4D N = 1 Supersymmetric Yang-Mills Theories on Kahler-Ricci Soliton

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1 Jounal of Mathematcs and Statstcs (4): , 0 ISSN Scence Publcatons do:0.344/mssp Publshed Onlne (4) 0 ( 4D N = Supesymmetc Yang-Mlls Theoes on Kahle-Rcc Solton Bobby Eka Gunaa Indonesa Cente fo Theoetcal and Mathematcal Physcs, Theoetcal Hgh Enegy Physcs and Instumentaton Reseach Goup, Faculty of Mathematcs and Natual Scences, Insttut Teknolog Bandung, Jl. Ganesha 0 Bandung 403, Indonesa Receved 0-0-3, Revsed ; Accepted ABSTRACT In ths study we consde fou dmensonal N = supesymmetc gauge Yang-Mlls theoy whose complex scala manfold s Kahle and defoms wth espect to a eal paamete. The defomaton of the geomety s govened by Kahle-Rcc flow equaton. Ths setup mples that some couplngs such as shftng quanttes, momentum maps and the scala potental tun out to be evolved wth espect to the flow paamete. We also dscuss defomaton of vacuum stuctues of the theoy n the context of Mose theoy. Keywods: Supesymmety, Yang-Mlls Theoy, Kaehle-Rcc Flow Scence Publcatons. INTRODUCTION The standad model of patcle physcs based on non- Abelan gauge theoy wth gauge goup SU(3) SU() U() has ganed seveal emakable success whch can be seen fom vefed expements n the enegy scale below 000 GeV ncludng the ecent dscovey of Hggs patcle. Despte ts success, t leaves many mpotant poblems. Fo example, the fst poblem s that the standad model neglects the gavty whch s descbed by the geneal elatvty. Secondly, t cannot explan the mass heachy. Also, t sevee fom quadatc dvegences. Thus, the standad model has to be extended. One of the good canddate fo the extensons of the standad model s supesymmety. In ode to get a easonable supesymmetc extenson of the standad model, ths extenton theoy must nhet the chal stuctue of the standad model. Thus, the only possble extenton s N = supesymmety because extended supesymmetes (N ) cannot accommodate the chal stuctue, fo a evew see, fo example, (Lous et al., 99). Although N = supesymmety has some phenomenologcal aspects, ou nteest s to study closely 44 to the mathematcal context. Fo example, we have pevous seal papes studyng soltonc solutons of fou dmensonal N = local supesymmety (supegavty) on Kahle manfolds satsyng Kahle-Rcc flow equaton (Cao, 95). Ou esults show that n the of both doman walls (Gunaa and Zen, 009a; Gunaa et al., 0) and black holes (Gunaa, 0) n geneal defom wth espect to a flow paamete elated to Kahle-Rcc equaton. Moeove, ths flow could change the natue of stablty of doman walls and geomety of black holes. We extend the studes n ths study to gd non- Abelan supesymmetc theoes, namely supesymmetc Yang-Mlls theoes n fou dmensons defned on Kahle-Rcc solton. Ths follows that some couplngs such as shftng quanttes, momentum maps and the scala potental defomed wth espect to the flow paamete, see Lemma. Moeove, vacuum stuctues whch can be vewed as solutons of feld equatons of motons ndeed evolve wth espect to the paamete. To see the latte, we smply consde a case whee at the level of equatons of motons all femons vanshes and the scalas ae fozen eveywhee such that the gauge felds ae tval. In ths case, the gound states can be thought of as supesymmetc ctcal ponts of the

