hep-th/0312175 PreprinttypesetinJHEPstyle-HYPERVERSION Localized tachyon condensation and G-parity 4 0 conservation 0 2 n a J 6 2 Sunggeun Lee and Sang-Jin Sin 3 Department of Physics, Hanyang University, 133-791, Seoul, Korea v E-mail: [email protected] 5 7 1 2 1 Abstract:Westudythecondensationoflocalizedtachyoninnon-supersymmetricorbifold 3 C2/Z . We first show that the G-parities of chiral primaries are preserved under the n 0 / condensation of localized tachyon(CLT) given by the chiral primaries. Using this, we h finalize the proof of the conjecture that the lowest-tachyon-mass-squared increases under t - p CLT at the level of type II string with full consideration of GSO projection. We also e show the equivalence between the G-parity given by G = [jk /n]+[jk /n] coming from h 1 2 : partitionfunctionandthatgivenbyG = {jk /n}k −{jk /n}k comingfromthemonomial v 1 2 2 1 i construction for the chiral primaires in the dual mirror picture. X r a Keywords: tachyon, orbifold, G-parity, c-theorem. Contents 1. Introduction 1 2. Equivalence of various GSO-projections 2 3. Equivalence of n(k ,k ) and n(1,k) 4 1 2 4. Conservation of G-parity under LTC 7 5. Proof of m-theorem 8 6. Discussion 9 1. Introduction After discovering interestingphenomenaonthetachyon condensationinopenstringtheory [1], it has been tantalizing question to ask the same in closed string cases. The simplest closed string tachyon is the case where the closed string tachyon is localized at the singular point of the background geometry. In this direction, Adams, Polchinski and Silverstein [2] considered localized tachyon for non-compact orbifold Cr/Z and argued that starting n from a non-supersymmetricorbifolds, there will bea cascade of tachyon condensation until space-time SUSY is restored. In a subsequent paper, Vafa [3] reformulated the problem using the mirror picture of gauged linear sigma model, which turns out to be an orbifolded Landau-Ginzburg theory, and confirmed the result of APS. Since the tachyon condensation process can be considered as a renormalization goup (RG) flow[4], it would be interesting to ask whether there is a quantity like a c-function. In non-compact orbifolds, the c-theorem[6] does not work[2, 5]. Therefore the authors of [5] tried to establish a closed string analogue of the g-theorem of boundary conformal field theory. It turns out that, if valid, it would give an explicit counter-example to the result of APS. On the other hand, in a related paper [8], Dabholkar and Vafa suggested that the minimal R-charge in the Ramond sector is the height of tachyon potential at the unstable critical point. In [9], it is argued that the g of [5] does not respect the stability of supersymmetric cl theory and suggested a modified quantity to replace it. In a subsequent paper [10], one of thepresentauthorssuggestedthattheminimaltachyonmasssquaredshouldincreaseunder the localized tachyon condensation. It turns out that this quantity is nothing but the GSO projected version of (negative of) minimal R-charge of Dabholkar-Vafa mentioned above. Later, the statement has been studied in a series of the papers[11, 12, 13], and it is proved – 1 – that the R-charge decreases at the level of conformal field theories before GSO projection. For type II theory, the proof was incomplete mainly due to the incomplete understanding of behavior of G-parity of GSO projection under the tachyon condensation. In related papers [12] the picture of Vafa was extended by working out the generators of daughter theories (the result of decay of the mother theory). The chiral rings and GSO-projection of orbifold theory was examined in more detail in [13]. The goal of this paper is to complete the proof of statement with