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JHEP07(2019)111 Springer July 2, 2019 July 18, 2019 May 14, 2019 : : : February 14, 2019 : Revised Accepted Published Received Published for SISSA by https://doi.org/10.1007/JHEP07(2019)111 c [email protected] , 3) in the presence of heterotic 5-branes. Gauge symmetry break- , . 3 = 2 and Tatsuo Kobayashi N 1901.11194 ( The Authors. a,b N Superstring Vacua, Brane Dynamics in Gauge Theories, Superstrings and c

Z We study the global structure of vacua of heterotic strings compactified on , / 4 [email protected] T Department of Physics andIrvine, Astronomy, California University 92697, of U.S.A. California, Department of Physics, HokkaidoSapporo University, 060-0810, Japan E-mail: Scranton Honors Program, Ewha WomansSeoul University, 03760, Korea b c a Open Access Article funded by SCOAP Keywords: Heterotic Strings ArXiv ePrint: nism provides top-down, stringy account toof the non-perturbative spectrum and vacua modular with invariance 5-branes. condition emitting Also and it absorbing takes ustheories instantons. from are one in This vacuum fact to meansthere connected another is that and by also are many transition inheritedtransitions vacua of from among twisted with perturbative instantons perturbative different vacua at vacua. as gauge the well. It fixed follows points. that Thus we can understand Abstract: orbifolds ing associated with orbifoldtransition is between described small by (zero instantons size in limit) the instantons field and theory. heterotic There 5-branes. is phase This mecha- Kang-Sin Choi Transitions of orbifold vacua JHEP07(2019)111 8 E × 8 E ]. This provides a 20 – 11 3 [ , = 2 12 ,N 16 N Z 8 / 4 12 T 13 6 orbifold 28 – 1 – 18 5 ’s N 8 22 Z ]. It recently attracted renewed attention in various E / 3 2 4 17 – ]. Such transition is important in understanding six 1 10 T 10 – orbifold 4 5 and 3 Z N Z / 4 4 R 22 20 5 1 14 29 In this paper, we use this phase transition to study vacua of heterotic strings, 4.1 Connected vacua4.2 of Communication between two 3.3 Example 3.4 Recombination of3.5 spectrum and anomaly Dual flow and3.6 inverse transitions Phase transition3.7 of twisted states Selection rule 2.5 Chirality of localized fields 3.1 Small instantons3.2 into heterotic 5-branes Modified modular invariance condition 2.1 Perturbative spectrum 2.2 Perturbative modular2.3 invariance Classification of2.4 perturbative vacua Instantons on 1.1 Global consistency1.2 condition Small instantons SCFT is associated withdynamics the of non-critical dynamics strings, of describingby M5-branes. fluctuation string of theory. It M5-branes, which is can also be tightly studied associated toand the SO(32), compactified on toroidal orbifolds Well-known is phase transition of smallsize instantons, and by become which heterotic instantons 5-branes maycontexts. [ shrink For to instance, zero geometry ofdescribe F-theory the helps us phase identify transitionsdimensional non-perturbative superconformal [ states field and theory (SCFT) whichdual is to based these on heterotic dynamics of 5-branes. M5-branes, In particular, the mysterious non-Abelian nature of such 1 Introduction 4 Connected vacua 5 Discussion 3 Non-perturbative vacua and their transitions 2 Orbifold vacua Contents 1 Introduction JHEP07(2019)111 ] 1 ]. V 17 24 ] so 12 , which , V 11 completely V in the mother group G into still broken part of 2 V H + 1 V = V , which approximates a local geometry at N Z / (which we usually call SO(32) for convenience). 4 – 2 – 2 R Z / , provides a flat (constant) background connection at N Z / 4 T or Spin(32) 8 E ]. In six dimensions, such vector bundle is a set of instantons [ . The non-perturbative model is described by the new shift vector × 23 2 8 V E ]. This information is parameterized by so-called shift vector 21 , then that unbroken group is the commutant , G 17 , 13 The low energy limit of heterotic string is described by supergravity. In the smooth An interesting consequence is that many different vacua, described by different shift In the heterotic orbifold theories, the worldsheet CFT is exactly solvable [ Instantons on non-compact orbifold . Thus we may hope that the above decomposition is possible and we can explain the 1 The gauge symmetry is brokeninstantons by is this instanton background:of if heterotic the string, structure group ofWhat the the shift vector describessubject should to be the following related global to consistency this condition. instanton background. It is also with the symmetry of internalgauge symmetry space. is In parameterized orbifold byassociated a compactification, shift with the vector it information and that on we obtain we the spectrum review by shortly. a projection compactification, the supersymmetry preservingstable solution vector bundle for [ the gauge field is the semi- By symmetry breaking and energyselection consideration, of we vacua. can seek a mechanism for1.1 dynamical Global consistencyTo condition obtain realistic vacua, we break the symmetry of heterotic string by associating it non-perturbative vacua in terms of perturbative ones. vectors or instanton data, canfrom be connected. one vacuum Chains of to phaseproblem. another. transitions For enable Whether instance, us given we to string may move understand vacua to are what connected extent is the an Standard important Model is unique. determines the twisted field spectrum.tive orbifold Roughly vacua speaking, to usinginstantons non-perturvative transition ones, from from we perturba- shift can vectorsand extract by instanton the decomposing part information it, aboutV small and systematic classification of vacua. that we can track the changesectors of in spectrum. which This the solvability fields includescannot are the spectrum localized be in at accessed twisted the orbifold by fixed field points, theory. which are singular In and stringy calculation a shift vector enters as an input in thethat string the worldsheet CFT. spectrum in It the issame an presence CFT outstanding of just observation heterotic by in modifying 5-branes ref. the5-branes. [ can zero be point However still energy, there be proportional to have calculatedare the using been empirically number the bottom-up of determined heterotic approaches, andIn in anomaly this which cancellation paper we quantitative was seek factors used top-down approach, for enabling consistency justification on condition. the calculation formulae 5-branes, opening moreleads possibilities us for to realistic understand global vacua structure for of the moduli Standard space. the Model. orbifold fixed It pointsinfinity also of [ new top-down way to study vacua of orbifold compactification in the presence of heterotic JHEP07(2019)111 (1.7) (1.3) (1.4) (1.5) (1.6) (1.1) (1.2) ) and 2 -field. π B (8 / 2 R -field. Suppos- B ) under the instan- 1.1 , . . = 0 2 = 0 = 0 F 2 2 T F F Tr 2 , Tr ) Tr 1 π a ] . 2 8 2 x . ) is translated into the relations among 1 π 1 π 8 26 − 8 ] − . , = 0 orbifold, whose details we will study in ) 1.3 ) can be decomposed into the bulk contri- = 0 x a − − T ( 25 N 25 x k . The total sum should be preserved so that k , a twisted instanton. are curvature two forms for the Lorentz and 2 = 0 , 1.1 2 U Z T (4) = 0, the last term in ( R k − 2 T − − 10 / F F 10 δ F 4 x F tr – 3 – n U − B ( T a Tr 2 k 2 X − d and 2 1 π 24 (4) − 8 1 π δ , with obvious notations. = 24 R 8 2 24 + n =1 k a − X dH B + 2 2 − 1 R d 2 k ) modified to [ tr R = = ) is, integrated over a compact manifold, integrally quantized 2 tr 2 1.1 k 1 π π 2 8 dH 1 π (8 8 / 2 labels different heterotic 5-branes. Integrating this over the entire . The orbifold geometry determines the second term tr F k -field and gives B . We note that the curvatures provide the magnetic source to the 8 , . . . , n heterotic string, the contribution on the gauge field can be decomposed into F E n 8 and the fixed point contribution . In the orbifold limit, there appear localized fields at the orbifold fixed points. E = 1 U ≡ ∧ × F a 2.4 n 8 Now we introduce a heterotic 5-brane. It provides magnetic source to We can regard a K3 as a blown-up If we take a background such that The global consistency condition is given by the Bianchi identity for the rank two F E where orbifold, we obtain a consistency condition [ which is Bianchi identity ( We may call the object contributing to Tr ing that they fill inspace, the six so dimensional they uncompact satisfy space the and equation look of point-like in motion the internal Integrating over the entire manifold, the relationnumbers ( section We will also see that thebution instanton number in ( is integrated to give 