JHEP07(2007)065 construc- ′ hep072007065.pdf June 8, 2007 July 16, 2007 July 25, 2007 h a spatially , D-branes. ce of spacetime Received: Accepted: Published: itical models in critical B string theory in ten- ′ perturbatively stable ex- on-supersymmetric string that can be constructed in in non-critical string theories r, all known type 0 t we are not expanding around y in perturbative string theory. y) interpretation and propose a Marie Curie ce. We discuss the D-branes that http://jhep.sissa.it/archive/papers/jhep072007065 /j ´ Ecole Polytechnique ∗ ´ Ecole Polytechnique, Published by Institute of Physics Publishing for SISSA † string theory without massless tadpoles in backgrounds wit ′ Superstring Vacua, Conformal Field Models in String Theory Is perturbative stability intimately tied with the existen [email protected] [email protected] Unit´emixte de Recherche 7644, CNRS — Unit´emixte de Recherche 7095, CNRS — Universit´ePierre et † ∗ 91128 Palaiseau, France E-mail: E-mail: Centre de Physique Th´eorique, GRECO, Institut d’Astrophysique de98bis Paris, Bd Arago, 75014 Paris, France SISSA 2007 c fermions in string theory in more than two dimensions? Type 0 dimensional flat space istheory with a a rare purely bosonic example closed of string spectrum. a non-tachyonic, Howeve n Abstract: tions exhibit massless NSNS tadpolesa signaling true the vacuum fact of tha amples the of theory. type In 0 this note, we are searching for varying dilaton. We present twothat examples with exhibit this four- property andcan six-dimensional be Poincar´einvarian embedded in thisthis context manner. and the We type also ofstring comment gauge theories on theories and the their embeddinggeneral holographic conjecture of (Little for these String the non-cr role Theor of asymptotic supersymmetr Keywords: Vasilis Niarchos Dan Isra¨el ° Tree-level stability without spacetime fermions:examples in novel string theory JHEP07(2007)065 3 9 1 3 6 13 15 losed string spectrum physics for the last two struction and a powerful s a closed string tachyon ed version of this theory, which may or may not be re supersymmetry has been has been analyzed [7, 8] (see e interested in string theory heories with this feature are is naturally driven at strong ype 0 closed string tachyon towards the true vacuum via nic, so supersymmetry in the on-trivial metric and dilaton. redient of the real world. At onal tadpoles for the dilaton as ertain non-supersymmetric large – 1 – = 2 Liouville theory 8 QCD. N N B theory [1 – 3]. However, typically in this case there are ′ B strings ′ B strings ′ 0 B strings in four dimensions B strings in six dimensions ′ ′ A well known example of a string theory with a purely bosonic c 2.1 Overview 2.2 A model without massless tadpoles but with tachyons 4 3.1 A few words on parities in 3.2 Type 0 3.3 Type 0 gauge theories, for instance large is type 0signaling string the theory, but perturbative in instabilitycan of this be the case projected vacuum. thewhich out is spectrum of The known contain the t as physical the spectrum type in 0 a non-orient 1. Introduction and summary Supersymmetry has been a major driving force in theoretical 2. Critical type or three decades.calculational It tool, is but is notthe most same only likely time, a also therebroken conceptually an is mildly appealing important an or con obvious ing isexamples interest altogether where in absent. the situations In spectrum whe of thisbulk closed note, is strings broken we is to will purely theexpected b boso to highest be degree. relevant for Weakly theN coupled holographic string description t of c Contents 1. Introduction and summary massless NSNS tadpoles at tree-level which drive the theory the Fischler-Susskind mechanism [4 –perturbative, 6]. the shifted In background this isIn new expected fact, vacuum, in to cases have where aalso the n [9] backreaction of for the a NSNScoupling. recent tadpoles discussion) Furthermore, at it higher was genus, foundan one immediate that obtains consequence the additi of system the absence of supersymmetry. 