full consideration of GSO projection, namely to prove the following statement for type II theory of C2/Z n orbifold: • Let m := max α′M2 . Then, m(UV) ≥ m(IR), under condensation of localized tachyon. (cid:12) (cid:12) (cid:12) (cid:12) Notice that Mm2in is negative and 0 0 increases under the tachyon condensa- m(IR) tion while m is positive and decreases m(UV) Mm2in asstatedabove. Figure1istheschematic diagram to clarify the content of this statement. Wecallthisasam-theorem Mm2in to prevent possible confusion with c- Figure 1: M2 increasesandm decreasesunder the min theorem or g-theorem. localized tachyon condensation. The rest of the paper goes as fol- lows. In section 2, we consider the consistency of various GSO-projections introduced by arbitrarily different authors using different logics. In section 3, we show that, n(k ,k ), 1 2 the Z orbifold with generator (k ,k ) equivalent to n(1,k) for some k which we will fix n 1 2 in detail. We call the latter as ”canonical representation”. In section 4, we prove that the G parity is conserved under the condensation of the localized tachyon given by the chiral primaries. This is the most important step in proving the m-theorem. In section 5, we finish the proof of the m-theorem. In section 6, we give a discussion on the implication of the theorem and conclude. 2. Equivalence of various GSO-projections Inthissection wefirstwanttounderstandwhetherorbifoldGSOchiralprojectionsrecently introduced in [3, 5, 10] are mutually consistent. Let k ,k be the generator of the orbifold 1 2 action of Z , that is, n x(1)(z) → e2πik1/nx(1), x(2)(z) → e2πik2/nx(2), (2.1) where x(1),x(2) are complex co-ordinate of C2. We represent C2/Z with generator (k ,k ) n 1 2 by n(k ,k ). In HKMM [5], k = 1 cases were discussed. Here we discuss their result 1 2 1 in the extended form, that is, with general (k ,k ). The chiral primary operators were 1 2 constructed from bosonized world sheet fermions ψ = eiHi as i (1) (2) X = X X , (2.2) j n{jk1} n{jk2} n n – 2 – where Xj = σjeinj(H−H¯), (2.3) n with σ being a twist operator. The Z action defining the GSO projection is given by j 2 n H → H +k π, H → H −k π. (2.4) 1 1 2 2 2 1 In untwisted sector it acts as (−1)FL and restricts both k and k to be an odd. In twisted 1 2 sector X has phase j Xj → eiπ(k2{jkn1}−k1{jk2/n})Xj := (−1)sXj, (2.5) where {x} = x−[x] with [x] being the greatest integer that does not exceed x. Note that s is an integer in general and especially when k = 1 and k =k 1 2 jk s = . (2.6) (cid:20) n (cid:21) In [3], Vafa reformulated the orbifold problem as Landau-Ginzburg theory by imbed- ding the orbifold geometry in the gauged linear sigma model[14] and subsequently taking the mirror dual. The superpotential coming from the vortex contribution can be written as W = un+un+et/nup1up2 (2.7) 1 2 1 2 (−1)FL should be defined by requiring W → −W. This can be achieved by defining the Z action on u by 1 n i u → eiπk2/nu , u → e−iπk1/nu . (2.8) 1 1 2 2 As a result, up1up2 → (−1)p1k2/n−p2k1/nup1up2 := (−1)s′up1up2. (2.9) 1 2 1 2 1 2 So by identifying p = n{jk /n}, i= 1,2, we get i i s = p×k/n = s′, (2.10) and consequently, two GSO actions are completely consistent. Notice that s is always an integer. We also see that up1up2 in the mirror LG theory correspondsto X of the operator 1 2 j construction. We now want to see whether the GSO projection coming from the partition function [10, 13] is also consistent with above two. The result of ref. [10] shows that up1up2 is 1 2 projected out if G = [jk /n]+[jk /n] (2.11) 1 2 is even (odd) for cc ring (ac ring). For our purpose, it is enough to show that G ≡ smod 2. (2.12) 1Wehaveto makemodifications of discussion of [3] on GSO projection. – 3 – j cc G s ca j cc G s ca j cc G s ca 1 (2,4) 0 0 (2,1) 1 (1,2) 0 0 (1,3) 1 (1,2) −1 −1 (1,3) 2 (4,3) 1 2 (4,2) 2 (2,4) 0 0 (2,1) 2 (2,4) −2 −2 (2,1) 3 (1,2) 3 0 (1,3) 3 (3,1) 1 1 (3,4) 3 (3,1) −2 −2 (3,4) 4 (3,1) 4 2 (3,4) 4 (4,3) 1 1 (4,2) 4 (4,3) −3 −3 (4,2) Table 1: Comparison of G and s in 5(2,4)(left) 5(1,2)(middle) and 5(1,−3)(right). 