VEV of Q-operator in U(1) linear quiver 4d gauge theories

Transcription

1 Amenan Jouna of Physcs, 208, vo., ssue, pp VEV of Q-opeato n U() nea quve 4d gauge theoes Gabe Poghosyan A.I. Akhanyan Natona Scence Laboatoy, 2. Akhanan B. Steet, 0036 eevan, Amena E-ma: gabepoghos@gma.com Receved 0 Mach 208 Abstact: Lnea quve N 2 4d gauge theoy n Ω backgound s consdeed. It s poved that the patton functon n a smpe way s eated to the expectaton vaues of Baxte s Q opeato (at specfc dscete vaues of the specta paamete) n the gauge theoy wth the fst node emoved. Expct expessons fo the VEV of the Q opeato n tems of geneazed Ape s functons ae found. Keywods: N2 supesymmetc gauge theoy, Defomed Sebeg-Wtten equaton, Baxte s Q opeato.. Intoducton Embeddng N 2 gauge theoy n Ω -backgound was nstumenta n a deveopments eated to the nstanton countng wth the hep of equvaant ocazaton techncs.,2 ae the Ω - backgound paametes. Futhemoe, sendng both Ω -backgound paametes,2 to 0, one gets the standad Sebeg-Wtten theoy [,2]. It s nteestng that even the case of U () gauge goup, n contast to the case wthout Ω -backgound, the theoy s non-tva. A chaactestc featue of ths case s that the nstanton sums become tactabe, and fo Nekasov patton functon, one obtans cosed fomuae. In ths pape t s shown that not ony the patton functon, but aso a moe efned quantty, namey the expectaton vaue of the Q -obsevabe can be computed n cosed fom. It was shown n [3] that the anaog of Baxtes Q opeato n puey gauge theoy context natuay emeges n Necasov-Shatashv mt ( 2 0 ) [4] as an ente functon whose zeos ae gven n tems of an aay of "ctca" oung dagams, namey those, that detemne the most mpotant nstanton confguaton contbutng to the patton functon. Ths obsevabe encodes pefecty not ony nfomaton about patton functon (whch s smpy eated to the tota sum of coumn engths of J oung dagams) but aso the ente cha ng [5] constucted fomt Φ, J 0,, 2, (Φ s the scaa of vecto mutpet). In pesent pape, smpe U () case n 4d settng s anayzed. The coespondng expectaton vaues of Q n cosed fom ae found. The souton s expessed n tems of a geneazaton of Appe's functon. The est of matea s oganzed as foows. Chapte 2 s a shot evew of 4d nea quve gauge theoy: the Nekasov patton functon and mpotant obsevabe Q ae ntoduced. An extended quve wth specfc paametes at the exta node s ntoduced and ts eaton to the Q - obsevabe s anayzed. Chapte 3 specazes to the case of U () theoy. Expct expessons fo the Q obsevabe n tems of geneazed Appe and hypegeometc functons ae found. Chapte 4 s the concuson.

2 VEV of Q-opeato n U() nea quve Amenan Jouna of Physcs, 208, vo., ssue 2. Genea settng The nstanton patton functon of the 4d, A + nea quve theoy wth gauge goup U(n) s gven by (see Fg. fo the setup) (,..., ) q Z q (2.) The sum n (2.) s ove a possbe -tupes of aays of n oung dagams. k s the tota numbe of boxes n the k-th aay of n oung dagams and Z s defned as: Z Z ( a, a,, a ),, 0 + n Z (, a, a ) Z (, a, a ) Z (, a, a ) bf 0, u, v, v bf, u, u 2, v 2, v bf, u, u +, v Z (, a, a ) Z (, a, a ) uv, bf, u, u, v, v bf u, u, v, v,, (2.2) Fo a pa of oung dagams λ, µ the bfundamenta contbuton s gven by [6,7]: Z ( λ, a μ, b) ( a b L () s + ( + A () s ) ( a b+ ( + L ()) s A () s ). (2.3) bf s λ μ λ 2 λ μ 2 s μ Aso, Aλ and Lλ known as the am and eg engths espectvey, ae defned as: f s s a box wth coodnates (,j) and λ (λj ) s the ength of -th (j-th) coumn (ow), then: L () s λ, A () s λ j (2.4) λ j λ Fgue. The nea quve U(n) gauge theoy: cces stand fo gauge mutpets; two squaes epesent n ant-fundamenta (on the eft edge) and n fundamenta (the ght edge) matte mutpets whe the ne segments connectng adjacent cces epesent the b-fundamentas. q,, q ae the exponentated gauge coupngs, the n-dmensona vectos a0,, a + encode espectve (exponentated) masses/vev's and 0,, + ae n-tupes of young dagams specfyng fxed (dea) nstanton confguatons. 35

