aa r X i v : . [ h e p - t h ] J a n Normalization of D-instanton Amplitudes
Ashoke Sen
Harish-Chandra Research Institute, HBNIChhatnag Road, Jhusi, Allahabad 211019, India
E-mail: [email protected]
Abstract
D-instanton amplitudes suffer from various infrared divergences associated with tachyonicor massless open string modes, leading to ambiguous contribution to string amplitudes. Ithas been shown previously that string field theory can resolve these ambiguities and lead tounambiguous expressions for D-instanton contributions to string amplitudes, except for anoverall normalization constant that remains undetermined. In this paper we show that stringfield theory, together with the world-sheet description of the amplitudes, can also fix thisnormalization constant. We apply our analysis to the special case of two dimensional stringtheory, obtaining results in agreement with the matrix model results obtained by Balthazar,Rodriguez and Yin. 1 ontents
D-instantons give a class of non-perturbative contributions to string amplitudes. One charac-teristic of these contributions is the presence of an overall multiplicative factor e − C/g s where g s is the closed string coupling and C is a constant. Besides this factor, the amplitudes admitusual perturbation expansion in powers of g s . The contribution to an amplitude at any givenorder in g s can be computed using the standard world-sheet approach by including Riemannsurfaces with boundaries ending on the D-instanton, but at each order one encounters certaininfra-red divergences [1–3] that render the amplitudes ambiguous. At any given order, theseambiguities can be encoded in a set of undetermined constants. String field theory [4–9] pro-vides an unambiguous procedure for determining these constants, by identifying the physicalorigin of these infrared divergences and rectifying them based on this understanding [10–13].So far this procedure has been applied to two dimensional string theory, for which there is adual matrix model description that can be used to check the results.However previous analysis left one constant undetermined – namely the overall normaliza-tion of the D-instanton amplitude. Formally this is given by the exponential of the annulusamplitude, with D-instanton boundary condition at the two boundaries and no other vertexoperator insertion. However the annulus amplitude is divergent due to the presence of mass-less and tachyonic open string modes on the D-instanton. In conventional string perturbationtheory, such diagrams are part of bubble diagrams and drop out in the computation of physicalamplitudes. However for D-instanton amplitudes the situation is somewhat different since the2-instanton contribution to the amplitude has to be first added to the perturbative amplitudeand then the sum needs to be divided by the sum of perturbative and D-instanton contributionto bubble diagrams. Therefore the overall normalization is physically relevant, and one expectsthat it should be possible to compute this in string theory. Since string field theory is capableof making sense of infrared divergences in the amplitudes, the natural expectation would bethat string field theory should be able to give an unambiguous result for the normalizationconstant. However, when one tries to compute this using string field theory, which in this caseis a theory of open and closed strings, one finds that there is no internal consistency require-ment within string field theory that can be used to fix this normalization, since this can bechanged by adding a field independent constant to the string field theory action that does notviolate any constraint coming from the requirement of gauge invariance.To overcome this problem, we shall take the viewpoint that the world-sheet approachalready fixes the normalization as the exponential of the annulus partition function, and thejob of string field theory is to simply give physical interpretation of the divergences of theamplitude and render them finite based on this interpretation. We show that the world-sheetresult may be regarded as the gauge fixed version of a path integral in string theory with aspecific normalization, and the divergences that we encounter arise due to breakdown of thegauge choice. However the ‘gauge invariant’ form of the path integral, expressed as an integralover the full classical string field divided by the volume of the gauge group, yields unambiguousresult. We apply this procedure to the case of two dimensional string theory, and find that thenormalization of the one instanton amplitude determined this way agrees with the results ofthe matrix model computed in [3] following the general formalism developed in [14].The rest of the paper is organized as follows. In § § § §
