Stable Vacua for Tachyonic Strings

PHYSICAL REVIEW D 103, 106026 (2021)

Stable vacua for tachyonic strings

Justin Kaidi * Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, New York 11794, USA

(Received 27 March 2021; accepted 3 May 2021; published 28 May 2021)

In addition to the familiar string theories with spacetime supersymmetry, there exist a number of nonsupersymmetric ten-dimensional superstrings. Nearly all of these theories have closed string tachyons, indicating a misidentification of the vacuum around which they are to be quantized. In this work, we identify (meta)stable vacua for all known tachyonic superstrings. These vacua are all lower dimensional, with most of them being familiar two-dimensional string theories. However, there are also intriguing examples of solutions in dimensions 6, 8, and 9, with respective gauge groups E7 × E7, SUð16Þ, and E8. These special vacua have positive vacuum energy and no moduli.

DOI: 10.1103/PhysRevD.103.106026

I. INTRODUCTION presence of a tachyon is not itself indicative of an incon- sistency in the theory. Instead, it simply means that the One of the long-touted virtues of string theory presumed vacuum (typically ten-dimensional Minkowski is its apparent uniqueness. Famously, the Type IIA/B, space) is not a legitimate solution. In other words, these 32 heterotic SOð Þ, heterotic E8 × E8, and Type I super- additional strings theories should not be interpreted as string theories have all been shown to be different describing fluctuations around ten-dimensional flat space, perturbative descriptions of a single underlying but rather as fluctuations about some other, possibly lower- theory. These five descriptions are typically given as dimensional background. As long as some stable vacuum expansions around ten-dimensional flat space, where exists, these theories are well-defined quantum gravitational they are tachyon free and have manifest spacetime theories and are thus expected on general grounds to fit into supersymmetry. the string duality web. In this paper, we will find such (meta) However, it has long been known that there exist stable vacua for the tachyonic theories listed above, using additional, nonsupersymmetric superstring theories, which techniques developed in Refs. [13,18–22]. lie outside of the standard duality web. These include the Seemingly unrelatedly, it has also been known that there Type 0A/B theories, eight Pin− Type 0 theories [1–4], and a exist a number of consistent two-dimensional string theories. series of seven nonsupersymmetric heterotic strings [5–7]. These include three heterotic strings [23–27], two oriented Though some preliminary work has been done on bringing Type 0 strings [28,29], and a total of eight unoriented Type 0 these theories into contact with the more familiar super- strings [30,31]. Though much progress has been made in strings [1,2,8–13], as well as exploring their connections to understanding these theories via e.g., matrix model tech- phenomenology [14–16], it is safe to say that this is not yet niques of Refs. [32,33], these two-dimensional theories have a completed endeavor. remained largely disconnected from the space of ten-dimen- One of the reasons that these additional string theories sional strings. Nevertheless, since these theories describe have remained relatively understudied is that almost all of consistent quantum gravity theories (albeit in two dimen- them are plagued by closed string tachyons.1 But the sions), by the string uniqueness principle, they are again expected to be continuously connected to the space of higher-dimensional string theories. There is one situation in which such a connection has been concretely realized— *[email protected] 1Notable exceptions are the Pinþ Type 0 strings [4], the namely, in Ref. [13], it was shown that closed string tachyon Oð16Þ × Oð16Þ heterotic string [5], and the Type I˜ string [17], all condensation in the ten-dimensional oriented Type 0 theories of which are tachyon free. leads to a dimension-reducing cascade, with the end point being the two-dimensional Type 0 theories. Many of the Published by the American Physical Society under the terms of stable vacua studied in this paper will be exactly of this type; the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to i.e., we will see that most of the tachyonic ten-dimensional the author(s) and the published article’s title, journal citation, strings admit a dynamical transition to one of the two- and DOI. Funded by SCOAP3. dimensional (2D) theories previously studied in the

2470-0010=2021=103(10)=106026(14) 106026-1 Published by the American Physical Society JUSTIN KAIDI PHYS. REV. D 103, 106026 (2021) literature. This paves the way toward bringing these 2D are obtained by identifying the left- and right-moving spin strings into contact with the usual duality web.2 structures. The simplest nonsupersymmetric heterotic − There are, however, some interesting cases in which string has torus partition function Z ¼ ZNS ZR, where tachyonic ten-dimensional string theories do not admit a stable 2D vacuum, at least not via the simplest condensa- 1 η −16 0 8 0 32 − 0 8 0 32 tion mechanism. For those cases, we will instead identify ZNS ¼ 2 j j ½ðZ0Þ ðZ0Þ ðZ1Þ ðZ1Þ ; stable vacua in dimensions d>2. Concretely, for the 2 1 −16 1 8 1 32 heterotic E8, Uð16Þ, and ðE7 × SUð2ÞÞ heterotic strings, η ZR ¼ 2 j j ðZ0Þ ðZ0Þ : ð2:1Þ we will identify stable vacua in dimensions d ¼ 9,8, and 6.3 All of the resulting lower-dimensional theories are free of perturbative anomalies, and interestingly all have The minus sign in the Neveu-Schwarz (NS) sector partition positive one-loop cosmological constant. Because these function comes from the fact that the superghost ground theories have no moduli (and in particular the dilaton has an state is fermionic, and the overall factor of 1=2 is added for effective mass), the dilaton tadpole implied by the positive the sum over spin structure. We use the usual notation that cosmological constant is not expected to signal any sort of perturbative instability. Hence, we suspect that these cases 1 ϑαβð0jτÞ 2 represent metastable solutions of string theory with positive α Zβ ¼ η τ ; ð2:2Þ vacuum energy, at least in the weak-coupling regime. ð Þ This paper is organized as follows. In Sec. II, we discuss nonsupersymmetric heterotic string theories. After review- with ϑαβðzjτÞ the Jacobi theta functions and ηðτÞ the ing the six tachyonic ten-dimensional theories in Sec. II A Dedekind eta function. as well as the three two-dimensional theories in Sec. II B, The spectrum of the theory in (2.1) can be easily read off we proceed to a discussion of tachyon condensation. In Z by considering the level-matched partition functions NS;R, Sec. II C we consider condensation to theories in d>2, obtained by integrating over τ1. This gives study local and global anomaly cancellation in the resulting theories, and compute the cosmological constant. In −1 1 Z 32 ¯ 2 4032 188928 ¯ 2 … Sec. II D, we consider condensation of the remaining NS ¼ ðqqÞ þ þ ðqqÞ þ ; heterotic theories to d ¼ 2. In Sec. III, we consider tachyon 2 Z ¼ 8388608qq¯ þ 3019898880ðqq¯Þ þ …: ð2:3Þ condensation of oriented and unoriented Type 0 string R theories, where we will see that all cases can be condensed to known two-dimensional strings. From the first line, we see that there are 32 tachyons as well as 4032 massless bosonic degrees of freedom. These II. HETEROTIC STRINGS correspond to a graviton (35), B-field (28), dilaton (1), and a remaining 496 gauge bosons of SOð32Þ. There are no A. Heterotic strings in d =10 massless fermions, so it is clear that this theory is not We begin by reviewing the nonsupersymmetric heterotic supersymmetric. string theories obtained in Refs. [6,7]. Our construction will We may obtain other nonsupersymmetric strings from be somewhat different from that of the original works. this SOð32Þ theory as follows. We begin by noting that the 5 First, recall that in the free fermion formulation, the system of 32 left-moving fermions admits a ðZ2Þ global worldsheet theory of heterotic strings consists of 10 symmetry, with generators bosonic fields Xμ, 10 right-moving fermionic fields ψ μ, and 32 left-moving fermionic fields λ˜a. The supersymmet- g1 ¼ σ3 ⊗ 12 ⊗ 12 ⊗ 12 ⊗ 12; ric heterotic strings are taken to have independent spin structures for the left- and right-moving fermions, both of g2 ¼ 12 ⊗ σ3 ⊗ 12 ⊗ 12 ⊗ 12; which are summed over in the Gliozzi-Scherk-Olive (GSO) g3 ¼ 12 ⊗ 12 ⊗ σ3 ⊗ 12 ⊗ 12; projection. In contrast, nonsupersymmetric heterotic strings g4 ¼ 12 ⊗ 12 ⊗ 12 ⊗ σ3 ⊗ 12;

