Fermionic Zero Modes of Supergravity Cosmic Strings

Fermionic zero modes of supergravity cosmic strings

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Please note that terms and conditions apply. JHEP06(2006)030 May 2, 2006 hep062006030.pdf June 1, 2006 June 15, 2006 rings could be itons Monopoles Received: Accepted: d generically occur in Published: ing models. In particular, ino, some (but not all) zero ngs arising in supergravity. realistic model to study the ion are not subject to vorton the existence of fermion zero eaking on cosmic D-strings. index theorem for the number bridge ts can give rise to stable string cherche au associée CNRS gly constrained, allowing many Hounsfield Road, Sheffield ant complications in the presence properties of cosmic strings in flat http://jhep.sissa.it/archive/papers/jhep062006030 /j Published by Institute of Physics Publishing for SISSA String theory and cosmic strings, Supergravity Models, Sol Recent developments in string theory suggest that cosmic st [email protected] [email protected] [email protected] [email protected] CEA–Saclay F–91191 Gif/Yvette cedex,E-mail: France Department of Applied Mathematics,S3 University 7RH, of U.K. Sheffield, E-mail: DAMTP,Centre for Mathematical Sciences, UniversityWilberforce Road, of Cambridge, Cam CB3 0WA,E-mail: U.K. Instituut-Lorentz for Theoretical Physics Postbus 9506, NL–2300 RAE-mail: Leiden, The Netherlands Service de Physique CEA/DSM/SPhT, Théorique, Unitéde re SISSA 2006 c Carsten van de Bruck Anne-Christine Davis Stephen C. Davis Abstract: Philippe Brax formed at the end ofproperties of brane strings inflation. arising in Supergravity branespace-time provides inflation. have a been Whilst extensively the studied thereof are signific gravity. We studyFermion the zero effects modes ofsupersymmetric are gravitation models. a on The cosmic common corresponding stri loops massless feature (vortons). curren of The vorton cosmic densitytheories strings, in with our cosmic an universe strings ismodes to stron on be cosmic ruled strings out. inof supergravity We zero theories. investigate modes A is general derived.modes We disappear. show that by Thiswinding including weakens number the one the gravit cosmic constraints D-strings inconstraints. on models We cosmic of also brane str discuss inflat the effects of supersymmetry br Keywords: and Instantons. ° Fermionic zero modes of supergravity cosmic strings JHEP06(2006)030 5 7 1 3 5 12 13 13 14 15 17 20 21 22 22 23 24 13 14 20 23 26 ls of brane sing from recent results in deed, in string theory a generic on of lower dimensional branes, – 1 – 1 branes, or D-strings, favoured [3]. Consequently, in mode D 3 and D 3.1 The Dirac3.2 equation Generic mass3.3 matrix Block off-diagonal3.4 or Dirac-like mass Zero matrices mass matrix 4.1 D-term strings 4.2 F-term strings 5.1 Fermions in5.2 supergravity Gravitino field5.3 equations Index theorem 6.1 10 D-strings 6.2 Non-BPS D-strings 6.3 D-strings with6.4 spectators Massless spectators6.5 on D-strings Non-BPS F-term strings 7.1 Extending zero7.2 modes to massless Two dimensional currents effective action 23 1. Introduction There has beenfundamental a superstring resurgence theory of (for interestfeature reviews of see in D-brane [1, cosmic 2]). anti-brane stringswith annihilation In ari is the producti Contents 1. Introduction 2. Gravitating cosmic strings 3. Zero modes and the index theorem 4. Supersymmetry examples 5. Fermion zero modes in supergravity 6. Supergravity examples 7. Fermion currents 8. Conclusion JHEP06(2006)030 smic strings can arise rings. In section 5, we ology of cosmic strings ro modes in the string he Dirac operator in the section 2 we consider the he D-term case, this has igated and shown to give studied to date. Here we ive supersymmetry break- S configurations led to the strings arising from the D- ic examples are reviewed in obal supersymmetry [9]. An phase transition (for reviews the case of supergravity. An e solve for the most general er the most general form of on, and subsequent annihila- . Ordinary cosmic strings are ise [5] and, in certain classes [4]. Depending on the details rticle content required by the [15, 16]. The presence of zero s [9], where the D and F refer from supergravity which need y via the emission of gravita- 1 rent carriers [14]. These stable ss the question in detail. ly relevant since general brane in such models have many fea- ide insight into those formed at sed later when calculating string act D-strings [11]. Consequently, nt ing network. However, when the gests that conductivity is reduced ology the distinction between super- and hen in fact they are actually just perfect ermion zero modes, but the fermions would s true for both types of current). e cosmological properties of conducting strings – 2 – In general supergravity contains both D- and F- terms, and co Cosmic strings in supersymmetric theories have been invest Strings arising in supersymmetric theories have fermion ze Such strings are widely referred to as ‘superconducting’, w 1 either long strings ortional closed radiation, loops. resulting in Thestring a carries latter scaling a usually solution current deca the ofloops, loop the or can str vortons, be put stabilisedmodes by constraints the on on cur cosmic the strings underlying hasimportant theory been question well to studied ask in the isanalysis case of whether supersymmetry of these transformations gl results in go ref.for [17] over supergravity sug to cosmic strings. In this paper we will addre from both classes ofinvestigate theory, both though D- only and theinflation F-term former theories. models have contain This been both isterm types particular would of be terms. BPS states.gravitational Of background course, The paper created only is bymetric arranged cosmic appropriate as to strings. cylindrical follows. symmetry, which In Here willzero be w modes. u In sectioncosmic 3 string we background prove includingmass a general matrices gravitation. index that theorem can Wesection for arise consid 4, t in where such we theories. apply Supersymmetr the index theorem to D- and F-term st follow from the fact that the currents are conserved (which i conductors. To be superconductingalso they have would to not only formperfect need into conductivity f Cooper is pair not very bound important, states since [13]. most of For th cosm inflation where a periodtion, of of inflation a is D-brane causedof and by the anti-brane, the D-strings attracti theory, form fundamentalof naturally strings, models, or axionic F-strings localtures can strings in also common [6]. with ar cosmic Thesee strings D-strings formed [7, arising in 8]). a cosmological However,to there be are addressed. added complications arising core [9]. This issupersymmetry an algebra. inevitable result In ofing the certain [12]. couplings cases and the pa Thedrastically zero since presence modes they of result surv fermion in the zero string modes carrying changes a curre the cosm rise to two sorts ofto strings, the called type D-term of andbeen F-term potential extended string required to to supergravityconjecture [10], break that the and the D-term an strings symmetry. of analysisan supergravity For were of analysis in t f the of BP stringsthe in end supergravity of theories brane should inflation. prov JHEP06(2006)030 (2.3) (2.2) (2.5) ) = 0. is related to the ∞ ( , from the broken α u 1) in supergravity. B T − n breaking, although any cted can produce cosmic ing number of the string. I ngle in the far away metric onstraints found for chiral taneous breaking of gauge es to the issue of