2 Bobby Eka Gunaa /Jounal of Mathematcs and Statstcs (4): , 0 scala potental. Takng the assumpton that the gound states to be nondegeneate, we fnd that Mose ndex of a gound state s affected by Kahle-Rcc flow. In othe wods, the flow possbly changes the popetes of supesymmetc vacua, see Theoem. Ths fact gves us an example of defomed Mose theoy. The applcaton of ths study has two mao dectons. The fst case s the dynamcs of monopoles o soltonc solutons of N = supesymmetc gauge theoes wth espect to Kahle-Rcc flow. We would lke to see how the flow changes the stablty of solutons. Fo example, ths aspect has been obseved n the case of doman walls n chal N = supegavty (Gunaa and Zen, 009b). The second case s the evolutons of eal and complex vacuum submanfolds of Kahle manfolds wth espect to Kahle-Rcc flow. Ths aspect s elated to the study of evolutons of mnmal submanfold unde Rcc flow, see fo example (Tsats 00).. BRIEF REVIEW: KAHLER-RICCI SOLITON The devoted to assemble some facts about Kahle- Rcc flow equaton whch s useful fo ou analyss n ths study. Ths flow equaton was fstly ntoduced n (Cao, 95). A complex Kahle manfold M endowed wth metc g(τ) s sad to be Kaehle-Rcc solton f t satsfes Equaton : g = R whee,, =,., dm M, τ s a eal paamete and R denote the -ank Rcc tenso of M. The smplest soluton of () s when the ntal geomety at τ = 0 s Ensten, namely Equaton : Scence Publcatons () R 0 = g 0 () whee, s a eal constant and nonzeo. Then, we have Equaton 3: g τ = τ g 0 (3) whose Kahle potental has the fom Equaton 4: 44 K( τ ) = ( τ ) K( 0) (4) wth K K( z, z; ) τ. As we have seen above, fo the smplest example, thee exsts sngulaty at τ = / whee the flow shnks to zeo. Ths ndcates that the sngulaty could be occus n geneal cases, see fo example (Toppng, 006; Cao and Zhu, 006). Anothe nteestng soluton of () s when the ntal geomety satsfes Equaton 5: R 0 = g 0 + Y 0 + Y 0 (5) whee, Y (0) s a vecto feld geneatng a dffeomophsm whch can be expessed n tems of a eal functon P( z, z ) on M as Equaton 6: Y g P z, z = (6) Such a soluton s called gadent Kahle-Rcc solton (Cao, 996; 997). In geneal, (5) can be splt nto thee cases as follows. Fo > 0 the solton s shnkng, wheeas fo < 0 the solton s expandng. In the case of = 0 we have a steady gadent Kahle-Rcc solton. 3. 4D N = SUPERSYMMETRIC YANG- MILLS THEORY ON KAHLER-RICCI SOLITON We focus on the popetes of the defomed N = supesymmetc gauge theoy n fou dmensons on Kahle-Rcc solton. The spectum of the theoy conssts of vecto felds A and spn-/ gaugnos λ wth = 0,,3, =,,n v, coupled to complex scala felds z and spn-/ femons x wth =,,n c. The complex scalas ( ) z, z span a Kahle geomety M of dmenson n c. The constucton of the N = supesymmetc gauge theoy on Kahle-Rcc solton follows closely (Gunaa, 0; Gunaa and Zen, 009a; 009c). Fst, we consde the chal Lagangan n (D Aua and Feaa, 00), say L ( 0) at τ = 0, whee the metc of the scala manfold s statc. Then, eplacng all geometc quanttes such as g τ, the bosonc pats the metc g 0 by the solton of the on-shell N = chal Lagangan can be wtten down as Equaton 7:

