Wilson loops and chiral correlators on squashed spheres

Journal of High Energy Physics, Nov 2015

Abstract We study chiral deformations of \( \mathcal{N}=2 \) and \( \mathcal{N}=4 \) supersymmetric gauge theories obtained by turning on τ J tr Φ J interactions with Φ the \( \mathcal{N}=2 \) superfield. Using localization, we compute the deformed gauge theory partition function \( Z\left(\left.\overrightarrow{\tau}\right|q\right) \) and the expectation value of circular Wilson loops W on a squashed four-sphere. In the case of the deformed \( \mathcal{N}=4 \) theory, exact formulas for Z and W are derived in terms of an underlying U(N) interacting matrix model replacing the free Gaussian model describing the \( \mathcal{N}=4 \) theory. Using the AGT correspondence, the τ J -deformations are related to the insertions of commuting integrals of motion in the four-point CFT correlator and chiral correlators are expressed as τ-derivatives of the gauge theory partition function on a finite Ω-background. In the so called Nekrasov-Shatashvili limit, the entire ring of chiral relations is extracted from the ϵ-deformed Seiberg-Witten curve. As a byproduct of our analysis we show that SU(2) gauge theories on rational Ω-backgrounds are dual to CFT minimal models.

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Wilson loops and chiral correlators on squashed spheres

Received: July Wilson loops and chiral correlators on squashed F. Fucito 0 1 J.F. Morales 0 1 R. Poghossian 0 1 2 0 Alikhanian Br. 2, AM-0036 Yerevan , Armenia 1 Via della Ricerca Scienti ca , I-00133 Roma , Italy 2 Yerevan Physics Institute We study chiral deformations of N = 2 and N = 4 supersymmetric gauge the N = 2 super eld. Using localization, we compute the deformed gauge theory partition function Z(~jq) and the expectation value of circular Wilson loops W on a squashed four-sphere. In the case of the deformed N = 4 theory, exact formulas for Z and W are derived in terms of an underlying U(N ) interacting matrix model replacing the free Gaussian model describing the N = 4 theory. Using the AGT correspondence, the J -deformations are related to the insertions of commuting integrals of motion in the four-point CFT correlator and chiral correlators are expressed as -derivatives of the gauge theory partition function on a nite In the so called Nekrasov-Shatashvili limit, the entire ring of chiral relations is extracted from the -deformed Seiberg-Witten curve. As a byproduct of our analysis we show that SU(2) gauge theories on rational -backgrounds are dual to CFT minimal models. Nonperturbative E ects; Wilson; 't Hooft and Polyakov loops; Supersymmet- - aI.N.F.N | Sezione di Roma 2, Contents 1 Introduction and summary 2 The gauge partition function on R The instanton moduli space and the equivariant charge The deformed gauge partition function Chiral correlators The gauge theory on S4 Wilson loops Perturbative expansion The N = 4 deformed theory Two point correlators htreC tr'J i The CFT side The conformal eld theory Integrals of motion The gauge/CFT dictionary The gauge theory side AGT duality: chiral correlators vs integrals of motion Chiral relations: 1; 2 nite AGT duals of minimal models The minimal models Degenerated states vs. critical masses The partition function and the four-points conformal blocks The gauge partition function on S4 vs the four-point correlator Introduction and summary extracted from the asymptotic fall o of a Wilson loop made of two straight lines. The correspondence. involves both the gauge eld Am and its scalar superpartners 'i W = htr eCi C = i (Amx_ m + 'i y_i)ds ent fractions of the original supersymmetry [1{3]. with Z(a; ) the gauge partition function on R4. Here Z is evaluated on an such that 1 = 2 = nite, so the gauge theory lives e ectively on a non-trivial in the limit where the mass of the adjoint hypermultiplet is sent to zero. Gromov-Witten invariants for certain complex manifolds. Indeed in presence of an background the zero mode solutions to the equations of motion xm Am with loop is given by the integral tr eC = dN a (a) tr e 1 e N V (~;a) prediction S = ln W e for the minimal area of a string worldsheet on the gravity dual theory as a function of a 4-dependent e ective t'Hooft coupling e . As a bonus, in this case, it will also be possible to nd the correlators of the Wilson loop with tr 2 and