 1G ^ 1G 1 1 1 1 1 A 1 A QA jt=1 = 1G ^ A 1 1 ^ 1G 1 A 1 1 A 1M 1 ^ 1G 1 1M 1 1 1 1 A A A 1 1 1 A ; 1M 1M 1 1 1 A 1 A 1 ; 1 ; A A (3.73) (3.74) : (3.75) where we have de ned The Q B term vanishes when contracted with QA jt=1 since Q is nilpotent. Substituting the B term into the variation, while switching to the degree grading, we obtain the 1M 1 A 1 1 A ; (3.77) 1 1 1M A 1 A A 1 1 1 1 1 1M 1 1 1 A 1 1 A A A A A 1 1 1 1 A A A 1 1 1 1 A A ; (3.76) QSA = 0: and the action contains the expected BRST gauge symmetry. 3.3 Higher potentials and the Maurer-Cartan equation We would now like to get a better understanding of the nonvanishing eld strengths of the lifted A1 theory. We already know that they are covariantly constant in the direction. But since F vanishes, the covariant derivative r is nilpotent. In fact, it turns out that the eld strengths are trivial in the r cohomology: (3.78) (3.79) (3.80) (3.81) 1 A(t) (3.82) More generally, Aij (t) 1G^ ( 1)(deg(i)+1)(deg(j)+1) + + 1 1 1 1 A(t) A(t) where the sign is chosen for later convenience. The string eld Aij will be called the 2-potential, and is given by the formula and used graded antisymmetry of the symplectic form. Equation (3.76) can be further simpli ed to QSA = We will prove this in appendix A. Note that we can drop the factor of G^ from (3.78) because G^ is cyclic. Then the projector 2 acts directly on the object in parentheses. However, the object in parentheses has no two-string component | it contains at minimum a tensor product of three string elds. Therefore Ft = FtQ = F Q = FQQ = r At ; r AtQ; r A Q; r AQQ: Fij = ( 1)( (i)+1) (j)+1 r Aij ; 1 1 1 1 A(t) 1 1 A(t) A(t) ai(t) : ai(t) 1 1 A(t) 1 1 A(t) The string eld aij (t) will be called the little 2-potential, and is de ned Interestingly, the 2-potential Aij looks like a two-index generalization of the potential Ai given in equation (3.44), and the little 2-potential aij looks like a two-index generalization of the little potential ai given in equation (3.45). Computing r Aij reproduces the eld strength Fij as a result of the identity 0 = aij (t) + ( 1)(deg(i)+1)deg(j) 1Di + ( 1)deg(j)+1 1Dj 1 1 A(t) 1 1 ai(t) A(t) 1 1 which is a kind of two-index generalization of the identity (3.48). The 2-potentials have not played a role so far since they do not appear in the action. But there are other expressions for the action where 2-potentials do appear. For example, if we try to express the action in the standard WZW-like form (3.21), we nd an additional term proportional to the 2-potential AtQ: 1 2 SA = A similar generalization of the standard WZW-like action has also been discussed in the context of heterotic string eld theory [6]. It turns out that the story does not end with 2-potentials. By factoring r Bianchi identity for the nonvanishing eld strengths, we learn that the 2-potentials must satisfy the identity: riAjk + ( 1) (j)( (i)+ (k))+ (j)+ (k)rj Aki + ( 1) (i)( (j)+ (k))+ (i)+ (j) rkAij + ( 1)( (i)+1)( (j)+ (k)+1) r Aijk = 0; where Aijk is a new object called the 3-potential. Acting on this equation with the covariant derivative ri and symmetrizing, one can further introduce a 4-potential, and so on. To clarify the structure of the hierarchy, it is helpful to invoke the language of di erential forms.6 For each derivation @i we formally introduce a corresponding basis 1-form: 6The author would like to thank S. Konopka for discussion which clari ed the nature of this hierarchy. (3.83) HJEP02(16) (3.86) (3.87) Note that we do not introduce a 1-form dual to . The basis 1-forms carry Grassmann parity and degree and ghost and picture number (dxi) deg(dxi) (i) + 1; mod Z2; gh(dxi) 1 picture(dxi) 1 We consider the algebra of \string eld-valued" di erential forms, de ned in the obvious way by tensoring the wedge product with the open string star product, with the appropriate signs from (anti)commutation of string elds and forms. We de ne the exterior derivative 1 2! d The exterior derivative is nilpotent and commutes with , since the @is commute among themselves and with . Also, d acts as a derivation of the wedge/star product. Next, we introduce di erential forms corresponding to the n-potentials:7 A(0) A ; A(1) dxiAi; A(2) dxi ^ dxj Aij ; A(3) dxi ^ dxj ^ dxkAijk; : : : : 1 3! Note that A is a scalar, while Ai are components of a 1-form. Therefore A interpreted as the 0-potential and Ai as the 1-potential, and the pattern completes to higher potentials. Accounting for the Grassmannality, ghost and picture number of the basis 1-forms, the n-potentials and exterior derivative are Grassmann odd, carry ghost number 1 and picture 1. On the basis of previous calculations it is easy to check that the potentials satisfy 0 = A(0) A(0) A(0); 0 = dA(0) + A(1) 0 = dA(1) + A(2) 0 = dA(2) + A(3) The rst two identities are equivalent to the statement that the eld strengths vanish in the direction. The third is equivalent to the statement that the nonvanishing eld strengths are trivial in the r cohomology, and the fourth is implied by factoring r Bianchi identity. It is clear that these identities arise as components of a Maurer-Cartan equation: (d + )A A A = 0; (3.96) 7The eld strength as de ned in (3.24) does not have the right symmetry properties in the odd directions to de ne a 2-form. This can be xed by multiplying the eld strength by the appropriate sign, but we did not bother since it would contradict the