24, the Euler number of the K3. Thus we have For that of each gauge connections, respectively, ande.g. a shorthand notation for the wedge product iston used, background Tr instanton number antisymmetric Here Tr is the trace over adjointsponding representation dual of Coxeter the number. gauge group, Also normalized by the corre- JHEP07(2019)111 , ]. N 27 Z then / 4 k T ]. Small . SU(2) for V 5 , ]. ], where the ' 3 4 , 31 1 , 30 , 21 , 19 heterotic 5-branes become n , is restored [ G ) group or its subgroup, from an n SU(2) holonomy on the torus. They Sp( ⊂ × N ’s are separately localized at the end of the 8 Z E – 4 – due to half representation. This can be easily 1 2 -function, the corresponding contribution of δ ). At the intersection between D9/O9 and D5, we have . As will see shortly, if instantons shrink to zero size ]. Two -function. n δ k 29 , and becomes 28 , n 2 5 , F 4 string, there is no additional gauge group from the 5-branes, but there ) as a conserved invariant, we may ask whether there can be transfer 8 E 1.7 , which is also explained by instantons in refs. [ . This is not possible for the Riemann curvature because the corresponding n V × 8 SU(2); the multiplicity is E × H In the previous works, emphases have been made on the effect of heterotic 5-branes In what follows, we will apply this phenomenon to strings on toroidal orbifold In the In the SO(32) heterotic string, there emerges extra gauge group Sp(1) In this paper, we make extensive use of this phase transition to study the global Regarding ( or instantons separately and thethe phase phase transition transition was idea notphase applied. to transition In understand making this small the paperin instantons connection we into the between apply heterotic shift them. 5-branes vector.perturbative we When effect. can This there track provides the is a change a top-down approach to understand the effect of non- are spread over theembedded internal instantons are manifold, localized as ata the vector shift fixed bundles. vector points. Their Whentransition data between we are smooth usually have described and orbifoldgroup by singular limit, is manifold the is Cartan described. subalgebra corresponding In to this non-vanishing limit components the of structure interval, which are interpreted as anothersuch 9-branes. heterotic 5-branes There are can be pulled a out tensor into branch the in bulkformed which and by become a M5-branes [ discretecan be rotational regarded element as of a singular limit of the K3 manifold. In the smooth limit, instantons open string connecting between them.provides an Six important dimensional clue gauge in anomaly understanding is its nontrivial non-perturbative and structure. it is another interesting newwith effect. a new We dimension can [ go to strong coupling limit to have M-theory understood in type Ibranes dual, are respectively in mapped which to theto D9/O9 Sp(1) and spacetime gauge D5 of group branes. the due Eachall to SO(32) heterotic coincident the 5-brane theory it projection gives is and from rise enhanced O9 heteroticthe to and 5- Sp( when bifundamental representation under SO(32) instantons are codimension four andThe undergo behavior phase is transition slightly into different heterotic in 5-branes two [ heterotic stringeach theories. small instanton.under We have also a hypermultiplet transforming as a bifundamental structure of orbifold vacua. 1.2 Small instantons An instanton, which weshrink need for to the zero reasonsingular, size discussed above, becoming the has broken ‘small continuous group, instanton.’ size. identical It Although to may the structure resulting group profile becomes between the two numbers so that the curvature Tr contributes to term has the opposite sign to the JHEP07(2019)111 8 ]. Γ 12 (2.2) (2.3) (2.4) (2.1) × , 8 twisted 11 c th twisted 2 or Γ j N/ generators of 16 twist acts as b T N . k shift vector [ 2 and N . j N S = 0. It also includes the 1 ) N 3. With the complexified j 0 , ,... ( 0 − ˜ L E . We associate this rotation , to yield six non-compact di- = 1 = 2 P 1 + N N ) j N Z ( ]. Then we have unbroken gauge / , which is shifted, in the ˜ is the order , ` N from the shift, thus the states with = ]. Here we consider the invariance 4 ) 32 , j T [ ( 1 + V N 12 ˜ 8 N , 2 in the torus, the order ) , φ E mod 1 11 ) 9 = 2 x × jV z 3 1 2 2 8 φ ` + , where N 0 mod 1 E πi/ πiφ 2 V – 5 – P 2 e ( ≡ ≡ + , e + V V = 1 P · · 8 z 2 L 1 x P P → m 0 = πiφ requires the existence of twisted strings, which are closed 2 α P 2 e ( S , z 7 → x = 0 : 3 ) 0 2 and the oscillator number ˜ πi/ 0 L 2 , z α e 1 z gauge bosons: ( + 6 belong to sixteen dimensional even and self-dual lattice Γ x untwisted matter: -th twisted sector the mass shell condition for the string left mover is j modulo an integer can survive. This will be important property of six = 1 NV 1 z N and ) transformation of modular parameter [ ≡ P Z , Modular invariance under The untwisted sector spectrum is obtained by The current algebra is described by the weight vector V · with the string tension sector. This comes from the constraint of Virasoro generator SL(2 condition in the absence of heterotic 5-branes first. string up to orbifold action.open Later strings, we also whose see ending thatsectors. hypersurface twisted strings is In share interpreted the many as properties branes. as We have dimensional toroidal orbifold models. 2.2 Perturbative modularWe invariance require modular invariance ofreparametrization string invariant partition theory. function to It divergence is free, generated worldsheet by so-called With appropriate combinations from the right-moversIn we can particular, make the orbifold right invariant state. moverP has the phase depending on the gaugegroup theory and spectrum SO(32) which or are invariant under the orbifold projection. This preserves half of the supersymmetry. with a shift of theBoth weight vector We compactify heterotic stringmensions. on We toroidal are orbifold particularlycoordinates interested in prime cases First we review perturbativewith orbifold orbifold vacua. projection Theunderstanding guided on stringy by what spectrum is modular is happening invariance. obtained in terms by However2.1 of CFT instantons. we can Perturbative gain spectrum physical 2 Orbifold vacua JHEP07(2019)111 1). . It , 2 (2.9) (2.5) (2.6) (2.7) (2.8) (2.10) (2.11) 1 N 2 , because most of ] 12 V, φ , . , 11 1 2 ) proportional to 2 + φ ) ]. The GSO phase is trivial ) j . ( 0 2 − 1 ) 1 φ − E 2 to let it lie in the interval [0 43 a − , N for a given 2 V a n + . = 0 mod 1 jφ ( V 42 ) ( 1 2 1 1) jφ j 1 2 N 1 2 − ( P, s − − − φ (1 · ) a + N N N φ . ( 0 2 ) needs the following necessary condition + jφ − ) s = 16 (2.12) L ( 2 0 mod j 2 ... 2.8 − =1 ) jφ = 2 a X n V φ ≡ · 1 – 6 – 1 0 + ) 1 2 2 n ˜ − 2 + L V s =0 1 ]. 2 , we have localized fields at the fixed points. Here ( j X φ N s + s 0 ( 34 P n 1 + = − – 0 − − 2 2 R 32 N+( 2 ˜ N = V − 1 m N 0 ) ˜ N + j α ( 0 = 2 cannot cancel the terms πi E ) 2 V 1 e N V = 0 : 2 + comes from the level matching condition [ 0 P L T = 0. The zero-point energy is given as ( j we subtract an appropriate integer from ) j ( a is the Virasoro generator. With the right-mover, the whole state is subject to φ 0 L Invariance under In our notation, the superscript in the shift vector denote the repeated entries. 1 with 2.3 Classification ofIn perturbative the vacua orbifold embedding, the brokenconsider gauge SO(32) group heterotic is string. parameterized by The a shift shift vector. vector of First a type And the resulting vacuawhen are the sufficiently massless anomaly condition free is satisfied [ for modular invariant perturbative models. In general there isthe no terms solution proportional for the to is weight vector necessary to requireinvariance the condition former to be also proportional. This is so-called the modular Consider the twisted sector. The condition ( generalized GSO projection, by requiring this phase to be one [ with a spacetime SO(8)again weight vector Here, by Also with the right movers, satisfying untwisted sector as JHEP07(2019)111 . 