tr 4 and check our results against those of [20].1 1We thank the referee of the paper to point out this reference to us. for a nite background. We carry out the explicit computation in the SU(2) theory partition functions are matched against the dual CFT correlators. Witten curve in the limit in which one of the two parameters of the -background is sent masses and rational -backgrounds and CFT minimal models. The gauge partition function on R Sclass = 1 Z Fclass( ) + h:c: + : : : the scalar ' in the vector multiplet has a non-trivial vacuum expectation value. In this the classical prepotential of the general form for some integer p and Fclass( ) = = ' + Fclass( ) = i tr 2 is recovered after setting J>2 to zero. with gauge group U(N ) and fundamental or adjoint matter. The instanton moduli space and the equivariant charge [N + 2k] 2k matrices 0; M characterizing the instanton gauge connection and the satisfy the super-ADHM constraints M + M = 0 To localize the integral one rst introduces the equivariant charge on M the moduli space via degrees of freedom in , M leaving a 4kN -dimensional superspace Mk;N .Fundamental hypermultiplets contribute an extra [k Nf ] matrix K of fermionic moduli that can be connection can be written in terms of the [N+2k] [N] matrix U de ned by U = U = 0 U U = 1l and reads Am = U @mU Q = Q + d + i = M Q K = h Q M = 0 Q h = 0 and i a contraction de ned by i dxm = i dMi = an element of the symmetry group SO(4) U(Nf ) rotating moduli and spacetime coordinates. The combined action can then be written as = M + d Q K = h We notice that Q2 = , so Q is nilpotent at the xed points of . The charge Q can be for localization, is the construction of an equivariant eld F de ned by U + QU QU + [U QU; U QU ] = r(U QU ) F = r(U dU ) components. Indeed, recalling that rotates both the instanton moduli and the spacetime variables xm one can write [33] modU + QU QU + [U QU; U QU ] + xm Am = ' + In the limit ` to zero, ! 0, '~ reduces to ' and the equivariant correlator computes operators are built out of '~ rather than '. after replacing m by dxm and evaluating the elds on the instanton background. Indeed, background can be written in the explicit Q -invariant form Q F = ( + gauge)(U Q U ) Sinst(~) = 2 J ! C2 level and instanton contributions The one-loop partition function Zone loop is given by with [17, 32] Zognaeugeloop = Zomnaetter loop = Z(~) = Zone loop Zinst+tree(~) Zone loop = Zognaeugeloop Zomnaetter QuN;v=1 2(av m) 2(auv + m + ) mu) 2(au mv+N + ) fund: Here 2(x) is the Barnes double gamma function2 and = 1 + 2 The instanton partition function is de ned by the moduli space integral Zinst+tree( ) = is given by the localization formula [12] = ( 2 ) D2 X with D the complex dimension of the space M , x0 labelling the xed points of take M = C2, = P1 J=2 2 JJ! tr F this action is the origin and the localization formula leads to a rotation in the Cartan of the SO(4) Lorentz Sinst(~) = 2 J ! C2 tr F J = 1 2 J=2 J tr '~(0)J To perform the integral over the instanton moduli space we take element of the Cartan subgroup of the symmetry group U(N ) = e Sinst(~) and 1 > 0; 2 > 0) 2(x) = Y i;j=0 x + i 1 + j 2 or adjoint I the Cartan of U(k) and 1;2 are Lorentz breaking parameters that deform the R4 spacetime geometry. For a gauge theory on at space 1;2 should be sent to zero at the end of the computation. For nite 's the integral describes the partition function on a non-trivial gravitational background, the so called -bakcground. Fixed the U(k) Cartan elements I . Explicitly (i;j) = au + (i 1) 1 + (j Zinst+tree( ) = ZY = = Zgauge ZYmatter with ZY the inverse determinant of and SY (~) the 0-form of the instanton action (2.22) evaluated at the xed point. For ZY one nds [11, 12, 32] ZYmatter = ( QuN;v ZYu;Yv (auv + m) QuN;v=1 Z;;Yv (mu mv+N ) fund ZYu;Yv (x) = i) + 2 (1 + k~ui (x + 1(1 + kuj exchange mi $ mj can be obtained by replacing mu+N ! mu+N + SY (~) = 1 2 J=2 J tr '~0J Y = 1 2 J=2 with '~0 = U U the zero form of the equivariant super eld F and OJ;Y the chiral primary operator evaluated at the xed point. To evaluate OJ;Y , one rst notices that for a speci ed by the Young tableaux data Y one can nd in nitely many solutions U (k;l) to the de ning equation U = 0. The the chiral operators [14, 33]. Y = V ez (k;l) = V u=1 (k;l)2=Yu reproduce the classical result hez' ez 2 ) in order to of z one