de nition used in previous papers. This is the origin of the sign factor relating the 2-potential and the eld strength in (3.81). where A is given by the formal sum A A(0) + A(1) + A(2) + A(3) + : : : : (3.97) We will call A the multi-potential. The Maurer-Cartan equation can be thought of as an \equation of motion" which determines the potentials and their higher-form descendants as functionals of the interpolation (t). Once we have solved this equation, we can write down a WZW-like action. Actually, for this purpose we need to assume an additional regularity condition: an arbitrary variation of the dynamical eld produces an arbitrary A at t = 1: HJEP02(16) Regularity Condition : arbitrary ! arbitrary A jt=1: (3.98) This needs to be true, otherwise the variation of the WZW-like action (3.69) does not imply the correct equations of motion. Thus, for example, A = 0 would not be a regular solution for the purposes of de ning a WZW-like action. At least perturbatively, the potential B of the Berkovits theory and the potential A of the lifted A1 theory are regular, since, to leading order in the string eld, they are proportional to B(t) and A(t), respectively. Berkovits' superstring eld theory provides a solution to the Maurer-Cartan equation in the form B = B + dxiBi: Since the eld strengths vanish, the higher potentials can be set to zero. In the lifted A1 theory the solution to the Maurer-Cartan equation is more interesting. We introduce di erential forms corresponding to the little potentials: dxi ^ dxj aij ; a(3) dxi ^ dxj ^ dxkaijk; : : : : 1 2! 1 3! For convenience we have de ned the little 0-potential to be a(0) = A(t). All little npotentials are degree even, ghost number 1 and picture 1 once we account for the basis 1-forms. We de ne the little multi-potential as the formal sum and postulate a solution to the Maurer-Cartan equation in the form a a(0) + a(1) + a(2) + a(3) + : : : ; A = 1G ^ 1 1 a : Here the tensor algebra of \string- eld-valued" di erential forms is de ned in the obvious way by tensoring the wedge product of the basis 1-forms with the tensor product of string elds, with the appropriate signs from (anti)commutation of the string elds and forms. Note that (3.102) agrees with our earlier expressions for A ; Ai; Aij once we extract the zero, one and two form components. The Maurer-Cartan equation for the multi-potential A translates into an equation for the little multi-potential a: 1( + D) 1 1 a = 0; (3.99) (3.100) (3.101) (3.102) (3.103) where D dxiDi. Note that this produces (3.48) and (3.84) when we extract the 1-form and 2-form components. The solution to this equation is not unique, but extrapolating from the expressions for ai and aij , we propose a solution of the form Z t 0 a(t) = ds 1( + D) 1 1 a(s) This de nes the little potentials and their higher-form descendants recursively; the n-form component determines the little n-potential in terms of products of little k-potentials for k < n. To show that this formula solves (3.103), take the t derivative of the left hand side of (3.103) and substitute a 1 a 1 1 a a a 1 1 a 1 1 a(t) : Note that (D+ )2 vanishes since d and are nilpotent and mutually commuting derivations of the star product. Bringing the last two terms to the other side of the equation gives 0 = d dt 1( + D) 1 1( + D) A 1 1 1 a a + 1( + D) 1 1 1 a 1 a Since 1( + D) 1 1 a vanishes at t = 0 (because This is a rst order homogeneous di erential equation in the string eld 1( + D) 1 1 a . A(t) vanishes at t = 0), it must vanish for all t. Therefore (3.103) is satis ed, and (3.102) gives a solution to the Maurer-Cartan equation. One advantage of the Maurer-Cartan equation is that it gives a clearer understanding of the symmetries implicit in the choice of potentials used to express the action in WZW-like form. In particular, solutions can be modi ed by an in nitesimal \gauge transformation" mcA = (d + ) is a sum of n-form gauge parameters (3.104) (3.105) 1 1 1 1 a ; 1 1 a a : (3.106) 1 1 a (3.107) (3.108) (3.109) In general can be a functional of the interpolation (t), and is Grassmann even, ghost and picture number zero. Note that this \gauge transformation" alters the choice of the potentials | not the dynamical string eld on which the potentials implicitly depend. It is interesting to see how this transformation e ects the WZW-like action. For this purpose it is enough to consider how the 0- and 1-potentials transform: mcA = r (0); mcAi = ri (0) + r i : Following a similar computation as in the previous section, it is not di cult to show that 0 dthAt; QA iL = h (0); QA iLjt=1: From this we make two observations. First, while the 1-form parameter potentials, it does not change the action. Second, the 0-form parameter i alters the (0) changes the action in the same way as a variation of the eld. Thus the action after the transformation is the same as the original action with the replacement + , where is determined ! by solving the equation A = (0): Thus the Maurer-Cartan gauge transformation alters the WZW-like action at most by a eld rede nition, and then only if the zero-form component of the transformation parameter is nonvanishing. Furthermore, it is clear that a Maurer-Cartan gauge transformation of the form (0)jt=1 = Q B + r B will leave the action invariant. By equating this with A , this determines an in nitesimal gauge transformation of the elds. In this sense, the