2 Additional two-dimensional string theories include the 2D g5 ¼ 12 ⊗ 12 ⊗ 12 ⊗ 12 ⊗ σ3: ð2:4Þ Type II [34–36] and Type I [31] theories. It is natural to suspect that these, too, can be obtained via dynamical transitions from tachyonic ten-dimensional strings. Indeed, for the Type II Each of these acts on 16 of the 32 left-moving fermions λ˜a theories, a potential candidate in ten dimensions is the with a minus sign and acts as the identity on the others. n ;n 6; 4 element of the “Seiberg series” of Ref. [13]. ð even oddÞ¼ð Þ We may now choose to gauge some or all of this global Since these tentative ten-dimensional starting points are not as n well known, though, we will not discuss them further. symmetry. Gauging ðZ2Þ for 0 ≤ n ≤ 5 means that we 3The first of these has already been studied in Ref. [22]. should replace the above partition function with

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1 10 η −16 0 8 0 − 0 8 0 TABLE I. Tachyonic heterotic strings in d ¼ . ZNS ¼ 2 j j ½ðZ0Þ L0 ðZ1Þ L1; 1 n Tachyons Massless fermions Gauge bosons η −16 1 8 1 ZR ¼ 2 j j ðZ0Þ L0; ð2:5Þ 0 32 0 496 1 16 256 368 where La are twisted left-moving partition functions,4 2 8 384 304 b 3 4 448 272 1 4 2 480 256 L0 Z0 16 Z0 16 2n − 1 Z0 16 2n − 1 Z1 16 ; 5 1 496 248 0 ¼ 2n ð 0Þ ½ð 0Þ þð Þð 1Þ þð Þð 0Þ 1 0 0 16 0 16 n 0 16 n 1 16 L1 ¼ ðZ1Þ ½ðZ1Þ þð2 − 1ÞðZ0Þ þð2 − 1ÞðZ0Þ ; 2n cautious reader should take all of our statements at the level 1 1 1 16 1 16 0 16 0 16 of the algebra. L ¼ ðZ Þ ½ðZ Þ þð2n − 1ÞðZ Þ þð2n − 1ÞðZ Þ : 0 2n 0 0 0 1 An intuitive explanation for the first four of the above ð2:6Þ gauge groups is as follows. As said before, each of the 5 generators gi of ðZ2Þ acts as −1 on 16 fermions and as It is easy to verify that the full partition function, i.e., the identity on the remaining fermions. Under any given n 5−n − subgroup Z2 , 2 fermions remain invariant, while Z ¼ ZNS ZR, is modular invariant for all n. ð Þ 25 1 − 1 With these expressions, we may now read off the spectra ð 2nÞ change sign. We thus expect that in general one so 32 so 25−n so 25 1 − 1 by considering the level-matched partition functions, breaks ð Þ to ð Þ × ð ð 2nÞÞ. This leads to an expected sequence of gauge algebras soð16Þ × soð16Þ, −1 1 1∶ Z 16 ¯ 2 3008 168192 ¯ 2 … so 8 so 24 so 4 so 28 so 2 so 30 n ¼ NS ¼ ðqqÞ þ þ ðqqÞ þ ð Þ × ð Þ, ð Þ × ð Þ, and ð Þ × ð Þ. Indeed, the soð25−nÞ factor is always present in the Z ¼ 2048 þ 8912896qq¯ þ … R final answer [recall suð2Þ × suð2Þ ≅ soð4Þ and uð16Þ ≅ −1 1 5 1 2∶ Z 8 ¯ 2 2496 157824 ¯ 2 … so 2 su 16 so 2 1 − n ¼ NS ¼ ðqqÞ þ þ ðqqÞ þ ð Þ × ð Þ], but it turns out that the ð ð 2nÞÞ Z 3072 9175040 ¯ … factor generically gets enhanced to a larger algebra of the R ¼ þ qq þ same rank. A crucial point is that in all cases, the tachyon is −1 1 3∶ Z 4 ¯ 2 2240 152640 ¯ 2 … 5− 5− n ¼ NS ¼ ðqqÞ þ þ ðqqÞ þ a vector of soð2 nÞ, hence its contribution of 2 n to the Z 3584 9306112 ¯ … partition function. We will discuss this further in Sec. II C. R ¼ þ qq þ −1 1 It will be useful to have more detail on the chiral matter 4∶ Z 2 ¯ 2 2112 150048 ¯ 2 … n ¼ NS ¼ ðqqÞ þ þ ðqqÞ þ content of each theory. In self-evident notation, we find the Z 3840 9371648 ¯ … following massless fermions, R ¼ þ qq þ −1 1 5∶ Z ¯ 2 2048 148752 ¯ 2 … n ¼ NS ¼ðqqÞ þ þ ðqqÞ þ 32 ∶ Z 3968 9404416 ¯ … SOð Þ none R ¼ þ qq þ ð2:7Þ 16 ∶ 128 1 1280 1 Oð Þ × E8 ð ; Þþ; ð ; Þ− This gives us the data in Table I.Thesetheoriesare, Oð24Þ × Oð8Þ∶ ð24; 8vÞ ; ð24; 8cÞ− 16 8 24 2 2 þ respectively, the Oð Þ×E8, Oð Þ×Oð Þ, ðE7 ×SUð ÞÞ , 2 5 2 ∶ 56 2;1 1 ⊕ 1 1; 56 2 Uð16Þ,andE8 nonsupersymmetric heterotic string theories. ðE7 × SUð ÞÞ ð ; ; Þþ ð ; ; Þþ; Let us note that we are not being particularly careful about ð56; 1; 1; 2Þ− ⊕ ð1; 2; 56; 1Þ− the global structure of the spacetime gauge group here. The 16 1 ∶ 120 2 ⊕ 120 −2 SUð Þ × Uð Þ ð ; Þþ ð ; Þþ;

4One could contemplate obtaining new theories by allowing ð120; −2Þ− ⊕ ð120; 2Þ− for discrete torsion in these sums. That is, we allow for some extra 2 n−1 E8∶ 248 ; 248−; ð2:8Þ phases dictated by an element of H ððZ2Þ ;Uð1ÞÞ. This is þ analogous to what was done for the supersymmetric E8 × E8 string to obtain the nontachyonic Oð16Þ × Oð16Þ heterotic string. We could in fact go further and modify the twisted where the subscript represents 8s;c of the spacetime ℧2 Z n−1 ≔ partition functions by a general element of SpinðBð 2Þ Þ Spin(8). Adding the fermions in each line gives the Spin n−1 HomðΩ2 ðBðZ2Þ Þ;Uð1ÞÞ, much in the spirit of Refs. [3,4]. cumulative results of Table I. We save this analysis for a separate work. Perturbative anomalies are canceled as follows. The 5Note that we could also obtain most of these theories via the SOð32Þ and E8 theories have nonchiral spectra, so these covariant lattice formulation of Refs. [37–39]. We do not use that theories are trivially nonanomalous. For the Oð16Þ × E8 construction here, though, since it does not give the E8 theory. This is because in that case the fermions cannot all be bosonized and Oð24Þ × Oð8Þ theories, anomaly cancellation follows to lattice bosons. simply from the relations