3, by this factor, we get agreement with the matrix model result. In § Our goal is to compute the normalization constant N appearing in the D-instanton amplitudes.In the world-sheet description it is given by: N = ζ exp[ A ] , (2.1)where A is the exponential of the annulus partition function: A = Z ∞ dt t T r ( e − πtL ) . (2.2)Here T r denotes trace over all states of the open string projected into the Siegel gauge by theprojection operator b c , weighted by ( − F where − ( − F denotes the grassmann parity ofthe vertex operator corresponding to the state. The extra minus sign multiplying ( − F is areflection of the fact that bosonic (fermionic) open string modes correspond to grassmann odd(even) states in the world-sheet theory. ζ in (2.1) is the multiplier factor that depends on howthe steepest descent contour associated with the D-instanton fits inside the actual integrationcontour [22–24]. In particular exp[ A ] represents the one loop contribution to the path integralfrom the full steepest descent contour passing through the instanton solution and ζ reflects themultiple of the steepest descent contour that forms part of the actual integration contour. Weshall see for example that in the two dimensional string theory, ζ = 1 / The constant N given in (2.1) is the overall multiplicative factor that appears in the instan-ton induced effective action of the closed string fields [27]. This is related to the normalizationconstant N introduced in [3, 28], appearing as a multiplicative factor in the S-matrix, via therelation N = i N . In our analysis, we shall not be careful in fixing the sign of N , since thiswill be fixed at the end using separate considerations. In string field theory, (2.1) may be justified by demanding that at the tachyon vacuum [25] N must be 1 sothat we get the usual perturbative closed string amplitudes. Since the boundary state vanishes at the tachyonvacuum [26], we have A = 0 and therefore e A = 1. Furthermore it will be seen in §
4, Fig. 1 that ζ = 1 at theperturbative vacuum. Therefore (2.1) should not have any additional factor.
4e can express (2.2) as A = Z ∞ dt t "X i e − πth bi − X j e − πth fj , (2.3)where { h bi } and { h fj } are the L eigenvalues of the grassmann odd and the grassmann evenstates of the world-sheet CFT. If we assume that the total number of bosonic modes equalsthe total number of fermionic modes so that the integrand is finite as t →
0, and furthermorethat the h bi and h fj are positive so that the integrand falls off exponentially as t → ∞ , thenthe integral (2.3) is finite. In this case it gives the result A = 12 ln Q j h fj Q i h bi . (2.4)Substituting this into (2.1), we get N = ζ s Q j h fj Q i h bi . (2.5)In the system that we shall analyze, h fj ’s come in pairs of equal values so that we can writethis as N = ζ Q ′ j h fj pQ i h bi , (2.6)where Q ′ j corresponds to the product running over only one member for each pair.We can express this as the result of integration over the bosonic variables b i and fermionicvariables f j , e f j as follows: N = ζ Z Y i db i √ π Y j df j d e f j exp " − X i h bi b i + X j ′ h fj e f j f j . (2.7)Equality of (2.1) and (2.7) is an identity when all the h bi ’s and h fj ’s are positive, but we shalltake (2.7) to be the defining expression for N even when this condition fails. In particular, weshall apply this formalism to D-instanton system for which some of the L eigenvalues vanishand / or are negative. A justification for this may be given as follows. Instead of studyingopen strings on a single D-instanton, we can take a system of two D-instantons separated alongthe Euclidean time direction and analyze the states of the open string