Mauer-Cartan equation plays a role for the WZW-like action somewhat analogous to the role of cyclic A1 algebras in the A1 action: it is an algebraic structure that is covariant under eld rede nition and implies the existence of a gauge invariant action. The interpretation of the WZW-like action suggested by the Maurer-Cartan equation is slightly di erent from the earlier interpretation in terms of potentials and eld strengths. For example, from the Maurer-Cartan perspective A is a scalar, and so should not de ne components of a eld strength. Nevertheless, the statement that the eld strengths vanish in the direction is a convenient way to characterize the essential information in the Maurer-Cartan equation for the purposes of writing the action. We will translate back and forth between both interpretations as convenient. 3.4 Little potentials The action of the lifted A1 theory can be formulated in a di erent way using little potentials. In (3.12) the lifted A1 action was expressed in the form 0 dt !L (3.110) (3.111) (3.112) (3.113) Note that the little potentials satisfy where the second equality follows from (3.48). Thus the A1 action can be expressed This is reminiscent of a WZW-like action, but note that the little potentials do not come from a solution to the Maurer-Cartan equation. Instead, the relevant equation for the little HJEP02(16) potentials is (3.103). To make the description in terms of little potentials more familiar, we can de ne a few objects by analogy with the potentials and eld strengths of the WZW-like action. First, de ne the little potential in the direction to be the little 0-potential: a a(0) = A(t): fij ( 1)(deg(i)+1)deg(j)+1 aij : Second, we de ne a \little eld strength:" Unlike the WZW-like formulation, it does not make sense to de ne components of the little eld strength in the direction, for reasons that will be clear in a moment. The little eld strength is graded antisymmetric, ; 0 dt !L(at; aQ): Note that this last identity is not a special case of the previous one after choosing Di to be and setting fij to zero. This is why it doesn't make sense to de ne components of the and constant in the ( 1)deg(i)deg(j)fji; fij = 0: Recall that the potentials and eld strengths satisfy (3.28) Little potentials satisfy analogous identities which allow us to swap the index of a coderivation with the index of the little potential. In particular, (3.84) can be expressed 1Di 1 1 a aj 1 1 a = ( 1)deg(i)deg(j) 1Dj ai 1 1 a + fij ; (3.122) where switching i and j adds the little eld strength fij . Meanwhile (3.48) can be expressed ai = ( 1)deg(i) 1Di 1 1 1 a 1 a : (3.115) (3.116) (3.117) (3.118) (3.119) (3.120) (3.121) (3.123) !L(at; aQ) = !L( a_ ; aQ) + !L a_ ; 1M Using cyclicity of M one can show that 1 1 a 1 1 a !L a_ ; 1M a 1M We prove this in appendix A. Therefore !L(at; aQ) = !L( a_ ; aQ) = !L( a_ ; aQ) = !L( a_ ; aQ) + !L ( a_Q; a ) ; d = dt !L(a ; aQ): !L !L 1M 1 d dt 1M 1 1 a 1 a Integrating from 0 to 1 gives the variation of the action This implies that the equations of motion can be expressed which using (3.123) is equivalent to little eld strength in the direction. Except for this small di erence, the formulation is quite similar to that of the WZW-like action. Now let's compute the variation of the action. Take the variation of the integrand: (3.124) ! ; (3.127) (3.128) (3.129) (3.130) (3.131) 1 1 a 1 1 a a a_ a_ ; a 1 1 ; 1 a 1 1 a 1 a : (3.125) ; a : (3.126) ; a ; SA = !L(a ; aQ)jt=1: aQjt=1 = 0; 1M 1 = 0: 1 1 A A Q 1G 1 = 0: !L(at; aQ) = !L( at; aQ) + !L(at; = !L( a_ + f t; aQ) + !L at; 1M 1 1 a 1 1 a + f Q where in the second step we used (3.122). Note that the identities (3.122) and (3.123) simplify quite a bit in the case of the derivations and d=dt since they commute with the cohomomorphism G^ . We can drop the little eld strengths f t and f Q since they are constant in the direction (in fact f t already vanishes identically). Moving in the second term onto the rst entry of the symplectic form and using (3.123) gives This is the standard expression for the A1 equations of motion. By contrast the WZW-like action gives the equations of motion in a \Berkovits-like" form These two formulations are related by conjugation with the cohomomorphism G^ . 0 0 dt hAt; QA iL; dt hBt; QB iL; where the potentials Bt and B for the Berkovits theory were de ned in (3.60). In earlier work, the relation between the Berkovits and A1 theories was described by partially gauge xing the Berkovits theory to the small Hilbert space. In this context, the eld rede nition between the theories was given by equating the respective potentials [1]: Unfortunately this condition is not enough to specify a eld rede nition between B in the large Hilbert space. It leaves a residual ambiguity related to an overall transformation. There are a number of ways to x this ambiguity. We will take a particular approach which at rst seems orthogonal but has interesting implications. Instead of equating the potentials, we will equate the t potentials: We now search for a eld rede nition relating the dynamical eld theory to the dynamical eld B of the Berkovits theory. This eld rede nition should A of the lifted A1 transform the WZW-like action of the lifted A1 theory into to the WZW-like action of the Berkovits theory (4.1) (4.2) (4.3) A and gauge (4.4) (4.5) (4.6) (4.7) A and B at t = 1. In fact, in a sense we will