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2n 2n 2n 2n 2n 2 0 Z Tr8v F ¼ Tr8s F ¼ Tr8c F ; Tr128F ¼ Tr128 F : This worldsheet theory enjoys a ð 2Þ symmetry, with each element of the group acting on 16 of the left-moving ð2:9Þ fermions with a sign and leaving the remaining 8 invariant. n 2 Gauging ðZ2Þ for 0 ≤ n ≤ 2 then gives rise to the Finally, for the E7 × SU 2 and U 16 theories, anoma- ð ð ÞÞ ð Þ following partition functions, lies can be canceled by the addition of appropriate Green- Schwarz terms. 1 1 0 − 0 1 ZNS ¼ 2 ðL0 L1Þ;ZR ¼ 2 L0; ð2:12Þ B. Heterotic strings in d =2 where La are now defined as In addition to the nonsupersymmetric heterotic strings b reviewed above, there are three heterotic strings in two 1 L0 Z0 8 Z0 16 2n − 1 Z0 16 2n − 1 Z1 16 ; dimensions. Two of these strings were introduced in 0 ¼ 2n ð 0Þ ½ð 0Þ þð Þð 1Þ þð Þð 0Þ Refs. [23–25], while the remaining one was discussed in 1 Refs. [26,27]. L0 Z0 8 Z0 16 2n − 1 Z0 16 2n − 1 Z1 16 ; 1 ¼ 2n ð 1Þ ½ð 1Þ þð Þð 0Þ þð Þð 0Þ In order to ensure worldsheet anomaly cancellation in 1 two dimensions, we consider solutions with a linear L1 Z1 8 Z1 16 2n − 1 Z0 16 2n − 1 Z0 16 : 0 ¼ 2n ð 0Þ ½ð 0Þ þð Þð 0Þ þð Þð 1Þ dilaton. In particular, we take ϕ ∝ X1 with the proportion- ality constant chosen such that the system has central ð2:13Þ 13 charge cX1 ¼ . The right-moving theory is composed of the fields X0;1 together with superpartners ψ 0;1, giving a As in (2.10), there are no right-moving contributions to the partition function, since in two dimensions they are total central charge cR ¼ 15. The left-moving theory is a bosonic string with X1, X0, and an additional 12 bosons, for completely canceled by ghosts. a total central charge of c ¼ 26. The 12 extra left-moving An alternative interpretation of these partition functions L 1 bosons must be compactified on an even self-dual lattice to is as follows. First, the n ¼ gauging can be interpreted as 24 give a sensible two-dimensional spacetime interpretation. starting with the Oð Þ theory, splitting the left-moving The candidate lattices are the root lattices of Oð8Þ × E8 fermions into groups of 8 and 16 and giving them 24 and Oð24Þ. separate spin structures. This naively breaks Oð Þ to 8 16 16 The two-dimensional heterotic strings may be given a Oð Þ × Oð Þ, but the Oð Þ experiences an enhancement free fermion interpretation as follows. We begin by to E8, as evident from the fact that the partition function replacing the 12 left-moving bosons with 24 free fermions. contains a factor of the E8 characters, Then, as in ten dimensions, different theories correspond to 1 1 different gaugings of discrete symmetries of the system of Z ¼ ððZ0Þ8 − ðZ0Þ8Þχ ðq¯Þ;Z¼ ðZ1Þ8χ ðq¯Þ: NS 2 0 1 E8 R 2 0 E8 fermions. As before, we may begin with the ungauged case, which in other words corresponds to assigning all fermions ð2:14Þ a single spin structure. This gives rise to the partition function Hence, this theory has Oð8Þ × E8 gauge group. By the abstruse identity, in this case, the full partition 1 1 function vanishes, i.e., Z Z − Z 0, and is thus 0 24 − 0 24 1 24 ¼ NS R ¼ ZNS ¼ 2 ½ðZ0Þ ðZ1Þ ;ZR ¼ 2 ðZ0Þ ; ð2:10Þ trivially modular invariant. The vanishing of the partition function implies that the spectrum of the theory is super- where the sign in the NS partition function comes from symmetric. However, the linear dilaton spoils spacetime right-moving superghosts. It is a convenient modular supersymmetry. The level-matched partition functions are − 24 accident that Z ¼ ZNS ZR ¼ , which is trivially modu- concretely lar invariant. The level-matched partition functions are 1∶Z 8 Z 8 n ¼ NS ¼ ; R ¼ : ð2:15Þ Z 24 Z 0 NS ¼ ; R ¼ : ð2:11Þ This corresponds to eight massless bosons in the 8v of Oð8Þ, The spacetime content of this theory is thus a set of 24 eight right-moving fermions in the 8s, and eight left-moving massless bosons transforming in the fundamental of Oð24Þ, fermions in the 8c. The corresponding vertex operators are with the gravitons and Oð24Þ gauge bosons not contrib- known [25], though we will not need them here. uting any propagating degrees of freedom in two dimen- As for the n ¼ 2 gauging, it can be interpreted as sions. There are, however, discrete graviton and gauge gauging left-moving worldsheet fermion number ð−1ÞfL fields, where by “discrete” here we mean that the corre- in the Oð24Þ theory. At the level of the partition function, sponding vertex operators exist at only discrete values of this can be seen as follows. Since the right-movers do not momenta (namely, p ¼ 0) [25]. contribute to the partition function, the gauging of this

106026-4 STABLE VACUA FOR TACHYONIC STRINGS PHYS. REV. D 103, 106026 (2021) rffiffiffiffi sffiffiffiffiffiffiffiffi symmetry changes the 24 massless scalars of the Oð24Þ Xrþ1 2 β 2 T aðXÞ¼ e Xþ MaXi; β ≔ ; ð2:21Þ theory to 24 chiral fermions. Hence, the partition function α0 i α0γ2 1 i¼2 is − 2 times the original Oð24Þ partition function, i.e., − 1 24 −12 Z ¼ 2 ð Þ¼ . It can be checked that this matches a T 1 … T r 0 where Mi is a matrix chosen such that ¼ ¼ ¼ the result obtained from (2.12) and (2.13) with n ¼ 2. 2 1 along some unique locus X ¼ … ¼ Xrþ ¼ 0. In particu- From this result, it follows immediately that the level- lar, as long as r ≤ 8, we can simply take matched partition functions are Ma ¼ mδa : ð2:22Þ 2∶Z 0 Z 12 i i−1 n ¼ NS ¼ ; R ¼ : ð2:16Þ Introducing such a tachyon gives rise to the scalar potential The Oð24Þ gauge symmetry remains unbroken, and we see 1 that we have 24 chiral fermions transforming in the m2 Xrþ V ¼ e2βXþ ðXiÞ2 fundamental. This matter content is seemingly anomalous, 4πα0 but as discussed in Ref. [26], the anomaly is canceled by a i¼2 two-dimensional analog of the Green-Schwarz mechanism. im Xr − βXþ λ˜a ψ aþ1 β aþ1ψ þ We will return to this point in Sec. II D. 2π e ð þ X Þ; ð2:23Þ a¼1