stretched between thepair of D-instantons. In this case L will get a non-vanishing contribution from the tension5f the stretched open string and the manipulations carried out above will be well definedfor sufficiently large separation. We can recover the original system of interest by analyticcontinuation of this result to zero separation and using the fact that in this limit the spectrumof open strings with two ends lying on different D-instantons coincides with the spectrum ofopen strings with both ends lying on the same D-instanton. Of course (2.7) is not well definedin this limit due to the appearance of zero eigenvalues in the bosonic and fermionic sectors,and so it does not lead to a finite unambiguous result for N at this stage. However, we shall seein § b i may be interpreted as the bosonic open string fields on the D-instanton, the variables f i , e f i may be interpreted as the fermionic open string fields on theD-instanton and the argument of the exponential may be interpreted as the quadratic part ofthe action of the open string field theory in the Siegel gauge. To see how this arises, we nowreview some basic aspects of string field theory.The off-shell open string field describing the degrees of freedom of a D-instanton is takento be an arbitrary element | Ψ i of H – the vector space of states of the open string, includingmatter and ghost excitations. Let {| φ r i} be the set of basis states in H . Then we can expand | Ψ i ∈ H as | Ψ i = X r χ r | φ r i . (2.8) { χ r } ’s are the degrees of freedom over which the path integral is to be performed after suitablegauge fixing. Even though we have referred to the χ r ’s as fields, they are actually zero dimen-sional fields – ordinary variables – since on the D-instanton the open strings do not carry anycontinuous momentum labels. Therefore it is more appropriate to call them modes. χ r haseven (odd) grassmann parity if the ghost number of φ r is odd (even). The kinetic term of theBV master action of string field theory takes the form: S = − h Ψ | Q B | Ψ i , (2.9)where Q B is the world-sheet BRST operator. The minus sign in front of the action is unusual,but has been introduced keeping in mind that we shall be using a convention in which theEuclidean path integral is weighted by e S .In the BV formalism the open string modes multiplying states of ghost number ≤ ≥ {| ϕ r i} in the ghost number ≤ {| ϕ r i} inthe ghost number ≥ h ϕ r | ϕ s i = δ rs = h ϕ s | ϕ r i , h ϕ r | ϕ s i = 0 , h ϕ r | ϕ s i = 0 , (2.10)and expand the string field as, | Ψ i = X r ( ψ r | ϕ r i + ψ r | ϕ r i ) , (2.11)then we call ψ r a field and ψ r the conjugate anti-field up to a sign. The path integral is carriedout over a Lagrangian submanifold. For our analysis it will be sufficient to consider a specialclass of Lagrangian submanifolds in which, for each pair ( ψ r , ψ r ), we set either ψ r to 0 or ψ r to0. The path integral can be shown to be formally independent of the choice of the Lagrangiansubmanifold. The Siegel gauge corresponds to the choice of the Lagrangian submanifold inwhich we impose the condition: b | Ψ i = 0 . (2.12)In this gauge the action (2.9) takes the form: S g.f. = − h Ψ | c L | Ψ i . (2.13)If we choose the basis states {| φ ( n ) r i} of ghost number n in the Siegel gauge, satisfying b | φ ( n ) r i = 0 , h φ (2 − n ) r | c | φ ( n ) s i = δ rs for n ≤ , (2.14)then by expanding | Ψ i in this basis and substituting in the action (2.13), we recover theexponent in (2.7) if we identify the variables b i , f i and e f i as the coefficients of expansion of | Ψ i in this basis.This shows that (2.7) may be given an interpretation as path integral over the open stringfields in the Siegel gauge. Note however that (2.7) comes with a specific normalization of theintegration measure that will be important for us. String field theory, by itself, cannot fix theoverall normalization of the measure, since this corresponds to adding a constant to the stringfield theory action, and the requirement that the action satisfies the BV master equation doesnot fix this constant. 7 ‘Gauge invariant’ path integral Let us now focus on the specific case of (1,1) D-instanton in two dimensional string theory. Inthis case we have [28]: A = Z ∞ dt t (cid:0) e πt − (cid:1) . (3.1)Comparing this with (2.3) we see that the contribution from all states with L > N , we can drop the integration over all states with L >