describe, this equation provides a eld rede nition between the entire interpolations Let us solve equation (4.4). Substituting Bt we obtain a di erential equation for The solution is provided by a Wilson line d dt e B(t) = At e B(t): g(t2; t1) = P exp ds At(s) ; Z t2 t1 where P denotes the path ordered exponential in sequence of decreasing s. To emphasize this, we have written g(t2; t1) so that the left-most argument t2 is viewed as a later time than the right-most argument t1. Thus we have e B(t) = g(t; 0) = P exp ds At(s) ; Z t 0 e B = g(1; 0) = P exp ds At(s) : 0 This is our proposed eld rede nition between A and To make sense of this eld rede nition we must clarify an important point of interpretation. At face value, the Wilson line (4.8) expresses B as a function of the entire path A(t), not only of A at t = 1. Therefore we need to be more speci c about the meaning of A(t) when t 6= 1. We will assume the dynamical eld which is subject to boundary conditions A(t) = fA(t; A); fA(0; A) = 0; fA(1; A) = A(t) is given as some time-dependent function of HJEP02(16) (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) Using this, we can compute the proposed eld rede nition out to rst order in the elds. To rst order, the t potential of the lifted A1 theory takes the form At = f_1(t) A + higher orders: Therefore, to rst order, the proposed eld rede nition takes the form 1 + B + higher orders = 1 + ds f_1(s) A + higher orders; = 1 + f1(1) f1(0) A + higher orders: (4.14) Under this assumption, then indeed the Wilson line (4.8) expresses B as a function of A. However, for this to be a valid eld rede nition, it must be invertible. To simplify analysis of this question, we will make an additional assumption about the interpolating function. The interpolating function will have an expansion in powers of the string eld: fA(t; A) = f0(t) + f1(t) A + f2(t) A A + : : : ; where the linear maps fn(t) : H n ! H are n-string products. We will assume that the zero string product in this expansion vanishes: f0(t) = 0: The boundary condition (4.10) implies f1(1) = I and f1(0) = 0. Therefore the proposed eld rede nition is simply B = A + higher orders: (4.15) This is obviously invertible at rst order, and small corrections in higher powers of the eld will not change this fact. Therefore, at least perturbatively, the Wilson line (4.8) is a valid eld rede nition between A and B. If we relax the assumption (4.12), the proposed eld rede nition will map the vacuum con guration A = 0 to a nite, pure gauge con guration for B. The question of invertibility in this case is more complicated, and we will not consider it. Now let's return to equation (4.7), which determines the Berkovits eld B(t) when t 6= 1. With a given choice of interpolating function for A(t), equation (4.7) expresses B(t) as a function of A. In turn, we can express A as a function B by inverting the eld rede nition, so in fact (4.7) gives an interpolating function for the Berkovits theory, B(t) = fB(t; B); which implicitly depends on the choice of interpolating function fA(t; A) of the lifted A1 theory. This is the sense that the equation At = Bt provides a \ eld rede nition" between For the sake of being concrete, let us compute the eld rede nition between A and B out to second order in the string eld. Expanding e B and the path ordered exponential to second order gives the expression + higher orders; B = A + 2 f1(s) A; f_1(s) A + 2 f_1(s) A; f1(s) A + f_1(s) A f1(s) A where 2 is the gauge 2-product of the A1 theory [ 5 ] and f1(t) is the 1-string product in the interpolating function (4.11). Let us restrict to a particular class of eld rede nitions which can be written exclusively in terms of ; and the open string star product. This implies that f1(t) can take the form f1(t) = x(t)I + y(t) ; where x(t); y(t) are number-valued functions of t satisfying boundary conditions x(0) = y(0) = 0 and x(1) = 1 and y(1) = 0. While there are an in nite number of possible choices of x(t) and y(t), reparameterization invariance of the eld rede nition implies that most choices are equivalent. By inspection of (4.17), it is clear that the only reparameterization invariant quantity that can appear at second order is C = ds x(s)y_(s); since the other possible combinations xx_ and yy_ are total derivatives which are xed by boundary conditions. Explicitly, we nd that the eld rede nition takes the form B = A + + C [ If we change the constant C, the eld rede nition will change by a term proportional to + 1 3 [ 1 3 1 2 1 2 A; A]: (4.16) 1 2 (4.17) (4.18) (4.19) (4.20) (4.21) One can check that this term is annihilated by , and therefore represents an gauge transformation. It is interesting to note that the eld rede nition is not completely arbitrary, despite having a free parameter C. The most general eld rede nition at second order constructed out of ; , and the star product actually has ve free parameters (only four if we require that the elds are equal at linear order). Perhaps the most signi cant appeal of this approach is that it gives the simplest possible proof of equivalence of the actions. All we have to do is prove that the are equal: B = ( e B(t))e B(t); g(t; 0) g(t; 0) 1; = = = = Z t Z t Z t 0 0 0 ds g(t; s) ds g(t; s) d = A (t)g(t; 0)g(t; 0) 1; = A : At(s) g(s; 0) g(t; 0) 1; d A (s) [A (s); At(s)] g(s; 0) g(t; 0) 1; g(t; s)A (s)g(s; 0) g(t; 0) 1; Since At = Bt and A = B , the actions are identical. According to the discussion of section 3.3, the eld rede nition must be equivalent to a Maurer-Cartan gauge transformation of the potentials. Here we should note that Maurer-Cartan gauge transformations do not transform the elds | rather they transform the potentials, and therefore the action, while keeping the string eld xed. But the net e ect is equivalent to keeping the action xed while transforming the string eld. With this understanding, the eld rede nition between the Berkovits and lifted A1 theories is equivalent to a nite Maurer-Cartan gauge transformation h i B0 = (d + )U U 1 + U BU 1 ; (4.22) (4.23) (4.24) (4.25) where B is the multi-potential of the Berkovits theory and B0 is the transformed multipotential, and the nite gauge parameter U is Z t 0 U (t) = P exp ds At[ B](s) e B(t); where At[ B](t) is the t-potential of the lifted A1 theory evaluated on the Berkovits string eld. The net e ect of this transformation is to replace the group-like element parameterized by e B with the group-like element parameterized as P exp hR0t ds At[ B](s)i. The Berkovits action will then be replaced with the lifted A1 action SB[ B] ! SA[ B]; after which we may as well rename B as A. Note that because U only has a zero form component, the Maurer-Cartan gauge transformation does not generate expectation values for the higher potentials. Therefore the transformed multi-potential B0 gives a representation of the WZW-like action for the lifted A1 theory where all eld strengths vanish. So far we have described the eld rede nition between the Berkovits and lifted A1 theories by equating the t-potentials. However, an equivalent characterization can be found by equating the little t-potentials. This naturally leads to a formula for the eld rede nition which inverts the Wilson-line of the previous section. First we need to describe the little potentials of the Berkovits theory. They are de ned implicitly by the formula where the little multi-potential b is a sum of little n-potentials B = 1G ^ 1 1 b b(0) + b(1) + b(2) + b(3) + : : : ; which can be described using the basis 1-forms dxi as b(0) = b ; b(1) = dxibi; b(2) = dxi ^ dxj bij ; b(3) = dxi ^ dxj ^ dxk bijk; : : : : (4.28) This is precisely analogous to (3.102) of the lifted A1 theory. We can invert (4.26) to express b in terms of the multi-potential of the Berkovits theory: 1 3! Multiplying this equation by G^ 1 and projecting onto the 1-string component of the tensor algebra gives Important special cases are 1 2! 1 1 B b = 1G^ 1 = 1 = G^ 1 1 1 1 B B B B 1 1 ; 1 1 1G^ 11 b ; : 1 B : b = 1G^ 1 bt = 1G^ 1 ; ; : (4.26) (4.27) (4.29) (4.30) (4.31) (4.32) (4.33) (4.34) (4.35) Note that the higher little potentials of the Berkovits theory do not vanish, even though the higher potentials do. The little multi-potential satis es by analogy with (3.103). 1( + D) 1 1 b = 0; hBt; QB i = !L = !L = !L = !L bt; 1M 1G^ 1Bt 1G^ 1 1G^ 1 1 1 1 1 1 1 B B 1 1 B ; 1G^ 1Q 1 1 1 1 1 B B 1 B ; 1MG^ 1 ; 1M ; 1 1 B 1 1 ; ; where we substituted the de nition of the little potentials. Next we use (4.35) to note that Integrating t from 0 to 1 therefore expresses the Berkovits action in the form This is the Berkovits action expressed in terms of little potentials. Now it turns out that equating the t-potentials of the Berkovits and lifted A1 theories is equivalent to equating the little t potentials. To see this, note that 1 A At 1 1 A = 1 1 B Bt 1 1 B : where we used the fact that At = Bt implies A = B . The left hand side can be expanded Next let us explain how to express the Berkovits action in terms of little potentials. Consider the integrand hBt; QB i = !L(Bt; QB ); = !L 1Bt 1 1 B ; 1Q 1 1 B ; (4.36) where Bt is the coderivation corresponding to Bt regarded as a zero-string product. Since the cohomomorphism G^ 1 is cyclic, we can use (3.13) to write 1 0 Bt Bt ; 1M 1 = bQ: SB = dt !L(bt; bQ): 1 1 A ^ G 1 1 = G^ = G^ 1 a 1 1 1 1 ^ 1G 1 1 ^ 1G 1 1 A(t) 1 at a at 1 1 a 1 A(t) 1 A(t) ; 1 1 = a : 1 B 1 at = bt: A At 1 Therefore Multiplying this equation by G^ 1 and projecting onto the 1-string component then implies ^ 1G 1 1 A(t) ; QQfA(t; A) Q Q Q s 3 T (t;0) T (1;0) g(1;0) A(t) B(t) + k Q g(t;0) Q Q fB(t; B) Q Q T (t2; t1) ds bt(s): and lifted A1 superstring eld theories. Since at(t) = _ A(t) this equation is easily integrated to express A in terms of solution of this equation is de ned by the integral Q Z t2 t1 Z t 0 0 The interpolating eld of the lifted A1 theory is therefore, and the eld rede nition from the Berkovits to the lifted A1 theory is A(t) = T (t; 0) = ds bt(s); A = T (1; 0) = ds bt(s): which is the string eld we started with. (4.44) (4.45) (4.46) (4.47) Similar to the previous section (but in reverse), this expresses A as a function of an entire path B(t) in the Berkovits theory. For this to be a eld rede nition between A and B, we assume that B(t) is speci ed by an interpolating function fB(t; B) whose zero-string product vanishes. Equation (4.45) determines the interpolating function fA(t; A) of the lifted A1 theory in terms of the interpolating function fB( B; t) of the Berkovits theory. We summarize the di erent elds, interpolations, and mappings between them in gure 1. We should emphasize that this is the same as the Wilson line eld rede nition of the previous section, but inverted. To see this, suppose we express the Berkovits interpolation B(t) in