3 Bobby Eka Gunaa /Jounal of Mathematcs and Statstcs (4): , 0 L = g D z D z Σν Σν + R F F + I F F% V τ Σ ν Σ ν (7) Now, we can wte down the dynamc equatons of the shftng quanttes (3) and the scala potental (). Lemma Equaton 4: whee the functon V(τ) s the scala potental of the theoy whch has the fom Equaton : Σ V = g W W + R P PΣ ( τ ) () Hee, g = K( τ ) s Kahle metc, wheeas R Σ and Scence Publcatons I Σ ae espectvely the eal and magne pats of holomophc gauge knetc functons F Σ. The covaant devatve D z s gven by D z = z k A + whee k s a holomophc Kllng vecto geneatng sometes of M satsfyng Equaton 9: Γ k,k f k Σ = (9) Σ Γ The feld stength F ν s defned as Equaton 0: F = A A + f A Σ A Γ (0) ν ν ν ΣΓ ν whle ts dual feld s F% ν = ε νρσ F ρσ. The holomophc supepotental W W(z) s ab- P τ P z, z; τ has tay. The eal momentum maps the fom Equaton : P τ = k K τ () The Lagangan (7) s nvaant unde supesymmetc tansfomaton (up to fou-femon tems) Equaton : δχ γ ε + τ ε = D z N, ν δλ = F γ ε + N τ ε, () ν δ χ ε δ λ γ ε + z =, A = h.c whee the shftng quanttes N (τ)and N (τ) ae gven by Equaton 3: ( τ ) = N g K ( τ ) = Σ N R P Σ (3) 443 N N K g, P R, ( τ ) = Σ Σ ( τ ) = V 4 Poof P Σ Σ = R W W + R P (4) One can use (), () and (3) n a staghtfowad way. 4. DEFORMATION OF VACUUM STRUCTURES We dscuss vacuum stuctues of the theoy whch can be vewed as the soluton of feld equatons of motons deved fom the Lagangan (7). The equatons of motons can be obtaned by vayng (7) wth espect to z and A. The femons vansh at the level of equatons of motons. Then, we have Equaton 5: ( F F F F% ) Σν Σν D D z = g τ RΣ ν + IΣ ν g V τ τ ( F + F% ) = D R I g k D z g k D z Σν Σν ν ν Σ Σ (5) Now, let us consde the soluton of (4) as follows. When the scalas z become fxed, namely z = z 0 whch follows A = 0. Moeove, (4) becomes smply Equaton 6: V ( τ ) = 0 (6) then we have Equaton 7: W = 0, P τ = 0 (7) Fom supesymmetc vaaton (), the condton (7) descbes N = supesymmetc vacua. In geneal, the vacuum geomety N 0 M s eal and defoms wth espect to τ. In ths study we patculaly consde N 0 to be dscete. In ode to chaacteze the gound states, we have to consde the second-ode devatve of the scala potental () wth espect to ( z, z ) called Hessan matx

4 Bobby Eka Gunaa /Jounal of Mathematcs and Statstcs (4): , 0 evaluated at p z, z Scence Publcatons τ = τ τ, whose nonzeo component has the fom Equaton : 0 k l = g V p ; τ τ Σ kw W + R k l τ k τ Σ () whee we have used (). Note that all quanttes n () have been evaluated at p 0. In the est of the study we smply consde the case of Kahle-Ensten metc satsfyng (3). So, equaton (7) becomes Equaton 9: k l V p ; τ = σ τ g 0 W W 0 k l + σ τ Whee Equaton 0: Σ R k ( 0) k ( 0) Σ (9) σ τ (0) and g ( 0 ) s a Kahle-Ensten metc. In (9) t s easy k l to see that when the flow becomes ll-defned at τ = /, the theoy also tuns to be sngula. Then (9) leads to the followng statements. Theoem Let the scala potental () be Mose functon and τ /, so that the detemnant of (9) s nonzeo. Suppose that p 0 (τ) = q 0 s an solated gound state (nondegeneate) of Mose ndex λ fo τ < / wth > 0. Then, thee exsts eal local coodnates X (τ) aound q 0 wth =,,n c such that Equaton : ( τ ) = ( 0 τ) λ( τ ) + λ X V V p ; X... X X nc () Takng the assumpton the eal and the magnay pats of the followng quanttes Equaton : ( 0 ) ( 0 ) V p τ ; τ σ τ V p τ ; τ () ae postve fo all τ and,. Let p τ = qˆ be anothe 0 0 solated gound state fo τ > /. Then, Kahle-Rcc solton changes the ndex λ to n c -λ such that nea ˆq 0 we have X X τ τ and Equaton 3: 444 τ = τ + τ + + λ τ λ+ τ V V p ; X... X X 0... X fo τ > /. Poof nc (3) Fst of all, we defne the Hessan matx of the scala potental () of the theoy as Equaton 4: H V V σ τ ( p ) V 0 V V (4) whee, p 0 s an solated ctcal