0, 16 shift > (2.14) (2.15) (2.16) (2.18) (2.19) 8 0 n case 2 Z case there is 2 Z ) (2.17) 1 − 0 N n 1) ). − ). In the 2 , 2.13 N − , ( 2 , n 2 N ... m U(1) , 1 +2 0 1 × 1 SU(2) cases, the same argument as above . n n is a spinorial representation of Γ − ) 1 2 3 1 2 × , 0 0 Z n N 8 7 0 = 8 . n 2 n ) (2.13) 0 i . E E + NV 1 ). There is no independent spinorial shift n : (0 )(0 1 n and 1 1 3 16 ) : SO(16) ): 1 n ... = 16 7 8 − 1 2 − 2 =1 8 i j X ∼ N N 2 Z (0 2 1 n : U(16) SO(2 SU( m m – 7 – 1 2 (2 0 = 2) 1) 3 2 2 1 2 i × × 2 1 =1 N Z − 1 n j − ) ) X ∼ 0 0 1 m ∼ 2 n n − ) N 2 1 =1 ) ( i 4 X N 6 − 0 0 4 2 1 2 ... N − (1 : SO(2 : SO(2 2 (1 1 1 2 n = 2 3 1 2 n 2 1 orbifold it can be always brought to vectorial shift by Weyl Z Z V 1 0 3 n n 1 Z ) can be partially explained by successive applications: if (0 0 n case, we form sixteen component vector by direct sum of two Γ 1 N 8 (0 2.14 orbifold however the spinorial shift vector cannot be reduced; it always 1 3 E = ). Note that the untwisted and twisted spectrum is the same if we use 2 × ∼ V Z 2.2 8 ) 2 E n 2 1 n 1 0 For the There is another set of shift vectors, where n (0 1 3 It is because up to Weyl reflections, we have three inequivalent groups in the tells us that the only vectorial shift vector combinations are sufficient with vector. In the gives the unbroken gauge group vectors, either vectorial or spinorial. In the In the specialreflections case accompanied of by a lattice shift by ( Up to Weyl reflection and lattice translation, it can be generally expressed as with an equivalent shift vector undergauge Weyl symmetry reflection ( accompanied by lattice translations. The a gauge symmetry enhancement due to self-conjugate, giving ( according to ( is called vectorial. This vector yields unbroken gauge group JHEP07(2019)111 ) ) we = 0. 2.14 (2.22) (2.23) (2.21) (2.20) n 2.11 ) of rank in ( 2.14 V and is regarded as carried by heterotic N 8 Z E ’s. / I 4 for which we have × R H 8 1 E , , . , , U(1) = 0. The shift vector for inequivalent, . I 2 mod 6 × n H . U(1) SU(3) ≡ I ) has eigenvalues of the generator of the V 2 0 × × , n ], which we discuss first. In the field theory 2 mod 6 with 8 7 6 ], as listed in table orbifold =1 2.11 16 ]. Thus the forms of the shift vectors ( case. We can do similar classification, yielding I ( X 2 E E E ≡ 32 30 + 4 – 8 – , 3 N 37 2 E 1 0 V ): ): ) : SU(9) ): ) : SO(14) – Z Z n 22 8 6 7 5 3 n / = 0 0 0 4 (0 35 + 2 2 4 + 4 , A 1 3 T 2 (2 0 . 1 (1 n 19 1 3 n 4 3 1 (2 1 (2 1 orbifold is locally described by 1 3 1 3 + 4 and N 1 N n Z ALE space [ / Z SO(16) maximal torus subgroup of 4 1 / × T − 4 and table N R 3 . The shift vector A ]. The gauge group from heterotic string is broken by this instanton N case − 3 21 , Z = = 3, the condition becomes ], but the total instanton number should be preserved. Inside 17 string we have a similar condition, E N 8 30 · , E ]. They are generated by Cartan subalgebra elements ) are sufficiently general in the E 19 is the exceptional divisor from the blown-up singularity, satisfying self-intersection × 21 8 E 2.17 In the orbifold limit, the unbroken group is enhanced to the group ( Each fixed point of Considering modular invariance condition the possibility reduces further. For SO(32) E relation SO(32) group or SO(16) string [ sixteen [ limit, the following constant background cannot be gauged away due to orbifolding Here The effect described by orbifoldtheory projection is limit captured [ by instantonbackground. background in the field singular limit of the perturbative vacua are classifiedand [ ( the spectrum in table 2.4 Instantons on modular invariant shift vectors for SO(32)For [ yielding also five combinations listed in table string with Without further symmetry breaking, e.g. by Wilson line, we have only five inequivalent and five in the JHEP07(2019)111 2 N Z Z dis- / / 4 4 N (2.24) (2.25) (2.26) (2.27) (2.32) (2.28) (2.29) (2.30) (2.31) T ]. The T d 22 contribution . I orbifold below 1 V N − ` − Z 2 / , ` 4 ) I invariant fixed points, T V and 1 ) = , N ` , − F ) T ]. We may interpret that Z . , Z ) ) j / (1 N 2 ( 4 I I Z 30 n , V V k R ( + 13 − + . =1 [ 16 , 1 2 I N X 1 U n I (1 , n Z n ( ) , χ 2 N / j N · · H 3 ( 4 Z I , , + 3 1 2 2 3 k 1 V T = subtracted by an appropriate lattice vec- · · 1 , n i by relating Euler numbers of fixed points = n T I = 2 =1 1 2 1 3 , n 16 I 2 X N · · N + 4 jV Z F Z + 3 2 1 2 1 2 2 ,N contribution on one N ), we have contributions at each fixed point / k i is ) 3 N · · 4 I K Tr ) N N – 9 – N j K T V d d ( − I Z 2.11 2 + 2 + 3 1 V / = = ] than that in the reference. ALE 4 = 9 = 16 2 3 1 T U , which are the number of R m , , k k K K − ( 21 i n , N N N N : : N χ d Z Z = = 1 2 3 we have A 13 k N k = 2 [ − 2 3 Z Z Z 2 =0 d i N N Z Z X N 2 m is equally distributed on the sixteen fixed points [ F d d k k at the fixed points as well, in the orbifold limit. 1 π w . : : 2 3 2 1]. In the presence of Wilson lines, the degeneracy from N 8 i/N ) = = = Z , 2 3 N 3. Here again T U Z Z = ∈ = , Z k k / T U N , , 4 Z N N = 2 k T localized is a multiple of a fractional number, since the spectrum is (equally if ) is the usual second Chern class from the flat connection from the Z Z ( k k N χ U k from the structure of the generator 2.25 shift vector of SO(32) in ( ) is the index of the localized fields at the tip of the ALE space. It can be F 24 = in ( N Z 2.25 U , eigenvalue for each N Z i in ( k n For a We see that Now we wish to collect all the above local contributions to that on the compact T , We use a different convention 2 N . It is contained in the weights Z In the absence of Wilson lines, all the fixed points are equal, thus in the bulk we have tor to lie it in theappears interval [0 and we have independent contribution from local shift vectorsthere at is each no fixed Wilson point. orbifold, lines) the distributed Euler to number abulk number contribution of is fixed points. For instance, in which are valid for and are determined by global structure of Therefore we have the total instanton numbers on an arbitrary number in thecompletely determines field it. theory, but the CFT of the global orbifold. To this end, weN should know the multiplicity of fixed points of possible orders of Here infinity. Roughly speaking, in on the other k have JHEP07(2019)111 ). 2.6 (2.35) (2.33) (2.34) ) and the ), we may ) and ( 2.10 2.17 2.4 ) gauge group is 0 n . 0 2 n ) has similarity with the + 3 0 1 2.10 , n 2 ) 2 , 0 1 + 3 ) ) as well. . In the untwisted sector, usually n F . 2 0 8 3 ). To sum up, in the perturbative v n R 0 E K x n + 4 2.29 ). Then the anomaly is related to the + (tr 0 2 0 , 1 R = 9 n 1 K n y in ( n ( 2 1 4 1 3 T + k 4 , k = 16 + + 2 F X 2 2 3 n v – 10 – K K for the twisted strings. tr , k for twisted fields + 3 ). 1 = = T 8 R n 1 n R k 2 3 x E n Z Z , = 2 k k = + 4 k : : T + 3 4 2 k 2 3 3 + F K . In the twisted sector, if a field is charged under the both, Z Z 1 K 8 R k E tr of localized fields. Essentially it is determined by the twisted ) for each = = 16 = 9 1 1 k R of gauge symmetry and Lorentz symmetry, respectively, whereas be representations of SO(2 k k 2.26 : : R 2 3 ). We note that, although the condition ( ), their contents are different. The shift vector and twist vector contain both describes SU( Z Z , because field theory cannot take into account the stringy nature of the 2 3 1.1 1.1 T string, the shift vector plays the same role: each component parameterizes k breaking 8 or in E 1 3 R × x is over the vectorial representation of SO(2 . To sum up, we have 8 8 v E E Interestingly, for the SO(32) string, anomaly consideration of SO(2 In the twisted sector, the contribution from twisted fields to the instanton number There exists no index theorem that automatically gives the spectrum in the twisted This establishes the relation between the modular invariance condition ( For depends on the type T where tr model we have sufficient to fix the instanton number k sector spectrum. Let coefficient internal manifold. However in thefixed toroidal points orbifold, and we they have carry twisted instanton fields number localized atsector, the yielding twisted strings. We should calculate the twisted fields from CFT formulae ( the Bianchi identity containsrelated because the the information shift/twist about vectors completely the determine spectrum. the2.5 spectrum. They Chirality are of naturally In localized the fields smooth compactification, all the 24 small instantons reside in the bulk geometry of Bianchi identity ( Bianchi identity ( information of the representation becomes multiplicity.second We have essentially the same counting for the where we count only thea states charged field under is the charged first under one components the rotation in the Cartanuse subalgebra the direction. same Thus formula with ( the shift vector ( Note that the instanton number is a gauge invariant quantity, as it should be. Either JHEP07(2019)111 ) 3 ). , n 2.42 (2.38) (2.39) (2.40) (2.36) (2.37) (2.41) (2.42) (2.43) = 2 ) 2 n Z ) . ) into funda- n 2 n ) ) for ) ) with 0 n n n 2 ) , 2 SU( , 2 ). 1 ) F , 0 1.4 1, as seen from ( 2 n 2 , n ) − 4 F ) ) may have negative con- n v 0 ≥ n 12)(tr 1 (tr ) 2 5 − SO(2 , − , − N F 2 n , n ) of SO(2 n n ) is zero for SU( 2.32 2 2 0 ) = 0 ) · n ) − 4 n T ) anomaly plays an important role. 2n 1 + (3 F . 