The spectrum of $(p,q)$ bound states of F- and D-strings has a distinctive
square-root tension formula that is hoped to be a hallmark of fundamental
cosmic strings. We point out that the Bogomol'nyi-Prasad-Sommerfield (BPS)
bound for vortices in ${\cal N}=2$ supersymmetric Abelian-Higgs models also
takes the square-root form. In contrast to string theory, the most general
supersymmetric field theoretic model allows for $(p,q,r)$ strings, with three
classes of strings rather than two. Unfortunately, we find that there do not
exist BPS solutions except in the trivial case. The issue of whether there
exist non-BPS solutions which may closely resemble the square-root form is left
as an open question.Comment: 4 pages; v2: references adde

Jackson, Mark G.

English

arXiv

The spectrum of (p, q) bound states of F- and D-strings has a distinctive square-root tension formula that is hoped to be a hallmark of fundamental cosmic strings. We point out that the BPS bound for vortices in N = 2 supersymmetric Abelian-Higgs models also takes the square-root form. In contrast to string theory, the most general supersymmetric field theoretic model allows for (p, q, r) strings, with three classes of strings rather than two. Unfortunately, we find that there do not exist BPS solutions except in the trivial case. The issue of whether there exist non-BPS solutions which may closely resemble the square-root form is left as an open question

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arXiv:hep-th/0610059v1   5 Oct 2006FERMILAB-PUB-06-365-AA Note on Cosmic (p, q, r) StringsMark G. JacksonParticle Astrophysics Center, Fermi National Accelerator Laboratory, Batavia IL 60510(Dated: January 11, 2007)The spectrum of (p, q) bound states of F- and D-strings has a distinctive square-root tensionformula that is hoped to be a hallmark of fundamental cosmic strings. We point out that the BPSbound for vortices in N = 2 supersymmetric Abelian-Higgs models also takes the square-root form.In contrast to string theory, the most general supersymmetric field theoretic model allows for (p, q, r)strings, with three classes of strings rather than two. Unfortunately, we find that there do not existBPS solutions except in the trivial case. The issue of whether there exist non-BPS solutions whichmay closely resemble the square-root form is left as an open question.PACS numbers: 98.80.Cq, 11.27.+d, 11.30.PbINTRODUCTIONThere has been recent interest in the cosmic rehabili-tation of fundamental strings [1]. This has led to the ex-citing possibility that the properties of a cosmic networkmade of fundamental strings may have observationallydistinct signatures from more mundane solitonic objects.There are two smoking guns. The first is the probabilityof reconnection which scales as P ∼ g2s for fundamentalstrings [2], while it is essentially unity for abelian vortices[3, 4]. Recently it was shown that non-abelian vorticesmay give rise to a probability P < 1 but the velocitydependence differs substantially from the fundamentalstring case [5].The second smoking gun [6] is the existence of bothF- and D-strings in warped IIB compactifications whichform a distinctive spectrum of bound states with tensionµ(p,q) =√p2 µ2F + q2 µ2D. (1)A simple consequence of these bound states is the exis-tence of 3-string junctions, with angles dictated by chargeconservation and the tension formula (1). Apects of net-work formation and gravitational lensing of such junc-tions was studied in [7, 8].It is rather simple to construct field theories which ad-mit bound states of vortices and the corresponding 3-string junctions. Examples include vortices charged un-der discrete symmetries [9] and multiple abelian gaugegroups [10]. However, none of the field theoretic modelsstudied so far reproduce the stringy spectrum (1). Thepurpose of this short note is to show that the generalBogomolnyi bound in gauge theories