5(2,4)(left) provides an examplewheres6=Gmod2. However,s≡Gmod2forallelementsin5(1,2)and5(1,−3). We first notice that for special case k = 1,k = k, 1 2 s = [jk/n] = G. (2.13) In the next section we will prove that all n(k ,k ) have equivalent representation n(1,k) 1 2 for some k. So the above result is enough if we consider only a given theory. However, we will need to consider the case where k 6= 1 when we consider the ‘decay’ of n(1,k) by 1 explicitly specifying the daughter theories. So let’s consider the general cases. For type II, we need to have k +k = even [13]. If both k ,k are odd integers, the equivalence can 1 2 1 2 be readily seen by considering s+G with the help of the following identity. s = k {jk /n}−k {jk /n}= −k [jk /n]+k [jk /n], (2.14) 2 1 1 1 2 1 1 2 so that s + G is even, which is enough for our goal. If both k ,k are even, then s is 1 2 even and G is not necessarily equivalent to s. In table 1, we give an explicit example for this case. However, if we further restrict ourselves to the case where n,k ,k are mutually 1 2 co-prime, we can restrict ourselves to the case where k ,k are odd. Later in section 4, we 1 2 will only need to consider the case where k = 1 or both k ,k are odd. 1 1 2 3. Equivalence of n(k ,k ) and n(1,k) 1 2 Now, we want to show that n(k ,k ) is equivalent with n(1,k) for some k. We can choose 1 2 a convention where k > 0, since n(k ,k ) = n(−k ,−k ) even after GSO projection. 2 1 1 2 1 2 First, notice that (k ,k ) and (1,k) should generate the same spectra if k /k = k and 1 2 2 1 n,k and n,k are relatively co-prime, since the spectrum is nothing but the modulo-n- 1 2 rearrangement of j(k ,k ) for j = 1,···,n−1. In fact, any of the element of the spectrum, 1 2 thatis,anyofj(k ,k )moduloncanbethegeneratorofthesamespectrumset. Therefore, 1 2 without GSO projection, the equivalence of the two is quite obvious. For type 0 case, the GSO projection does not eliminate any variety of chiral primaries in the following sense: if an operator with a certain charge is projected out in cc ring, there is a surviving operator in aa ring with the same charge. This can be seen from the fact that j-th element of cc-ring and (n −j)-th element of aa ring have the same charge but different G-parity if k +k = odd [13]: 1 2 n(1−{(n−j)k /n}) = n{jk /n}, G(n−j) ≡ G(j)+k +k mod 2. (3.1) i i 1 2 2Onecan see this from [−jki/n]=−[jki]−1 regardless of thesign of ki. – 4 – Similar relations hold between ca and ac rings. Therefore if two type 0 theories have the same spectrum before GSO projection, so do they after GSO. Hence from now on, we concentrate on the type II case, where k +k = even. We 1 2 first have to specify k more precisely. Let k−1 be the multiplicative inverse of k in Z so 1 1 n that there is a unique integer a depending on k such that 1 k−1k = na+1, for any given k . (3.2) 1 1 1 Then, k is equal to k−1k modulo n. So there exists an integer l such that 1 2 k = k−1k +ln, and −n < k < n. (3.3) 1 2 In fact, there are two such l’s, since the length of range is 2n. They are consecutive. In order for n(1,k) to be a type II string theory, we require that k =k−1k +ln = odd. (3.4) 1 2 For odd n, this fixes l uniquely, since k 6= k±n mod 2. However, for even n, the ambiguity will beremoved only after wetake account theG-parity ofk−1 and l morecarefully. Before we proceed, we give some examples to give some feeling on how things work. • 8(3,−5): It is the same with 8(1,1) before GSO projection, since −5 ·3−1 ≡ 1 in Z . However, these are NOT equivalent after GSO as one can see from the table 2. 