terms of the lifted A1 interpolation A(t) using the Wilson line (4.7). This allows us to write Bt = At and B = A , and substituting into (4.45) gives 1 1 1 A (s) 1 At(s) A(s) ; ; A(t) = ds 1G^ 1 = = = Z t 0 Z t Z t 0 0 ds 1G^ 1G^ ds _ A(s); A(t); Let us describe an unrelated proposal for the eld rede nition suggested by Y. Okawa.8 This approach does not consider the interpolations B(t), but focuses on the dynamical elds A and B at t = 1. The starting point is the condition Since a jt=1 = A, this gives As mentioned before, this relation does not fully constrain the eld rede nition. However, this can be remedied with a few additional choices. Equating the -potentials is equivalent to equating the little -potentials: Suppose we assume that the lifted A1 the above relation implies that the lifted A1 eld satis es eld satis es the gauge condition A = 0. Then A = b jt=1 if A = 0: This is not quite a eld rede nition between A and B since it is not invertible | the eld A always satis es the gauge condition A = 0. We can x this by adding the -closed term A(t) = B(t) + b (t): 8The author would like to thank him for sharing this idea. (4.48) (4.49) (4.50) (4.51) (4.52) (4.53) (4.54) A(t) (4.55) Taking of this expression implies a jt=1 = b jt=1, as desired. A nice property of this eld rede nition is that it is compatible with the natural gauge xing to the small Hilbert space. In particular, xing theory, and vice-versa: 1 3 A = 0 in the lifted A1 theory xes B = 0 in the Berkovits A = b jt=1: 1 A = B + b jt=1: A = 0 B = 0: This is not generally true for the eld rede nition based on the Wilson line. Expanding the eld rede nition to second order gives B = A + [ A; A] [ A; A] + higher orders: Note that this is not a special case of (4.20) for some choice of the constant C. Therefore the eld rede nition cannot be realized as the endpoint of a pair of interpolations and B(t) related by At = Bt. This makes the proof of equivalence of the actions less direct. First let us generalize the eld rede nition to intermediate t by taking With this identi cation the -potentials are equal for intermediate t but the t-potentials are not the same. Instead we have A (t) = B (t); At = B~t; where B~t is de ned 1G 1 1 _ B(t) + b_ 1 1 (4.58) HJEP02(16) (4.56) (4.57) (4.59) (4.60) (4.61) (4.62) (4.63) which is not the same as Bt. Therefore applying the eld rede nition to the lifted A1 action gives 0 SB = dt hB~t; QB iL: This is not the Berkovits action as it is usually written. However, as noted in [1], B~t usual Berkovits action. Another way to understand this observation is that (4.59) is an expression of the Berkovits action using a nonstandard set of potentials, related to the usual Berkovits potentials by a nite Maurer-Cartan gauge transformation h B~ = (d + )e (1) i e (1) + e Be (1) ; where the 1-form gauge parameter (1) is This Maurer-Cartan gauge transformation leaves B invariant, while it transforms Bt into B~t: We can compute the gauge parameter t as follows. Consider where in the second step we traded b_ with bt. Now pull out so it acts on the entire Bt Bt = 1G = 1G = 1( 1 1 1 1 1 1 b bt) + bt bt) b bt) 1 1 1 b 1 b + 1G ^ 1 : 1 1 1 1 b Bt; 1 b Bt; Bt; b t 1 1 b Bt; (4.64) Bt = 1G ^ = 1G 1 1 1 1 b _ B(t) + b_ _ B(t) + b t (1) = dt t: B~t = Bt + r t In the second step the second term cancels with Bt by the de nition of the little potentials. What remains can be interpreted as r t, where t is t = 1G 1 1 bt) : (4.65) Therefore relating the interpolations through (4.55) and performing a Maurer-Cartan gauge transformation turns the WZW-like action of the lifted A1 theory into the standard WZWlike action of the Berkovits theory. It is interesting to contrast the variety of eld rede nitions we nd in the large Hilbert space with the seeming uniqueness of the eld rede nition found in the small Hilbert space [1, 2]. The reason for this discrepancy is that the operations , ; m2 used to construct the eld rede nition in the large Hilbert space can also implement gauge transformations of the eld rede nition. In the small Hilbert space the gauge invariance is not present, and and m2 alone cannot implement interesting gauge transformations in the small Hilbert space. 5 Mapping gauge invariances the lifted A1 of (4.8) produces eld We will now use the eld rede nition to determine how the gauge symmetries of the lifted Let us rst consider the Wilson line eld rede nition (4.8). To get the information we're after, we must compute the change of the Berkovits eld B induced by a change in A and/or the interpolating function fA(t; A). Taking the variation 0 e B = dt g(1; t) At g(t; 0): Using (3.28) we can switch the variation with a time derivative: d e B = dt g(1; t) rtA + F t g(t; 0); g(1; t)A g(t; 0) + dt g(1; t)F t g(t; 0); 0 Z 1 0 = A jt=1e B e B A jt=0 + dt g(1; t)F t g(t; 0): This is the expected formula for the variation of a Wilson line. We assume that A vanishes at t = 0 because the interpolating function is required to satisfy the boundary condition fA(0; A) = 0. Let us see what to do with the eld strength integrated along the curve. We can express the eld strength in terms of the 2-potential 0 dt g(1; t)F t g(t; 0) = dt g(1; t) r At g(t; 0); = = dt g(t; 0) 1 r At g(t; 0); dt g(t; 0) 1At g(t; 0) : 0 e B e B (5.1) (5.2) (5.3) (5.4) Therefore, the variation of the eld rede nition is e B = A jt=1e B e B dt g(t; 0) 1At g(t; 0) ; 0 is constrained