C. Tachyon condensation to d =6; 8; 9 which gives rise to an exponentially growing mass þ ≔ βXþ i ψ i 2 … 1 We now consider condensation of the tachyon in the ten- mðX Þ me for X and ði ¼ ; ;rþ Þ, as well ˜a dimensional nonsupersymmetric heterotic string theories, as λ ða ¼ 1; …;rÞ. Thus, as Xþ → ∞, fluctuations in following the general strategy of Ref. [22]. If we denote these directions are highly suppressed, and we expect to the left-moving fermions transforming in the vector of obtain a theory in spacetime dimension d ¼ 10 − r. SOð25−nÞ by λ˜a for a ¼ 1; …; 25−n, then the tachyon In the case of the heterotic theories being studied here, 25−n appears in the worldsheet action via a superpotential term, we would ideally like to condense all components of the tachyon. Clearly, this will only be possible using the ≥ 2 X25−n ansatz (2.22) if n ; otherwise, we would naively obtain a 10 − 25−n 0 W ¼ λ˜aT aðXÞ: ð2:17Þ theory in d ¼ < dimensions. Accepting this γ a¼1 restriction for the moment, we may now fix the parameter in terms of n as follows. As Xþ → ∞, we may integrate out This superpotential induces a scalar potential, which takes the fields Xi and ψ i ði ¼ 2; …; 25−n þ 1Þ, as well as λ˜a the form ða ¼ 1; …; 25−nÞ. This leads to quantum corrections to the rffiffiffiffi dilaton and metric, and as it turns out, these corrections 1 X i α0X appear only at one loop. The corrections can be computed T a 2 − ∂ T a λ˜aψ μ V ¼ 8π ðXÞ 2π 2 μ ðXÞ : ð2:18Þ exactly using the methods in Refs. [18,20] and are found a a to be rffiffiffiffi The linearized equation of motion for the tachyon is 2 23−n 24−n found to be Δϕ ¼ Xþ; ΔG ¼ −ΔG−− ¼ : α0 γ þþ γ2 2 μ a μ a a ð2:24Þ ∂ ∂μT − 2v ∂μT T 0; 2:19 þ α0 ¼ ð Þ In particular, we see that we have obtained a linear shift of where vμ ¼ ∂μϕ is the dilaton gradient. For reasons to be the dilaton. This is consistent with worldsheet anomaly seen, we will choose a lightlike linear dilaton profile of cancellation—the contributions to the worldsheet central the form charge lost by integrating out fields must be made up for by a linear dilaton. In particular, the total central charge lost γ i ψ i λ˜a 3 25−n ϕ ¼ − pffiffiffiffiffiffiffi X−; ð2:20Þ upon integrating out X , , and is 2 × , and hence 2α0 we require

1 0 1 4− pffiffi γ 0 6α0 3 2 n where X ¼ 2 ðX X Þ and > is a constant to be cϕ ¼ Gμνvμvν ¼ × : ð2:25Þ fixed momentarily. There are certain solutions to (2.19) which give rise to Using the results in (2.24), this fixes tachyon superpotentials that are exactly marginal. One such 2−n γ 2 2 solution is to consider a lightlike tachyon profile [18–21], ¼ : ð2:26Þ

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Note in particular that for this value of γ, the lower- TABLE II. Heterotic vacua in d>2. dimensional dilaton (i.e., that obtained after integrating out the massive fields) simplifies as follows: ndMassless fermions Gauge bosons rffiffiffiffi 3 6 112 266 3− 2−n 4 8 240 255 γ 2 2 n 2 2 ϕ ¼ − pffiffiffiffiffiffiffi X− þ Xþ ¼ pffiffiffiffi X1: ð2:27Þ 5 9 248 248 2α0 α0 γ α0

In other words, the dilaton gradient becomes spacelike. since this results in a two-dimensional vacuum, we will This will be an important point in what follows. Before wait until the next subsection to discuss it. moving on, let us emphasize that (2.20) and (2.21) In more detail, the spectrum of each theory is as follows. α0 constitute -exact solutions to string theory. For n ¼ 3, we have a six-dimensional (6D) tachyon-free Having constructed the relevant dimension-changing theory with gauge group E7 × E7 and with massless solutions, we now study the lower-dimensional theories fermions in the ð56; 1Þ and ð1; 56Þ .Forn ¼ 4,wehave þ → ∞ þ − obtained as X . The partition functions for these an eight-dimensional tachyon-free theory with gauge group lower-dimensional theories are obtained by simply inte- SUð16Þ and with massless fermions in the 120 and 120−. i ψ i 2 … 25−n 1 λ˜a þ grating out the fields X , for i ¼ ; ; þ and Finally, for n ¼ 5, we have a nine-dimensional (9D) 1 … 25−n for a ¼ ; ; . Doing so, we straightforwardly obtain tachyon-free theory with gauge group E8 and with a single massless fermion in the adjoint. It is easy to see how this 1 −16þ26−n 0 8−25−n 0 0 8−25−n 0 chiral matter descends from that in (2.8). All theories also Z ¼ jηj ½ðZ0Þ L0 − ðZ1Þ L1; NS 2 have gravitons, B-fields, and dilatons for the appropriate 1 6−n 5−n number of spacetime dimensions. These bosonic fields are Z η −16þ2 Z1 8−2 L1; 2:28 R ¼ 2 j j ð 0Þ 0 ð Þ massless in the sense that there are no mass terms in the low-energy effective action, but due to the presence of the a where Lb are now given by linear dilaton background, the effective mass operator gets shifted [40] (i.e., there is nonzero vacuum energy), leading

1 5− to the effective masses observed in (2.30). Note finally that L0 Z0 16−2 n Z0 16 2n − 1 Z0 16 0 ¼ 2n ð 0Þ ½ð 0Þ þð Þð 1Þ in six dimensions, the B-field is nonchiral. þð2n − 1ÞðZ1Þ16; 0 1. Anomalies 1 5− L0 Z0 16−2 n Z0 16 2n − 1 Z0 16 Let us now discuss the anomalies of the putative d ¼ 6, 1 ¼ 2n ð 1Þ ½ð 1Þ þð Þð 0Þ 8, and 9 string theories identified above. The 9D theory is n 1 16 þð2 − 1ÞðZ0Þ ; clearly not subject to any perturbative anomalies, so we 6 1 5− focus on the cases of d ¼ , 8 for the moment. In both L1 Z1 16−2 n Z1 16 2n − 1 Z0 16 0 ¼ 2n ð 0Þ ½ð 0Þ þð Þð 0Þ cases, anomalies can be canceled via appropriate Green- Schwarz counterterms, which descend from their ten- 2n − 1 0 16 þð ÞðZ1Þ ; ð2:29Þ dimensional counterparts. We begin with the 6D E7 × E7 theory. Recall that the cf. Eqs. (2.5) and (2.6). The case of n ¼ 5 was studied chiral content of this theory consisted of a left-moving originally in Ref. [22]. fermion in the (56, 1) as well as a right-moving fermion in Integrating over τ1 gives the following level-matched the (1, 56). Because there are equal numbers of left- and results, right-movers, pure gravitational anomalies cancel automati- cally. The remaining anomaly 8-form is given by 1 3 3∶ Z 1080 ¯ 4 31360 ¯ 4… n ¼ NS ¼ ðqqÞ þ ðqqÞ 1 4 4 Z 224 276480 ¯ … I 8 ∝ ðTr 56 1 F − Tr 1 56 F Þ R ¼ þ qq þ ð Þ 24 ð ; Þ ð ; Þ 1 5 4∶ Z 1566 ¯ 8 76440 ¯ 8… 1 n ¼ NS ¼ ðqqÞ þ ðqqÞ − 2 − 2 2 24 ðTrð56;1ÞF Trð1;56ÞF ÞtrR ; ð2:31Þ Z 960 1711104 ¯ … R ¼ þ qq þ 1 9 5∶ Z 1785 ¯ 16 108500 ¯ 16… with F the curvature of the E7 × E7 bundle. We may split n ¼ NS ¼ ðqqÞ þ ðqqÞ F ¼ F1 ⊕ F2 in terms of curvatures for the individual E7 Z 1984 4058880 ¯ … R ¼ þ qq þ ; ð2:30Þ factors. As is well known, exceptional groups have no order-4 Casimir invariants, and hence the terms of type 4 ≔ 4 from which we read off the data in Table II. Note that the trFi Tr56Fi can be factorized. Indeed, one computes methods discussed here also apply to the case of n ¼ 2,but from e.g., the Appendixes of Refs. [41,42] that