0 , k 2 . SU( 1 n + 6(tr ) 1 + 3 n 2 SU( n ] − F ) − ) ) and ( − n F N n 1 2n i 14 ]. From the contributions of gaugino 4 SU( n = 24 − n F 4 N 1 21 2.12 SO(2 T − 4 SO(2 fundamentals, respectively, to cancel SO( n − k + 6(tr F =1 1 . F i X ) gauge anomaly, with the chiral matter + 3(tr ) ) N v n ) (or ( n − v 0 + n n ) 0 2 n 1 – 11 – n tr − n U term [ n 4 4 − 2 5 k SU( SU( 0 8) tr + 54) tr , 4 4 − SU( ) we have [ n F F − 1 U n = n − F n F − k 2 tr k 8) n 17 N − n − n 8 + 32 8 + 2 − − 8) tr = 2 tr = 0 − − + 2 = (2 = 2 ) ) n − n 1 ). The reason is that the total instanton number is related n n . The coefficients for tr ) ) = 24 = = ( n n n (2 n n 1 2 8) vectors and 2 T 4 4 k SO(2 SO(2 + 2.36 4 SU( = ( = − 4 SO(2 F F F n ) ) F n v 1 n n − Tr tr → n Tr 4 4 2 SU( SU( F F tr 2n ) ) ) under SU( , 1 2 ) 0 − − . Roughly 16 spinorial degrees of freedom contribute 2 n n n ( T 2n )( 6 n k , 1 1 − tr ) anomalies. Some other representations may contribute as follows. For the n − ( n n N n tr It should be noted that the spinorial representation of SO(2 In understanding the consistent vacua, the SO(2 This cancels six dimensional SO(2 + 1 n mental of SU( thus the corresponding groups are anomaly free. tribution to We will see later that this makes some phase transitions difficult. We have similar decompositions for the spinorial and antisymmetric of SO(2 The absolute normalization is understood in the branching of vector SO(2 antisymmetric representations of SU( we see that weand need (2 SU( In six dimension, the chiralityfermions of in gaugino the hypermultiplets, in so theSix that vector dimensional the multiplet gauge matter is anomalies contents always can areanomaly opposite be constrained polynomial to by cancelled, has anomalies. the up vanishing to tr Green-Schwarz mechanism, if the This proves the relationto ( the unbroken gauge group. Here the only unbroken part is SO(2 It is because we can show the following using ( ( as long as the total instanton numbers should satisfy the condition ( JHEP07(2019)111 7 n E E can (3.3) (3.1) (3.2) of V orbifold. 56 2 . Some of Z V 0 ) of the E 1 ). Here the instanton n + ) group is SO(10) with 0 2 2 n V 2.20 1 2 contributes 1 and 6 + . 2 E 2 2 V V ) and ( · , which describes also the recov- of from matter states. 2 1 2 . 1 , or even SO( 0 V T V 8 + 2.19 k 27 E E 2 + , modular invariant theory parameterized V 2 2 · V V + ∆ 1 · ), we have 0 + V . Thus formally we have the same relation P E 1 1 2.4 + V + 2 – 12 – . Here the maximal SO( = V ˜ ˜ N + N + T · 10 k V + + P perturbative, + 2 2 ) ) ≡ 1 1 0 16 V V 2 2 E + + → ∆ ) group. For instance, P P 0 ( ( . Decomposing 27 n 1 = = V 2 L m 0 ) but different groups listed in ( α 0 2 1 n , with instantons emitted. Then the extra piece from the expanding 1 V heterotic string vacua have a number of distinct features. There is no relation 8 E ). We have no gauge symmetry enhancement from heterotic 5-branes, and any × 8 )-type subgroups, or exceptional subgroups of . Expanding the mass shell condition ( Consider the twisted sector of a E The instanton number is different from other anomaly coefficients, for example of n 2.34 V We may regard this formula asvacuum the with mass shell condition for thebe twisted regarded sector for as a modification daughter in the zero-point energy by the emitted instanton componentsered part. are described by The information about instantoninstantons embedding can is be encodedunbroken emitted group. in into the The bulk shift resultingvector unbroken to vector that group we become should shall be heterotic call also 5-branes, described recovering by another a shift larger seemingly unconnected vacua are actuallyrameterized connected. by the Since shift the vector,the effect we difference of can of instantons trace the is the shiftthe pa- information vectors zero of on point vacua. instanton energy, the exchange From this number from of we can 5-branes3.1 prove and many difference relations of among Small the instantons shift into vectors. heterotic 5-branes 3 Non-perturbative vacua andNow their we transitions introduce heterotic 5-branes.sition As between alluded small in instantons the introduction, and there 5-branes. is phase tran- Applyting this, it turns out that many contribution comes from vectorial componentsthose of of a the representation, when maximalcontributes we SO( 2 branch to it the into instantonwhich number we have branching as ( subgroups are anomaly free, thus the anomaly freeness is less restrictive. SU( However there is instanton number contribution for between anomaly and instantondo numbers. not Technically yield the SO( zero entries of the shift vector JHEP07(2019)111 1 G (3.7) (3.8) (3.4) (3.5) (3.6) ) by removing 3.4 means that the . P ) on . 0 3.3 E . = 0 mod 1 ]. The 5-branes can only affect )+∆ 1 2 1 2 N φ 17 − . , the only consistent way is to keep all the 2 )[ 2 1 0 2 V N + E ( V 3.2 1 2 1 2 − − , . + ∆ ). They are to recombine into the representation 2 0 mod φ + 0 · ) 0 ) 2 φ E 2.4 φ E ≡ = 0 V 2 + · 2 + 2 s 2 = 0. In general they form incomplete multiplets of ( φ V 1 s − 2 , as we see in the next section. However the resulting ( · – 13 – V 1 1 from ( V − V ) can be expanded as V · · 2 P − ≡ ) 0 , we expect their cancellation gives an integer sum, V 1 P 0 0 E 2.7 V + E E /N + 1 ) is invariant under Weyl reflection. Indeed it is reflec- 0 mod 1 we can make them projected out by ( . P V ∆ 0 + ∆ 3.6 6≡ = E 2 N+( 2 2 V ˜ N + ∆ V − V · ˜ N + P 2 πi ) is different for various representations. If all the twisted states ) 2 e V terms. Thus we require the modified modular invariance condition 3.2 2 + 2 φ P ( ]. of the daughter theory with and 17 -dependence on the zero point energy ) 1 P 2 0 mod 1 which is not strictly G V 3.2 ‘condensates’ in the CFT description. This justifies the shift of the zero point ≡ 2 2 . For the states with V 1 V · 3 G term in the phase. However we are not aware of the mechanism or condition to project out the state P Note that the combination ( Note that the dependence of modified zero point energy ( 2 If we allow V 3 · states from the mother theoryof with the group spectrum is (i)group anomalous and/or (ii)P cannot recombine intowith the complete representations ofthus are the inconsistent. gauge expect the perturbative case, we may considercondition level ( matching condition from the modified mass shell Since each term is proportional to 1 energy in ref. [ 3.2 Modified modularThe invariance change condition of mass shell condition,into now account with non-perturbative the correction effects, to affects the the zero-point energy modular taking invariance condition. As in the tion that obviously preservesdoes the not inner affect product. the internal Sinceincluding geometry that the of of inclusion orbifold, the of we right havevector heterotic mover. no 5-branes As change a in result, the the spacetime part part, of instantons described by the shift The spectrum can still bethe calculated using same the formula same as CFT thatthe in of the CFT presence the on of perturbative the 5-branes, case5-branes using fixed ( should point be only related quantitatively. to We ∆ may also expect that the number of satisfy the zero point energy becomes constant The extra phase is also simply expressed by ∆ mass shell condition ( Also the generalized GSO projector ( JHEP07(2019)111 ) ), ). 8 14 0 0 2.10 8 2 (1 (1 1 3 1 3 ), but this = = 1 V V 2.10 ]. In the SO(32) het- , 17 , , although the unbroken 2 n , heterotic string, most vacua are ) and . 8 8 , 0 ) as a necessary condition. In the ) 1 E 7 6 modular invariance condition ( n = 0 (8 oscillators) SO(16), whose root vectors are 3.8 × 0 (8 oscillators) , , 1 0 1 0 × 8 ) E 2 , 6 ). E ) , − 8 + SO(16) with the shift vector . 7 SO(28) with the shift vectot 0 3.8 1 0 1 ˜ × × = 1 N 6 − 3.7 = U(8) + , a 1 0 1 – 14 – 2 ¯ a = (1 0 = ( perturbative G ) a − = 0 giving the previous condition ( ) depends on it. V P P G = U(8) =U(2) 8 ) also turns out to be sufficient condition. Unfortu- 0 1 ), + G 2.4 ): ): ): E G 1 1 P = (1 = (0 2.4 2.10 , , ( 16 1 , 1 2 P P ( 28 8 ( ( ): ): 1 , . 