8 Then, it may look like a counter example. However, for even n, both n(k ,k ) and 1 2 n(k ,k ±n) represent the same type of theory regarding to whether they are type 0 1 2 or type II [13]. Here ± is chosen such that −n< k ±n < n is satisfied. Therefore we 2 should also consider 8(1,−7) instead of 8(1,1). Remarkably, 8(3,−5) and 8(1,−7) have the same GSO projected spectrum as one can see from table 2. • 7(3,5): It is equivalent to 7(1,−3) and also to 7(1,4) before GSO projection. But 7(3,5) and 7(1,−3) is a type II while 7(1,4) is type 0. Therefore in this case there is a unique representation in the same type. One can explicitly check that 7(3,5) has identical spectrum with 7(1,−3) after GSO from table 3. Now let us come back to the general argument. Consider G-parity for n(k ,k ) and 1 2 n(1,k) with k given in eq.(3.3). We call them as A and B orbifold theory respectively. The G-parity for j-th element of A is G (j) = [jk /n]+[jk /n] and the G-parity for j-th one A 1 2 of B is G (j) = [jk/n]. We remind that the cc- and ca-rings of A and B are the same (as B sets) before GSO projection. Let the j-th element of A theory appears as the j′-th element for B so that j′ = n{jk /n}. (3.5) 1 For our purpose, it is enough to show that G (j) ≡ G (j′) mod2. (3.6) A B – 5 – j cc G s ca j cc G s ca j cc G s ca 1 (5,5) −1 5 (5,3) 1 (1,1) 0 0 (1,7) 1 (1,1) −1 −1 (1,7) 2 (2,2) 0 2 (2,6) 2 (2,2) 0 0 (2,6) 2 (2,2) −2 −2 (2,6) 3 (7,7) −1 7 (7,1) 3 (3,3) 0 0 (3,5) 3 (3,3) −3 −3 (3,5) 4 (4,4) 0 4 (4,4) 4 (4,4) 0 0 (4,4) 4 (4,4) −4 −4 (4,4) 5 (1,1) 1 1 (1,7) 5 (5,5) 0 0 (5,3) 5 (5,5) −5 −5 (5,3) 6 (6,6) 0 6 (6,2) 6 (6,6) 0 0 (6,2) 6 (6,6) −6 −6 (6,2) 7 (3,3) 1 3 (3,5) 7 (7,7) 0 0 (7,1) 7 (7,7) −7 −7 (7,1) Table 2: cc-,ca-ring of 8(−3,5) (left), 8(1,1) (middle), 8(1,−7) (right). G,s for each element are given for comparison. Both8(1,−7)and8(1,1)areequivalentto8(−3,5)beforeGSO.Butonly8(1,−7)issoafterGSO.This isageneralphenomena: ForevenntypeII,kisnotdetermineduniquelyfrom(k1,k2)beforeGSO.Thisambiguity orfreedomwillbeessentialtofindcorrect k withGSOprojectionconsidered. j cc-elements G ca-elements j cc G ca 1 1 1 (3,5) 0 (3,2) 1 (1,4) −1 (1,3) 2 (6,3) 1 (6,4) 2 (2,1) −1 (2,6) 3 (2,1) 3 (2,6) 3 (3,5) −2 (3,2) 4 (5,6) 3 (5,1) 4 (4,2) −2 (4,5) 5 (1,4) 5 (1,3) 5 (5,6) −3 (5,1) 6 (4,2) 6 (4,5) 6 (6,3) −3 (6,3) Table 3: cc-, ca-rings for 7(3,5) (left) and 7(1,−3) (right). Notice that 7(1,4) is a type 0 theory and it is not tabulated here. ForoddntypeII,k ischosenuniquelyfrom(k1,k2). Since k is an odd integer, G = [(jk /n−[jk /n])k] ≡ [jk /n]+jk (k−1k +ln)/n mod 2. (3.7) B 1 1 1 1 1 2 Using eq.(3.2), one can easily show that G ≡ G +jk l+jk a mod 2. (3.8) B A 1 2 If both k ,k are even and n is odd, then our job is done. If both are odd, then G ≡ 1 2 B G +j(l+a) modulo 2. For even n, one of the consecutive l’s (see below eq.(3.3)) can be A chosen such that l+a is even and this condition removes the ambiguity in the choice of l (hence in k) as mentioned before. For odd n, l is fixed as follows: If k−1k is already odd, then l should be even not to 1 2 change the type 0/type II. If the former is even, then l should be odd. For k−1k odd case, 1 2 using eq. (3.2), (na+1)k = k ·odd ≡ k mod 2. (3.9) 2 1 1 If k ,k are both odd, na+1 should be odd. Since n is odd, a must be even. Therefore if 1 2 k−1k is oddandk ,k have thesamenumberof factor 2, l+ais even as desired. Similarly, 1 2 1 2 for k−1k even case, 1 2 (na+1)k = k ·even ≡ k mod 2. (3.10) 2 1 1 If k ,k are both odd, then na+1 must be even, hence a must be even. Therefore in this 1 2 case also l+a is even as desired. Hence we proved the following – 6 – • Lemma: For any orbifold n(k ,k ), we can represent it by n(1,k) with GSO projec- 1 2 tion properly considered. 