to vanish by boundary conditions, and (5.5) simpli es to rede nition between what we found in (4.21). Now let us see how Right multiplication of e B by an -closed string eld is an in nitesimal gauge transformation in the Berkovits theory. Therefore, a change of the interpolation only e ects the eld A and B by an gauge transformation. This is consistent with B responds to gauge transformations in the lifted A1 theory. First consider the gauge transformation in (3.10). We obtain 0 e B = A jt=1e B e B g(t; 0) 1At g(t; 0) : We can simplify the rst term using equation (3.71): Plugging in we obtain A jt=1 = r Bjt=1; = g(1; 0) g(1; 0) 1 B g(1; 0) g(1; 0) 1: e B = e B 0 dt g(t; 0) 1At g(t; 0) : HJEP02(16) (5.5) (5.6) (5.7) (5.8) (5.9) (5.11) e B = e B g(1; 0) 1 B g(1; 0) dt g(t; 0) 1At g(t; 0) : Therefore, the gauge invariance of the lifted A1 theory maps into the of the Berkovits theory. Now consider the BRST gauge transformation Q in (3.11). From the computation of (3.73), we nd 0 Z 1 0 1M 1 1 A = 0: Qe B = B e B e B dt g(t; 0) 1At Q g(t; 0) : (5.10) Left multiplication of e B by a BRST closed string eld is an in nitesimal gauge transformation in the Berkovits theory. It follows from the computation at the end of subsection 3.2 that left multiplication of e B by B is also a symmetry of the action, even though B is not BRST closed. However, note from (3.75) that B vanishes when the equations of motion are satis ed: Therefore left multiplication by B must represent a trivial gauge transformation [23]. The upshot is that the BRST gauge transformation of the lifted A1 theory maps into a combination of a BRST gauge transformation, an gauge transformation, and a trivial gauge transformation in the Berkovits theory. Now let's consider the reverse question, namely, how the gauge invariances of the Berkovits theory map into those of the lifted A1 theory. For this purpose it is useful to consider the inverse eld rede nition as described in section 4.2. Taking the variation of (4.46), one nds that a change of the Berkovits eld and/or interpolation changes the lifted A1 eld through A = = 0 dt bt; d b = b jt=1 0 b t ; dt b t ; where we used (4.35) to interchange with a d=dt, which produces a term proportional to the little 2-potential. If we change the interpolating function of the Berkovits theory, the boundary term at t = 1 drops out and what remains is an gauge transformation of The BRST and gauge transformations of the Berkovits theory can be written Qe B = Q B e B ; e B Be B ; Before B and B were de ned as functions of A and the gauge parameters A and A of the lifted A1 theory, but now we view them as independent variables de ning the gauge parameters of the Berkovits theory. The potentials corresponding to these variations are B Q jt=1 = Q B; B jt=1 = r Bjt=1: Let us rst compute the little potential b jt=1 from (4.33): B 1 B B B (5.12) (5.14) (5.15) (5.16) (5.17) b jt=1 = Plugging this into (5.12), we nd where 1G^ 1 1G^ 1 1G^ 1 1 1G^ 1 1 1 1 B B B r m2) 1 1 B 1 1 1 1 B 1 B 1 B t=1 B B t=1 t=1 t=1 ; A = A dt b t ; 1G^ 1 1 1 B 1 1 B t=1 Therefore the gauge invariance of the Berkovits theory maps into the of the lifted A1 theory. Note that (5.17) is the inverse of the formula (3.71) expressing B as a function of A. Therefore we have a \ eld rede nition" relating the gauge parameters in the two theories. Now consider the BRST gauge symmetry: (5.18) t=1 t=1 t=1 ; ; (5.19) (5.20) (5.21) (5.22) (5.23) B 1 1 1 B 1 1 1 B 1 A 1 t=1 B 1 1 B 1 1 1 t=1 A 1 B QB B B B 1 1 1 1 QB 1 B t=1 dt b Qt ; 1 1 B t=1 1G^ 1 1G^ 1 1 1G^ 1 = 1M 1 1 1G^ 1 1 B QB B QB QB 1 1 1 1 B t=1 1 B t=1 1 B t=1 1 B t=1 1 1 1 1 1G^ 1 B B B 1 Using where we therefore obtain Q A = A A A = 1G^ 1 1M 1 1G^ 1 1G^ 1 1 1 1 B 1 A 1 1 B 1 1 B 1 B A B QB 1 1 B Note that (5.21) is the inverse of the formula (3.74) expressing B in terms of A. Also note that A vanishes on shell, and so must represent a trivial gauge transformation. Therefore the BRST gauge transformation of the Berkovits theory maps into a combination of a BRST, an , and trivial gauge transformations in the lifted A1 theory. Similar conclusions follow using the eld rede nition proposed by Okawa, since in this case the variation takes the form A = b jt=1 + B b jt=1 : which, aside from unimportant di erences in the closed term, is equivalent to (5.12). Acknowledgments The author would like to thank T. Takezaki and Y. Okawa for collaboration, and S. Konopka and Y. Okawa for comments. This work was supported in parts by the DFG Transregional Collaborative Research Centre TRR 33 and the DFG cluster of excellence Origin and Structure of the Universe. A Some computations involving cyclicity In this appendix we provide a few missing calculations referred to in the text, in particular as pertains to cyclicity of the A1 products and cohomomorphism G^ . These calculations are simpli ed with the help of the \triangle formalism" of the product and coproduct, introduced in appendix A of [2]. Here we review this formalism and provide the missing calculations in the text. algebra into a pair of tensor algebras: The tensor algebra has a coproduct, which is a coassociative linear map from the tensor and group-like elements satisfy where we use the symbol 0 to distinguish from the tensor product used to construct T H. The coproduct is coassociative and acts on tensor products of states as 4A1 : : : An = : : : Ak) 0 (Ak+1 : : : An); where at the extremes of summation 0 multiplies the identity of the tensor product 1T H. Note that 1T H is not the identity with respect to 0. Coderivations and cohomomorphisms satisfy 4 : T H ! T H 0 IT H)4 = (I 0 4)4; n X(A1 k=0 4D = (D 4H^ = (H^ 0 IT H + IT H 0 H^ )4; 0 D)4; 4 1 1 A 1 1 A 0 1 1 A 0 T H ! T H; m+n =4 h m 0 n 4: (A.1) (A.3) (A.4) (A.5) (A.6) (A.7) m+n (A.8) By taking variations we can derive the action of the coproduct on more general states. In addition, the tensor algebra has a product which operates by replacing the primed tensor product 0 with the ordinary tensor product . The central formula of the triangle formalism is an expression for the projector onto the (m + n)-string component of the tensor algebra: For further elaboration, see appendix A of [2]. h!j 2H h!j 2H 1 1 1H 1H 1 1 1H = h!j + h!j + h!j Now let us revisit the derivation of (3.13), which is also featured in appendix A of [1]. A cyclic cohomomorphism H^ satis es Consider this formula acting on a particular element of the tensor algebra: where A is degree even and B; C are arbitrary string elds. We nd On the other hand, we can replace 2 =4 h 1 0 1 4 on the left hand side and act with 1 B 1 B 1 A 1 0 1 4H ( 1 0 1)(H^ 1 1 1 A 1 A A B 1 1 1H A A 1 1 1 A 1 1 1 1H 1H A 1 1 B 0 A A B B 0 1 1H 1 A 1 1 1 1 A 1 1 1H A 1 1 1 1 A C C 1 A B 1 1 C A 0 C A 1 1 A B 1 1 1 A 1 1 1 A 0 A B 1 1 1 C 1H B 1 1 1 1 1 1 1 A A = h!j 2 = h!jB C C 1 1 B 1 C C 1 A 1H B 1 1 1 1 A A B 1 C 1 1 A B 1 1 1H 1 C C A C 1 A 1 A C 1 1H 1 1 A 1 1 1 1 1 A A A 1 1 A A (A.9) (A.10) 1 ; (A.11) C (A.12) The rst and last terms cancel by antisymmetry of the symplectic form. The second term, however, remains. We have therefore shown Now suppose that the string elds A and B happen to take the form = !(A; B): (A.13) ; 1H 1 1 A 1 1 A 1 1 A C A = 1D1 B = 1D2 for some coderivations D1 and D2. Plugging into the above formula then gives 1D1 ; 1D2 Now let's prove the equivalence of equations (3.78) and (3.76). The left hand side of (3.78) is 1 0 1 4 and acting with the coproduct produces the expression 1 1 1H which reproduces (3.13). h!Lj 2G A.2 A 1 1 A 1 h!Lj B 1G 1 + !L 1G 1 1 A A 1 A ; 1G A 1 1M 1M 1 1 1M 1 A 1 1 A 1 A 1M 1 A A A 1 1 1 1 1 1 1 1 1 A 1 1 1 1 1 A A A A 1 0 1 1 A 1 1 1 1 A A 1 1 1 1 A 1 1 A A 1M 1 1 1 1 1M 1 0 1 1 1 1 1 1 1 1M A A A 1 1 A A A 1 1 A A 1 1M 1 1M 1M 1 1 1 1 A 1 A A ; 1G 1 1 1 1 1 1 A A A A A A (A.14) ; (A.15) 1 1 1 1 0 1 1 1 1 1 1 (A.16) (A.17) A A A A 1 1 1 1 1 1 1 1 A A A A 0 1 1M 1 1M 1M 1M 1 1 1 A 1 1 1 The rst and last term simplify to 1 A (A.18) and they cancel by antisymmetry of the symplectic form. Meanwhile, the second and third terms in (A.18) produce equation (3.76). This lls the missing steps between (3.76) and (3.78). Now let's prove (3.126): a 1 1 0 = h!Lj 0 = h!Lj 2M a 1 !L 1 a 1 1 a 1M 1 1 1 a 0 IT H +IT H 0 1 a For this purpose consider the identity which vanishes because M is cyclic with respect to the large Hilbert space symplectic form. 1 0 1 4 and acting with the coproduct gives 1 1 a 1 1 a 0 1 1 a a 1 1 a 0 = h!Lj + h!Lj + h!Lja + h!Lj a_ 1M 1 1M 1 1 a 1 1M 1M a 1 1 1 1 a a 1 a 1 1 a 1 a 1 1 1 a 1 1 1 a a 1 a a 1 1 a a 1 a 1 1 1 a a Some cross terms drop out since 1 acts on the tensor product of two or more states. What is left is The second and third terms cancel out by antisymmetry of the symplectic form, while the rst and last terms reproduce (3.126). Open Access. 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. [1] T. Erler, Y. Okawa and T. Takezaki, A1 structure from the Berkovits formulation of open superstring eld theory, arXiv:1505.01659 [INSPIRE]. [2] T. Erler, Relating Berkovits and A1 superstring eld theories; small Hilbert space perspective, JHEP 10 (2015) 157 [arXiv:1505.02069] [INSPIRE]. [3] N. Berkovits, SuperPoincare invariant superstring eld theory, Nucl. Phys. B 450 (1995) 90 [Erratum ibid. B 459 (1996) 439] [hep-th/9503099] [INSPIRE]. [4] N. Berkovits, A new approach to superstring eld theory, Fortsch. Phys. 48 (2000) 31 [hep-th/9912121] [INSPIRE]. 1 a ; a : (A.19) 1 a 1 1 a a (A.20) 1 1 0 1 a 1 a (A.22) (2014) 150 [arXiv:1312.2948] [INSPIRE]. 08 (2014) 158 [arXiv:1403.0940] [INSPIRE]. theory, arXiv:1506.06657 [INSPIRE]. HJEP02(16) superstring eld theory, arXiv:1512.03379 [INSPIRE]. formulation in open superstring eld theory, JHEP 03 (2014) 044 [arXiv:1312.1677] xing, ghost structure and propagator, JHEP 03 (2012) 030 [arXiv:1201.1761] [arXiv:1201.1769] [INSPIRE]. Superstring Field Theory, JHEP 04 (2012) 050 [arXiv:1201.1762] [INSPIRE]. formulation, Prog. Theor. Phys. Suppl. 188 (2011) 272 [arXiv:1201.1763] [INSPIRE]. Actions in the Witten Formulation and the Berkovits Formulation of Open Superstring Field arXiv:1508.00366 [INSPIRE]. arXiv:1508.05387 [INSPIRE]. [5] T. Erler , S. Konopka and I. Sachs , Resolving Witten`s superstring eld theory , JHEP 04 [6] N. Berkovits , Y. Okawa and B. Zwiebach , WZW-like action for heterotic string eld theory , [7] T. Erler , S. Konopka and I. Sachs, NS-NS Sector of Closed Superstring Field Theory , JHEP [11] M. Kroyter , Y. Okawa , M. Schnabl , S. Torii and B. Zwiebach , Open superstring eld theory


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP02%282016%29121.pdf

Theodore Erler. Relating Berkovits and A ∞ superstring field theories; large Hilbert space perspective, Journal of High Energy Physics, 2016, 121, DOI: 10.1007/JHEP02(2016)121