3. Non-critical type 0 4. Discussion JHEP07(2007)065 n ction. isolated examples -wave mode s 9 orientifold, ′ ) is projected out, 1 rofile. In section 3, bona fide S 6-plane at an equal ′ sider in section 2 type st es motivates the study s. Although the ground n O in ten-dimensional string le closed string spectrum g the trivial it is, in general, to laton, that gives a univer- atures? The possibility of and/or winding remains. It also situations with space- ifold. models at hand correspond re are tachyonic modes that diagonal GSO projection in ed string spectrum and full rbatively stable. Similar to ittle String Theory. We find he standard examples of type r, the spectrum in this case is 0A bulk that contains a closed nly possible deep in the stringy fivebranes only an 0 string theories in the presence odel corresponds to the double t, in the near-horizon geometry string theories in non-critical di- sed string tachyons from the con- ong the linear dilaton direction). e). ′ B and fivebranes/orientifold configu- ′ 6-plane does not succeed in fully projecting with a modified worldsheet parity projection ′ 1 – 2 – S × 1 , 8 that exhibits full tree-level stability. R . This example involves a space-filling O 1 S i.e. vacua arise as an equivalent description of the near-horizo string theory that is at the same time non-tachyonic and does ′ ′ are tachyon- and tadpole-free). Are these examples special vacua with a linear dilaton is discussed in the concluding se ′ i.e. string theories in backgrounds with a non-trivial dilaton p ′ As a simple attempt to get rid of the massless tadpoles, we con This correspondence between non-critical type 0 The generic presence of dilaton tadpoles in the above exampl The vacua presented in section 3 are, to our knowledge, the fir lies in the presence of the background charge of the linear di ′ constructed in such astate way of the that NSNS it sector (with sources zero only momentum and massive winding RR alon field 0B string theory compactified on that acts non-trivially on the of weakly coupled stringtree-level stability theories ( with a purely bosonic clos rations leads to aof number of more interesting than observations. two Firs parallel fivebranes, the O distance between them. of the tachyon remains and this is projected out by the orient a subset of more massive statesturns with out non-vanishing that momentum forcannot any value be of projected the out. compatificationobtain an radius This example the paradigm in demonstates typenot 0 how exhibit non- massless tadpoles, of type 0 mensions, with an asymptoticthe linear ten-dimensional dilaton, model thatfilling discussed orientifolds are in sourcing pertu section onlynon-tachyonic. massive 2, The RR these orientifold fields. projection are removes Howeve tinuous the representations clo (closed stringsThe that models propagate al we consideris have tachyon-free. no localized The tachyons,0 origin so of the this who crucial difference with t out the tachyon. Perturbative stabilityregime, in these for models a isto o pair fivebranes/orientifold of configurations immersed fivebranes. instring a tachyon. Secondly, type In the the near-horizon non-critical decoupling limit of two we present the main result of this note, namely type 0 limit of NS5-branes configurations inof ten-dimensional type orientifold planes.scaling For instance, limit the of six-dimensional a m pair of parallel fivebranes in type 0A with a sal positive mass shiftnon-critical to superstrings (for the the spectrum, six and dimensional exampl the non-trivial cases, or part ofmore a general more type general 0 construction with the same fe To address this issuetheory we and discuss their the relation embeddingthat with of our fivebrane configurations these non-critical and vacua type L 0 JHEP07(2007)065 ) N e.g. = 1 ( (2.1) U string N ′ -branes ′ 9-plane, which . ′ NSNS 1 r i ¯ ψ 0 r ψ −| = →− r, which is part of the e unstable and give rise r 6, 15, 17] and references ne the true vacuum ( ¯ denotes the right-moving upersymmetric gauge the- ψ e general role of spacetime ich is similar in spirit with ce-filling O equires the cancellation of ype 0B strings is modded hat we are not expanding r y two D9- and D9 rther examples of this sort, c gauge theory with ¯ NSNS ory see [19]). However, the ψ fferent dimensionality. Non- F use the true vacuum may be i al gauge theories like ring sector as pecific examples presented in g theory and propose a new nal non-critical type 0 0 l string