0. Therefore we shall introduce arestricted string field | Ψ R i given by a linear combination of basis states with L ≤
0. Beforegauge fixing, | Ψ R i has the following expansion: | Ψ R i = ψ c | i + ψ c c | i + ψ c | i + ψ | i + ψ c − c | i + ψ c − c c | i + ψ c α − | i + ψ c c α − | i , (3.2)where | i is the SL(2,R) invariant vacuum, c n , b n are the usual ghost oscillators and α m are theoscillators associated with the Euclidean time coordinate X , satisfying [ α m , α n ] = m δ m + n, .In the α ′ = 1 unit the X ’s satisfy the operator product expansion ∂X ( z ) ∂X ( w ) = − z − w ) . (3.3)This leads to the following state operator correspondence: c α − | i = i √ c (0) ∂X (0) | i . (3.4)The basis states in which we have expanded the string field in (3.2) are normalized accordingto (2.10) provided we choose: h | c − c c | i = 1 . (3.5)In this case the { ψ r } ’s label fields and the { ψ r } ’s label the conjugate anti-fields in the BVformalism.In the Siegel gauge, the modes that survive are ψ , ψ , ψ and ψ . Of these, ψ and ψ arebosonic modes and ψ and ψ are fermionic modes. Therefore, (2.7) may now be written as: N = ζ Z dψ √ π Z dψ √ π dψ dψ e S . (3.6) We shall use the standard doubling trick in which we regard ∂X as an analytic function over the fullcomplex plane, with the understanding that ∂X ( z ) for z in the lower half plane actually represents − ¯ ∂X ( z ). ψ . We avoid this problem by choosing the ‘gauge’ in which ψ = 0 but ψ = 0.In this case all the anti-fields are set to zero and we we integrate over all the field modes ψ , ψ , ψ , ψ . Now we have three bosonic modes ψ , ψ , ψ and one fermionic mode ψ . Theaction (2.9) now takes the form: S = − (cid:20) −
12 ( ψ ) − ( ψ ) (cid:21) . (3.7)This shows that we still have a pair of zero modes – one bosonic zero mode ψ and one fermioniczero mode ψ over which we need to integrate. Since ψ is a grassmann odd variable, naivelythe integral would vanish. However, the mode ψ is the ghost field associated with the stringfield theory gauge transformation generated by θ | i for a parameter θ , and the integration over ψ can be interpreted as division by R dθ , with the integral running over the volume of thegauge group [11, 13],. This allows us to express (3.6) as, N = ζ Z dψ √ π Z dψ √ π dψ e S (cid:30) Z dθ . (3.8)In the BV formalism the equivalence of the ‘gauge invariant’ form of the path integral, wherewe set all the anti-fields to zero, to the Siegel gauge fixed version, is usually proved at the levelof correlation functions [18–20] for which the overall normalization of the path integral cancels.Since the normalization is important for us, we have shown in appendix A that the equality of(3.6) and (3.8) can be understood using the standard Faddeev-Popov formalism.We shall now show that (3.8) leads to finite unambiguous result. Let us first carry out theintegral over ψ and ψ by taking the integration contours to be the steepest descent contours.Both of these lie along the imaginary axis, and the final result takes the form: N = − ζ √ Z dψ (cid:30) Z dθ . (3.9)The minus sign in (3.9) is the result of the product of two i ’s, one from having to integratethe tachyon ψ along the imaginary axis and the other from having to integrate ψ along theimaginary axis. However in the open string field theory, the reality condition on the mode ψ is ( ψ ) ∗ = − ψ [31], indicating that we should carry out the path integral over the variable iψ instead of ψ . This would remove one factor of i from (3.9). There is however a similar factor Consequences of this for tree amplitudes have been discussed in [11, 29, 30]. i involved in the integration over the gauge transformation parameter θ in (3.8). Theseeffects cancel each other, and so we shall proceed with (3.9) without removing any factor of i .This has been discussed in footnote 4.The mode ψ is related by field redefinition to the collective mode corresponding to thefreedom of translating the D-instanton along the Euclidean time direction. If e φ denotes thecorrectly