massless he metric is regular at the ) (2.4) 2 s the number of zero modes . This is achieved by having th scale. When there is only → ction 8. uations onto Dirac equations tor, although more generally es are considered in section 6. volves the energy momentum or BPS D-strings the number is the string’s winding number. dθ lindrically symmetric metric, as 2 ar fields have a profile . n α ) . ) θ i θ ρ ρ + φ T ) (2.1) θ θ T ) 2 . r u µ + ( s i + κT dρ A ρ T ρ t φ t u s T + + T = ( ( 2 2 iT iθT B (0) = 1. We also impose ) B 0 0 2 e 2 dz − α B →∞ in the global case to 2( µ αe ( r – 3 – + | ∂ 2 )= κ − καe u θ 2 n is an overall conformal factor and A dt = = = = ( r, θ u B ( i − 0 i 00 00 T ( ) φ 0 φ α B B µ 2 2 e is the electric charge and D αB (0) = 0 and ( α Q = is some linear combination of generators, 2 s T ds where (0) = 0, 0 nQ B = s T The gravitational effects of a cosmic string lead to a deficit a for a cosmic string configuration. Thisdiscussed is by the Thorne most [18]. general cy Notice that deficit angle of thetensor cosmic and strings. reduces The to Einstein equation in gauge group. In thisThe paper covariant derivative we of will the define scalar it fields to is include the wind The simplest string gaugesolutions fields can will have only several involve gaugeone one generator, fields, genera each with it own leng of spacetime. In the following we will consider the metric This must vanish at infinity to ensure a finite energy solution These equations can beorigin solved and numerically satisfies by imposing that t consider the supergravity case.amenable Here to we a map treatment the as gravitino inWe eq section find 3. that Supergravity the exampl on presence supergravity of cosmic the strings. gravitinoof In generically chiral particular, reduce zero we modes find that are f reduced from 2 Thus winding number 1theories D-strings [16]. evade the In stringentcurrents section vorton on c 7, the we cosmic extend string world–sheet. our results We conclude on in2. zero se mod Gravitating cosmic strings We consider a cosmicsymmetries. string configuration The created simplest by example the of spon this would be a U(1) symmetry breaking whose vacuumstring manifold solutions. is In not the simply cosmic conne string background, the scal where the string generator JHEP06(2006)030 ). ), 1 ρ ρ C T (2.8) (2.7) (2.6) (2.9) (2.11) (2.10) (2.12) + − is zero. t and far t (1 T r B ( π κ ≈ is identically = C B 2 m ll only require the le ∆ = 2 ng a U(1) symmetry . ∗ rsymmetries in its core leave 1/2 of the original the origin B he top hat approximation 2 rested in this paper, are an + zero and therefore side the cosmic string with a 2 consider the outside solution. string will only be BPS if the dθ e. First of all, notice that the ¶ 2 0 dθ on-zero potential energy for the 2 . Let us define C 0 ˜ θ ρ θ mρ 2 1 + T dρ i C 2 φ ≈ sin i ∗ ). The bosonic fields satisfy first order + dr ρ ρ r ρ B Q . ) ρ . 2 ( T s a 1 ˜ ρ Z + . Hence an approximate solution will be , and the conformal factor T )+ d C ρ na r ρ mρ C 2 − 2 m θ m µ ρ T + we obtain 1 + δ ρ dz ρ − 2 0 n sin( T = – 4 – dρ + C ∓ 4 dz κ µ B 2 = mρ = e = + A ≈ ). dt α i 2 ) the metric becomes α = − φ 1 /r B ( r dt ln sin and is given by ∂ (1 B dr − /C 2 i µ 0 ρ O e φ ρ = i ρ C 2 + = 2 m + 0 κT 2 nQ ds ρ C − ds = + i = = ( r φ 1 s ρ B T C = C 6= 0, the solution is a cosmic string solution with a deficit ang 1 C Let us characterise the BPS cosmic strings. We are consideri In supersymmetric theories, a cosmic string breaks all supe Using the change of variables In order to determine of the number of fermion zero modes we wi breaking, so we take equations zero. As we will discusssuperpotential in vanishes for more the detail in stringstring subsection solution. comes 5.1, The from the only the n D-term corresponding to in general. BPSsupersymmetry objects unbroken. are D-strings, an inexample exception of which this. to we will These this strings be rule, have inte vanishing as they from the string which is the metric we will use for the rest of this paper. Near near infinity. Indeed, after defining ˜ then the metric inside reads satisfying the boundary conditions atIn the this origin. region, the Let energy us momentum now tensor is approximately A good approximation for small while form of the string solution at large and small When sufficient for our analysis.whereby the scalars We are assumed willjump to look be at constant the for inside string and solutions radius.energy out momentum using Let tensor us t is almost consider constant the with solution insid Now we also find that JHEP06(2006)030 (3.2) Weyl (2.14) (2.16) (2.17) (2.13) (2.15) f n µ cannot be w 1), we must n ground. The < 1 j C χ 4. In particular we / i µj ) = 0 and b J 0 ¯ σ itive ( nsion is higher than the a + B ) (3.1) the U(1) gauge symmetry . σ k bject to both the electro- c . χ . − tional to the winding num- j umber of fermion zero modes µ b φ ¯ σ µ i xistence of 1/2 supersymmetry. + (c e consider a family of a ξA φ ˙ D i α tions. These equations are conse- σ lies that the winding j i jk K ¯ χ i )+ j = ( ij , with unit charge. Notice that BPS Q φ +Γ φ M µ ab i j ˙ ξ . ξ . α i B θ σ | | D X and the spin connection term χ ¯ χ j A n n i j 1 ) 2 a µ − K u ± A − | − | ξ + T − 1 ( ¯ j α  = – 5 – u µ = 1 ¯ φ χ = 1 ≤ 0 µ µ D iA 1 1 C D D C C = ¯  − i ˙ ¯ αα µ a K χ ( r ¯ σ is less than the BPS case C i ) i ∂ 2 ˙ 1 α i µ n ¯ χ C w = ∓ are flat indices and ij B µ + g A µ = a, b ∂ L = ( 2. i / ), which gives 3 χ n µ iσ 2, where D − / ) is a sigma-model metric depending on the cosmic string back sign( ab = ¯ φ σ − 12 φ, = σ µab ( is the Fayet-Iliopoulos term which triggers the breaking of ij w is the Kahler potential. The Einstein equations reduce to ± g ξ − K We will also consider the non-BPS configurations where the te In order to make sure that the deficit angle of the string is pos = µ where fermions with a Lagrangian choose Notice that the deficitber. angle Restricting ourselves of to thearbitrarily positive large. tension string strings is imp directly propor BPS tension, this implies that We will see that this changeon in a the string. deficit angle can alter the n 3. Zero modes and the index3.1 theorem The Dirac equation When fermions live inmagnetic a interaction cosmic and string the background, gravitational they interaction. are W su w where and find that and The simplest BPS configuration will just have one cosmic strings are solutions of first order differential equa quences of the Killing spinor equations when requiring the e where covariant derivative involves the gauge fields JHEP06(2006)030 . µ J such (3.6) (3.5) (3.3) (3.8) (3.9) 3. a i e , where . ≡ µ K J z j ¯ . ∂ a ontributions i χ ∂ a . q c = 2 and ˆ χ = ¯ ij k ≡ ¯ g a φ ˆ θ µ χ igma-model manifold s D 1, T j c th the Kahler manifold alars in the curved case. are zero. For the super- e osmic string background. An example where such a using the vielbein ≡ he Levi-Civita connection t ¶ J rmions having a flat sigma- ) (3.7) ted to the scalar fields and r . ¯ k 0 r metric ributions are included in c ¯ φ . ∂ 0, ˆ C d, we absorb it into ∂ to take into account the repara- and ) (3.4) ≡ = ¯  µ z ¯ + (c ˆ t φ ˆ r i jk a j ˆ J θ b µ e θ ¯ χ D . + ¯  ab b j c ¯ b j K e . χ M e a i k a a a i e − e ¯ φ χ χ , w j ab µ i a 0 1 2 ab φ e δ D B µ M j c + – 6 – = − e D = = a i j χ ¶ ij = χ ij K µ g k ˆ ( z ˆ r ∂ M D 4 1 z ∂φ µ w ¯ σ = µ i a j a µ = e indices is understood and χ J ˆ t ˆ has only three non-zero entries r t + = ¯ w a a, b µab L χ µ w D to diagonalise the string generator, so that a → χ a χ ) is the Kahler potential of the dilaton. In