4. Conservation of G-parity under LTC The final most important step in proving the m-theorem is to show that G parity is con- servedunderthelocalizedtachyoncondensation. Namely,ifaparticularchargeisprojected out in the mother theory, its daughter image under T is also projected out in a daughter p theory. Inthepreviouswork[12],weshowedthatthedecayofn(k ,k )underthecondensation 1 2 of localized tachyon with weight p = (p ,p ) is 1 2 n(k ,k ) → p (k ,s)⊕p (−s,k ) (4.1) 1 2 1 1 2 2 wherep = {jk /n}, p = {jk /n} and s = p×k/n =k {jk /n}−k {jk /n}. Its G-parity 1 1 2 2 2 1 1 2 is given by jk jk 1 2 G = [ ]+[ ] mod2 (4.2) p n n Here [x] means the integer part of x while {x} means the fractional part of x. In order for (p ,p ) to survive, G should be an odd integer. On the other hand, by the result of last 1 2 p section, we only need to consider the decay of a canonical representation: n(1,k) → p (1,s )⊕p (−s ,k), (4.3) 1 p 2 p with p = n{jk /n},s = p × k/n = (p − kp )/n. In order to fix the ambiguity in i i p 2 1 the daughter-theory-generators, we use the fact that for type II theory, the bulk tachyon is projected out and it can not and should not be regenerated by the localized tachyon condensation process. That is, type II can decay only to type II. Then s and k should be both odd. This conditions are already satisfied: the orginal theory is type II hence k is odd and p as a surviving element of cc-ring must have odd G (= s ). If any of p , p is p p 1 2 odd, that can not be added or subtracted to the generator, since it convert the type II to type 0. If any of them are even, it can be added to the generator and we have ambiguity in the determination of the daughter theory. However, as we will see shortly, this can not be so and we will see that the eq. (4.3) describe the tachyon condensation properly even after the GSO projection. For a moment, we assume that eq. (4.3) is true. We need the map which gives the G-parity value [lk/n] when the data (q ,q ) = 1 2 (l,n{lk/n}) of a charge are given. This is given by the observation: G = [lk/n] = (kq −q )/n = q×k/n. (4.4) q 1 2 Under the condensation of (p ,p ), q is mapped to q′ by the tachyon map T [11]. If q′ 1 2 p belongs to up-theory of daughter theory, 1 0 q q q′ q′ = T+q = 1 = 1 = 1 . (4.5) p (cid:18)−p /n p /n(cid:19)(cid:18)q (cid:19) (cid:18)p×q/n(cid:19) (cid:18)q′ (cid:19) 2 1 2 2 – 7 – Then, q′ −q′s 1 1 Gq′ = 2 1 p = (p×q−q1(p×k)) = (q×k) = Gq, (4.6) p np n 1 1 which proves the conservation of G-parity for the up-theory spectrum. Similarly we can prove the same statement for the down-theory using the equivalence of G-parity with s proved before. Notice that both s and k are odd, so that the result applies here. Since q′′ = T−q = (−p×q/n,q ) := (q′′,q′′), (4.7) p 2 1 2 we have q′′s+q′′k 1 1 Gq′′ = 2 1 = (−(p×q)k+q2(p×k)) = (q×k) = Gq, (4.8) p np n 2 2 finishing the proof of G-parity conservation. 3 An important remark is in order. The decay rule eq. (4.1) or eq.(4.3) is given at the conformal field theory level. After GSO, since n(k ,k ) and n(k ,k ±n) are different 1 2 1 2 theories, it seems that the decay rule has the same ambiguity: that of adding ±p , ±p to 1 2 s and k respectively in the right hand side of eq. (4.1): why the following process is not 2 allowed? n(k ,k ) → p (k ,s±p )⊕p (−s,k ±p ). (4.9) 1 2 1 1 1 2 2 2 If p (k ,s ) is replaced by p (k ,s ±p ), then a computation shows that 1 1 p 1 1 p 1 Gq′ = Gq ∓q1. (4.10) This means that an operator that is projected out in the mother theory can be resurrected in daughter theory under the localized tachyon condensation(LTC). This is not physical, because what changes under the LTC is not the operator uq1uq2 but the Lagrangian: 1 2 W = un+un+et/nup1up2. (4.11) 1 2 1 2 Namely, each operatorremainsthesameandonlythecoefficient et/n ofcondensingtachyon operator changes so that the measuring method of the weight of operators uq1uq2 changes. 