theories considered in e from the NSNS sector. This 3-plane in ten-dimensional flat as can read off the true properties | the presence of an open string ing theory were first discussed ic supersymmetry generalizing ′ ′ lized on D-branes without open ′ ineer odd dimensional (flavored) Ω , r and Ω ψ r Ω, where ¯ ψ ¯ F r = ) ψ ′ − model. We comment on the various possi- Ω → r ′ ¯ r ψ ′ ¯ = ( ψ – 3 – r Ω ′ ψ , r ¯ ψ = D3-branes on top of an O ′ Ω N r ψ ′ Ω , n B strings ′ α 0 = ¯ ′ Ω n α ′ Ω In the last part of the concluding section, we elaborate on th Another motivation behind this work is holography for non-s By definition Ω acts on fermion bilinears as 1 spacetime, one obtains on the D-branes a non-supersymmetri ories. For instance, by placing fermions and supersymmetryconjecture in that perturbatively links stableprevious perturbative strin arguments stability in and [10 – 12]–[13] asymptot this in note. accordance with the s gauge group. The AdS/CFTthe correspondence type for 0 this AdS/CFT theory, wh proposalin of the [14], presence was discussed ortherein. in absence [15]. D-branes of Fu in bulk thethis tachyons, type note can IIA can be variant be of foundSQCD used the [18]. in to non-critica [1 Investigating engineer the interesting D-branes four of dimension the four-dimensio theory of section 3to four-dimensional we non-supersymmetric find gauge thattachyon. theories Nevertheless, the in it corresponding is D-branes possiblenon-supersymmetric in ar gauge this theories context by tosupersymmetric using eng four-dimensional D-branes gauge of theories a canstring di be tachyons rea in the six-dimensional type 0 bilities in section 3. 2. Critical type in [1 – 3].out In by the the original modified example, worldsheet the parity worldsheet Ω theory of t 2.1 Overview Tachyon-free type 0 models in ten-dimensional critical str worldsheet fermion number. This parity acts in the closed st is a source forthis the tadpole. non-dynamical RR This ten-form. can Consistency be r achieved with the addition of thirt by implementing the Fischler-Susskind mechanism)naturally at and strong beca coupling, it is unclear to what extend we (for a detailed descriptionaddition of of D-branes D-branes leads intadpole to type an does 0B uncancelled not string dilatonaround tadpol the render a the true theory vacuum of inconsistent, the but theory. shows Since t it is hard to determi and projects outtype the 0B tachyonic spectrum. ground state In in spacetime, the this action NSNS introduces secto a spa JHEP07(2007)065 (32) (2.2) (2.3) U ] b a [ 4 4 the modular ¯ (32). ¯ η ϑ closed strings ] 4 U b τ a ′ η [ 4 of , which does not ϑ 1 2 S vacua? Is it always Z a similar example on o-form field, which is 496 ∈ ′ oth of these questions × ture in these examples. X s tadpoles exist and no 1 a,b ⊕ , 8 2 ionic spectrum from open ) celed dilaton tadpole. Are tical type 0 heory of unoriented closed R s to a very interesting non- cations has been performed and one is usually left with 2 onic truncation of the type 496 wR type 0 re absent and non-tachyonic neralized Ω projection. There or a more recent discussion of ) fundamental domain. The − ty of such theories, we present t the zero modes of the closed Z -functions. n R gelantonj. , xamples, one finds twisted sector ( η 2 1 B but with a massive RR tadpole, ¯ ′ q This theory has the torus partition 2 ) 2 . 2 - and , wR 1 ϑ S + Ω ¯ n R F × ( ) 2 1 1 − , is the PSL(2 q ( 8 s – 4 – . At the massless level, the former give a R ′ F −∞ = = ∞ -branes one obtains a bosonic spectrum from open X ′ P the overall volume diverging factor, and n,w 10 8 ¯ η V 1 πiτ 8 2 η e ) are the standard 2 τ / orientifold of type 0B theory on ( 9 = η ) ′ 2 radius, q 1 τ ), 2 τ 1 π ]( S b a (8 [ . On the D9-, D9 ′ ¯ τ ϑ 2 τ 4 dτd F Z denotes the 10 2 V R = An exhaustive analysis of tachyon-free orbifold compactifi The presence of massless NSNS and RR tadpoles is a generic fea Despite this annoying feature, the above construction lead As an attempt to obtain a model similar to type 0 The example in this subsection was suggested to us by Carlo An T 2 function exhibit any massless tadpoles,string but tachyon. fails to fully project ou 2.2 A model without masslessLet tadpoles us but consider with type tachyons 0B string theory on