120 64 , 1 ( 1 ( that fails to satisfy V ) is not symmetric under the exchange ) is. It is rather a condition about the absolute size of the shift vector. The 2.21 2.14 to the non-perturbative vacua from the perturbative vacua of The mother theory has gauge group The modular invariance condition has preferred basis, so not every Weyl equivalent We obtained the modular invariance condition ( A shift vector • • invariant moduli fields from gravitons. The first twisted sector is the only twisted sector 3 which survive projection conditionZ with appropriate right movers.and has The the last mass ones shell are condition two ( where we allow permutationstwisted for sector are underlined components. The matter states in the un- They are found in table 3.3 Example Through this section, we consider an exemplar transition shift vector satisfies the samedition modular ( invariance condition.group The modular ( invariance con- twisted sector mass shell condition ( anomaly free even without perturbative modular invariance. perturbative case that conditionnately, ( this is not thesatisfying case for the the condition non-perturbative we case.selection sometimes Even rule though have and we anomalous counterexample had vacua. in shift vectors section We will come to a further fails in this case. Instead we require the formcan ( nevertheless describe aerotic consistent string non-perturbative vacuum theory, [ fields a coming failure from of the heterotic anomaly 5-branes. cancellation In can the be remedied by extra chiral In the perturbative case we have ∆ JHEP07(2019)111 (3.9) (3.13) (3.12) (3.11) (3.10) = 6. In the 1 , describing a n 1 as V 2 4 = 3 V are going to become U 2 k V , , 1 3 2 3 . − − ), there is modification as is the same for all the inherited , ) . ) 8 2 , = = ) survive. This is consistent with 7 9 8 SU(2). It is described by the ) 0 V 3.9 ). We have more inherited states . 2 2 0 · V V − 14 6 1 3 × , 3.13 · · 6 P 1 ) and an extra in ( 3.10 8 = = + 2 1 0 , 1 0 0 1 0 1 ) 2 (0 . G 6 V U(1) − E ) · 13 . 1 3 − 1 3 ) ) ) ,P = 0 in ( ⊂ of 1 16 1 . ) 1 0 − 1 16 10 2 V 1 0 2 3 1 − 10 0 V , ) + 1 . − − ,P ), we find the spectrum 0 V Z + · ) 1 6 14 = (0 P = ( 1 P 10 · G = (0 0 3.2 = 1 = ( (1 0 − ). 1 3 2 1 3 3 1 0 0 ,P ,P · , a ,P (1 1 3 ,P ) with the massless right-movers. The multiplic- – 15 – 3.2 − ¯ a SO(28), with untwisted matter = (1 0 = ( 1 0 1 3 1 2 1 3 a 1 2 − 1 · = 0 = × 2.7 P P G = 0 = − = ˜ ˜ N N 1 0 2 ˜ ˜ ): ): ): N N 6= 0 in ( V − 1 1 1 + 2 ): ): = 0 + 2 , , 28 ): ): + V 1 1 , 1 1 = ( = (0 0 − 0 . Only the state 1 1 · , , ( ( 1 2 , , V E 1 1 ( V = P 1 ) because the phase ( is decomposed into ∆ 9( 0 28 = ,P ,P 18( E 3.4 18( G 9( V = 0 = 0 of 2. ˜ ˜ , N N V = 1 , a 3 , which is eventually related to the instanton number / 1 1 V a − in the projector ( α 2 V · P ), The unbroken gauge group is U(2) The shift vector 3.6 We see that these states are inherited from the mother theory ( 4 and so on. They cannotremove form the complete representations ofstates SO(28). from We the may mother projectnot theory out having these with states the if massless we state with in ( Plugging these into the mass shell condition ( Instantons are embedded in theshift structure vector group twisted sector, on top of the usual zero-point energy non-perturbative vacua with the gauge group Upon phase transition the small instantonheterotic components 5-branes. described by Therefore we are left with the remaining component They survive the projection conditionity ( 9 is fromdirections the number of fixed points, while the extra factor 2 is the possible oscillator Its solutions form with the zero point energy JHEP07(2019)111 . 2 9 = (3.15) (3.18) (3.14) (3.16) (3.17) 2 0 2 V 1 2 . For this, + G 0 2 V · 0 . This also lead 1 2 9 V = = . are related by Weyl ) 2 0 2 1 0 SO(22) model to non- 1 E V , 1 2 V 6 × , + 2 ( 2 and V , + · 1 ) in the untwisted sector. There ) ) 1 V 2 1 1 V , , V 16 contribute to anomaly. We see that , 1 6 , , = + 8 . . 1 2 0 ) 1 are completely converted to as many 0 ) ) SO(16) ( V E E 11 2 . , 11 11 G, H ). ) ) 0 V × +( + ). With the total 20 vectors, we can check ) ) 4 15 ⊂ ) = 16 1 1 36 , , , 2.38 11 ; 6 6 0 U(6) (0 1 (2 0 , , 36 28 4 1 3 1 3 1 2 , – 16 – , the spectrum branches as × , 1 ( ( 1 = = H ( ( . + 0 0 (2 1 (1 1 1 1 0 0 (1 0 0 0 1 0 1 2 1 2 ) ) + 1 1 3 1 3 1 3 → V V U(2) = 0 happened, which can be relaxed in the follow- 1 ) + G , 16 1 = = = G 2 ≡ , , V 2 1 1 15 1 ) and V , , , · H V V and 2 1 4 1 36 ( ( ( G V ; 1 → → → , ) ) ) 2 1 1 , , 16 , SO(24) model via 8 28 64 ( ( ( × = 0. T k ). The biggest gauge group Sp(18) is generated if all the heterotic 5-branes 1.7 can be explained simply by branching and recombination from those of 1 From the symmetry breaking In the above example, As a result, the 18 instantons described by G = 18 heterotic 5-branes by phase transition, so that the total number is conserved to be Note that only vector representations ofthe SO(16) only SO(16) vector representation is the multiplet ( which is a subgroup of both 3.4 Recombination ofLet spectrum us and analyze anomaly the flow spectrumFirst change, we see in that the the previous vectorof example, representations during (in the the hypermultiplets) phase in transition. it the untwisted would sector be convenient to consider a common subgroup Remarkably the zero-point energy is corrected inThe the same two amount ∆ modelstransformation, have we the should same always have spectrum. the same ∆ As long as This gives the modification ofus zero-point to anomaly-free energy vacuum. ∆ Alternatively, we can take another combination have hypermultiplets in ( that gauge anomalies of SO(28) cancel from ( ing. There canperturbative be U(4) a transition between the perturbative U(5) twisted sector n 24 as in ( become coincident. Besides the vector multiplets of both groups SO(28) and Sp(18), we Since there is no SO(28) vector or spinor, thus no instanton contribution comes from the JHEP07(2019)111 , ) 2 ). 2 × r ) G V 0 n ) up q + ≡ 2.32 (3.21) (3.19) (3.20) 1 1 14 . 0 V . p ) became 2 1 2 SO(2 4 = (0 q 1 3 1 2 + → V in eq. ( SO(20) p and ) and = 2 = ( × 1 m 2 3 n 2 ) V V 1 , U(6) and SO(2 15 1 of SO(28), responsible × ). Exchanging the role , → n 1 28 SO(28), we have recombi- . +4) 3.11 . Thus the decreased small ), two are recombined into n = 8). Since U( q × H ) + ( 1 , r 2.12 SO(16) . , , 1 ) of ) × V 15 1 = 6 · , , 16 = U(2) , the emitted instanton data is con- 1 1 1 , , q 2 1 , V 1 V 1 , 1 2 G . 2 = 2 ) ) + ( + → p ’s are un-higgsed and became the part of 1 16 should be multiple of 2 thus the number of ) increases by 2 , V 0 H ) + ( 120 16 q · of SO(16) in ( 6 , n 1 , and thus the number of necessary heterotic 5- 2 , 1 1 – 17 – , V 6 q 16 , 1 = 2 ). Again due to the even nature of the lattice, the ) states in the twisted sector and, after the branching 0 ) + ( 1 ) + ( E , 16 ) + ( 1 2.31 , , 1 28 6 dual , , 6 36 in another way by looking at the spectrum change. ∆ 1 , , 2 2 1 n ( ( + ( ). The other six in eq. ( 1 ← ← n and ) ) ) from the field strength. Here, we can understand the multiplicative 3.19 orbifold. The emitted instantons are described by 1 28 2 n , 378 16 more vector representation to compensate the anomaly of gauginos, 2.32 Z 2 , ( 1 − ( = 16 (In the above case, we have , which should explain the coefficients 3 in front of q r , we can view the transition as that of U(8) , thus the zero-point energy is 2 + 2 V q V of SO(28) in ( + + 4) in the p ) are going to be recombined to the vector representation and m 1 28 For this dual transition, let us come to the example with ( The same argument is possible for the transition from SO(2 We can also keep track of anomaly flow during phase transition. We have derivedThe the emitted instantons are described by a part of the shift vector From the gauge symmetry enhancement 1 ), the number of vectors reduces by , p V 28 of 3.5 Dual and inverseThe transitions above phase transition always has aif dual we process. take From the a decomposition tained daughter in model using the shift vector SO(2 to permutation. The numberby of 4, vector reduces the by coefficientnumber 4 of of but extracted the instantons necessary is instantons always increase a multiple of 8, as seen in tables U( branes for the anomaly cancellationinstanton of is SO(2 3 Due to even rootextracted property instantons of should weight be vector, always a multiple of 6, as checked in tables instanton number ( factors 3 in front of where Among the eight vectortwo representations the adjoint of SO(28).we Thus need the 12 = number 28 ofthus 18 vector in representations total. is5-branes. decreased They are by provided 6, by and 18 emitted instantons becoming as many heterotic ( for anomaly. There are also nineand ( recombination, they are projected out except ( nation is no vector representation in the twisted sector. This multiplet and the untwisted state JHEP07(2019)111 , V ) and (3.30) (3.27) (3.28) (3.22) (3.24) (3.25) (3.29) (3.26) 1.4 . ) . 