this case the two c µ ¯ S D is non-vanishing. + θ S J ln( − The Lagrangian can then be written as For a cosmic string, We have also included the Levi-Civita connection Γ In the following we will diagonalise the sigma-model metric Notice that the Weyl fermions live in the spin-bundle of the s of the scalar manifold, although this is not always the case. = i jk from the non-flatness ofMore the generally, sigma if any model part metric of cancel the in connection the is c not cancelle K We also choose the and redefine the fermions that where hatted quantities are flat indices and we identify and only metrisation of the sigma-model scalar fields. Any other cont Given the form ofgravity the D-strings string discussed solution, in we section 2 will we assume find that The covariant derivative becomes cancellation does occur has been given in ref. [6] for a Kahle For a wide range ofΓ models this cancels the term arising from t defined by the scalarWe field, can hence have we more have fermions asmodel than many scalar metric. fermions fields as by sc consideringdefining This fe the happens set in oftheir scalar the sigma-model fields. metric supersymmetric is The setting flat. gauginos wi are not associa where summation over the JHEP06(2006)030 0. ≡ (3.13) (3.14) (3.15) (3.12) (3.11) ab M must all be (for negative a . q a − χ lower left corners = 0 (3.10) = 0 1 ∗ iteness of the norm ∗ ) n ) b + b − × χ of the bosonic fields, it χ isable ( . around the origin. This 2 ( n ab o ab π ose the fermions using the es is obtained by analysing ilθ iM stems, and analyse each one iM ∓ agonal. By this we mean that + 2 ermion charges e . . Close to the origin the metric − ral choice of norm rs or half integers. We will deal to the above rule is if + ) θ 2 r | a + ceptions which require a different ( a − . a directions cancel. We will look for χ ∗ → . χ ) χ a t ± | r θ 1 2 ( V B ¸¶ ¸¶ 2 θ ab θ + ∓ J and J Z M ilθ z + θ + ) eigenstates. The upper component spinors ∈ ± u θ b u θ e l q 3 ) A . drdθCe upper-right and the A + σ r u − a u a ± ( – 7 – 2 q Z a ( χ a ± iT n i iT e U are not simultaneously block diagonal. If they are, a l,q − × − n X 0 = 1 0 a θ µb + n C a = J C ab a i 2 iq ). This will be the case if system has only Dirac mass i 2 q 2 e f M || − n + C χ and 1 θ θ || √ ∂ = ∂ ab · · 2 = M i n C i C a ± (for positive chirality), and the lower ones by + χ + − 1 a + r r n χ J J normalisability of the + 2 + r r L ∂ ∂ µ µ factor is constant at the origin and at infinity, we see that fin B As the mass matrix is obtained using algebraic combinations Gauge invariance implies that the mass matrix must be factor The zero modes can be separated into In both these cases thewith charges are them not in automatically later intege subsections. 3.2 Generic mass matrix The index theorem giving thethe number normalisability of of normalisable the zero modes mod is at the simply origin the and flat atansatz infinity metric in polar coordinates. One can decomp terms. It is also relevant for D-strings. Another exception of the matrix (with either integers or half integers, althoughtreatment. there One are exception some is ex ifthe the only mass non-zero matrix entries is are block in off-di the we simply separate the fermions into several independent sy individually. is a single-valued matrix. In general this implies that the f We can assume that both is equivalent to chirality). The Dirac equations now read will be denoted by We require that the fermions are single valued when and similarly implies that As the Notice that the spin connection parts along the normalisable solutions of these equations, and use the natu JHEP06(2006)030 r 2 / , the +1 (3.22) (3.18) (3.23) (3.19) (3.17) a ∞ q solutions = f r ≥∓ n l at solutions for a (false) = 0). I ab r are half-integer, or M a q ¶¾ 1 2 behaviour of the fermion + deriving will not be valid mined by the mass matrix. r nish at infinity, and so be . a q l . = 0 (3.16) are non-zero, the 2 (true) = 1 and = 0 − ¶ 1 I b a ≥∓ a ± 1 b ± q demands that either 2 l V λ = 0 (3.21) = 0 (3.20) l/C C ± U r µ r − ab a a ± ± ab r which diagonalise I ˜ ˜ V U ∼ |≤ M a a l ∼ i M + V | a i ± a iλ iλ µ ± ¶ + + ˜ V I 2 1 + a + ± a z and is an integer when the ± 2. The total number of normalisable, large a a U ± n ± − / l V ] ˜ ˜ U 1 ] V U a – 8 – a , V − a q r r q l , q 1 1 f l l + 1 ± a n >C C C can be neglected. ∓ q l | ≤± l [ l/C l ± µ [ solutions at small | + − r l r J r 1 )= r 1 a a = 0 then the leading order behaviour of those solutions l ± µ a ± ± ∼ ( ∼ ˜ ˜ − V a χ U I + r ± a r ∞ λ ± a is subdominant near the origin, as it is frequently the case ± a ½ ± a ∂ ∂ ˜ ± . The total number of normalisable, small vanishes, so to leading order U N U l U µ f V r =1 u J n r i X ∂ ∂ T u θ (1), this implies that the solutions behave as A )= l O is therefore ( l ± = 0 N ab 2 for a given M / to be linear combinations of 1 ˜ V − are the eigenvalues of the mass matrix. If all a ˜ U, q is the characteristic function of each interval (so a is then λ I l ≤ ± Close to the origin On the other hand if some Normalisability of the above Now since At infinity the behaviour of the fermion fields is mainly deter l solutions for a given One of these will be normalisable if is instead the converse. We will now use the approximate large and small field equations to determine the number of zero modes. will have exponential behaviour,normalisable. and exactly half of them va or and given where we have again assumed that where These two conditions are satisfied provided where asymptotic Dirac equation reads in supergravity. If it is not subdominant, the results we are (although the approach we are using can be generalised). We have also assumed that Defining JHEP06(2006)030 and odes. (3.26) (3.24) (3.25) (3.27) copies + lutions + n z n ...q . malisable at and . (3.19). If we ¶ ≥ . 2 1 − a q + 1 q + ¶¾ ly the number zero 1 2 − a ˆ q + − tions at the origin and ¶ ˜ sible there will be addi- q 1 2 ed this possibility in the o that zero mode solutions for a of the intersection of these − 1 for a cosmic string with a ≥∓ eparate the winding numbers a × = N − = 0, we find that the number of real a + , the presence of massless fermions in a µ 2 ) (3.29) 1 q l l − n + I a a ( £ + 2 ), we obtain z n q ) ± + 2 z +1 n z + µ ∗ n n + = ¤ N + n I X + − ¤ − a = + + n +1 − l a q − a − n − = 0 if q X ,n + − n + ˆ X + ¶ – 10 – are all the negative = − = n ,n 2 1 ¶ + a a . . . − q − 1 2 + a a ± 1 £ ˆ + q q n ) (3.26) evaluates to zero. Hence we can ignore all a min( N l − ∗ − + 2 = a ( =1 £ n a a q , we can rewrite the above sum as + a X a + | + 2 q 0 ∗ − P − a =1 N n , and ˆ a X q + . ≤ a = min( = 2 ¤ q ± l 0 ∗ − a + q n N = 2 ≤ | ≤ | ). n z n ...n + ¯ (1) n and respectively. ˆ q 1 = 1 η n , . At large a . . . 