Canceling the RR tadpolesa requires theory the of addition open ofmassless and D-branes closed tadpoles strings at in the tree presence level of an an uncan essential property of all in [21 – 23]–[24] (for aclosed review string see tachyons [25]). that cannot Typically in beare these projected e certain out by cases, any ge however,vacua can where be twisted constructed sector [25]. tachyons a in more than twoD-branes dimensions, need where to no bebackgrounds tachyons with and added a no for constant massles consistency.in dilaton. the To We illustrate next are the subsection not rari a aware 0 of parameter of the torus, where necessary to add ais certain no. number In of the D-branes? next The section, we answer present to two b examples of non-cri of the theory from the above construction (see however [20] f tadpoles and vacuum redefinitions in string theory). supersymmetric model. Instrings the with bulk, this a is masslessIIB a spectrum, spectrum. purely which bosonic Theprojected is t only out essentially difference by a is Ω bos the absence of the NSNS tw gauge theory and the latter a Majorana-Weyl fermion in the strings stretching between the same type of branes and a ferm strings stretching between a D9 and a D9 modular functions we want to mod out this theory with the worldsheet parity JHEP07(2007)065 ) d ii (2.6) (2.8) (2.4) (2.7) (2.5) t be pro- . In that Ω include 1 and ( ′ R where both I . ± , and winding ) ) R n = ) ) il it il it ] ( ( w ](2 1 a (2 4 Ω and 1 a [ , that parametrizes 4 [ η 4 ) the antisymmetric I er η 4 i R = 0 and the other at ϑ ϑ Ω case, the orientifold bute. Since all of these a e radius a ontribute. In the direct with radius ) I X . ) re ( − Ω, which can be obtained − ( ′ 2 X ( 2 1 I 2 R Z rldsheet parity Ω, the parity Z ∈ X ∈ a . X n the transverse crosscap chan- a 2 i 1+ (with momentum 2 2 R 4 w n R 2 − 9 orientifold by looking at a similar Ω and ′ w − πt I = 2 πl (1) boson n, NSNS 1 − − | U i theory [26]. Both e e ± πR . = 2) = +1 (1) boson of radius +1 + ′ 2 n n +1 n U ) n, w /R Z α ) | 2 9-plane presented here has no massless tree X − X ′ ∈ − ( ( , m w B theories that will be presented below, share – 5 – Z ′ → 2 ) , the ∈ 4 X n il R i→ X X ( 1 ) 8 : it η 1+ n, w 1 s | (2 8 − dl -functions and the Poisson resummation formula we can : η η = ∞ 2 0 / 1 and zero winding. The winding (respectively momentum) 1 P e 9 R sectors appear. Using the well-known modular transfor- Z ) ± ± e R t Ω parities of the 2 acts on the states 9 1 = 2 s ) - and = 2 w and they have two fixed points, one at π π P ϑ R n m (8 X 2(2 projects out all the NSNS ground states with even momentum an t are simultaneously positive. Hence, the bulk tachyon canno 2 dt f 10 NS and 2 P 1 V → − ∞ ± ) as 0 f NS − 1 = Z X 2 n S = 10 is again given by (2.4), and the parities m 2 e V K s ) t 1 . So, both parities involve a pair of O0-planes. In the = is a shift symmetry acting on 2 and part of the background: s K 1 = πR 1 ± l It is clear from these expressions that there is no value of th In this channel only odd winding modes in the RR sector contri In the Klein bottle amplitude only states with zero winding c We can gain intuition about the geometry of this O In the NSNS sector, In particular, S = = around the 2 w Ω, where X the involution by T-duality from the Ω, context, one can formulates four basic parities: the usual wo modes are massive, we conclude that the O channel one obtains jected out fully in this example. where many similar features with this ten-dimensional example. m the combination of the modes with zero momentum and winding numb level tadpoles. The non-tachyonic, type 0 example in the conformal field theory of a single states with momentum w easily deduce the expression of the Klein bottle amplitude i nel ( mation properties of the where only the modes have mass squared (in our conventions zero winding. The lightest allowed modes in the NSNS sector a JHEP07(2007)065 = 1 c Ω. We B ˆ ¯ F ′ l matrix Ω parity, ) s − le-wrapping Ω orientifold, / itical strings. Type 0 ther parities, there is a genus expansion is then hic dual in this case is a ancel the massless one-loop dea to look for stable non- ckground, where the string e interesting for a number of ll known class of examples dimensional examples which