106026-6 STABLE VACUA FOR TACHYONIC STRINGS PHYS. REV. D 103, 106026 (2021)

1 To see that it does, we begin by reexpressing the trace in trF4 ¼ ðtrF2Þ2;i¼ 1; 2; ð2:32Þ i 24 i the antisymmetric 2-tensor representation in terms of the trace in the fundamental. For general SUðNÞ, one finds which allows us to rewrite the anomaly polynomial in the 1 1 form iF iF 2 2iF Tr 2 e ¼ ðtre Þ − tre ; ð2:38Þ ½ 2 2 1 1 2 1 1 2 ∝ 2 − 2 − 2 − 2 Ið8Þ 2trR 24trF1 2trR 24trF2 : ð2:33Þ which in particular gives

3 − 4 3 Since the anomaly polynomial does not factorize completely, Tr½2F ¼ðN ÞtrF ; the anomaly cannot be canceled by a Green-Schwarz term 5 5 2 3 Tr 2 F ¼ðN − 16ÞtrF þ 10trF trF : ð2:39Þ involving a single two-form field. Fortunately, in the current ½ theory, the B-field is nonchiral—in other words, it can be Remarkably, we see that for SUð16Þ, the trF5 term is absent split into self-dual and anti-self-dual pieces B, which can 5 120F then be used to write two separate Green-Schwarz terms, ala` from the last equation, and hence Tr does indeed Ref. [43]. In particular, if we take factorize. We thus have 1 1 1 1 5 3 2 2 2 2 2 I 10 ∝ Tr120F − Tr120F trR dH trR − trF ;dH− trR − trF ; ð Þ 60 36 þ ¼ 12 1 ¼ 12 2 1 1 ð2:34Þ 3 2 − 2 ¼ 3 trF 2 trF trR : ð2:40Þ for H ¼ dB, the relevant Green-Schwarz terms are simply Then, modifying the B-field such that Z 1 1 2 − 2 ∝ ∧ 2 − 2 dH ¼ trR 2 trF ; ð2:41Þ SGS Bþ trR 12 trF1 Z 2 1 2 we conclude that the appropriate Green-Schwarz term is − B− ∧ trR − trF : ð2:35Þ 12 2 Z ∝ ∧ 3 SGS B trF : ð2:42Þ Note that under spacetime parity transformation, we effec- tively interchange the two factors of E7.Thisisconsistent with the fact that the chiral spinors charged under each E7 This completes the perturbative anomaly analysis. are also interchanged. Incidentally, the opposite signs We now briefly discuss global anomalies. Beginning 6 between the two Green-Schwarz terms confirms that the with the case of d ¼ , the full set of possible global B-field must nonchiral, as opposed to having two B-fields of anomalies is captured by the same chirality. ℧7 ≔ ΩSpin 1 Now, consider the eight-dimensional (8D) SUð16Þ SpinðBE7 × BE7Þ Homð 7 ðBE7 × BE7Þ;Uð ÞÞ: theory. The chiral content of this theory is a left-moving ð2:43Þ fermion in the 120 and a right-moving fermion in the 120. In eight dimensions, there are no (perturbative) pure For our current purposes, it suffices to treat BE7 as the gravitational anomalies, and the anomaly 10-form is Eilenberg-MacLane space KðZ; 4Þ, in which case we can 1 use the results of Ref. [44] to conclude that ∝ 5 − 5 Ið10Þ ðTr120F Tr120F Þ 120 ℧7 0 SpinðBE7 × BE7Þ¼ : ð2:44Þ 1 3 3 2 − ðTr120F − Tr120F ÞtrR ; ð2:36Þ 72 Hence, this theory is free of global anomalies. Things are somewhat less straightforward in the with F the SUð16Þ curvature. In the current notation, F is remaining cases. For the d ¼ 8 theory, the relevant group anti-Hermitian, so we have ℧9 16 ≅ ℧9 ⊕ ℧˜ 9 16 is SpinðBSUð ÞÞ SpinðptÞ SpinðBSUð ÞÞ. Using 2nþ1 2nþ1 the results of e.g., Appendix B of Ref. [45], the reduced Tr120F ¼ −Tr120F : ð2:37Þ dual bordism group is seen to vanish,

In order to use the Green-Schwarz mechanism, we must ˜ 9 5 ℧ ðBSUð16ÞÞ ¼ 0; ð2:45Þ thus check that Tr120F factorizes. Spin