1 , 1 , = 0 1 2 T F Tr ) + 2( 2 1 . π 16 , 8 ). For instance, the transition 6 ) 1 54 , − 2 ). = 12 heterotic 5-branes. It should 2 2 2 2 . = 3.28 2 U V n V 3.21 above to the mother theory with 1 2 1 9 F 2 1 3.15 1 ) = 0 ) + ( may undergo transition into the same − V = + Tr 2 n , 2 1 2 . 2 1.6 V 2 . ) 16 1 V V 0 V , − 1 π 1 V · ). This should be the result of combining − · , 1 8 E yielding the same vacua, there can be many 1 2 T 1 ) and ( 6= 0 , 6 k V V V ∆ 1 1.7 , 2 − T + V 2 − − − ) k − – 18 – ) and ( = 1 3.14 a V ∆ U 1 = = = x ) + ( · k 1.3 V ) changes not only the untwisted instanton number 0 ) + ( 2 − , the small instanton emission is accompanied by the 1 − 10 E 4 V x , 0 120 – ( , 6 rev = 24 6 1 , 1 (4) 0 ∆ , 2 (1 δ . The phase transition takes place preserving the sum 1 E 1 3 T n ∆ =1 k = a X ) (3.23) ) + ( 1 1 ) + ( − , V 1 16 2 , , 15 1 6 R , , , ) to 1 4 1 tr ( ( ( 11 2 0 1 π ← ← ← 8 4 ) ) ) 1 , 20 (2 1 , 190 1 3 15 , 6 ( ( 1 = ( V Reverse process, from a daughter theory with but also the twisted one U This is the result of thenon-perturbative combining condition perturbative condition with including twisted heteroticthe states ( ( correspopnding Bianchi identities ( from k We also find phase transition of twisted states In general, as we canchange see of in tables the twisted states, resulting in the change ( Thus the process should be reversible. 3.6 Phase transition of twisted states then we can change the shift vectors of the mother and the daughter theories be noted that, however, fordifferent one dual transition vacua, as in the examples ( is also possible. If we express Again, we verify that the modified zero-point energy ( This verifies also the transition of small instantons into number of heterotic 5-branes. Accordingly, another recombination takes place vacua. Six small instantons now described by JHEP07(2019)111 T k are U k ) n ). 3.2 9 = 18 vectors from the × vacuum we have obtained . It is because the twisted T k (heterotic 5-branes . SO(28) and the instanton number . × , to heterotic 5-branes, there is also = 0 = 18 U or equivalent in the twisted sector. We ) k U 28 k , n non-perturbative ). , n ) ) 14 14 0 0 3.28 – 19 – 2 2 . The (1 (1 1 1 3 1 3 vacuum with (instantons = = V has V 3.3 ) T k perturbative . 1 = 6 are the same. (twisted states . There are phase transitions among small instantons, heterotic 5-branes and twisted 1 n The transition between fixed point states and 5-branes happens in general: if we have The modification to zero-point energy affects the twisted field spectrum. The We may also have transition between untwisted and twisted instatons, not involving = 3 U in the non-perturbative model, satisfying ( a transition between smallchange in instantons, the changing twistedspectrum sector is spectrum changed in since general, the changing mass shell condition is changed as in ( essentially counts the number vectorhave multiplets SU(2) doublet in each ofamong nine them. fixed Therefore points. in the Withouttwisted perturbative Wilson sector. vacuum, line we This we remove should do 2 not give rise distinguish to the same number of vectors from heterotic 5-branes The entire untwisted sector istions the for mass same and in projection. bothk The cases, gauge because group we is U(2) use the identical condi- However there is also a heterotic 5-branes. Since wewe have no cannot control prove over the thecompletely direct twisted determined sector transition. by fields the via However,We shift field noting may vector, theory, the compare we untwisted perturbative can sectorSuch and indirectly an and non-perturbative use example vacua the can having chain befrom the of found the same in transitions. transition table shift in vector. section fields. We have transition among small instantons, twisted5-branes states as at in orbifold fixed figure points, heterotic Figure 1 JHEP07(2019)111 ), ) 3.2 ). In (3.34) (3.32) (3.33) (3.31) ], in a 3.6 = 3, 1.7 47 and there , N 2 46 V + 1 V ) with vacua. They are -field, in which the = 3 B 2.32 Z V . m 3 . K ! − 2 3 . , n, K 2 3 = 0 − = = 24 2 n 3 n m 2 K + ∆ n = 24 2 ∆ 2 n − n + 3 2 takes place in a special form, as in eq. ( , 2 m 1 , 1 3 + 3 n + 3∆ 1 1 – 20 – n 1 = 1 n + 3 n of heterotic 5-branes subject to global consistency m 3 + 3 ∆ n 3 K , i − + 3∆ m i K

3 n = 9 1 2 orbifold theory K − = 9 ). The instanton number is given in ( m = i k k 9∆ n 0 2.11 E = i ∆ n ∆ . There should be ) can be regarded as the magnetic equation for the T , 3 1.2 ). We have supposed that this vacuum is inherited from a perturbative Z k 1.7 ≡ 3 K In the simplest case where the shift vector is simply decomposed Let us calculate the modification in the zero-point energy of We can view the above exchange as dynamics of 5-branes. The dual description of but also the chirality of twisted sector spectrum. When small instantons get transition 1 is no overlapping elements, then the change of zero-point energy is obtained from ( V into heterotic 5-branes, both are related. Subtracting them we have theory. The change of each contribution is described as follows, Note that, the change of instanton number is not only due to the change of the components modular invariant theory with where the superscripts indicate that the corresponding quantities are those of perturbative where condition ( addition, this phase transition in namely that should be parameterized byconstrained the by shift a vector. selection Therefore the rule transition that is we further shall seedescribed here. by shift vectors ( instantons and fixed points are all the magnetic sources. 3.7 Selection rule We have seen phasethe transitions important constraint that is are the understood preserved number by of exchange 5-branes as of in 5-branes, the in condition ( which small instantons is D5-branesalso on be regarded top as of open(5 D9-branes. + strings 1)-dimensional whose Twisted hypersurface ends are strings onsimilar interpreted sense on which as that the the branes. open twisted orbifold strings Theseseen strings that are can are that are localized localized eq. localized on ( a [ NS5 or D5-brane. Moreover, we have JHEP07(2019)111 = of 0 T k E (3.35) (3.37) (3.38) (3.36) SO(8) × = 0 from 3 K vacuum with ) contributing 2 1 , n ) to a non-perturbative 1 2 0 ) and the change ∆ 12∆ thus we need ∆ 14 0 cannot be changed. 2 3.33 − (1 n 1 1 3 > ) + 18( , s n 2 2 8 = n another perturbative n , ) provides a selection rule. This is 3∆ 1 ) is not a sufficient condition in the V . − 6= 0 we may eliminate it to have 9∆ 3.36 1 , 3.8 = − 2 2 n = 0 1 3 6= 0. Also we assumed that there is no n 3 N N n shows that we would have a U(12) ) + 18( K 6 6 2 K 1 ’s. We will come back to this in the example 2 i V , 3∆ ; otherwise the modification of the zero point = V 9∆ n · 2 0 – 21 – − V 66 + ∆ 1 − E 2 V 2 n n n ∆ and = ∆ 3∆ 1 0 of heterotic 5-branes in ( vacua. V − E n 8 3 ∆ E K ) is forbidden. This also shows that not every model satis- × ) states that some transition is not possible. For instance, the 6 9∆ ) as follows. In case ∆ 8 0 − E 3.36 10 3.34 = )). If we have only vector representations in the twisted sectors, it (1 n 1 3 2.4 ) is not possible. There is no change in ∆ = 0 or not. Thus, the relation ( ) should only be dependent on the number of heterotic 5-branes. It follows 4 = 1 0 n V 12 3.35 ). A stringy calculation using 6= 0. Note that we have no way to predict the change and rearrangement of has different dependence on ∆ (1 3 0 1 3 3.36 K E = = 3 here. This vacuum should be independent of transition, thus the modified zero 2 = 0, then the instanton described by entries with 6= 0. However everything should be same if the transition is commutative. This N V This selection rule is based on some assumptions. First, assumed that all the fixed This selection rule ( We may relate the number 3 2 K n 9 = 9∆ complication due tooverlapping the components possibility between of energy ∆ below. Since this selection ruleenergy, it relates the holds number valid of for instantons and the shift of zero point sentations. If we obtained the vacuumbe from well-branched the under transition, the the spinor transitionthe representation (technically cannot twisted it sector can be ( is seen likely from that the mass we formula have in well-defined non-perturbative vacua. points are equivalent. This may be relaxed if we have Wilson lines. There can be a further the twisted fields in theresulting field spectrum theory is limit anomalous, sovacuum the too. with selection rule The isfying would-be necessary the chain condition. modified involving The modular anon-perturbative invariance non-perturbative vacua. condition The ( main reason is the anomaly contribution of spinorial repre- phase transition from thewith perturbative vacuum with the rule ( vacuum