loop amplitude, but leave a ivial physics takes place. In s explicitly with the addition ue of the condensate. With sheet and matrix model point pling region is effectively cut- turbative and the higher loop d. In this section, which con- symmetric vacua (at tree level just in the asymptotic region. atively and non-perturbatively ckgrounds that possess asymp- ing coupling constant falls off ng theory on these backgrounds ] that there is no type 0 ave a weak coupling asymptotic y with its T-dual, an mechanism. From the worldsheet ounds appear in the near-horizon g Theory [30]. [33, 34]. For the 0B pagating degree of freedom, is a massless Ω case they have opposite tension. In the ′ I – 6 – = 1 Liouville potential is odd under ( N B strings ′ Interestingly enough, it was pointed out in [33] that the dua 3 part of the theory, we can think of as a combination of two circ 1 S Another well known example is that of two dimensional non-cr Backgrounds with an asymptotic linear dilaton direction ar The tachyon in two dimensions, which is the only physical pro 3 scalar field. planes have the same tension, whereas in the will see that this obstructionwe can now be proceed evaded to in discuss. a set of higher case analyzed in this subsection, we are dealing essentiall string in two dimensions, because the model appears to describedivergences have string been cancelled propagation via the inpoint Fischler-Susskind of a view shifted one ba couldof attempt the to cancel appropriate the number tachyon ofdivergence tadpole D1-branes in [34]. the The leadingfinite D1-branes volume uncanceled c leftover diverging [35]. piece It of was pointed the out one- in [33, 34 one finds no masslesstachyon tadpole. tadpoles from the Klein bottle. For all o which, in the O1-planes with opposite RR charge. 3. Non-critical type 0 strings in two dimensions have beenof discussed view from in the [31, world 32].stable. In Orientifolds this in case this context string have theory been is discussed both in perturb Given the natural presence of dilaton tadpoles in non-super and beyond), it is,supersymmetric as vacua explained with in atains non-trivial the the dilaton introduction, main results backgroun a oftotically good this note, a i we linear will dilaton. concentrate onregion In ba where general, these the backgroundsexponentially tree-level h there) analysis and is athe reliable strong precise (as coupling setup that end the will whereoff str be non-tr by analyzed the below, addition thecontrolled of strong by cou a an non-trivial effective momentumappropriate condensate. tuning string string coupling The theory related inbackreaction these to is vacua accordingly the becomes suppressed val per everywhere and not reasons. It has been arguedis on general holographically grounds [27] dual that stri arises to in a type II non-gravitational stringgeometry theory. theory, where of linear dilaton A NS5-brane backgr we non-local, configurations non-gravitational [28, theory known 29]. as Little The Strin holograp JHEP07(2007)065 . y is 2 k +1 2 d = 2 q (3.2) (3.3) (3.1) tot c N = has the Q ϕ . † The theory. We Φ = 2 Liouville ′ 4 2 k ars right-moving . is the compact † q N n slope − e M = 15, where ψ, ψ (1) boson † rg models). vacua in non-critical ¯ θ se to a type 0 string U ′ tot d c † . = 4 and another one at ), so the real parameter 2 d ely stable theory. In this pe 0 Q z dθ · · · , ds to a type 0 2 the concluding section. The Γ d en-dimensional type II string ted theory. This chiral GSO + unstable. Our objective here round state is really a tachyon ric cousins these theories pos- / 2) worldsheet supersymmetry. 2) worldsheet supersymmetry compact ory on (3.1) with at least 2 ] , , Z that were first considered in [36] ¯ θF ate orientifold in pears as an equivalent description † π θ µ 2 = 3(1 + = (2 = (2 × M c + 2 + and two fermions N ¯ N Φ and Γ: one at = 2 Liouville theory. On the worldsheet ψ ¯ 2 k θ 2 r, ϕ N = 2 conventions) reads q M ′ √ − i α + ¯ θ e – 7 – = 2 superfield θψ = 2 Liouville) 2 z dθd N 2 √ N i d [( , such that the superpotential has the right periodicity + Z × 1 ,... π iϕ , µ 2 2 1 = 2 Liouville theory is , + one can construct spacetime supercharges which are mutuall − d + r N Q † R = 1 will be used to denote complex conjugation in