with multiple U(1)gauge groups includes the string spectrum (1). In fact, weshall see that there is a maximum of three different typesof supersymmetric vortices, with the tensions boundedbyµ =√k21µ21 + k22µ22 + k23µ23 (2)where ki are integer gauge winding charges. If we choosea field theory without the third type of vortex, then thismimics the IIB string theory spectrum of cosmic strings.Although we find that no nontrivial BPS solutions existwhich have this square-root spectrum, it is possible thatnon-BPS solutions exist which could closely resemble it.THE BOGOMOLNYI BOUNDIn theories with N = 2 supersymmetry, the real D-term and complex F-term are unified into a triplet, trans-forming under an SU(2)R R-symmetry. This existence ofthis triplet is responsible for the three different tensionsappearing in (2). Recall that matter lives in a hypermul-tiplet, consisting of two complex scalars φ and φ˜ trans-forming in conjugate representations of the gauge group.For a single scalar charged under a single U(1) gaugegroup, the D- and F-terms in the scalar potential [12]are fixed by N = 2 supersymmetry to be,V =e22(|φ|2 − |φ˜|2 − r3)2 +e22|2φ˜φ− r1 − ir2|2. (3)Here e2 is the gauge coupling constant. There are threevacuum expectation values r1, r2 and r3 allowed by su-persymmetry (often referred to as a Fayet-Iliopoulos pa-rameters). The SU(2)R symmetry of this potential canbe made manifest by defining the doublet ωT = (φ, φ˜†)and writingV =e22(ω†~σ ω − ~r)2 (4)where ~r = (r1, r2, r3) and ~σ are the triplet of Pauli ma-trices.Consider now a U(1)N gauge theory with gauge cou-pling e2a, a = 1, . . . , N . We couple N hypermulti-plets ωi with integer charges Qia under the ath gaugegroup. The covariant derivatives are given by Dωi =∂ωi− i(∑Na=1QiaAa)ωi. The energy functional for static2(∂0 = A0 = 0) configurations isE =N∑i=1|Dωi|2 +N∑a=112e2aB2a +e2a2(N∑i=1Qiaω†i ~σ ωi−~ra)2.(5)with Ba the magnetic field for the ath gauge group. Wechoose detQ 6= 0 to ensure that in the ground state,defined by∑iQiaω†i ~σωi = ~ra, the U(1)N gauge group isfully broken and the theory exhibits a mass gap.Lowest energy vortex states may be found by the usualBogomolnyi method. We search for straight strings, ex-tended in the x3 direction, by setting ∂3 = A3 = 0 andwritingE =N∑i=1|D1ωi − i(~m · ~σ)D2ωi|2 +N∑a=112e2a(~mBa − e2a(N∑i=1Qiaω†i ~σ ωi − ~ra))2− Ba ~m · ~ra.The above decomposition holds for any unit vector ~m.The last term yields a topological charge when integratedover the plane transverse to the vortex string:∫d2xBa =−2πka. Noting that the first two terms are squares, wederive the bound on the tensionµ =∫d2x E ≥ 2π∑aka ~m · ~ra. (6)This is maximized by choosing ~m parallel to∑a ka~ra.The resulting bound is the tension formula (2), with µα =2π∑a ka(~ra)α for α = 1, 2, 3. In IIB string theory thetension-squared for a string with integer charge vectorka = (p, q) is expressed asm2 =∑a,b=1,2(M−1)abkakbwhere Mab is the metric on the IIB auxiliary torus ofmodular parameter τ . We obtain the same spectrum bydefining (M−1)ab = 2π~ra · ~rb, where now a, b = 1, 2, 3.The bound is saturated by solutions to the equations~mBa = e2a(N∑i=1Qiaω†i ~σ ωi − ~ra)and D1ωi = i(~m · ~σ)D2ωi (7)where, as explained above, ~m is the unit vector paral-lel to∑a ka~ra. When all ~ra lie parallel, for example~ra = (0, 0, ra), these reduce to the usual coupled vortexequations studied in [11]. They have solutions only whenthe winding ni of all scalar fields with non-zero expec-tation value, defined by ni =∑aQiaka is non-negative.