106026-7 JUSTIN KAIDI PHYS. REV. D 103, 106026 (2021) Z and thus there are no pure gauge nor mixed gauge- 1 d2τ Λ ¼ ðZ − Z Þ; ð2:48Þ gravitational anomalies. There can, however, be pure d d d=2þ1 R NS 2ð2πÞ2 F τ gravitational anomalies, corresponding to the nonzero dual 2 spin bordism group where we have included the factors of ð2πτ2Þ coming from the integral over bosonic zero modes. Though this ℧9 Z 2 SpinðptÞ¼ð 2Þ : ð2:46Þ integral cannot be evaluated analytically, we may evaluate it numerically, giving the following results for the d ¼ 6,8, In fact, one part of this group is familiar from the classic and 9 theories, — work of Alvarez-Gaum´e and Witten [46] physically, it Λ ≈ 0 164 Λ ≈ 0 079 Λ ≈ 0 054 corresponds to the fact that a theory of an odd number of 6 . ; 8 . ; 9 . ; ð2:49Þ Majorana fermions is anomalous in eight dimension. Since α0 2π −1 in the present case we have an even number of Majorana in units where ¼ð Þ . In particular, we see that the one-loop cosmological constant is positive in all cases. fermions, we clearly do not run afoul of this Z2 anomaly. It is useful to contrast the current situation with that of As for the remaining Z2, this is expected to capture an 3 the Oð16Þ × Oð16Þ heterotic string. The Oð16Þ × Oð16Þ analogous anomaly for an odd number of massless spin- 2 string was also tachyon free and nonsupersymmetric, and fields, of which we have none.6We thus expect the 8D its one-loop cosmological constant was evaluated in theory to be free of global anomalies. 10 Ref. [5], giving a result Λ10 ≈ 0.037 of a similar order d 9 ℧ BE8 Finally, in ¼ , we consider the group Spinð Þ.A of magnitude to the values above. However, in that case, classic result of Stong [47] gives us there was a crucial conceptual difference in the interpre- tation of this cosmological constant. A positive cosmo- ℧10 ℧10 ⊕ ℧˜ 10 Z 3 ⊕ Z 2 SpinðBE8Þ¼ SpinðptÞ SpinðBE8Þ¼ð 2Þ ð 2Þ : logical constant generally signals the presence of a dilaton tadpole, which for a massless dilaton gives rise to an ð2:47Þ instability of the vacuum. In the Oð16Þ × Oð16Þ heterotic string, the dilaton was indeed massless, so despite the fact In this case, we see that there are both potential gauge and that the theory was tachyon free the positive cosmological gravitational anomalies. As before, one of the pure gravi- constant indicated a vacuum instability (see e.g., Ref. [50] tational anomalies was already discussed in Ref. [46] and for potential stabilization mechanisms). On the other hand, corresponds to the fact that a theory of an odd number in the case of the lower-dimensional heterotic strings being of Majorana fermions is anomalous in nine dimensions. studied here, the dilaton is actually massive. As such, a Since we have an even number of fermions, the current tadpole for it only serves as a small perturbation around the 7 setup is not subject to this anomaly. By an argument Λd > 0 vacua, at least in the weak-coupling regime. analogous to that in Footnote 6, there is also an anomaly Where exactly is the weak-coupling regime? Since in the 3 involving an odd number of spin-2 fields, which causes no lower-dimensional theories the dilaton has a spacelike trouble here. It remains to fully study the other potential gradient, see (2.27), we conclude that the weakly coupled global anomalies. region is X1 ≪ 0. On the other hand, recall that the lower- dimensional vacuum is really only observed at Xþ ≫ 0, where the integrating out of massive fields is valid. Thus, 2. Cosmological constant the region, which is well described by a lower-dimensional Because the d ¼ 6, 8, and 9 theories identified above are theory with positive cosmological constant, is Xþ → ∞ tachyon free and nonsupersymmetric, one expects them to and X1 → −∞, i.e., late times and toward the left of the have a finite and nonzero cosmological constant. The one- X1-axis. loop contribution to the cosmological constant may be We should note that, since the lower-dimensional the- obtained by simply integrating minus the torus partition ories have no “Liouville walls,” strings can cross over into F function over the fundamental domain [48,49], the region to the right of the X1-axis. In this region, tadpoles may lead to strong-coupling effects which ruin the vacuum, 6A rough explanation for why this anomaly should be related and we should ideally switch to an S-dual description. One 3 to spin-2 fields is as follows. Consider compactifying the theory could certainly imagine deforming the lower-dimensional on a closed 8-manifold down to zero dimensions. If there are theory by adding a Liouville wall, thereby keeping the an odd number of fermionic zero-modes on the 8-manifold, we will encounter the zero-dimensional anomaly captured by ℧1 Z 7 SpinðptÞ¼ 2. An 8-manifold admits two multiplicative genera, The intuition is that a tadpole gives rise to modifications of the g which correspond to the indices of two distinct Dirac operators. vacuum of order 2. If the particle with a tadpole is massless, this 1 M One linear combination of these is the usual spin-2 Dirac operator, destroys the vacuum. But if the particle is massive, then as long as and the other is the Rarita-Schwinger operator. I thank Kantaro we are at sufficiently weak coupling, these corrections are small, Ohmori for this argument. and the vacuum is not appreciably affected.

106026-8 STABLE VACUA FOR TACHYONIC STRINGS PHYS. REV. D 103, 106026 (2021) fields confined to the weakly coupled region with positive Likewise, beginning with the Oð16Þ × E8 tachyonic cosmological constant. However, it is unclear what such a theory, we condense eight components of the tachyon, wall would uplift to in the full, time-dependent solution. which breaks the gauge group to Oð8Þ × E8. The remaining To summarize, it appears that the lower-dimensional eight tachyons become massless, giving rise to the eight Z 8 8 string theories admit perturbatively stable vacua with contributions in NS. It is easy to see how the s and c positive vacuum energy in some appropriate region, and fermions descend from the data in (2.8), upon noting the with no moduli. This is at the cost of having a linear dilaton. branching 128 ¼ð8s; 8sÞ ⊕ ð8c; 8cÞ under the decomposi- 16 ⊃ 8 8 1280 It would be interesting to see if the positivity of Λd persists tion Oð Þ Oð Þ × Oð Þ, and likewise for . Finally, when higher-loop corrections and nonperturbative effects in the case of the Oð8Þ × Oð24Þ theory, we again condense are taken into account. eight components of the tachyon, breaking completely Oð8Þ. In this case, we have actually removed all tachyons, D. Tachyon condensation to d =2 and hence we are left with an empty NS sector. There is, however, a single chiral fermion in the fundamental of In the cases of n ¼ 0, 1, the ansatz (2.22) cannot be used 24 Z Oð Þ. This gives the factor of 12 in R. to condense all components of the tachyon, since there exist Incidentally, note that the technique of reducing down to more tachyons than physical spacetime dimensions. We two dimensions in this way cannot be applied to the heterotic must therefore resort to an alternative mechanism. theories with higher-dimensional vacua, i.e., those with One particular mechanism is to simply condense eight n>2. Indeed, it is simple to check that reducing those components of the tachyon in the previous manner to theories to two dimensions in the above way gives rise to reduce to a theory in two dimensions. This automatically negative degeneracies in the tentative partition functions. gives rise to a stable vacuum since, due to the coupling to This is in line with the fact that the three 2D heterotic strings the spacelike dilaton gradient, the masses of the remaining discussed above are the only such consistent theories. components of the tachyon are renormalized to zero. Said Finally, let us address the issue of anomalies in the 2D in a different way, in two dimensions, the boundary theories obtained here. Since the field content of the n ¼ 0 conditions at X1 → −∞ (where the linear dilaton theory and n ¼ 1 theories is clearly nonchiral, there are no is weakly coupled) for normalizable states is such that even perturbative anomalies. On the other hand, for the n ¼ 2 relevant operators do not give rise to normalizable modes case, one has 24 chiral fermions, which on their own do 8 that grow exponentially. have a chiral anomaly. In order to cancel this, one would With this in mind, we condense eight components of the expect the presence of a Green-Schwarz term. At first sight, tachyon. Integrating out the relevant fields from the this might seem rather mysterious from the higher-dimen- partition functions in (2.5) and (2.6), we obtain precisely sional perspective, since as discussed at the end of Sec. II A the partition functions (2.12) and (2.13) of the known 2D the original Oð8Þ × Oð24Þ theory had no Green-Schwarz heterotic strings reviewed in Sec. II B. For convenience, we term. However, inR two dimensions the Green-Schwarz term recall that the level-matched partition functions for these ∝ is simply SGS B, which can in fact descend from ten theories are dimensions.9