1 2 Onceanoperatorisprojectedoutatthemomentoft → −∞,itisnotpossibletoresurrectit byasmoothdeformationoftakingt → ∞. Thesameargumentholdsforthep (−s ,k±p ). 2 p 2 Therefore the original rule eq.(4.1) or eq.(4.3) remains its form and the modified rule eq.(4.9) is not allowed. 5. Proof of m-theorem Accordingtotheresultof[11],thespectrumofdaughtertheoriescanberegardedasimages of certain linear mapping T for cc ring and F for ca or ac rings given by T+(q) = (q ,p×q/n), T−(q) = (−p×q/n,q ), p 1 p 2 F+(q¯) = (q ,p −p×q/n), F−(q˜)= (p +p×q/n,q ). (5.1) p 1 1 p 2 2 3We can also use the inverse representation n(k′,1) → p (k′,s′)⊕p (−s′,1), to prove the G-parity 1 p 2 p conservation for the down theory spectrum. Here k′ := k−1 in the finite field Zn, and s′p = p×k′/n = (p1−k′p2)/n. Using G=[lk′/n]=(q2k′−q1)/n, we haveGq′ = q2′(−ps′p2)−q1 = n1p2(−q2(p×k′)+p×q)= n1(q2k′−q1)=Gq. – 8 – Here, superscript + is for up-theory, − is for down-theory and q¯= (q ,n−q ) ∈ ca ring, 1 2 q˜= (n−q ,q ) ∈ ca and p′ = (p ,−p ). T is a mapping for tachyon condensation which 1 2 1 2 exists due to the worldsheet N = 2 SUSY, while F is not a mapping describing tachyon condensation and its existence is due to the conformal symmetry of the final theory. For moredetail, see[11]. Themostimportantpropertyof T andF istheir monotonicity, which enables the proof of m-theorem possible. Theconservation ofGparitymeansthatthesamelogic canbeappliedtothespectrum of type II even after GSO projection. The worst thing that can happen is the case where the operator with minimum R-charge of the Mother theory is deleted in the mother theory but resurrected in the daughter theory. The G-parity conservation guarantees that such phenomena can never happen. See figure 2. For completeness, we describe some of the de- tail here. Let p be the condensing tachyon and q′ be the element with minimal R-charge of the min daughter theory, i.e. the smallest one among all charges in up and down theories. If q′ belongs min toccringofup-theory,thenitisanimageofanele- ment q in the mother theory under T+. q must be- p long to the cc-ring of the mother theory due to the G-parity conservation. If q is the minimal ele- min ment of the mother theory, R[q ], the R-charge min of q , is smaller than R[q] by definition. Then min our desired statement, R[q ]≤ R[q′ ] (5.2) min min comes from the monotonicity of T+, namely, R[q]≤ R[T+(q)] = R[q′ ]. (5.3) p min Figure 2: A candidate phenomena that If qm′ in belongs to cc ring of down-theory, then Tp− candestroythem-theoremintheabsence replaces T+ for above argument. If q′ ∈ ca-ring of the G-parity conservation. Filled dots p min ofdaughtertheories,thentheentireargumentscan represent operators surviving under the be repeated by replacing the map T± by F± to GSOprojectionandemptydotsarethose projected out. finish the proof. 6. Discussion Our proof was actually motivated from the numerical work we performed, which showed that the theorem holds for all n(k ,k ) with n ≤ 100 and −n < k < k < n, which 1 2 1 2 provided significant evidence to believe that the m-theorem is true. The m-theorem we just proved implies that the one loop cosmological constant of the non-supersymmetric orbifold is a monotonically decreasing quantity under the localized tachyon condensation. This is because, the former is defined as the integral of one loop – 9 –