(This is simply the statement that there is no holomor-phic vector bundle of negative degree). In this case thereis no attractive force between vortices. In contrast, whenthe ~ra do not lie parallel and the vortices in differentgauge groups are coupled through the scalars ωi, one mayexpect bound states to form.In both the field theoretic and string theoretic con-texts, the Bogomolnyi bound is expected to receive cor-rections at the scale at which the protectorate supersym-metry is broken. In warped IIB compactifications, su-persymmetry is broken from 16 supercharges (in the ori-entifold background) to 4 at the compactification scale,with subsequent low-energy breaking at the TeV scale.NON-EXISTENCE OF BPS SOLUTIONSNow decompose each field ωi into eigenvectors of ~m ·~σ,writingωi = ψi |~m+〉+ ψ˜†i |~m−〉. (8)Then the covariant derivatives becomeDz ψi ≡ (∂1 − i∂2)ψi − i∑aQia(Aa1 − iAa2)ψi = 0,Dz¯ ψ˜†i ≡ (∂1 + i∂2)ψ˜†i − i∑aQia(Aa1 + iAa2)ψ˜†i = 0.From this, we see that both ψi and ψ˜†i transform withcharge Qia under the U(1)a gauge group. Taking thecomplex conjugate of the second equation, this ensuresthat both ψi and ψ˜i are covariantly holomorphically con-stant, i.e.Dzψi = Dzψ˜i = 0 (9)where now ψi has charge Qia while ψ˜i has charge −Qia.In other words, as the notation suggests, these are therotated form of φi and φ˜i. We can now look at the firstBogomolnyi equation. Dotting with the unit vector ~mtells usBa = e2a(N∑i=1Qia|ψi|2 −Qia|ψ˜i|2 − ~m · ~ra). (10)Equations (9) and (10) are now in the form of the usualcoupled vortex equations described, for example, in Mor-rison and Plesser [11]. The criterion for the existence ofsolutions is that for each scalar field ψi we can definethe winding ni =∑aQiaka, while for each ψ˜i we haven˜i = −∑aQiaka. Clearly ni = −n˜i. Solutions to (9)and (10) exist if ni is non-negative for each ψi that gainsan expectation value. (If ψi has no expectation value forsome i then it may remain zero throughout the solution).Similarly, each n˜i must be non-negative for each ψ˜i whichis non-zero. Clearly, since ni = −n˜i, either ψi or ψ˜i isallowed an expectation value, but not both.There are two further real equations that come fromdotting the first equation in (7) with ~lα where ~lα · ~m = 0,3for α = 1, 2. We write ~l = ~l1+i~l2. There is an ambiguousphase to the vector ~l, associated to rotating the basis~l1 and ~l2, but we can always pick a basis so that theremaining two real equations combine into the complexequationN∑i=1Qiaψ˜iψi =~l · ~ra. (11)Thus we see that in order for either ψi or ψ˜i to bezero, the vector∑a(Q−1)ia~ra must be perpendicular to~l, making it proportional to ~m. Since ~m ∝∑a ka~ra, thisrequires ka to be proportional to (Q−1)ia (and of courseka must be integer-valued). That is, in order for only nito be nonzero we select ka to be the ith entry of Q−1.There is also the trivial solution when all ~ra lie parallel, sothat each ~ra is proportional to ~m and thus perpendicularto ~l, but this does not produce a square-root spectrum.While these are the necessary conditions for BPS so-lutions, we find they are not sufficient. Consider theradially-symmetric field ansatzωi =(ψiψ˜†i)= (1− qi(ρ)~m · ~σ)(sis˜i)einiθ (12)where we use polar coordinates (ρ, θ) in the (x1, x2)-plane. The asymptotic boundary condition isqi → 0 as ρ→∞making the vacuum selection at infinity∑iQia (s∗i , s˜∗i )~σ(sis˜i)= ~ra. (13)The ansatz for the gauge potential isAaµ =ǫµνxνρ2Aa(ρ), Aa = −ka + ρfa(ρ) (14)where we similarly require fa → 0 as ρ → ∞. The fieldstrength is given by the simple expressionBa = ∂xAay − ∂yAax =1ρ∂Aa∂ρ. (15)To determine the long-distance behavior of the fields, weinsert the ansatz into the equations (7) and expand tolinear order in qi and fa to obtain(faρ+ f ′a)= −2e2a∑iQia(|si|2 + |s˜i|2)qi, (16)(~m · ~σ∂ρ −∑aQiafa)(1− qi(ρ)~m · ~σ)(sis˜i)= 0.The second equation can be solved (to linear order!) togiveq′i = −∑aQiafa. (17)Differentiation of these first-order equations then pro-duces the modified Bessel equationsf ′′a +1ρf ′a −1ρ2fa −∑bL2abfb = 0,q′′i +1ρq′i −∑jM2ijqj = 0where the mass-squared matrices are given byL2ab = 2e2a∑i(|si|2 + |s˜i|2)QiaQib,M2ij = 2(|si|2 + |s˜i|2)∑ae2aQiaQja.As should be expected from a BPS solution, the gaugeand matter mass-squared matrices L2ab and M2ij