0∶Z 24 Z 0 n ¼ NS ¼ R ¼ III. TYPE 0 STRINGS 1∶Z 8 Z 8 n ¼ NS ¼ R ¼ In this section, we study tachyonic Type 0 string theories. 2∶Z 0 Z 12 n ¼ NS ¼ R ¼ ; ð2:50Þ We will show that all such tachyonic strings admit two- dimensional stable vacua. For the case of oriented Type 0 where we have included the n ¼ 2 case, which had a total strings, this was already shown in Ref. [13]. In particular, it of 8 tachyons. was found there that a cascade of tachyon condensations In more detail, beginning with the ten-dimensional leads to a gradual reduction of dimension, ending even- SOð32Þ theory, we condense eight components of the tually with the two-dimensional Type 0 strings introduced in Refs. [28,29]. In the current section, we will extend that tachyon, which transforms in the fundamental of − SOð32Þ. Doing so breaks the SOð32Þ and gives rise to a result to the eight Pin Type 0 strings. 2D theory with gauge symmetry SOð24Þ. The remaining 24 Of use to us will be the recent understanding that string tachyons become massless due to their coupling to the theories with different GSO projections can be thought of Z dilaton, giving rise to the 24 in NS. Since there were no massless fermions in the original SOð32Þ theory, there are 9Note, by the way, that the interpretation of this Green- again none in the resulting 2D theory. Schwarz term is rather different from its counterpart in higher dimensions—in the current case, we require that dH ¼ 1 and hence that there is a spacetime-filling source for the B-field. The 8I thank Simeon Hellerman for very useful correspondence role of this source is played by a “long string”; see Ref. [26] for about this point. details.

106026-9 JUSTIN KAIDI PHYS. REV. D 103, 106026 (2021) as differing by the addition of certain topological terms to for some r. This profile then gives rise to exponentially the worldsheet action [3,4]. In the presence of boundaries, growing masses for the 2r coordinates Xi with these topological terms give rise to anomalous edge modes, i ¼ 2; …; 2r þ 2, as well as their fermionic superpartners which in turn lead to extra structure (e.g., Clifford bundles) ψ i. Consequently, as Xþ → ∞, we obtain a theory in d ¼ carried by the end points of open strings. This extra 10 − 2r dimensions, localized at X2 ¼ … ¼ X2rþ2 ¼ 0. structure modifies the K-theoretic classification of branes In this way, we may condense ten-dimensional N ¼ in the different GSO-projected theories. ð1; 1Þ theories to any even number of dimensions (for the From this point of view, the fact that the classification of case of an odd number of dimensions, see Ref. [13]). 2D string theories mimics that of tachyonic ten-dimensional However, by explicitly integrating out the appropriate fields (10D) theories is in some sense obvious. Beginning with a in the ten-dimensional partition functions given below, it is given 10d worldsheet theory, condensing the tachyon leads simple to check that a new tachyon appears if we condense to solutions in which certain fields are integrated out. But to any dimension d>2. One thus obtains a cascade of this process has no effect on the topological term present in tachyon condensations, the stable end point of which is a the worldsheet action. Thus, the lower-dimensional string two-dimensional vacuum. We now show that these are should also admit variants distinguished by such terms. precisely the known two-dimensional strings. When considering open strings, these terms demand the same anomalous edge modes in two dimensions as in ten B. Oriented Type 0 strings dimensions, and hence the K-theory classification of branes (in the applicable range of world volume dimensions) must 1. d = 10 be the same. We will give slightly more detail on this below. We begin by reviewing the known case of oriented Type To recapitulate, for many known two-dimensional closed 0 strings. These strings have the same worldsheet content as superstrings, we will identify a ten-dimensional tachyonic the Type II strings, namely, ten left- and right-moving string which admits a dimension-changing dynamical μ μ bosonic fields X , together with their superpartners ψ transition to them. It should be noted that this hierarchy μ and ψ˜ . The defining feature of Type 0 strings is the fact can be continued upward indefinitely, i.e., the ten- that left- and right-moving worldsheet fermions are taken to dimensional theory can itself be understood as descending have the same spin structures. Because of this identification from condensation of a supercritical theory, and so on. The of spin structures, the torus partition function is only special thing about ten dimensions is that a linear dilaton is not needed for worldsheet anomaly cancellation. 1 η −16 0 16 0 16 1 16 1 16 Z ¼ 2 j j ðjZ0j þjZ1j þjZ0j jZ1j Þð3:4Þ A. Tachyon condensation in the notation of (2.2). Above, we have included a possible We begin by reviewing the general framework of choice of sign in front of the final term. This sign has no tachyon condensation for N ¼ð1; 1Þ superstrings, which 1 effect on the partition function since Z ¼ 0, and the level- differs slightly from that in the heterotic case. 1 matched result is For the known tachyonic N ¼ð1; 1Þ superstrings, there is always a single tachyon. Condensation of the tachyon is −1 1 Z qq¯ 2 192 1296 qq¯ 2 49152qq¯ … : expected to give a (1,1) superpotential, ¼ð Þ þ þ ð Þ þ þ ð3 5Þ

W ¼ T ðXÞ; ð3:1Þ in both cases. We see that both theories have a single tachyon. There are no spacetime fermions in these theories, which gives rise to a scalar potential, so the 192 massless degrees of freedom can decomposed into a graviton (35), a B-field (28), a dilaton (1), and 0 0 α μν iα μ ν V ¼ g ∂μT ðXÞ∂νT ðXÞ − ∂μ∂νT ðXÞψ˜ ψ : ð3:2Þ remaining 128 massless bosonic degrees of freedom which 16π 4π end up being Ramond-Ramond (RR) form fields. Unlike in (2.18), this scalar potential depends only on the The difference between the two signs is in how the derivatives of the tachyon. Hence, we will have to take the 128 degrees of freedom are organized into separate RR tachyon profile to be quadratic, as opposed to linear, in form fields. As was discussed in Refs. [3,4], this may be the fields that we want to give a mass to. Taking the dilaton understood by interpreting the sign as arising due to the to be as in (2.20), the simplest tachyon profile (subject to presence of an extra topological term in the worldsheet Σ some additional discrete global symmetry constraints) is of action. Concretely, given a worldsheet and a spin σ the form [20] structure , we may add to the action a factor of the Arf invariant,10 Xrþ2 2m β T X e Xþ XiXrþi 3:3 10 ð Þ¼ α0 ð Þ We consider only the cases of zero or one copy of Arf since it i¼2 is a mod 2 invariant.