with the twisted− sector fields 9( regardless of ∆ nontrivial because, for acan given be shift obtained. vector, In we∆ cannot particular, if expect there what is kind no of change twisted in fields the chirality in the twisted sector and We may have this∆ same non-perturbative vacua from means that we should always have the zero point energy ( with point energy ( JHEP07(2019)111 Sp(6) Sp(12) ) ) ) × 3 ) × 13 11 orbifold are 0 0 0 SO(6) 15 3 SO(22) 10 2 4 Z × / × (2 0 4 1 3 SO(26) (2 1 (2 1 (2 1 SO(30) T 1 3 1 3 1 3 × × U(11) U(5) vacua of both heterotic 3 U(3) U(1) Z . Each heterotic string on this orbifold. Among five perturbative and 8 3 2 E Z Z / 4 × T 8 E Sp(6) Sp(18) Sp(12) ) ) ) ) ) × × × 2 for ) 14 12 10 10 Sp(24) 0 SO(4) 0 0 0 0 3 16 SO(16) × × 14 2 4 6 8 (0 × – 22 – 1 3 (1 (1 (1 (1 (1 SO(20) 1 3 1 3 1 3 1 3 1 3 SO(28) SO(24) . The transition maps for the 4 × × × – U(14) U(8) SO(32) 1 is indicated by solid arrows. Transition for dashed arrows can orbifold U(6) U(2) U(4) 1 9 3 Z = 0 E for SO(32) and in figure 2 ) 14 0 SO(28) 2 × (1 . Connected vacua of SO(32) heterotic string on 1 3 U(2) string theories are showndepicted in in tables figure orbifold has five perturbative,are disconnected modular and invariant admit vacua. nosector transitions is because In too the large. SO(32) instanton number vacua In of two these the of models, untwisted them we have large spinorial representations contributing of transitions of small instantons,another. heterotic 5-branes and twisted fields take one vacuum to 4.1 Connected vacua of We consider global structure of moduli space. All the be proven using the chain of such transitions. 4 Connected vacua The above discussion forces us to conclude that most of the vacua are connected. Chains Figure 2 models, bottommost onenearest in transitions each with ∆ column, three generate chains of connected vacua. Only the JHEP07(2019)111 3 Z / 4 T 9 T U 18 0 0 0 0 0 n k k − − ) 9 ) 18 1 ) 12 ) 9 ; 1 ) ) 9 ) 18 1 1 ; c 1 ) 33 ; ; = 24. 1 1 ) 42 ) 24 ) 15 , 1 2 , 1 , 1 1 ) 6 ) 12 ) 6 ) 18 , ) 0 1 1 1 , 3 , n , , 5 ) 0 ) 0 1 , , , ) 6 1 1 1 1 ) 1 1 1 1 12 24 12 36 ) 0 ; ) 9 1 1 1 1 s + ; ; ; ; ; ; ) 6 ; ) 3 1 1 1 1 1 1 1 1 1 , , ; 1 1 1 T 16 , , , , , , , , ; 1 1 1 k , 1 1 ) 12 ) 24 ) + 2( 6 6 3 2 ) + 18( 1 1 1 ) + 2( ) + 18( ) + 2( ) + 2( ) + 45( 1 ) 0 ) + 2( ) + 2( ) + 2( ) + 18( 1 ) 0 1 + 1 1 1 1 1 1 s ; ; 24 48 1 1 1 1 , , ; ; , ; 6= 0) vacua of SO(32) string on 2 ; ; , , , 1 ; 1 U 1 1 1 1 , ) + ( ) + ( ) + ( ) + ( , , 1 k , , 55 10 n ) + 18( ) + 18( ) + 3( ) + 3( ) + 18( ) + 2( 1 30 32 91 28 10 ) + 9( ) + 18( ) + 18( 6 3 ( ( 14 12 24 12 36 1 1 1 1 1 15 18( Twisted 1 1 1 1 2 1 2 ; ; ; ; , ; ; ; ; 28 , Untwisted ; ; 1 , ) + ( 1 1 , ) + ( ) + ( ) + ( 20 24 26 28 28 2 28 30 30 ) + 18( ) + ( ) + ( , , ) + 9( 11 ) + 9( ) + ( , , , , , 4 1 1 6 1 10 ) + 9( 9( 9( , 1 1 1 1 1 1 15 2 16 22 9( ; ; , heterotic 5 localized 28 22 ; ( ( ( ( 1 ; , , 9( 9( , , , 9( 1 2 1 2 1 2 1 2 14 8 5 2 24 26 1 11 1 26 20 , , – 23 – , , 9( 9( 9( 4 3 1 6 9( are respectively instanton numbers in the untwisted and twisted Sp(6) Sp(6) Sp(12) Sp(18) Sp(12) V )( )( )( )( )( )( )( )( , n 5 × × )( )( × × × 11 13 2 0 ) 2( 8 T 10 12 14 14 Sp(24) 0 0 0 15 SO(4) 0 SO(10) 0 0 0 0 16 SO(16) SO(22) SO(28) 10 4 2 , k × 8 × = 0) and non-perturbative ( 14 6 4 2 2 × (0 U × × × (2 0 (1 1 3 k (1 (1 (1 (1 (1 Group n 1 3 1 3 SO(20) SO(26) (2 1 (2 1 (2 1 1 3 1 3 1 3 3 1 1 3 SO(24) SO(28) SO(30) 1 3 1 3 1 3 × × × × × Shift vector U(14) U(8) U(5) U(2) U(11) SO(32) U(6) U(3) U(4) U(2) U(1) . Perturbative ( Table 1 orbifold. The parameters sectors, and the number of heterotic 5-branes, satisfying JHEP07(2019)111 T U ) ) 9) 0) 0) 2 2 0) 0) 9) 9) 3) 0) 0) 6) 9) 9) 9) 3) 9) 9) 0) 0) 0) 0) 0) 0) 0) , , , , , , , , , , , , , , , , , , , , , , , , 0 6 0 0 0 6 6 0 n 12 12 18 24 , k , k 1 1 k k heterotic string on ) (3 1 8 , ) (9 E 1 1 ) (6 , ) (0 ) (15 ) (6 1 × 1 3 , , , 1 1 8 ) (9 ) (18 , , 1 ) (0 1 1 1 1 ) + 2( , ) (6 ) (3 E , 1 1 1 , , , , , ) (0 1 1 1 1 1 1 , , ) (0 64 1 1 1 3 , ) (9 , ) (9 1 1 ) (0 1 ) (0 ) + 3( ) (15 , ) (6 ) (6 1 1 1 1 1 1 , , , , 1 1 1 1 , ) + 2( , , , , ) (0 64 9 1 1 1 1 ) + 2( ) + 3( , ) + 18( 3 ) (0 1 1 1 1 3 ) + 18( ) + 45( ) (0 , ) + ( ) + 4( ) + 2( ) + 18( 3 3 , ) (0 1 3 3 1 1 , , , 1 , 1 1 1 , , 14 , , , , 1 27 14 56 1 , 1 1 , , , , , 1 27 27 1 3 1 ) + 18( ) + 2( ) + ( ) + 18( ) + 3( ) + 18( ) + 3( ) + 3( , , , ) + 18( 1 1 1 1 64 9 ) + 18( , 1 1 1 3 1 1 1 6= 0) vacua of 1 1 1 1 18( Twisted ( 1 9( , , , , , , , , 1 14 Untwisted ( , 9( n , 1 1 ) + 9( ) + ( ) + ( ) + ( 36 84 1 27 27 14 56 56 ) + 18( ) + 9( ) + 9( ) + ( ) + ( , , 1 1 1 9( 1 1 18( 9( 9( ) + ( , , , 1 1 1 1 1 14 , , heterotic 5 localized , , , , 1 9( , ) + ( 1 1 1 64 56 14 56 14 , , 1 27 1 9( , , , 9( 9( – 24 – 3 1 84 56 , ) + ( 56 9( 1 , 27 14 U(1) U(1) × 8 )( )( )( 5 5 E SU(3) U(1) × 8 8 )( 5 0 0 SU(3) )( )( )( )( 6 0 )( )( V 8 7 5 E E = 0) and non-perturbative ( × )( 2 2 7 8 × × 0 8 8 SU(3) 0 × ) 2( 2 8 6 7 8 E n × × 2 8 SU(3) 6 )(0 2 × E SO(14) E E )(0 )(0 3 )(0 E × )(2 0 6 6 6 × 0 U(1) )(2 1 )(2 1 )(2 0 )(1 )(0 7 6 SO(14) 0 0 × × × × )(2 1 3 5 7 6 8 6 E × 4 0 6 0 0 0 U(1) U(1) × 8 )(2 1 2 2 E × Group (0 0 2 8 × 4 2 2 E (2 0 1 3 × × (1 (1 2 (2 0 × 1 3 (2 1 (1 (0 SU(9) 1 3 1 3 U(1) U(1) (1 7 7 1 3 U(1) 8 1 3 1 3 1 3 (1 Shift vector SU(3) 1 3 E E (2 1 (2 1 U(1) 1 3 × × E × 1 3 1 3 × 7 SO(14) SU(9) 7 × 6 E 7 E E E . Perturbative ( orbifold. SO(14) 3 Z / 4 Table 2 T JHEP07(2019)111 2 Z / 4 (4.1) T T U 0 0 0 n k k ) 8 ) 16 . ) 16 ) 8 ) 16 16 32 ) 16 1 1 ; ; 1 ) 24 ; ; ) 8 1 , 1 ) 24 ; s 1 1 1 , 24 28 , , , 1 1 2 , , 1 = 14 ) 0 2 ) 24 1 1 1 ) 0 ) 8 1 1 n ) 0 ) 0 ) 0 1 ) 0 ) 0 ( ( b 1 ; 1 1 1 N 48 , 1 1 2 1 2 ∆ ; ; , 16 1 s ; 6 ; 16 s + ; , 1 1 ) + 4( ) + 4( 1 ) + 4( s 2 ) + 32( ) + 4( ) + 4( ) + 4( 1 32 32 8 ) + ) + = 16 ( ; 16( 1 1 8( 16( 18( Twisted ( 20 8( 1 2 ; ; 28 28 2 120 8( Untwisted , 2 16 32 , , , a ; ; c V 24 28 4 120 ) 12 1 1 2 , , b 1 2 , , 8 4 heterotic 5 localized 8( 8 4 ( ( + 1 0 1 2 1 2 2 V – 25 – · 1 1 V Sp(8) = Sp(16) a V × 0 ) 2( )( )( )( )( U(2) model there is a transition between twisted fields × (0 3 2 ) 2( ) 4( E Sp(8) 10 12 14 14 - Sp(24) 3 1 SO(20) 0 0 0 0 × 16 SO(28) 16 ∆ × × 1 2 × 15 6 4 2 2 (0 = × ( 1 2 1 2 U(16) 1 2 (1 (1 (1 (1 Group SO(24) ( 2 SO(28) 1 2 1 2 1 2 1 2 1 2 in the twisted sector. After the transition, in the presence of V × × U(16) = 0) and non-perturbative vacua of SO(32) heterotic string on Shift vector T SO(32) SO(4) SO(12) k n SO(8) SO(4) are small instanton number, the number of heterotic 5-branes and the number of , n T , k U k . Perturbative ( Starting from one vacuum, we can go to another by emitting small instantons into time. In particularand in heterotic SO(24) 5-branes, whosetransitions, direction we have is the denoted relation by between dashed the arrow. shift vector In and emitted all instanton of number the above heterotic 5-branes. We can travel everyshift solid arrow vector by transition parameterized by auxiliary We may go directly to any of them by ejecting a multiple of six instantons at the same the instanton number heterotic 5-brane, we have largeto positive have shift still of the the spinorial zero representation, point making energy, transition hence impossible. it is difficult Table 3 orbifold. heterotic 5-branes. These are allup the to inequivalent Weyl vacua reflections. satisfying the modular invariance condition JHEP07(2019)111 ) 2 3 Z 0 / ) by to a 4 case, 8 10 T 2 8 V )(0 E ), giving 6 (2 1 48) in the 0 + 12 , 1 3 2 1 T U ) ) V 0) 2 2 (1 0) 0) 0) 0) 0) 0) 8) 0) , 16) 16) 1 3 , , , , , , , , 0 8 8 0 n , , 24 16 , k , k = 1 1 ) and k k = 2 V and SO(32) works, 1 0 1 1 1 1 0 V ) by Weyl reflection. 8 ) (8 − 14 8 ( 1 E , heterotic string on 1 3 ) (8 (1 1 )(0 1 1 3 × 8 , , 4 = 1 E 8 1 0 , , 2 ) (0 1 E × 1 4 1 V 8 ) (16 ) (0 , ) (0 (1 ) (8 ) (8 2 1 1 E 1 1 3 , , , 1 1 , ) + 4( , , ) (0 1 1 1 ) + 4( 1 2 1 1 2 ) (0 = , ) (0 ) (0,0) , 1 1 1 0 128 16 ; ; , 56 , , , 2 1 1 V ) + 4( 1 1 ) + 32( ) + 4( ) + 16( ) + 4( ) + 16( , 1 , 8( , 1 2 1 2 ) vacuum. 1 8( Twisted ( 1 , , , , 1 16 Untwisted ( = 0.) The reverse transition is possible so 128 8( ) are horizontally connected by chains of 14 , , 2 0 56 56 56 56 2 1 15 , V ) + ( 8( 8( 2 ) + ( 1 heterotic 5 localized · 1 1 , 8( (1 , 1 2 – 26 – (2 0 1 1 3 V , , 1 3 2 56 , = 56 1 V ) is equivalent to ]. 