2 , then the above to be the combi- . z ¶ (3.18), and so the ≤ Z n ≤ 1 2 and ¶ for f a (1) a ∈ 1 2 − η . q a − (1) 1 2 q η − − ¶ q n ...n (2) ± 1 = ˜ 2 q fine ˆ (2) C = a q . If2 1 (1) a (3.20) and (3.21). We define z + 1) q x ≤± n l> 2 y eq. (3.22). The total number , for which the charges satisfy l ≤± n µ 1 − η µ l I ¸ ¸ 1 − I 2 µ η η n 1 z f I z ) in a similar form to eq. (3.35) by ...n n ∓ ± l n = n 2 ( 1 1 =1 has eigenvalues + n n η 2 2 ± a X s + ∞ C C = 1 = ( η s T ¶ . N · · + ¶ a a 1 T is single-valued at the origin implying that 2 + 1 2 ) C . The winding operator can be decomposed (2) ∓ ∓ ± ¶ ) 0 2 r q s a ± 2 1 η η r + ( − n T ( χ ∗ , where ¯ . = ˜ a a − + + – 11 – ± ± n + Z (1) 2, i.e. ± = , and order them so that ˆ V U ˜ l< 1 1 2 2 q / 1 s (1) ± θ θ ∈ (1) a ) ) µ n q T ± ± l l (1) q ± η I q ∓ ± + 1 1 = ≥∓ a a = = z ∓ q q l Z ( ( f ≥∓ l n i i µ n satisfy eqs. (3.16) and (3.17), although with a restricted (1) (2) e ± ± e l ∈ ˜ ˜ q q + I a µ , we can express ( 2 η n z I Z V , we can choose some value × n 2 − copies of ˜ 1 ’s to be integers or half integers. We use the decomposition ∈ =1 n a + ) = ¯ a C X 2 (2) l a 1 q z η and ( n q √ ...n n has half-integer eigenvalues. Defining a ± ∞ ∓ )= 0 = l s l U N ) = ¯ ( solutions is T = 1 l a ± and ± ( 0 χ a ± 2 and a ± ∞ N / χ (1) a N . This implies that q for 1 2 + 1 to be the number of massless fermions with respectively 1 2 Z . The leading order behaviour of the solutions is given by eq. + n z a + n ] is the lowest integer which is strictly greater than Z ∈ a x q η ∈ l = We can now write Near the origin and + ± 1 z (2) a (1) a a nation of the approximate solutions are either exponentialof or normalisable are given b The number of massive fermions is then 2¯ in such a way that q There is no need for the q as q and proceed in a similar way to the previous subsection. We de introducing and where [ The total number of fermions is Using the fact that instead of eq. (3.14). We now impose that The behaviour of the spinorsn at infinity is determined by eqs. choice of total number of normalisable solutions there is expressions agree with eq. (3.24), and ˜ JHEP06(2006)030 2. / 2, or / (3.42) (3.41) (3.40) (3.43) +1 1 Z . ∈ , and order l>C will cover it Q ¶¾ (2) ± 2 1 q + . (1) , the compensating a 1 when ˆ q + Z i − ∓ ass matrix is zero. In . The complex solution ∈ s may occur at a phase (1) − copies of ˜ C l ˜ q 1 √ z − for the two chiralities. This / ˆ ). Again only half the terms in the above q q − l + = ( 2. At ± = ¸ ¶ / h ± a ± , N 1 2 a 1 to be the 1 2 – 12 – z χ q given by eq. (3.37). Hence N n + − =1 − l − + i a X (2) ¯ a n (1) ± q Q P q (1) + q (2) a · q ˆ q = 2 + ˜ = 2 ≤± = 2 ± a l ≤± q ± + N l h 2, with ˜ . This allows us to write N N / z f ≤ =1 n n a X 1 2 =0 if + + 1 ¯ (2) n r ˆ q + = 2 (1) ± ˜ ) could also be altered, and so q f . . . (1) + a ˆ q n 2) , N ≥ ∓ (1 ± ≥∓ q µ . Similarly we define ˆ )= l (2) 1 2 l I q z ( 0 is a special case of a block off-diagonal matrix, although we ½ n ± ∞ z n ≡ + N =1 + = 0, and there is only one charge 1 a X ¯ n z , and obtain ab η )+ n ± l + M ( It is interesting to note that the values of ˜ N = ± 0 z N string’s gravity. This isstring. not the case if all the fermion3.4 field Zero mass matrix Finally, we will determinefact the number ofseparately. zero In modes this when case theare the only the m gauge things fields determining and whethe gravity. We take hence may change iftransition). the If the string mass mass matrixwill has per depend zero unit on eigenvalues, then length ˜ t changes (a last expression is particularly usefuldiagonal in mass supersymmetry matrices. wh Let us apply this result to the case treated in ref. [21]. When equivalently if factor them so that ˆ Note that there are different definitions of the ˜ expression (3.40) are non-zero, which allows us to reorder t for In contrast to the earlier case, solutions for different n number of real solutions is will be normalisable at This is exactly equal to the number of integers satisfying 0 JHEP06(2006)030 ± φ zero (4.5) (4.3) (4.1) | n | 1. These ap- . The winding − λ n . Other fields may ≤ ξ decay exponentially. f the U(1) symmetry, √ . m χ F-term strings. As an θ = ≤ approximate form of the = 0 solution by applying 2) 2, we find that near the / ouples to moduli fields as φ / 1 lution. The fermion mass and 1 m les and count the associated − λ e m e the form − , − l r . n and the gaugino ( − i e = 0 and χ = 0 = m ) (4.4) φ = 0. In supersymmetry, the number = 0 (4.2) 2 − 1 m x must satisfy 0 − ξ . − (2) λ n − q r + m , φ − 2 | φ ∼ φ and + inθ φ ± n −| ( – 13 – e n , N ) φ = = r ( = , χ implies that the potential depends only on = 2 D (1) χ θ xf ξ q + 0, and defining W 2) which is charged under an Abelian gauge group. The / = N interpolates between φ +1 ± φ m φ n > ( i sets the scale of the U(1) symmetry breaking. The fields the two chiralities are interchanged, and there are 2 e m x n r is uncharged. This model can be used to give hybrid inflation, ∼ φ λ 1, and ± 0. For negative as the inflaton. In that case inflation ends with the breaking o φ Using the analysis of the previous section, we can obtain the n > have charges with which results in the production of cosmic strings. These hav and a U(1) gauge group. a supersymmetry transformation to the string solution. 4.2 F-term strings Consider the supersymmetric model with superpotential modes with negative chirality. zero mode solutions. Taking For the solution to be normalisable, the integer origin the positive chirality zero modes have the form proximate solutions are obtained from eq. (3.18). For large As was shown in ref. [9], it is also possible to obtain the abov The cosmic string is such that of zero modes is therefore given by [9] for 4. Supersymmetry examples In global supersymmetry, oneapplication can of find the index zero theorem, modes wezero review around modes. some D- known We examp and alsomight consider be the the case case where in an string F-term theory. string4.1 c D-term strings This model uses onepresence superfield of a Fayet–Iliopoulos term be present, but theymatrix in will global be SUSY is zero off-diagonal everywhere and for involves only the string so numbers for the index theorem are JHEP06(2006)030 ). and , λ (4.7) (5.2) (4.6) (4.8) µ A W and the i n D ¯ λ 2 ¯  − ¯ χ K/ i = e ) (5.1) . − c Qφ (2) χ . = ¯  − q i i χ , K + n m 2 + (c , µ √ = φχ al moduli fields, then it i ¯ interactions ψ rm, and the other index W + µ − 2 ories. This is complicated + (1) χ look at cosmic strings with , in the above Lagrangian. σ l gauge symmetries are still ¯  q rve the Lagrangian, and act − K/ portant to fix the gauge. We ¯ χ y Lagrangian is χ e µ + Λ ´ σ i j = λ ν χ . χ ¯ − σ 2 i µ | ¯ is no longer zero inside the string. µ φ / χ n ¯ 3 . 2 D ψ | i ij φ µ m √ φ ¯ σ i m ¯  ν = 2 = 0 . 2 1 ¯ χ + ), and one U(1) gauge superfield ( D ¯  − − i ¯  i i χ − = 0 . i + ,χ K = 0 solution is destabilised. A similar effect iK i χ φ µ , N , N = 0 this mass matrix is block off-diagonal, and 2 φ W µ | φ 1 ³ − ψ i – 14 – φ √ ¯ σ n µ | λ K µ − ¯ = 0 σ µ ¯ − ψ + i + + D λ = 2 χ will mean that µ , whose mass and kinetic terms have a different form µ m N + ¯ W ´ µ σ σ 2 φ i . Since i )¯ ¯ λ λ i ψ ∂ N i √ = 0. In the following we will not fix them. φ + ρλ ¯ φ = µ − 4. We have defined F + 2 / λ ν ) ρλ δψ √ ¯ W ψ µ ψ ¯ i µ σ i ρ i σ σ µν D ν − D + ¯ σ ¯ ν σ χ µ ¯ σ ¯ D − ψ − µ ( 2 φ ¯ with ψ µ ν / ³ ¯ 3 ψ σ µ W 1 2 − m j µνρλ σ + ² − Weyl zero modes of each chirality. D i | are zero. Hence we see that [9] = = (¯ n D | q 2 L are the partners of the µν i K/ σ e χ In supergravity, the supersymmetry transformations prese If our toy model is extended, for example to include addition = ij Working in Planck units, the fermion part of the supergravit other two as gauge transformations. Whendo analysing the this theory by it imposing is im is possible that their coupling to to those used in sectionan 3. arbitrary We number keep of the chiral discussion general superfields and ( There are To achieve this weIn have general incorporated this a does Lagrange notpresent multiplier fix corresponding Λ the to gauge ¯ symmetry fully. Residua The mass matrix willtheorem then (3.30) cease will apply. to It have tells a us block that off-diagonal now fo where m The mass matrix of the fermions contains the Yukawa and gauge so we use the index theorem (3.41). The winding numbers are was noted in ref. [12] for SUSY breaking. 