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S ¼ S0 þ iπnArfðΣ; σÞ;n¼ 0; 1; ð3:6Þ broad classes of unoriented Type 0 strings—those with Pin− structure on the worldsheet and those with Pinþ where S0 is the standard worldsheet action without topo- structure. Recall that the difference between Pin can be logical term. Concretely, the Arf invariant is defined such captured by the squaring properties of the orientation that [51] reversal operation R being gauged, i.e., 2 σ 2 σ 2 σ 0 ArfðT ; NSNSÞ¼ArfðT ; NSRÞ¼ArfðT ; RNSÞ¼ ; Pinþ∶ R2 ¼ 1; Pin−∶ R2 ¼ð−1Þf; ð3:8Þ 2 σ 1 ArfðT ; RRÞ¼ ; ð3:7Þ and hence the contribution of this topological term to the with ð−1Þf ≔ ð−1ÞfLþfR the total worldsheet fermion num- 2 torus partition function, namely, ð−1ÞnArfðT ;σÞ, reproduces ber. In the standard conventions, worldsheet parity Ω acts the sign in (3.4). In the presence of a boundary, the Arf on fermions as invariant leads to Majorana edge modes, which modify the K-theory classification of D-branes and RR fields.11 Ωψðt; σÞΩ−1 ¼ −ψ˜ ðt; 2π − σÞ; The upshot is that for the plus sign, the 128 degrees of −1 Ωψ˜ ðt; σÞΩ ¼ ψðt; 2π − σÞ; ð3:9Þ freedom are organized into two 1-forms and two 3-forms, while for the minus sign, they are organized into two scalars, Ω2 −1 f Ω two 2-forms, and a non-self-dual 4-form. This field content and hence ¼ð Þ . Thus, gauging alone gives rise to worldsheets with Pin− structure. Unlike for Type II is exactly twice that of the usual Type IIA and IIB theories, Ω and for that reason, these theories are usual referred to as theories, is a symmetry of both Type 0A/B and hence can be gauged in both to give two separate Pin− theories. Type 0A and 0B. The spectrum of branes in a given Type 0 Ω theory is twice that of its Type II counterpart. One can also consider gauging twisted by some additional discrete global symmetries. In particular, the Type 0 theories have a Z2 × Z2 symmetry generated by 2. d = 2 left-moving spacetime and worldsheet fermion parities, Two-dimensional analogs of the Type 0 theories have a ð−1ÞFL and ð−1ÞfL . We may then consider alternative F f long history; see e.g., Refs. [26,28,29]. In two dimensions, orientifoldings by ΩF ≔ Ωð−1Þ L and Ωf ≔ Ωð−1Þ L . 2 f the Type 0B theory has as its physical operator spectrum Note that ðΩFÞ ¼ð−1Þ once again, and thus gauging it one propagating massless bosonic field in the Neveu- in the Type 0A/B theories again gives two Pin− theories. In Schwarz-Neveu-Schwarz (NSNS) sector (often referred fact, each of the four Pin− theories mentioned thus far “ ” to as a tachyon, even though it is massless in two admits two variants, differing by the action of Ω on Chan- dimensions) as well as a pair of compact chiral bosons Paton factors; this is analogous to the difference between in the RR sector. As for the Type 0A theory, there is again a O9− and O9þ orientifolds of Type IIB. So, in total, there are single scalar in the NSNS sector, as well as a pair of 1-form actually eight Pin− Type 0 strings. fields (fixed at zero momentum). In Refs. [3,4], these eight theories were interpreted as In Ref. [13], it was shown that the ten-dimensional arising from the addition of a topological term to the Type 0A/B theory condenses to the two-dimensional worldsheet action. In particular, given a worldsheet Σ Type 0A/B theory upon choosing the profile (3.3) with with Pin− structure σ, one can consider adding to the 4 r ¼ . The massless NSNS sector scalar is the descendant action n copies of the so-called Arf-Brown-Kervaire of the tachyon in higher dimensions. The fact that the (ABK) invariant [57], much like the Arf invariant in spectra of RR fields match with their higher-dimensional the oriented case. The ABK invariant is a mod 8 invariant, 2 counterparts (for p-forms with p< ) follows from the fact and we have the following identifications, that the worldsheet factors of the Arf invariant in (3.6) are left unchanged by the integrating out of fields and lead to n ¼ 0; 4∶ ð0B; ΩÞ n ¼ 1; 5∶ ð0A; ΩÞ the same spectra of boundary Majorana fermions. n ¼ 2; 6∶ ð0B; ΩFÞ n ¼ 3; 7∶ ð0A; ΩFÞ; ð3:10Þ C. Unoriented Type 0 strings with the first element in parentheses denoting the 1. d = 10 starting theory and the second element denoting the We now move on to the new case of unoriented Type 0 operator being gauged. The difference between theories theories. In Refs. [3,4], unoriented Type 0 theories were differing by Δn ¼ 4 is the action of the parity operators classified, again using invertible phases.12 There are two on Chan-Paton factors. The massless and tachyonic spectra of all eight theories was determined in — 11This mechanism was anticipated in Refs. [52–54]. Refs. [3,4] for our purposes, we simply note that all 12For previous works toward such a classification, see e.g., theories have a single tachyon, and the spectrum of branes Refs. [1,2,55,56]. isasgiveninTableIII.

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TABLE III. Dp-brane content of the eight Pin− Type 0 theories. nnp −1 012 3 456 7 89

0 2Z2 2Z2 2Z 0002Z 0 2Z2 2Z2 2Z 1 Z2 Z ⊕ Z2 Z2 Z 0 Z 0 Z ⊕ Z2 Z2 Z ⊕ Z2 Z2 2 2Z 0 Z2 Z2 2Z 0 Z2 Z2 2Z 0 Z2 30Z 0 Z ⊕ Z2 Z2 Z ⊕ Z2 Z2 Z 0 Z 0 40 02Z 0 2Z2 2Z2 2Z 0002Z 50Z 0 Z ⊕ Z2 Z2 Z ⊕ Z2 Z2 Z 0 Z 0 6 2Z 0 Z2 Z2 2Z 0 Z2 Z2 2Z 0 Z2 7 Z2 Z ⊕ Z2 Z2 Z 0 Z 0 Z ⊕ Z2 Z2 Z ⊕ Z2 Z2

Before proceeding to two dimensions, we address the two distinct types of D1-branes. In the n ¼ 1 theory, there is remaining parity operator Ωf. It is clear that again a single propagating massless scalar field, as well as a single 1-form. The object charged under this is a single type 2 fL fL fLþfR f 2 ðΩfÞ ¼ Ωð−1Þ Ωð−1Þ ¼ Ωð−1Þ Ω ¼ð−1Þ Ω ¼ 1; of D0-brane. In the n ¼ 2 theory, there are two massless scalar fields, one of which is compact. There are Dð−1Þ- ð3:11Þ branes charged under the compact scalar. Finally, in the 3 so this gives a theory with Pinþ structure. In Ref. [4], this n ¼ case there is again a single propagating massless case was shown to be tachyon free, so it will not be relevant scalar field, as well as a single 1-form. The object charged for our discussion. under this is a single type of D0-brane. The (nontorsion) spectrum is periodic mod 4, so this completes the list. 2. d = 2 In all cases, the massless scalar field can be interpreted as − the descendant of the higher-dimensional tachyon upon The Pin theories discussed above condense to two- condensation. The spectra of branes is seen to match precisely dimensional unoriented Type 0 theories. The latter were with the nontorsion elements listed in Table III,inthe classified in Refs. [30,31]. As expected, they are obtained appropriate range of world volume dimensions. This again by starting with the 2D Type 0 theories of Sec. III B 2 and must be the case, since the worldsheet factors of ABK—and gauging an appropriate parity operation. — − hence the corresponding K-theory classification remain In Refs. [30,31], eight 2D Pin strings were identified, unchanged upon condensation of the tachyon. again interpretable as orientifolds of 2D Type 0A/B by Ω or ΩF, with some twofold choice of action on Chan-Paton ACKNOWLEDGMENTS factors. We may again label the theories by n ∈ f0; …; 7g,in accordance with the assignments in (3.10).InRef.[30],the I would like to thank Luis Alvarez-Gaum´e, Oren spectra of these theories were found to be as follows. In the Bergman, Simeon Hellerman, Per Kraus, and Kantaro n ¼ 0 theory, there is a single propagating massless scalar Ohmori for very useful discussions at various stages of field from the NSNS sector, as well as two 2-forms (of fixed this project. I am especially grateful to Simeon Hellerman momentum). The objects charged under these 2-forms are for information regarding two-dimensional strings.

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