8 44 )(0 , ) vacuum to 8 8 7 )( )( SU(2) 8 ) and 6 4 E E )( )( )( V 0 0 0 41 8 8 4 × 12 ) 4( we have always , SO(16) 0 × × 8 = 0) and non-perturbative vacua of 8 8 2 4 7 0 (2 0 1 E )(0 )(0 4 E 40 × n 1 3 (1 models, we have two groups of connected vacua. In the 6 6 4 = 0. If they are directly connected by virtual instanton exchanges, V )(0 )(1 )(1 0 0 × 1 3 8 6 6 SO(16) × )(1 2 8 (1 0 0 8 2 2 2 8 = Group (0 SU(2) SU(2) × 1 3 E V 2 2 E 1 2 (1 (1 (0 ). (For the transition from a mother theory with 8 1 2 SU(2) × × 1 2 1 2 1 2 (1 (1 V E × Shift vector 7 7 = 1 2 1 2 SU(2) × + E E 8 7 3.38 × 2 V E E 7 V E · 1 V . Perturbative ( = For the When all the 24 small instantons are emitted and become coincident we have the This argument may hold for two vacua with the same instanton numbers. Consider 0 E because all the spectrahas have one-to-one been correspondence. well-studied [ The brainy picture in type Ithe side shift vector Starting from either vector we can obtain the descendent model with we may expect that thecould bottommost be two connected vacua by of transition SO(32), as with well. maximal gauge group Sp(24). Thehalf biggest multiplet. ‘bifundamental’ representation Note is (32 that in this case, the duality between can go directly from for example transitions. We can describe∆ their direct connection using generalizing ( daughter theory with the transition is twoeven way. if Note they that are there not can directly be connected a by direct an arrow. transition between For two instance, vacua we have seen that we Table 4 orbifold. JHEP07(2019)111 8 8 E 8 ) ) E ) 8 8 E × 8 ) 8 8 × E × )(0 )(0 )(0 5 3 . )(0 × 6 0 0 2 U(1) 7 0 0 2 ). They all give 2 4 U(1) V × 8 2 SU(3) 1 2 (2 0 × (1 × 1 3 )(0 (2 1 (2 1 SU(9) 1 3 + 7 4 6 3 1 3 1 0 2 E E V SO(14) · 0 1 V = (0 0 1 1 0 orbifold. Among five perturbative 2 1 3 2 3 V Z 1 2 = / 4 0 2 × + T V 7 ) U(1) 2 U(1) 8 ) 6 E ) V 6 0 ) 6 E × · × 0 8 0 × ) 2 7 1 8 7 × 8 2 2 E ) and V E E )(0 8 )(1 6 )(0 )(1 5 × = )(1 0 × × 8 6 U(1) 0 7 )(0 0 U(1) U(1) 0 – 27 – 8 2 6 2 (0 × E E 2 1 3 × (1 (2 0 1 3 U(1) 1 0 7 (1 SU(3) 1 3 (2 1 1 3 E 1 3 × − heterotic string on × 7 8 6 SO(14) E E (1 E 1 3 × 8 = E 2 V ) ) 5 5 × 0 0 SU(3) ) 2 7 2 SU(3) × 6 U(1) × U(1) E × 6 )(2 1 )(2 0 )(2 1 × × 3 7 5 E 0 0 × 4 2 . Connected vacua of (2 0 1 3 SU(3) SO(14) (2 1 (2 1 SO(14) There are transitions between perturbative vacua, in which smallNote instantons that are there ex- is duality between two heterotic string theories. At the tip of the × 1 3 1 3 SU(9) 6 E changed to twisted fields. We have shown this using the chainchains of of dualities. each stringtheories theory, we are have dually the related. same field contents. So the two heterotic string respectively shift vectors the same zero point energy correction ∆ Figure 3 models, bottommost one in each column, three generate chains of connected vacua. JHEP07(2019)111 ) = = 6 0 3 (4.2) V 2 T U K under ) ) 3) 2 2 0) 9) 0) 0) 3) , )(1 , , , , , 6 6 n 12 , k , k 3 1 1 0 k k 4 ) (9 ]. By emitting (2 1 1 , 1 3 45 1 , orbifold, but passing . = 1 , 3 ) (9 the non-perturbative 1 1 1 Z V ) (15 , / 1 1 4 all , , SU(3) T 1 1 ), ) + 2( , ) to × 1 1 5 , 0 6 4.2 , chiralities can be exchanged. 8 ) + 2( 64 E 2 , E ) + 2( ) (0 ) (0 64 1 × , , 3 3 1 , , ) (0 , 1 1 no charged field under two groups )(2 1 1 1 1 = 0. However there are inconsistent 3 , , , 27 , 0 3 9 3 3 ) + ( ) + ( , , 1 ) is anomalous although the transition 9( SU(3) 4 , 1 1 1 Twisted ( K 7 , 1 14 Untwisted ( 9( 9( , × 1 14 heterotic string on 6 (2 1 , 1 3 ) + ( 1 8 E heterotic 5 localized )(2 0 , 1 E 3 , ) + ( 1 ’s = 0 1 1 8 – 28 – × , , ’s, and deserves further study [ 4 3 V 8 E 8 , ) + ( 84 E E 1 ’s. This provides a suggestive explanation on the ) applies well to supplement the modular invariance , 27 8 ) giving (2 1 1 5 , 1 3 E = 1 we have ∆ 0 3 3.36 , 2 The above anomalous model, which satisfy the modified 2 = seems possible according to the selection rule from ∆ n 27 0 1 V V ’s. 8 )(2 1 5 E and absorbing them in the other 0 8 U(1) 2 )( E 5 × )( )( U(1) SU(3) 0 7 7 V (2 1 × 2 × . can be exchanged. 1 3 6 ) to the above inconsistent vacua, so this may be inconsistent as a string 8 5 E )(2 0 )(2 0 5 3 5 E )(2 1 SO(14) . The desired representation does not arise in the twisted sector because 0 0 0 × 5 0 × V 2 4 2 SO(14) Group 2 × (2 1 (2 1 Shift vector SU(3) 1 3 1 3 (2 1 )(2 1 vacua even they pass both conditions; the resulting spectra are anomalous. They SU(3) 1 3 3 × . (Possibly) inconsistent vacua of 8 0 6 × SU(9) E 6 4 E = 0. It would be anomaly free if some twisted fields are branched from those in the One interesting observation is that, except this one with ( Another example is a seemingly consistent vacuum with For example, the model E 2 × n 8 (2 1 1 3 modular invariance condition andtwo pass groups the coming fromcommunication selection problem different rules, between requires two small charged instantons in field one Thus these two Note that this vacuum is anmodel intermediate although one it along is field the theoretically chain anomaly of free. transitionvacua in from theoriginating presence from of different heterotic 5-branes contain It is not anomalous because the unbroken gauge groups are anomaly free by default. from the above vacuum with ∆ vacuum with the modification of themass zero shell point condition. energy is too large for this representation to satisfy the is not possible becauseE while ∆ are listed in table condition. For example, transition from Table 5 the selection rule. 4.2 Communication between two In many cases, the selection rule ( JHEP07(2019)111 3, ’s. , 8 ). It E = 2 4.2 ’s, which N ]. Rather 8 E 17 orbifold, with heterotic string. N 8 Z E / 4 × T 8 E ] calculated the modular invariance condition – 29 – 10 ’s in the strong coupling limit of 8 E of heterotic 5-branes, this introduces modification of the zero point energy, n ’s. The technical reason making sense of this argument is that if we have non- 8 E We note that some vacua that satisfy the modified modular invariance condition in the The most striking feature is the connection between two perturbative vacua, in which In the presence of heterotic 5-branes, although the theory is non-perturbative, we In the strong coupling limit we open up a new interval that separates these two the modular invariance condition is notdo the not sufficient know condition fully for sufficient theThis condition consistency. is for We due still anomaly to cancellation lack in oforiginate, the the e.g. full presence from understanding of branching on 5-branes. theCFT of twisted a equation. fields; larger we In do representation not the but know smooth at how they best case, we ref. obtain [ them from nism. This is anotherconditions feature are that described twisted fields byfor heterotic behave the like 5-branes. open orbifold We fixed strings might points. whose have boundary a fuller brainy description presence of heterotic 5-branes give anomalous spectra. Contrary to the perturbative case, the modular invariance and Bianchi identity. some small instantons aresuggests exchanged that to orbifold fixed twisted pointstion sector among also fields. small behave instantons, like heterotic Comparison branes. 5-branesindirect of and chains Therefore twisted of vacua we states. transitions, also may Although it have we would transi- have be used interesting to study any direct transition mecha- obtained by instanton transition from aa perturbative solvable CFT. vacuum, which The is effect indeedthan of described heterotic previous by 5-brane bottom-up modifies the approach,phase zero-point we transition energy [ can between derive smallmodular the invariance instantons condition. modified and This 5-branes. zero-point gives us energy deeper With using understanding this, on we the relation derive between modified between small instantons andsuggests 5-branes. that most It of vacua doesof of not toroidal stringy orbifolds vacua only are in explain connected. the the landscape. This spectrum, hints at but the also evolution have still exactly solvable CFT in the worldsheet. This is because the 5-branes can be 5 Discussion We have studied vacua ofin heterotic the string compactified presence on of heterotic 5-branes. A new understanding comes from phase transition which makes hard to satisfy nontrivialvery mass difficult shell to condition. have a Inpartly charged this state situation, justify under typically the two it groups above is would inherited observation. from be different This also interesting wouldcommunicating for let two further us study to if rule this out transition the should vacuum always ( be involved in The 5-brane is interpreted ason M5-brane an if interval we and regardthe after this introduction. phase theory Therefore transition as M5-branes M-theory this canthe compactified be can two messengers be for emitted exchanging chiralityzero in between number the bulk as explained in JHEP07(2019)111 , H Nucl. Nucl. , , , → Nucl. 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