3 Poghosyan Amenan Jouna of Physcs, 208, vo., ssue The Q obsevabe pays an mpotant oe, n Nekasov-Shatashv mt 2 0 ths obsevabe satsfes Baxte s T-Q equaton [3], and s defned as: Q(, x λ) x λ x ( j ) 2 x ( ) ( j ) 2 (2.5) Geneazaton fo the case of genec Ω backgound (n both 4d and 5d cases) s due to [8]. The expectaton vaue of the Q -opeato assocated to the fst node, by defnton s n Q(, u,, u) (,..., ) u Qx Z x a Zq q (2.6) It was notced n [9] that such nseton of the opeato Q s equvaent to addng an exta node wth specfc expectaton vaues. Let s ook at a quve wth + nodes wth expectaton vaues at the addtona node (denoted as 0 ) specfed as (see Fg.2): a a δ (2.7) 0, 0, u,. u u Due to the specfc choce of a0, n ode to gve a nonzeo contbuton, the aay of n dagams assocated wth the speca node 0 has to be seveey estcted. Namey, the dagam 0, shoud consst of a snge coumn and the emanng n dagams 0,2,..., 0, n must be empty. Thee s a cose eaton between the Nekasov patton functon assocated to above descbed specfc ength + quve and the expectaton vaue of a patcua Q opeato n a genec quve wth nodes. Ths eaton s a consequence of the dentty: Fgue 2. The quve dagam wth an exta node, abeed by 0. Z a a a a q q q q 2 (, 0,,,, ) 0 2 0,, 0 + a a + n Q( a a +, ) Z ( a, a,, a ) q q q u 0, u, u 2 2 0,, u 2 0 0,, + a 0, a 0, u (2.8) 36

4 VEV of Q-opeato n U() nea quve Amenan Jouna of Physcs, 208, vo., ssue Whee 0,u fo u s a one coumn dagam wth ength and the est ae empty dagams (see fgue 2). The Pochhamme s symbo s defned as: ( a) ( a)( a+ ) ( a+ ) (2.9) 3. Q obsevabe fo U() quve theoy Fom now on we estct ouseves to the smpest case of the quve of U() s. Nekasov patton functon of such nea quve can be found fo exampe n [0] : Z p p j j ( a a )( aj+ aj 2) 2 (3.) Whee: p q (3.2) j j Expandng (2.8) n powes of q0 and takng nto account (3.), we get the fom of Qx ( ) fo. Whee x a (3.3) 0 2 Snce Q(x) and hence the ente LHS of the eq. (4.5) estcted up to an abtay nstanton ode s a atona functon of x, the fom must be vad aso fo genec vaues of x. We fnd: a0 x a a a a m+ m m... m mk m m Qx C p... pk a x m,..., m 0 m+ m m m!... m! 2 a x a a + + a a + + a x CF (,,..., ; ; p,.., p ) ( ) (3.4) Whee F s the Ape s F functon geneazed fo an abtay numbe of vaabes and C s the nomazaton facto, fxed fom the condton m Qx : 37 x

5 Poghosyan Amenan Jouna of Physcs, 208, vo., ssue ( a) ( b)...( b ) F a b b c x x x x (3.5) ( k ) m m m k mk m mk (,,..., k; ;,.., k) ( )...( k) m,..., m 0 ()...!...! k c m + + m m m k C 2 ( ) a a p 2 (3.6) Fo the smpest case, whee n (3.4) Qx becomes a hypegeometc functon: a a a0 x a a a x Qx () q 2 F 2, ; ; q (3.7) 4. Concuson Thus, statng fom a genea settng n chapte 2 we ustate the A + nea quve theoy wth gauge goup U(n). In chapte 3 we expcty epesent the VEV of Baxte s Q opeato (3.4) n tems of geneazed Ape s functons, whch s the man ogna esut n ths pape. Refeences [] N. Sebeg and E. Wtten, Monopoes, duaty and cha symmety beakng n N2 supesymmetc QCD, Nuc. Phys. B43 (994) , [hep-th/ ]. [2] N. Sebeg and E. Wtten, Eectc - magnetc duaty, monopoe condensaton, and confnement n N2 supesymmetc ang-ms theoy, Nuc. Phys. B426 (994) 9 52, [hep-th/ ]. [Eatum: Nuc. Phys.B430,485(994)]. [3] R. Poghossan, Defomng SW cuve, JHEP 04 (20) 033, [axv: ]. [4] N. A. Nekasov and S. L. Shatashv, Quantzaton of Integabe Systems and Fou Dmensona Gauge Theoes, n Poceedngs, 6th Intenatona Congess on Mathematca Physcs (ICMP09), axv: [5] F. Cachazo, M.R. Dougas, N. Sebeg, and E. Wtten, Cha ngs and anomaes n supesymmetc gauge theoy, JHEP 2 (2002) 07, [hep-th/0270]. [6] R. Fume and R. Poghossan, An Agothm fo the mcoscopc evauaton of the coeffcents of the Sebeg- Wtten pepotenta, Int. J. Mod. Phys. A8 (2003) 254, [hep-th/020876]. [7] U. Buzzo, F. Fucto, J.F. Moaes, and A. Tanzn, Mutnstanton cacuus and equvaant cohomoogy, JHEP 05 (2003) 054, [hep-th/0208]. [8] N. Nekasov, BPS/CFT coespondence: non-petubatve Dyson-Schwnge equatons and qq-chaactes, axv: [9] G. Poghosyan and R. Poghossan, VEV of Baxte s Q-opeato n N2 gauge theoy and the BPZ dffeenta equaton, JHEP (206) 058, [axv: ]. [0] L. F. Aday, D. Gaotto, and. Tachkawa, Louve Coeaton Functons fom Fou dmensona Gauge Theoes, Lett. Math. Phys., vo. 9, pp ,