normalized collective mode that measures the amount of translation along the timecoordinate, then the dependence of any amplitude on e φ should be of the form e − iω e φ where ω isthe total energy carried by all the external closed string states. Therefore the relation between ψ and e φ may be found by studying the coupling of ψ to an amplitude and comparing thiswith the expected coupling of e φ to the same amplitude. Let us begin with a disk amplitude ofa set of closed string states carrying energies ω , ω , · · · . Since the vertex operator of the stateassociated with ψ is given by i √ c ∂X , inserting this into this amplitude will correspond toinserting the integrated vertex operator i √ Z ∂X ( z ) dz . (3.10)Using the operator product expansion (3.3), and recalling that when we use the doubling trickmentioned in footnote 2, insertion of a vertex operator e − iω k X ( w k ) is implicitly accompanied byits image e iω k X ( ¯ w k ) , we get * i √ Z ∂X ( z ) dz Y k e − iω k X ( w k ) + = i √ X j Z dz "(cid:26) iω j z − w j ) − iω j z − ¯ w j ) (cid:27) *Y k e − iω k X ( w k ) + = − πi ω √ *Y k e − iω k X ( w k ) + , ω ≡ X j ω j . (3.11)Since we have not included any dependence on the string coupling in the quadratic terms inthe action, the open string vertex operator (3.10) should also carry a factor of the open stringcoupling g o ∝ √ g s . The precise relation between g o and g s was determined in [25] and takesthe form g o = g s / (2 π ) in the convention in which the instanton action is given by 1 /g s . Weshall proceed for now by ignoring the factors of g o since g o (in)dependence of N has alreadybeen understood in [32]. At the end of this section we shall briefly discuss g o dependence ofdifferent contributions to N and show how they cancel. (3.11) now shows that coupling of ψ
10o an amplitude with closed string states carrying total energy ω generates a factor of −√ πiω .On the other hand, since the dependence of an amplitude on the collective coordinate e φ is ofthe form e − iω e φ = (1 − iω e φ + · · · ), the coupling of e φ to an amplitude with closed string statecarrying energy ω generates a factor of − iω . This gives the identification of −√ πiωψ to − iω e φ , in agreement with the results of [25]. Therefore in (3.9) we can make the replacement: dψ = 1 √ π d e φ . (3.12)Integration over the collective mode e φ generates the usual energy conserving delta function2 πδ ( ω ) that is part of any amplitude and is not included in the normalization constant N .Therefore we can now express (3.9) as N = − ζ √ √ π (cid:30) Z dθ = − ζ π (cid:30) Z dθ . (3.13)We now turn to the evaluation of R dθ . Physically, this gauge transformation is related byfield redefinition to the rigid U(1) gauge transformation that multiplies any state of the openstring, stretched from the D-instanton to a second D-instanton, by e i e θ . Since e θ has period2 π , in order to determine the range of θ integral, we need to find the relation between θ and e θ . This in turn can be determined by comparing the string field theory gauge transformationlaw generated by θ to the rigid U(1) gauge transformation with parameter e θ for any state ofthe open string that connects the D-instanton to the second D-instanton. This is achieved asfollows:1. As in [13], we shall work with a particular mode ξ that multiplies the vacuum state | i of the open string stretched between the two D-instantons but the relation between θ and e θ is independent of this choice. The conjugate anti-field ξ ∗ of ξ will multiply thestate c − c c | i of the open string that connects the second D-instanton to the originalD-instanton.2. The vertex operators associated with the modes ξ and ξ ∗ are accompanied by Chan-Patonfactors (cid:18) (cid:19) and (cid:18) (cid:19) respectively, and the open string mode ψ , that connectsthe original D-instanton to itself, carries Chan-Paton factor (cid:18) (cid:19) .3. It follows from the gauge transformation laws of the string field theory that the gaugetransformation of ξ under the gauge transformation generated by θ is given by the second11erivative of the action with respect to ξ ∗ and ψ . The leading contribution comes fromthe ξ - ξ ∗ - ψ coupling in the action arising from