spacetime and b = n † ΦΦ Φ= , R and (3.2) is fixed by the criticality condition θ Q 4 n Q parametrizes a linear dilaton direction with linear dilato z d = 2 r d R Z 8 is an even positive integer, Γ is some discrete group and π 1 8 ≤ d = a product of supersymmetric minimal models or Landau Ginzbu The common factor of these backgrounds is In this expression Φ is the chiral In what follows, we will examine the possibility of stable ty One can also impose a non-chiral GSO projection that gives ri The boson By construction the background (3.1) exhibits S In what follows daggers 4 that appears in = 6. The possibility of generalizations will be discussed in e.g. the latter is a theory of two interacting bosons quantities on the worldsheet. Liouville action in superspace notation (and The central charge of the ( theory on (3.1). Assess their a ten-dimensional closed non-supersymmet stringis tachyon, to hence find they an are orientifold perturbatively that projects out the tachyon and lea spacetime supercharges. It is preciselyin this theory the that near-horizon ap geometrytheory of [38]. fivebrane configurations in t target space of some two-dimensional CFT with the total central charge of the worldsheet theory (3.1). The string theories in higher dimensions.that In that needs case to the NSNS benote g projected we will out focus in on(for a order a class to review of define see non-critical string [37]). a theories perturbativ These are string theories on k possible radii will focus on two specific examples with trivial properties. At the special radius where d non-trivial part of the exercise is to identify the appropri local with all theprojection vertex gives operators rise to of a a supersymmetric chirally type GSO-projec II string the JHEP07(2007)065 A of (3.7) (3.4) (3.5) (3.6) arity Ω (1) CFT can U current. In the / ) R R t that stable non- , (1) U † nic NSNS ground state, ψ take the simple form rmation arises when one 42, 39]. A distinguishing ψ . onto real values. However, hat appears in the previous ¯ obvious candidates for the ype (3.1) with appropriate → J There is a certain subset of → µ ¯ ψ J, , whereas B-type parities give , = 2 Liouville will be relevant. ¯ s re the ground with a few words ψ = 2 Liouville parities ¯ ϕ ∂ϕ , ing the left- and right-movers) though there are some interest- B , = 2 Liouville theory have been 2 N breaks spacetime supersymmetry, † Ω N ¯ √ ¯ ¯ J ψ ion ψ , i N iπ + → − → currents e ¯ F ¯ F R πi = , ψ is the right-moving , ψ e (1) ¯ ) ) directions are compact, then the asymptotic J 2) supersymmetry there are two basic types ¯ = 2 Liouville theory is the following. A-type J U , = 2 supersymmetry. They are known as A- = 1 = 2 Liouville worldsheet fields as , Liouville theory z, z 1 z, z e – 8 – N P (¯ N − † N d and = (2 Φ(¯ ∂ϕ , , Φ 2 R = 2 . In the nomenclature of [39] the basic worldsheet ϕ B N → ϕ . Usually in this case a closed string tachyon ap- √ → i Ω N ) ϕ ) ϕ ¯ z ¯ + z s z, z, F = = acts on the P = 2 Liouville theory. A general analysis of the orientifolds J A : Φ( N : Φ( B .Ω A Ω. B vacua. These are based on the Ω ′ B Ω ¯ F ) − = 2 Liouville interaction that appears in (3.2). The A-type p and Ω = ( N A B is the shift (2.4) acting on as ϕ = 2 Liouville theory and its mirror dual supersymmetric SL(2 s B In a two dimensional QFT with Concluding this short introduction we would like to point ou The B-parity is related to the standard worldheet parity Ω, t We will be interested in parities that project out the tachyo N parities are Ω orientifolds that are extended along theory and to analyze its properties. Orientifolds in be found in [39]. Before delving into specific examples, weon would certain like to parities prepa of the for the purposes of this work only B-type orientifolds of of parities that reversewhile the preserving the worldsheet holomorphy chirality of (exchang the pears infinitesimally close to the supersymmetric point, al ing exceptions [40]. If some of the flat background at infinity involves a higher dimensional torus. ¨he/ope structure deformationsK¨ahler/complex of this torus that but does not generate any closed string3.1 tachyons [41]. A few words on parities in the As explained inconstruction [39], of with type B-type 0 orientifolds there are two