5. Fermion zero modes in supergravity 5.1 Fermions in supergravity We will now extend our zero mode analysis to supergravity the Hence the zero modes are removed if the by the inclusion of the gravitino where ¯ JHEP06(2006)030 (5.7) (5.4) (5.3) (5.6) (5.8) (5.5) (5.9) (5.12) . hat they are = 0 (5.11) = 0 µ µ Λ (5.10) ψ ψ ) µ µ σ ¯ σ ρλ ¯  σ ¯ = ¯ m . This condition gives ρλ i ¯ ²  ¯ i in a cosmic string back- ¯ tions are χ χ = 0. Hence a non-zero iF K ² . ) to be a constant and an µ µ of ¯ ² 0 r r 2 ¯ µ σ σ ates. The conformal factor + ¯ ² i ( ∂ i ν ¯ B σ ies. However there exist BPS 0 2 √ W² 2 / ¯ ² σ ¯  m D metries. These configurations / given by i 3 ymmetry is preserved. ( 3 2 = 2 − erest in string theory. D φ i µ ¯  m uating some of the above fermion ν ¯ k r i √ ¯ σ im ¯ − χ D 1 2 K ¯ k ¯ ¯  ² + ¯  i 2 + = ´ ν + ¯ ² i K m ² ¯ = 0 and K/ ¯  ψ 1 φ 2 ¯ i e ¶ ¯ ², δψ χ θ 1 2 σ 2 i µν 0 B µ √ / ¢ K φ D ¯ 3 σ √ A B θ s θ 2 ² . − − i / m ) 2 σ ² T A − 3 ¯ µ  ) λ i i ¯ + µ + ψ iD + m only. Such solutions will be BPS cosmic strings. φ D i K ν – 15 – 2 σ µ θ µ θ )¯ 2 − φ σ + ¯ ², B ω − r µ 2 iw √ ρλ 3 ¯ ²D ∂ / i ¯ µν σ 2 + ¯ r σ 3 µ σ i and σ σ µ σ φ + − 12 ρλ m r ∂ ν 2 ³ µν 3 F λ i µ iF ( F = D √ µ σ ), the variations of the other fields for a cosmic string 2 i i ¡ 2 r ² D + ( 1 √ − i 1 µ √ 0 = ( = = 2 σ ² ¯ 0 D σ i = = = 2 i ( µ − B i δλ 2 1 θ θ/ − δχ ¯ λ 3 δψ δλ i δχ + δψ iσ φ e s λ T ψ 2 ρ √ D ) = i ν . From this we obtain the BPS equations (2.12)–(2.14). Note t ¯ σ 3 + r, θ i ( σ components of the gravitino variation lead to the equations ² χ µνρλ z µ is defined in eq. (2.15). ² D B µ µ A ¯ and σ i t Taking For the above theory, the fermion supersymmetry transforma In general, cosmic string solutions break all supersymmetr Let us now look for solutions of the Killing spinor equations must also be a constant. − B 2 gravitino mass is note compatible with the existence of BPS st background are The gravitino variation is and the spin 1/2 equations are whose only solution is obtained when both eigenstate of cosmic strings which preserveare half solutions of of the the Killing spinor originalvariations equations supersym to obtained by zero. eq BPS configurations are of particular int ground, where the fields depend on where The first order differential equations, and that 1/25.2 of the Gravitino supers field equations Using the previous Lagrangian, the gravitino equations are For a BPS cosmic string, these must all vanish for some choice the required field equations for the string solution. We take JHEP06(2006)030 (5.20) (5.14) (5.15) ¯ λ λ )Σ = 0(5.18) )Ψ = 0 (5.17) 12 . z z ¶ , simplifies the z F ¯ ∂ = 0 (5.19) t ψ ∂ B θ 3 z z z λ σ σ A = 0 σ ) σ i 2 z i ¯ k + ¯ and − ∂ + ¯ χ )Σ = 0 (5.16) ¯ t χ z t t + ) z z ¯ k ∂ ¯ 0 z ∂ σ ¯  ψ t Π + 2 ∂ k 2. These equations are t t ∂ z z C σ / ¯ χ σ z m + ¯ σ (¯ ¯ σ j 3 w (¯ d in the proof of the index B θ i ¯ ee equations which can be σ t i i z 2 φ ∂ − iσ θ ons. For the gravitino, more t ld has 3 components. For a K iA + ¯ rms of the three independent 2 (5.13) iσ t σ D t Π= / ∂ − (¯ 0 ¯ rms in the covariant derivatives − − Ψ+ t ∂ i )Π + i jk t θ 2 derivatives are σ B ¯ z λ 2 / ∂ σ (¯ i 1 − ∂ 3 / θ i (¯ 0 φ z µ σ i ]Σ + 2 +Γ , ¯ s m Σ D ¯ σ θ C − − i θ − = T 3 ˜ 12 ¯ + D χ ¯ 2 ψ − Π θ = F λ j θ iσ Ψ 2 t 3 ¶ θ i √ σ t / χ σ ∂ i 3 ı ¯ σ φ t B θ + ] ¯ φ − σ θ + m A ] θ − , D (¯ )+ θ r i i ∂ i 2 ¯ D  µ ¯ − = 0. The ψ χ θ ¯ D ¯ χ ) ψ r /C i θ − = r σ 0 , w – 16 – θ Ψ σ φ σ θ ¯ z ¶ w Σ+ 2 θ C s − D 2 / B θ A w ˜ θ − 0 / D T r i z + 3 ¯  σ A r ∂ i B Ψ= r r i ∂ 2 2 iσ iQ m ∂ r + K σ ( [ 2 σ r iσ − − r [ 2 0 0 + 2 √ σ 1 D ıj ¯ [ = i √ j , r i C C K z θ σ χ 3 3 )Π + 2 + ı ¯ 2 ( ¯ θ − 2 2 ψ w i ¯ φ iσ iσ λ θ √ D ν ) Σ+ σ i θ ]Ψ θ . We also have − − 0 − D θ φ σ D + ν θ θ ν ˜ B D θ ∂ ∂ σ r )Σ θ D σ ¯ ]+ θ ψ ıj ¯ µ µ σ 0 ν r + σ D K B σ − are θ = = 2 2 r ) 1 σ i ∂ + µ √ √ r z,t χ ¯ r ψ − σ θ − /C + Σ= w ∂ θ ] 0 ( [ 0 i D r D C + is given by eq. (2.15). B σ ( + i + B µ z,t r ∂ r A ∂ ∂ ( [ = r After eliminating the Lagrange multiplier, one obtains thr The fact that our cosmic string background is independent of In a similar manner the spin 1/2 equations are given by r σ σ [ ( z,t i i for the Π equation, and for the Σ equation and a combination which depend only on and cast in Dirac-like form.cosmic string After background gauge it is fixing,Weyl convenient fermions the to write gravitino them fie in te This set of equations cantheorem. be No mapped manipulation to is thework necessary Dirac is for equations needed. the use spin 1/2 fermi where above equations. In particular weD see that the connection te The gravitino equations can then be expressed as coupled in general. closing the system. We have defined JHEP06(2006)030 uss (5.26) (5.25) (5.21) (5.22) (5.23) hysically more ± . . Π gonal mass matri- 0 # )] . ¢ c 2 is the Lagrangian (5.1). . , B | ace (instead of a string , and so we can ignore able bound state. Note We choose a norm com- (c ± θ L Π further gauge fixing. It is ve the possibility of using Σ qual to − efore be applied directly to θ ± odes can still be determined − | on-negative norms. It is still ∂ . (5.16)–(5.18) have a different Ψ Ψ 2 nce we expect the string to be and 0 o derive the number of fermion | n this case the gravitino mass i ν that our analysis leads to some P C field theory ψ B r µν ∝ = 0∝ (5.24) +Ψ η Σ + Π µ Π ± ± ± | ¯ ψ P Σ Π Ψ + B 2 2 | ¸¶ ¸¶ ¸¶ + Π the gravitino Dirac equations reduce to a a a Ψ | Σ r ψ ψ ψ ¡ P iq solutions. As we will show, this does allow our iq iq 1 2 drdθCe [Σ r – 17 – + + + + 0 0 0 C Z 2 | C C C B i 1 2 2 i i i 2 2 2 χ | = ∓ ∓ ± i 2 θ θ θ X || ∂ ∂ ∂ drdθe µ " · · · ψ Z B i i i C C C || ) as the conjugate momentum, where 2 i 2 X ∓ ± ∓ 0 = dependence of the fields will have to be included. We will disc 0 r r ∂ ∂ t 2 C δ∂ C ( || µ µ / drdθCe µ + L ψ and r δ Z || ∂ z µ = X derivatives. For fermion currents on the string (which are p P t must be zero. For large and small 2 / 3 and m Let us now discuss the normalisation of the gravitino fields. We will start by extending the index theorem for block off-dia z The above norm for the gravitino part of the wavefunction is e We identify The zero mode solutions we are looking for depend on The full norm for the fermion states is then the interesting), the this in section 7. patible with the requirements of canonical quantisation in This must be finitethat if the positivity the of zero thestable, mode norm is any solution not normalisable is guaranteed. fermion to Howeverpossible bound si be that states they a should are normalis have gaugeunphysical n artifacts, zero and so modes. there isbackground) If a we we risk could had have removed beennot any working at unphysical modes all in with clear Minkowski howresidual sp to supersymmetries do to this further in work. a string background5.3 and Index we lea theorem We see that the Dirac-likeform to equations that for assumed in the eqs.zero gravitino (3.10), fields (3.11), modes. which were The used t models expressions which derived include in gravitinos. section However,by 3 the analysing number cannot approximate zero large ther m and small ces (3.41), as thisterm will be the relevant case for D-strings. I index theorem to be modified to include gravitinos. JHEP06(2006)030 ). . As (5.31) (5.30) (5.29) (5.28) (5.27) (which tends to 12 r F 3 σ . ) . 