the disk amplitude. Since two of thethree vertex operators – those associated with ξ and ψ are just identity operators, thecoefficient of this term is given by h | c − c c | i T r (cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) = 1 . (3.14)This corresponds to the presence of a term ξ ξ ∗ ψ (3.15)in the action if we ignore the factors of g o as before.4. Taking the derivative of (3.15) with respect to ξ ∗ and ψ , we see that the gauge trans-formation generated by the parameter θ takes the form δξ = θξ . Comparing this withthe infinitesimal rigid U(1) transformation δξ = i e θξ . we get θ = i e θ . This gives N = − ζ π (cid:30) Z dθ = ζ i π (cid:30) Z d e θ = ζ i π . (3.16)Finally we shall discuss the dependence of N on the string coupling. This has already beenfully understood in [32] but we include the discussion here for completeness. We denote by g o = p g s / (2 π ) the open string coupling [25]. We shall work in the convention in which thekinetic term of the open string fields has g o independent normalization, so that in the Siegelgauge the quadratic part of the action is g o independent, in agreement with the g o independentexponent appearing in (2.7). In this convention, each open string vertex operator carries afactor of g o . This introduces an additional factor of g o in (3.10), (3.11) and therefore a factorof 1 /g o in the right hand sides of (3.12), (3.13),(3.16). On the other hand the disk three pointfunction of three open string vertex operators now gets a factor of g − o from the disk, and afactor of g o from the three open string vertex operators, producing a net factor of g o . Therefore(3.15) gets a factor of g o , leading to the gauge transformation law δξ = g o θξ . Therefore we nowhave g o θ = i e θ , leading to an extra factor of g o on the right hand side of (3.16). This cancelsthe earlier factor of 1 /g o and leaves the right hand side of (3.16) unchanged. Therefore N is g o independent. This factor of i is the result of imposing wrong reality condition on the mode ψ or equivalently theparameter θ . We have not corrected it since this cancels the factor of i arising out of the wrong choice of realitycondition for the mode ψ . This has been discussed below (3.9). ψ = 1 /a complex ψ -plane ψ = 0 ⇒ ⇒×⇑ Figure 1: The integration contour in the complex ψ plane. We now turn to the determination of the multiplier ζ . For this we need to know how thesteepest descent contour / Lefschetz thimble passing through the saddle point representing theD-instanton fits inside the actual integration cycle that computes the full amplitude in stringtheory [22–24]. For the case of two dimensional bosonic string theory this was discussed in [27]where it was argued that the actual integration contour contains only half of this thimble. Inbrief, the argument can be stated as follows. After integrating out the massive open stringmodes, the tachyon effective potential on the D-instanton has a potential V ( ψ ) that has amaximum at ψ = 0 describing the D-instanton and a minimum at some positive value 1 /a describing the perturbative vacuum where the potential vanishes. The potential is unboundedfrom below as ψ → −∞ . Therefore the integration contour over ψ cannot be taken to bealong the real axis all the way to −∞ , but near the perturbative vacuum where the potentialhas a local minimum we expect the contour to lie along the real axis. If we model the potentialas V ( ψ ) = −
12 ( ψ ) + 13 a ( ψ ) + 16 a , (4.1)then one can easily see that the potential goes to + ∞ as we approach the asymptotic regionwithin three 60 ◦ cones, centered around the lines ψ = r , ψ = r e πi/ and ψ = r e − πi/ forreal positive r . Therefore we can take the integration contour to interpolate between the regions ψ = e − πi/ × ∞ and ψ = ∞ as shown in Fig. 1 or we can choose the complex conjugatecontour. On the other hand, the steepest descent contour for the saddle point at ψ = 1 /a ,13epresenting the perturbative vacuum, lies along the real ψ axis from 0 to ∞ , while thesteepest descent contour for the saddle point at ψ = 0, representing the D-instanton, consistsof a contour through the origin that interpolates between the regions around ψ = e − πi/ × ∞ and ψ = e πi/ × ∞ . Therefore the integration contour shown in Fig. 1 can be regarded as theunion of the steepest descent contour of the saddle point at ψ = a , and half of the steepestdescent contour of the saddle point at ψ = 0. This gives ζ = 12 , (4.2)and N = i π . (4.3)Let us now comment on the sign of N about which we have not been careful so far. Thisclearly depends on the choice of the full integration contour – if instead of the contour shown inFig. 