haveidentical eigenvalues, which can be seen by acting withQia as a similarity transformation. Denoting the masseigenvalues as λA (so that the mass-squared eigenvaluesare λ2A), the solution to (16) and (17) is then given byfa =∑ASaACAλAK1(λAρ),qi =∑a,AQiaSaACAK0(λAρ) (18)where SaA is the diagonalization matrix for L2ab and thecoefficients CA cannot be determined in the linear ap-proximation and would have to be fixed from numericalcomparison to the non-linear solution. We would expectthat only gauge fields charged under the given ka and thematter field ωi with nonzero winding ni should attain aprofile, but from (18) we see that the Qia mix the mat-ter and gauge fields into a basis such that non-chargedfields are excited. To prevent this it must be that the L2abandM2ij are diagonal, making the CA proportional to ka.This gives us a total of 6 constraints (3 each from settingthe off-diagonal components of a symmetric matrix tozero). These are precisely enough constraints to set theoff-diagonal components of Qia to zero, which now makesthe interaction trivial. The acceptable BPS windings arethen simply of the form ka = (p, 0, 0), (0, q, 0) or (0, 0, r),which will not display any distinctive square-root behav-ior.The situation is very reminiscint of supersymmetricquantum mechanics, with the doublet ωi = (ψi, ψ˜†i ) tak-ing the place of the bosonic and fermionic componentsof the wavefunction. The ground state (BPS) solution(when it exists) is given by first-order solutions whichallow one component or the other, but not both. It isstill likely that non-BPS solutions exist which will con-tain both components, but it will have energy greaterthan the BPS bound.4CONCLUSIONWe have shown the BPS spectrum for supersymmetricvortices exhibits the same square-root cosmic string spec-trum as superstring theory, including not just two butthree types of vortices. Unfortunately no BPS solutionsexist which actually exhibit this square-root spectrum.It is still likely that non-BPS solutions exist which wouldhave an energy higher than the BPS bound, but whichmight approximate the square-root BPS spectrum for acertain choice of parameters. It would be interesting tomake a full analysis of these solutions.AcknowledgementsI would like to thank D. Tong for suggesting thisidea and much assistance during its completion, and M.Strassler for useful discussions. I am supported by NASAgrant NAG 5-10842.[1] J. Polchinski, “Introduction to cosmic F- and D-strings,”[arXiv:hep-th/0412244].[2] M. G. Jackson, N. T. Jones and J. Polchinski, “Collisionsof cosmic F- and D-strings,” JHEP 0510, 013 (2005)[arXiv:hep-th/0405229].[3] E. J. Copeland and N. Turok, “Cosmic String Interac-tions,” Fermilab PUB 86-127[4] R. A. Matzner, “Interaction of U(1) Cosmic Strings: Nu-merical Intercommutation”, Computers in Physics, 2(5),51 (1988)[5] K. Hashimoto and D. Tong, “Reconnection of non-abelian cosmic strings,” JCAP 0509, 004 (2005)[arXiv:hep-th/0506022].[6] E. J. Copeland, R. C. Myers and J. Polchinski,“Cosmic F- and D-strings,” JHEP 0406, 013 (2004)[arXiv:hep-th/0312067].[7] E. J. Copeland and P. M. Saffin, “On the evolution ofcosmic-superstring networks,” JHEP 0511, 023 (2005)[arXiv:hep-th/0505110].[8] B. Shlaer and M. Wyman, “Cosmic superstring grav-itational lensing phenomena: Predictions for networksof (p, q) strings,” Phys. Rev. D 72, 123504 (2005)[arXiv:hep-th/0509177].[9] D. Spergel and U. L. Pen, “Cosmology in a string-dominated universe,” Astrophys. J. 491, L67 (1997)[arXiv:astro-ph/9611198].[10] P. M. Saffin, “A practical model for cosmic(p, q) superstrings,” JHEP 0509, 011 (2005)[arXiv:hep-th/0506138].[11] D. R. Morrison and M. Ronen Plesser, “Summing theinstantons: Quantum cohomology and mirror symme-try in toric varieties,” Nucl. Phys. 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A Note on Cosmic (p,q,r) Strings

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