pont nea whch the scala potental () can be expanded as Equaton 5: V (5) n V( τ ) = V( p 0; τ ) + ( p 0; τ) δx δx p,q= x x n c whee we have defned eal coodnates z x + x + such that δx x x 0. Snce we only consde nondegeneate case, the matx (4) does not have zeo egenvalues and s non sngula because τ /. Let us now ewte (5) as Equaton 6: ε σ n V( τ ) = V( p 0; τ ) + Vpq ( p 0; τ) δx δx σ τ p,q= Whee Equaton 7: fo τ < / ε( σ) fo τ > / (6) and then, one can defne new coodnates Equaton : V ( p ; τ) / n p pq 0 Y V ( p 0; ) x x p= + V p 0 ; (7) τ τ δ + δ () fo n c. Thus, we can ewte (5) n the smplest blnea fom () and () wth dentfcaton Equaton 9: Y X τ fo τ < / τ X τ fo τ > / (9)

5 Bobby Eka Gunaa /Jounal of Mathematcs and Statstcs (4): , 0 Some comments ae n ode. The extenson of Theoem fo degeneate vacua s n a staghtfowad way. It s woth mentonng that Theoem s the evdence of defomed Mose theoy elated to defomaton of (vacuum) submanfolds of Kahle geomety. Snce the flow () could change the ndex of a gound state, so n geneal t could ndeed affect the geometcal natue of the submanfolds. The latte aspects wll be consdeed elsewhee. 5. CONCLUSION So fa, we have constucted fou dmensonal N = supesymmetc Yang-Mlls theoy on Kahle-Rcc solton. As we have seen, ths setup mples that some couplngs, namely the shftng quanttes, the momentum maps and the scala potental evolved wth espect to the flow paamete, that s equaton (3) n Lemma. Moeove, we also have showed that the nondegeneate vacua of the theoy s evolved wth espect to the flow paamete. It s also possble that the Mose ndex changes caused by Kahle-Rcc flow, see Theoema. 6. ACKNOWLEDGEMENT The study of ths study s suppoted by Hbah Kompetens DIKTI 0 No. 7a/I.C0/PL/0. 7. REFERENCES Cao, H.D. and X.P. Zhu, 006. Hamlton-Peelman s poof of the poncae conectue and the geometzaton conectue. Asan J. Math., 0: Cao, H.D., 95. Defomaton of Kahle matcs to Kahle-Ensten metcs on compact Kahle manfolds. Inventones Math., : DOI: 0.007/BF03905 Cao, H.D., 996. Exstence of Gadent Kaehle-Rcc Soltons. In: Ellptc and Paabolc Methods n Geomety, Chow, B. (Ed.), A K Petes, Wellesley, MA, ISBN-0: , pp: -6. Cao, H.D., 997. Lmts of solutons to the Kahle-Rcc flow. J. Dffeental Geomety, 45: D Aua, R. and S. Feaa, 00. On femon masses, gadent flows and potental n supesymmetc theoes. J. Hgh Enegy Phys., 5: DOI: 0.0/6-670/00/05/034 Gunaa, B.E. and F.P. Zen, 009a. Kahle-Rcc flow, Mose theoy and vacuum stuctue defomaton of N = supesymmety n fou dmensons. Adv. Theoetcal Math. Phys., 3: Gunaa, B.E. and F.P. Zen, 009b. Defomaton of cuved BPS doman walls and supesymmetc flows on d Kahle-Rcc solton. Commun. Math. Phys., 7: DOI: 0.007/s Gunaa, B.E. and F.P. Zen, 009c. Flat Bogomolny- Pasad-Sommefeld doman walls on twodmensonal Kahle-Rcc solton. J. Math. Phys., 50: DOI: 0.063/ Gunaa, B.E., 0. Sphecal symmetc dyonc black holes and vacuum geometes n 4 D N= supegavty on Kahle-cc solton. Repots Math. Phys., 69: DOI: 0.06/S () Gunaa, B.E., F.P. Zen and Aanto, 0. N = supegavty BPS doman walls on Kahle Rcc solton. Repots Math. Phys., 67: DOI: 0.06/S ()600-9 Lous, J., I. Bunne and S. Hube, 99. The supesymmetc standad model. Hgh Enegy Phys. Toppng, P., 006. Lectues on Rcc flow. Cambdge Unvesty Pess. Tsats, E., 00. Mean cuvatue flow on Rcc soltons. J. Phys. A: Math. Theo., 43: DOI: 0.0/75-3/43/4/0450 Scence Publcatons 445