1 we choose the complex conjugate contour, the sign of N will change. The actual choiceshould be dictated by physical considerations, e.g. if a D-instanton induced amplitude leadsto violation of unitarity, it should be describable by an effective Hamiltonian with negativeimaginary part, reflecting loss of probability due to possible transition to states that have notbeen accounted for in the effective Hamiltonian. As discussed in [27], the choice of sign givenin (4.3) is the correct choice according to this consideration. Therefore (4.3) gives the finalresult for the normalization constant associated with single D-instanton amplitudes in twodimensional string theory. This agrees with the result obtained in [3] by comparison with thematrix model results for the instanton induced amplitudes, after we multiply this by a factorof i to compute the normalization of the D-instanton contribution to the S-matrix elements. The method described here can in principle be applied to other D-instanton systems, e.g. gen-eral ( m, n ) ZZ-instantons in two dimensional bosonic string theory [28], D-instantons in twodimensional type 0B string theory [33, 34] and D-instantons in type IIB string theory [2, 35].Part of the analysis that may be somewhat non-trivial is the computation of the multiplierfactor ζ , since this requires the knowledge of how the steepest descent contour / Lefschetzthimble associated with a particular D-instanton fits into the full integration contour. Forexample, in the context of ( m, n ) ZZ-instantons in two dimensional string theory, this willrequire the knowledge of how the different ZZ-instantons are represented as different extrema14n the configuration space of string fields. However since the multiplier factors are just ratio-nal numbers, we only need topological information on the locations of various extrema in theconfiguration space of string fields instead of requiring detailed dynamical information. There-fore we do not consider this to be an insurmountable problem. Similarly for computing theD-instanton contribution to the type IIB string theory amplitude, we need to understand howthe D-instantons, which in this case represent complex saddle points, fit inside the integrationcontour over the string fields. Acknowledgement:
I wish to thank Bruno Balthazar, Victor Rodriguez, Xi Yin and BartonZwiebach for many useful discussions. This work was supported in part by the J. C. Bosefellowship of the Department of Science and Technology, India and the Infosys chair professor-ship.
A Siegel gauge fixing in the Faddeev-Popov formalism
In our analysis the formal equality between the Siegel gauge fixed path integral (3.6) and thegauge invariant path integral (3.8) plays an important role. In this appendix we shall showhow this can be proved using the standard Faddeev-Popov formalism.Classical open string field | ψ c i is an arbitrary state of ghost number 1 of the open stringand the gauge transformation parameters describing the symmetries of the classical open stringfield theory correspond to an arbitrary state | θ i of ghost number 0. As in (3.2), we shall workwith the restricted string field carrying L ≤