2 for each of the θ ) +3 / l l 1 ± + , and . i ψ ψ nalysed both at the − 1 q q φ . ψ µ − ∓ lutions, whose leading ount for this by intro- ) erms. Since the string ( ± fermion equations near q . in section 3, some zero r i D ( from Σ. This is obtained ) e ψ +2 l , l ds then comes from their q O Π µ − C ± − behaviour. The behaviour ≥∓ minant, and solve for Π to hirality dependent effective − ψ the above conditions in the ψ tino fields is determined by U = √ q l q ξA behaviour of the solution ± = ore. In order to use the same ± lds are at least as well behaved f eq. (5.25) to be subdominant, r r , λ − xpression for Π into eq. (5.25), ( i ( (1) Π B µ Π= O O 2 and A / Π= Π= , , χ + 1 , , q θ ψ ) Σ 1 l +2 q , λ, l i ∓ ± + ψ , λ, ψ q i ψ ( q ≤± i q . For the gravitino fields we find the leading ∓ l − ± – 18 – r e = q , χ 2, Σ ∼ C l / ± (2) Σ 3 , χ − V √ Ψ l ± − +1 − 1 , V ψ ψ q 2. We have neglected − q Σ ± Σ= in eq. (3.2)), since they are all subdominant as ± ψ . For these solutions the gravitino fields are subdominant, , q q r is negligible there, so we can ignore it in eqs. (5.24)–(5.26 λ µ 1 ± nξ/ θ ≤± J r ∼ − ∓ l A , Ψ ∼ ± ψ and = , V θ q ) l and i l Ψ ± ψ , V + ∓ χ q = V Σ ψ ab ± ψ q q V ( ∓ M i (2) Ψ near the origin, we can obtain the leading behaviour of Ψ and Σ r q − e r ∼ Ψ C ± Π ∼ ± V √ 0 U B Ψ= . ∞ The gravitino is massless at infinity and so, as was discussed We also need to consider the behaviour of the other fermion so These equations are analogous to a Dirac equation and can be a We start with a solution whose leading order behaviour comes As in section 3, we look for the leading order behaviour of the and so the analysis of section 3 still applies. modes with low angularducing momentum will some be effective lost. winding numbers We can ˜ again acc Notice the shifts of one unit. behaviour comes from above three solutions toalgebra be that normalisable leads inside to thesame the string form expression c as (3.41), eq. wecharges (3.35). want to This is write achieved by introducing the c Hence we need respectively with the other twotheir solutions, couplings to the Yukawa behaviour terms. of We find the non-gravi Finally, we take theobtain right-hand the side third of independent eq.and solution. (5.26) using to Substituting this be e subdo would form part of Another solution is obtained by takingand the then right-hand solving side o forcoupling Ψ. to Ψ The via leading Yukawa terms. behaviour of Hence the other fiel 0 or origin and at infinity. Following the ansatz (3.34) we take by solving eq. (5.24).of other Eq. fields (5.25) comessolution from implies is their Ψ regular at coupling will the toas origin, have Σ we Σ similar find and there. the Ψ other Putting via fermion this fie Yukawa all t together gives the leading order where we have defined the origin. The gauge field JHEP06(2006)030 2 / +1 (5.34) (5.33) (5.35) (5.36) (5.37) (5.40) (5.32) (5.41) (5.42) ± (1) Σ ˜ q hey must 6= 0, then 2 / 3 ≥∓ l m 2, / gravitino solutions r ff-diagonal, we must +1 ± ino implies that we take (1) Ψ ˜ q the above solutions at the s by including some extra . . ψ when gravitinos are included ) (5.38) ≥∓ q , we can now use the index 1 o o o l must be integer or half integer. s ± by eq. (3.18). If nξ ) (5.39) = ilθ ilθ ¸ ilθ nd the analysis of subsection 3.2 l/C if Σ ¸ ψ , η ± ∓ r ± ± q η nξ (1) e Ψ e e (2) ± 2 q ∗ ∓ ∓ q ∼ Ψ Σ ∓ ± ± Π 1 ± ≤ V V 1 Ψ = 2 ± V ] C 2 C 1 1 + + and ˜ 3 , V + − C C · · Σ 1 ilθ ± ilθ Ψ , ilθ − − ∓ ∓ ∓ ∓ (2) Σ ± l/C e e q e ∗ ∗ η η (1 − ψ 1 1 Ψ Σ Π q ± ± ± I (3[1 – 19 – r − − U U U − , V I , we find that l/C ∓ = 0, then some of the large 1 1 2 2 ξ n n n ∼ 1 | − = ∓ 2 1 θ θ θ ± ± 2 n Ψ / nξ ± ψ ψ ψ l/C − 3 C C C 2 2 r r V iq iq iq = = nξ nξ − − | m √ √ √ − − − ± 1 ± e ∼ ∼ ∼ e e = = = (1) Σ (1) Ψ far away from the string. Following the reasoning behind ˜ ˜ Σ q Π Ψ q ≤ ± ± ± ± ± (2) ± 1 V U (1) V Σ (1) Ψ /r ˜ q ˜ ˜ q q Σ = Π = Ψ = 1 at infinity, and it is therefore not a normalisable bound stat 1 /Cdr ) 0 . For the negative chirality states Σ does not decouple, and s /C ψ C 0, we see that the field Σ also decouples for positive chiralit q 2+1 − − − (1 , and the second contains the gaugino ( = 1). Since the gravitino field is massless at infinity we need to n > ψ R nξr ± q ∓ (1) Ψ ∝ ψ q − exp[ q n ∝ = Taking The expressions (3.41) and (3.30) now give the number of ferm Since we have a BPS string, the = ) winding numbers. Instead of the expression (3.24), we use th ± r (2) ( Ψ (1) χ ± 0 q ˜ supersymmetry obtained in ref.² [17]. In order to satisfy the case so we also need to include winding ˜ ( 1/2 supersymmetry. In supergravity, the spinor tions (5.16)–(5.17) vanish. We seeso that we Π can decouples ignore from it. the Solving eq. (5.17) we findseparate Σ the = 0. fields into Theq fermion two mass groups. matrix The is first bloc contains the Hig finite norm. Howeverpositive, if and the so fieldwavefunctions some Π are of is dominated these by non-zerosolutions solutions the we whose leading may other cannot order fields, be behaviour be comes theThis from gauge cert Π suggests norm artifac which a that wil wethe may winding be numbers corresponding able to to Πto avoid from work, the counting and analysis. the so Ho gauge we will z consider both possibilities in6. what Supergravity fo examples We will now apply our indexto theorem the to models cosmic discussed strings in in super section 4, we6.1 must now D-strings include the g For the global SUSYfind equivalent a of fermion the boundidea D-string state can (see using be subsecti the triedby broken eqs. for supersymmetry (5.7)–(5.9). D-strings. tra For The transformations for a st q numbers (5.35)–(5.36). Withapply all to these models modifications, with gravitinos. the in Hence Ψ contrasts with the situationlocalised for on global the supersymmetry, string.index whe theorem. The disappearance of this zero mode JHEP06(2006)030 2 ψ / q 6= 0. (6.5) (6.4) (6.3) (6.2) = 0 and i = 1 c (2) ± ≥ N q m orm of this 2 1, / corresponding 1 − 1 and ˜ − q 0) 2 n ys exponentially, If this is not the = 0, the general m ± / . 1 − supergravity. It is 6= 0. These strings ψ ≤ r − r 0 q e solutions is n> 3 m − c B m = 0 r 3 t eaten by the gravitino. = = − c t the gravitino we would auge for BPS D-strings, (for clude the Π = (1) Π that in the broken ignored. ed. q Ψ behaviour st have , there will be an additional g a superHiggs argument, i.e. ermion wavefunction will now . , N nξ re that they have positive norm, ro modes, i.e. a complex spinor, ) , U sult is remarkable as it exemplifies = 0 2 nξ / 2+ , V − 2 +3 / > 2+ m = 0 mode as it cannot be normalisable r +1 ) is no longer zero. We see that the fields Π 2 > 1 m m c ] r r r C 1 2 , N T = c C − 1) Σ = − – 21 – , which is an integer. Near V − λ ψ q n ∼ (3[1 Ψ I 2+ = 2( / , V 1 + = 0 will be given by the above expression with all 2 / r − N 1) + 2 1 is increased by two. These extra zero modes do satisfy all the , V l − − 2 + − m / 1 n − N − n = r m 1 0) the number of zero modes is − = 2( c m n 0. Including it we lose the r + = 1 c n> N ≥ χ 0) is the only normalisable solution there. In general, the f U = m χ are constants. At infinity one combination of solutions deca ≥ ≥ i U c 1 ) term will be zero unless the string deficit angle is very big. m − . . . 