supersymmetric vacua can bedeformations constructed of in type theorieschanges II of the string the t theory asymptotic on radius (3.1). of One such defo constructed recently in [39]. and Ω parities give orientifolds that are localized in the direct section, by Ω exchanges the two superpotential terms in (3.2) and project and B-type and thefeature details of of these their parities in definition the can case be of found the in [ where but retain the asymptotic weak coupling region, where the JHEP07(2007)065 = = 2 k (3.9) (3.8) . (3.11) (3.10) N he form i b a h 2, i.e. ). ¶ √ τ ; ¯ and a two-form = n, w ′ w 0 trivial and Γ the 2 Q − ,C 0 M n C project out the NSNS p, ng theory, it is natural a double scaling limit a e ) can be found in [43, 44, P µ e torus partition function c ich is geometric and has a = 4, es of the graviton multiplet heory on , s the background is unstable ch d owest level spectrum of this ves on two orthogonal NS5- ) and i uous representation characters, τ ) region. It receives contributions b ( a τ P ¤ h ( . b a 3 ¶ £ η τ . ϑ ; 2 B w m Ω k 2 + × ϕ + 2 ¯ p = 2 Liouville theory and is proportional to p n π ) q 2 ) τ N √ p, τ (¯ i = (¯ 3 = 2 Liouville – 9 – µ − ¤ η i c e ) b a b a τ £ h represent the momentum and winding around the ( = ×N 3 ϑ ) -momentum. Both = 2 Liouville interaction and give space-filling orien- ) e η τ 1 P w , dp ch 2 τ ϕ ; 3 ) ( N 2 ∞ ¤ R τ 0 b a 2 and Z p,m £ ( π , eq. (3.6), projects out the tachyonic NSNS ground state Z = 2 Liouville theory (the former is broken by the ϑ c n ∈ P (8 2 ch X N n,w Z ∈ X of , two non-negative mass squared RR scalars × a,b = 2 Liouville theory has a background charge as T ϕ ¯ τ 2 k τ 0), but keeps all the combinations of tachyon zero-modes of t N as 4 , dτd e P is the right-moving F Z B strings in four dimensions ϕ . All these modes appear with any momentum and winding ( ′ 2 0 p 2 V ) = (0 C ), a tachyon = n, w T In this case, where ¯ The quantum numbers The type 0B torus partition function of string theory on (3.9 The worldsheet parity G, B, φ identity. In other words, we want to consider type 0B string t In this subsection we consider a special case of (3.1) with we can rewrite ground state, commute withtifolds. the In theclearer ensuing, spacetime we interpretation. will focus on the3.2 former Type parity wh angular coordinate the type 0B torus partition function on (3.9) reads 1 in (3.2).to conjecture Extrapolating that theholographic type arguments description 0B of of string [38]branes the theory to in Little on ten-dimensional type type String (3.9) 0the 0B provides Theory meaning string stri in of that theory. this However, li correspondence a is unclear. 18]. Here weassociated will to be the weakly interested coupledonly only asymptotic linear from in dilaton the the continuous leading representations of piece of th Liouville interaction intheory the can strong be found coupling in( region). table 1 of The appendix A l of [43]. It compris potential with ( defined for generic the infinite volume of the target space. In terms of the contin JHEP07(2007)065 P P / (3.15) (3.13) (3.12) (3.14) ore, it = 0, (1) was U . w / ations, but ) ) it R ) , . it (2 3 ¸ (2 = 2 Liouville] η 5-plane. Hence, it 1 a 2 ¤ ′ ogy of [39]. In that ) crosscap channel 1 a · t £ 2 ´ π ϑ ×N il ate associated with the ; 1 (8 , ) 2 3 w y show up in these sectors. it , R 0 (2 model in critical string theory, ³ dpoles. Using the well-known the mirror SL(2 tised, a tachyon-free spectrum. odd winding modes in the RR i c racters and the theta-functions 1 table at tree-level as it exhibits † ative consistency of the theory. 1 a pears at the end of the following nge 2 S . Therefore none of them appears example of a non-supersymmetric ch 5-plane presented here. For more √ ′ × Z Φ+Φ ´h 1 +1 , 5-plane in question does not exhibit d near the strong coupling region of − ′ 2 ∈ n 8 Z 2 , j R 0) – to be invariant. Hence, the parity X ∈ p, θ e , w 4 ³ 1 e R sectors contribute to the trace. Adding 2 c ) + 1 Q− ± e R z d ) il / k ch 2 ( il ϕ d ¤ ( p +1 1 a gives rise to a space-filling O 3 . This is an appealing feature, since it allows for ) = ( n – 10 – £ Z η ) n P ϑ − and ¯ f
JHEP07(2007)065 Construc- ′ Hep072007065.Pdf June 8, 2007 July 16, 2007 July 25, 2007 H a Spatially , D-Branes