0. In this case, it is evident from (3.2) that thegauge transformation parameter | θ i , carrying ghost number 0, satisfies the condition b | θ i = 0.Let us introduce basis states {| φ ( n ) r i} of ghost number n , satisfying b | φ ( n ) r i = 0. Then the fullset of basis states at ghost number n may be taken to be {| φ ( n ) r i} and { c | φ ( n − s i} . We shallfurther normalize these basis states for n = 0 and 2 such that h φ (2) r | c | φ (0) s i = δ rs . (A.1)We can now expand the restricted classical string field | ψ c i and the gauge transformationparameter | θ i as: | ψ c i = X r e ψ r | φ (1) r i + X s b ψ s c | φ (0) s i , (A.2) | θ i = X s θ s | φ (0) s i . (A.3)15he gauge invariant classical action up to quadratic order and the linearized gauge transfor-mation laws are given, respectively, by, S g.i. = − h ψ c | Q B | ψ c i , (A.4)and δ | ψ c i = Q B | θ i . (A.5)We now consider a gauge invariant path integral of the form: I g.i. = Z Y r d e ψ r Y s d b ψ s e S g.i. (cid:30) Z Y s dθ s (A.6)We can evaluate this by gauge fixing in the Siegel gauge, by setting b ψ s = 0. This introduces afactor of Q s δ ( b ψ s ) in the path integral, accompanied by the appropriate determinant. To findthe determinant we need to examine the gauge transformation law of b ψ s . Substituting (A.2),(A.3) into (A.5) and taking the inner product of the resulting equation with h φ (2) s | , we get δ b ψ s = h φ (2) s | Q B | θ i = M su θ u , M su = h φ (2) s | Q B | φ (0) u i . (A.7)The determinant entering the integrand of the path integral is det M . We can express this asa path integral over a pair of grassmann odd ghost variables { p s } , { q s } . This gives the gaugefixed path integral: I g.f. = Z Y r dψ r Y s dp s dq s e S g.i. + S ghost (cid:12)(cid:12)(cid:12) b ψ s =0 , S ghost = − X s,u p s M su q u = −h P | Q B | Q i , (A.8)where we have introduced | P i = X s p s | φ (2) s i , | Q i = X s q s | φ (0) s i . (A.9)The equality of I g.i. and I g.f. give in (A.6) and (A.8) in the context of two dimensional stringtheory gives the equality of (3.8) and (3.6). To see this note that in this case the master field | ψ i contains fields of all ghost number without any gauge condition, and the master action is − h ψ | Q B | ψ i . When we pick the Lagrangian submanifold in which we set the modes of ghostnumber ≥ ζ / (2 π )times I g.i. given in (A.6). On the other hand when we pick the Lagrangian submanifold by16mposing the Siegel gauge condition, the master action reduces to S g.i. + S ghost given in (A.8)and (3.6) may be interpreted as ζ / (2 π ) times I g.f. given in (A.8).The equality of I g.i. and I g.f. that we have established is formal, since in the context in whichwe apply this, there are component fields that multiply basis states of vanishing L eigenvalue.This means that the Siegel gauge action becomes independent of those fields, making the pathintegral (A.8) ill-defined. This can be traced to the fact that the Siegel gauge b ψ r = 0 is asingular gauge choice due to the appearance of zero eigenvalues of the matrix M introducedin (A.7). However the original gauge invariant path integral (A.6) is well defined. The pointof view we take is that (A.6) is the proper definition of the path integral, and the problem weface with (A.8) arising in the world-sheet formalism is due to illegal choice of gauge. The analysis described above also resolves an apparent puzzle that arises out of the equalityof (3.6) and (3.8). Since the integral in (3.6) has equal number of bosonic and fermionicvariables, it remains unchanged if we multiply S in the exponent by some constant C . Followingthe logic that led from (3.6) to (3.8), we can see that the effect of this rescaling is to multiplythe exponent in (3.8) by the same constant C . Since in this expression both ψ and ψ representbosonic variables, this will produce a factor of 1 /C on the right hand side of (3.9) that carriesover to the right hand sides of (3.13) and (3.16). So the question is: what compensates thisfactor?The answer to this question comes from (A.7) and (A.8). If the exponent of (3.6) ismultiplied by C , then the ghost action involving the modes p s , q s , which correspond to ψ and ψ in (3.6), also gets multiplied by C . Therefore the matrix M in (A.7) must be multiplied by C . This can happen if we include an extra factor of C in the gauge transformation law (A.5).If we denote by θ ′ the new gauge transformation parameter, then it is related to the old gaugetransformation parameter θ by θ = Cθ ′ . Therefore the R dθ ′ factor that now appears in thedenominator can be identified to R dθ/C , and we get an extra multiplicative factor of C onthe right hand side of (3.9). This cancels the extra factor of 1 /C coming from the integrationover ψ and ψ . 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