1 zero mode. Near the origin the leading order behaviour of th ( n − 1 (the last condition comes from the gravitino field). Withou I If the strings are very heavy, and 3(1 On the other hand if we do include the additional charges Let us be more explicit and show which zero mode disappears in = ≥ m requirements for normalisability, although we cannotand be they su may justsuggesting be they will gauge. also be The gauge corresponding in solutions this are case, g and should be and Σ no longer decouple, and the analysis is changed. If we ex for the field Π, the index to Π, we obtain almost the same result as for the BPS D-strings The have the same field content as BPS ones, but case the results ofinclude the small previous contributions section from hold, the although Π the and f Σ fields. Hence if the solution is to be normalisable everywhere, we mu need m close to the origin. The rest of the6.2 zero Non-BPS mode D-strings tower is preserv We now suppose that the above strings are no longer BPS and tha and (for where the solution to the fermion field equations has the leading order combination of solutions near sector corresponding to thehas positive disappeared. chirality, a In pairthe of ref. Goldstino ze [17] field this does resultOur not was calculations justify lead suggested this usin to heuristicthe zero reasoning. role modes This of re anymore supergravity as in the it counting is of zerohelpful modes. to introduce we find that (for Comparing this with the results for global SUSY (4.2), we see JHEP06(2006)030 . Σ V (6.8) (6.6) (6.7) 2. In 0, we 6 / alysed c 1 + − 1 positive n > 1 = − − 1 is an integer. Φ /C Q ) nQ ψ Qn q − 2 / ring core, in contrast vanish too. +1 i e string, the number of 0 and m ee here that the fermion ( ef. [22]. The above result is case. Taking r ver the above normalisable 5 site sides of the string loop n of SUGRA effects. ut ten times the size of the the deficit angle is increased, c iggs field to have zero modes currents will be less effective ring, which itself depends on positive chirality zero modes, Q > mmetry breaking. If we have ig, and such heavy strings are = ally outside the string core, the ∂W/∂φ e previous subsections. Applying 2 as its charge is Ψ Qn n/ n . − , where = = . , V − 2 Qn Φ q Φ +1 + 1 φ 2 a ψ /C q ) ψ = – 22 – , N q = − 2 W q for a BPS string. Hence there are , there is a solution to the fermion field equations / = 0 . This results in an additional zero mode, and so the ψ +1 + nξ ψ q q m ( N − − − r 1 = ∼ to 0) we find < Σ (1) − 1 (1) + q q n> C , V 1 and ˜ ξ 2 does not couple to any of the other fermions, and so it can be an − 1 and 3 √ associated with Φ has winding − instead, the equivalent calculation gives Φ ψ − e Φ q χ 4 χ c − = Qn = = m + χ ψ (1) + U q q − Note that these zero modes have power law decay outside the st Consider a fermion with winding The field ∼ λ = V irrespective of whether the string is BPS or not. to the usual exponentialat decay. stabilising This vortons. impliesstring corresponding The core. radius If of the zerooverlap a mode between typical wavefunctions decay the vorton exponenti wavefunctions is of abo fermion states on oppo The expression (3.42) will give the number of zero modes in th conductivity of the string is actually increased by supersy q The presence of thesealso chiral holds zero for modes global has SUSY, and been is obtained not6.4 in affected Massless r by spectators the on inclusio D-strings It is also possible foron fermion fields a which D-string. do notzero couple In mode to contrast a solutions H to iswhether sensitive fermions the to which string the are is deficit BPS massive angle or off of not. th the st chirality zero modes. On thewhich other alters hand for the a value non-BPS of string, ˜ independently. Its presence does not affectthe the index results of theorem th (with have ˜ Hence if wavefunctions have power law decaysolution outside is the only string. possibleruled if Howe out the by string astrophysical deficit observations. angle is6.3 very b D-strings with spectators We add to the D-strings a new field Φ and a superpotential which has the above form and is normalisable everywhere. We s the string background, we have Φ = 0, and the derivatives The fermion and near infinity it is JHEP06(2006)030 1 ± ndex 1, ∓ equations of to the fermion ± = 0, and so there V ) n 4.2. This has no − of the fermions must N z,t ure a ( q . We will show that for = ± of gravitinos means that ops with very large radii θ f h in this case the mixing producing particles which ero modes disappear. We + rality Higgsino zero modes ill be produced by a string rs, bilising the loop will decay, e fact that the norm is not mbers defined in eq. (5.31). , Σ and Π do not decouple, r law decay, this overlap will , and so we suspect it is also occur for our SUGRA model → N though the gravitino coupling = 0 (7.1) massless currents by adding a and dom. This appeared to be the ± i re much weaker than usual [16]. harges (5.31) reduce to r V χ ) z = 2. These extra zero modes arise ∂ z and , 0 and the gaugino is also chargeless. − σ n ± N + ¯ − U ) t , = ∂ n t + z,t σ ( N ± – 23 – f = (¯ , the above conditions hold, and so this extension λ ) ) → z z ∓ ∂ ± t ( z U arising from Π, we obtain σ iω e q + ¯ t ∂ )= t σ (¯ z,t ( ± f the mass matrix is not block off-diagonal, and we must use the i dependence of our zero mode solutions will still satisfy the 2 / θ 3 m and r dependent phase to the zero mode solution. t 1. As before the Higgsinos have charges ± We see from the field equations (5.19) and (5.20) that if we ens As an example, we extend the F-term string model of subsectio Using the index theorem we find that and ansatz (3.14). Taking then the are no fermion zero modes. 7. Fermion currents 7.1 Extending zero modes toThe massless zero currents mode solutions we have found depend only on motion. Let us make the changes theories without gravitinos thesez can easily be extended to from the inclusion ofpositive the definite Π suggests gravitino these field maycase in be with gauge the the degrees analysis. corresponding of modesthe Th free for case the here. non-BPS D-strings If we remove the then be all either half integer or integer. Fayet-Iliopoulos terms, and the three effective gravitino c and Since we are now usingmix a together generic to mass form matrix,discussed the massive a different states, similar chi and effectarose the from for an corresponding the extra z SUSY non-trivial(perhaps Yukawa term. F-term due This model, to could the althoug also effectsalone of is other enough sectors to ofnone the remove of theory), the the al zero global modes. SUSY zero Hence modes the survive. inclusion theorem for generic mass matrices (3.30). The winding numbe will be tiny. On thebe other hand large, if and the we wavefunctionsare have expect powe not the confined fermions to the toand string. scatter the Hence off vorton the each current willwill other, which collapse. still was sta be However, stable, wenetwork. but expect The only corresponding that a vorton some small constraints number are lo therefo of such6.5 vortons Non-BPS w F-term strings For non-BPS strings the threeand we gravitino need degrees to of includeFor freedom the non-zero Ψ corresponding effective winding nu JHEP06(2006)030 . ) ith (7.6) (7.7) (7.4) (7.5) (7.2) r, θ ( ± )Σ appear )Ψ z ∂ z,t z ( ¯ σ ± . At low energy, f z − k t ∂ ± t ino field. Extending )= as would be required σ µ = )Π does not vanish for x needs to have the same z t ( bsection 6.1), we try the ∂ k ± z tions. We see that it has e zero modes. A possible ¯ σ Ψ he speed of light along the lutions to massless currents e spectrum does not appear modes to currents runs into the string by considering the his case the only part of the . e extended to currents with a − )Σ and (¯ rality. t er the low energy(so behaviour we of ignore all fermion states z ∂ s cannot be met simultaneously . ∂ t , ) z σ ) σ = 1. In the following subsection we r, θ + ¯ r, θ ( | 6 ( t 1 ± ∂ ± h t | . λ )Π σ . ) = 0 (7.3) i z,t = 0 z,t φ ( ( = 0 2 µ ± 0 ∗ / ± where ), instead of the usual Π = 0 3 f – 24 – f D z B z µ k m k ( σ 1 h )= )= h µ µ = x x ≈ ( ( t t ± k ± k Π , λ ) r, θ ( ± )Ψ = 0, the ansatz works, and we again have a massless current w z χ ) ∂ z σ z,t ( , this suggests that + ¯ ± z t f k ∂ t σ )= 6= 0, then in order to satisfy the gravitino equation (5.16), Π µ Let us now consider what happens in the presence of the gravit However if we try to include the other gravitino fields Σ and Π ( 0 x ( B ± χ find further indications that this might be7.2 the case. Two dimensional effective action We will now investigate the naturetwo-dimensional of effective the action fermion there. currentsfermionic on We excitations will around just the consid cosmicapart string from background the zero modes and their corresponding currents) for small non-zero Π. This implies that and so for consistency we also need in the gravitino equationsunless (5.18), Σ (5.16). = 0. Both This constraint is only consistent if the scalars satisfy and In conclusion, we find thatis the only usual consistent extension in ofgravitino zero supergravity which mode for does so BPS not backgrounds. vanishto is In Ψ. contain t In any the masslessresolution non-BPS cases, of excitations, th this is even thatnon-standard though the dispersion it normalisable relation zero does modes hav must b phase as Ψ, and so If Now this is not generally consistent with eq. (5.18) as (¯ of our fermion zerodefinite mode chirality (like solution the alsostring. zero satisfies The mode), direction the and of field that the equa current it is moves determined atthe by t above its arguments chi to theansatz zero modes on a BPS D-string (see su Since (¯ for non-BPS strings), we findmajor that difficulties. the above First extension of of all zero we see that both (¯ definite chirality. JHEP06(2006)030 (7.9) (7.8) (7.14) (7.12) (7.13) (7.11) (7.10) . Each f . ¸ ¸ assless fermion. z (7.2) into the ¢ ¢ ), etc. to be one . 2 2 ) | | Π r, θ Π becomes ( r, θ ( ± ± − | ave seen that the four- − | ould-be negative norm f ± 2 2 string world sheet. The | | ed with two types of in- )Π direction in order unravel assless currents, and seem Π cts. This would guarantee sical relevance is not at all modes and their associated f a massless fermion, ffective action by extending i − z,t ve a positive norm, although ents. Here we investigate the ( ∂ rmion s states) can be obtained as a i Σ + Π Σ ± | ions. It could be that the nor- g currents. However this seems | s with the one used in the index ¯ σ f ¯ . f + + ) 2 2 2 | 2 | | | )= Ψ λ µ Ψ | | Ψ f | | i x . ¡ ¡ ( ∂ + z i 2 1 ± 2 1 2 k | t Π z ¯ + χ + fN drdθC | N N 2 2 | | + Z λ ± λ | | 2 – 25 – dtdz 1 2 , || = ) + + µ t Z = 2 2 ψ k | i | 2 r, θ || χ ( χ − || ( | | µ ± · · ψ C C )Σ || ) as arbitrary functions, and take Ψ dtdz B B 2 2 z,t Z z,t ( ( i ± ± − f f drdθe drdθe )= Z Z µ t z x σ σ ( ± = ¯ = ¯ t Σ z and N N , we find that the action does not describe a two-dimensional m z z,t N = 6= i t Starting with the BPS case (so Π = Σ = 0) we substitute the ansat Let us repeat the same analysis in the non-BPS D-strings. We h N It would be verycurrents interesting further. to investigate Indeed, theconsistencies. non-BPS the First zero non-BPS the case zeroresidual modes seems gauge are to symmetries not be might guaranteedstates. to plagu be Even ha enough if to theclear. gauge zero Indeed away modes they the are doto w of not have positive seem two-dimensional norm, to Lorentz their bemalisable breaking phy naturally zero dispersion extendible modes relat to in m the the absence non-BPS of case two-dimensional Lorentzunlikely, are as invariance all breakin a gauge BPS artifa D-stringlimiting (which case has of problem-free a massles the non-BPS physics string. of More zero work modes is in needed the in non-BPS case. this fermion action. We keep More precisely we find that the dispersion relation for the fe and Hence we have obtained an effective two-dimension action for As where of our zero mode solutions. We obtain BPS normalisable zero modenorm leads used for to the a gravitino istheorem massless positive (5.23). current definite on and coincide the dimensional equations of motion dobreakdown not of lead the to masslessness masslessthe condition curr ansatz at (7.2) the to level include of the e where Substituting this into the action gives JHEP06(2006)030 = 1 n space analogue of a couplings to the tion in zero modes in string models and take Postma for useful com- re is one less zero mode the Netherlands Organi- assless states. The pres- l case in the presence of n zero modes bound to a lds and gravity is enough n the case of supergravity ermions in the Lagrangian, ve power law decay outside r of zero modes. Physically symmetry breaking resulted fermions which are massless s only possible for quantised l, even if it includes massless C.D. and C.v.d.B. were sup- ings still have zero modes. It use of this the corresponding strings, or must it decay? tly because the gravitinos are s implies that there is no zero g confines them less effectively ts on conducting strings arise gravity strings implies that the nts on chiral vortons which their nt for supergravity). The presence so the resulting vorton constraints 1 string if it splits into several n> integer), and since the deficit angle at infinity – 26 – = 1. This is consistent with the results of ref. [17]. p n (with 1, it seems that D-term strings are difficult to construct p/n p < = − ξ = 1 1 < C Usually fermion zero modes are confined to the string by Yukaw Finally one would like to include a gravitino mass for D-term There are rather different physical reasons behind the reduc In particular for D-term strings in supergravity, we find the implies that 0 into account the effects ofFayet-Iliopoulos supersymmetry terms breaking. As this i when a gravitino mass is present. Acknowledgments We are grateful toported Renata by Kallosh PPARC. for S.C.D. usefulments, thanks discussions. and Rachel Jeannerot was A. supported andsation by Marieke for the Scientific Swiss Research Science (NWO). Foundation and strings. Does the wavefunction spread out over the different the D-term andthe F-term superHiggs cases. effect.supersymmetry The breaking The terms. former F-term There appears case itin was is to the found similar mixing that be super to ofwith a the zero F-terms, curved modes, globa the which gravitinoresulting aided mixes in the their the left destruction. absence of and zero I right modes. moving f string Higgs fields.to However confine in the some fermions casesoutside (this the the is string only string). gauge relevant The forthe fie wavefunctions models string of (as with such opposed zerocurrents to modes are the less ha usual effective exponential at decay).are stabilising string weaker Beca loops, than and usual. 8. Conclusion We have derived acosmic general string. expression This for index thefermions theorem and number is gravitinos of valid (both for possibilities fermio a beingof general importa massless mode eigenstates inthis the can mass matrix be reduces interpretedence the as of numbe gravitinos the also bound reducesmassless the states away number from mixing of the zero string with and modes,due free secondly firs to because m the their strin differentfrom kinetic the terms. presence of vortons, Sincevorton reduced strong constraints conductivity will constrain of be super relaxed. state than the correspondingmode model on without a gravitinos. cosmic Thi string with winding Consequently these D-strings evade theglobal stringent analogues constrai were subject to.is interesting Higher to winding ask number what str happens to states on a JHEP06(2006)030 B , 07 , , B 209 Phys. JHEP , ysics , JHEP (2004) 013 , ]. (2005) Phys. Lett. 06 , (2004) 043 46 , Cambridge (1995) 477–562 Phys. 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