nds OJ;Y = O2;Y = tra2 O3;Y = tra3 For the rst few values of J one nds u=1 (i;j)2Yu u=1 (i;j)2Yu O4;Y = tra4 (2 12 + 3 1 2 + 2 22 + 6 The deformed partition functions can then be written as Zinst+tree(~) = X ZY (~) = 1 2 J=2 energy associated to the deformed instanton partition function Fe (~) = 1 2 ln Z(~) Z( J = J;2) = e Zoneloop with q = e2 i and Ztree = e and Zinst = P Y ZY qjY j the instanton part. the tree level contribution to the partition function function. To this aim, we rst introduce the Q -invariant volume form 4 = dz1 dz1 dz2 dz2 i( 1jz1j2dz2dz2 + 2jz2j2dz1dz1) and write the chiral operator in the equivariant form d4x tr '~J = htr '~J i = or equivalently from (2.33) J ! htr '~J i = @ J ln Z(~) derivative of the prepotential in the undeformed theory in at space. On the other hand, the general multi-trace chiral correlators htr '~J1 tr '~J2 : : :iundeformed in the undeformed theory. The gauge theory on S4 The gauge partition function on S4 is given by the integral [7] ZS4 (~) = c dN a jZone loop(a) Ztree+inst(a; ~)j2 vevs and masses taken in the domains au =2 iR mu = mu = nant Zone loop. jZognaeugeloopj2 = jZomnaetteloropj2 = QuN;v=1 (av (x) = (x) = is an entire function satisfying x). It has an in nite number of single zeros at x = 0 integers. Finally the normalization c has been xed for later convenience to be4 Wilson loops A supersymmetric Wilson loop is de ned by the line integral with '1 = 12 (' 'y) and Am taken to be anti-hermitian matrices. We use complex where 1 2 = 1 and write 1 = tion (2.18) is given by the same formula with 2(x) replaced by x). We adopt units one nds c = q 21 m3(m3+ )+ 12 m4(m4+ ) C = i (Am x_ m + jx_ j '1)ds z`(s) = r` ei `s L = 2 n1 = and the Wilson loop can be written in the suggestive form jx_ j2 = 12jr1j2 + 22jr2j2 = 1 C = i xm + '1) ds = precisely match. C0 = i'~0 = iU U is anti-hermitian and that correlators of '~(s) do not depend on s, so the Wilson loop operator on the instanton background Y reduces to = tr ei L '~0 Y = S4 = jZone loop(a)Ztree+inst(a; ~) j2 Plugging (3.16) into (3.15) one nds, for the leading J -correction to the Wilson loop expectation value at weak coupling, the result S4 = dN a (a) e 12 2 Im tra2 tr aJ tr a2 + : : : the correlators tr eC tr '~J1 '~J2 : : : S4;undef: tation value is computed by the correlator S4 = tr eC tr '~J S4;undef: Perturbative expansion the weak coupling regime Im ! 1. In this limit, the integral (3.13) is dominated by the one nds (a) = Zinst = 1 + : : : The N = 4 deformed theory at our disposal: the N = 4 theory. of N = 2 theory where the N and matter multiplets. located at m = 1 or m = 2. Indeed, for any choice of J and instanton number the 1. Indeed using the double Gamma function identity V (a; ~) = W = J tr(ia)J + h:c: dN a (a) tr e 1 e N V (a;~) one nds tion function reduces to the U(N ) matrix model integrals jZoneloopj2 = (a) = u6=v Z = methods [5]. One de nes the resolvent w(x) = u=1 x w(x) = 4fp 2(x) where fp 2(x) a polynomial of order p 2 determined by the condition that w(x) the square root, so we can write with S the union of the cuts and (x) the density w(x i0) = (x) = w(x) = 1 V 0(x) x1 for large jxj and therefore w(x) is fully determined. As an example, let us consider the quartic potential with Qp 2(x) a polynomial of order p 2. Indeed the number of unknown variables in b2 = V (a) = g4 = 4 Im( 4) x1 for large jxj one nds w(x) = (1 + 4 g4(x2 + 2b2))px2 4 2 -line where Wilson loop operators C( ) are inserted. Plugging (3.31) into (3.30) and expanding for large x and small one nds [5] x w(x) = N n=0 = 1 + with 2n external lines, p propagators and k four-point vertices. In gure 1 we show the the analytic form [5] x w(x) = 1 + X X1 ( 12g4 2)k n (2n)! (2k + n x2n n! (n 1)! k! (k + n + 1)! For the Wilson loop one then nds W = N n=0 = 1 + X = p n=1 k=0 n=1 k=0 e = = gY2 M N n1n2 0 and n1n2 = 1 one nds the familiar N = 4 formula [1, 4, 6] YM = gY2MN On the other hand in the limit of large with g4 2 kept nite one nds the same ending on the loop is given by ln W so one expects S = ln W providing a precise test of the duality. Two point correlators htreC tr'J i the expansion of the exact formula (3.34) in the limit where 2 0 i.e. the undeformed theory. correlator are traceless, so the diagrams in gure 1 and gure 2 with loops starting and YM from the quartic vertex contract among themselves. The correlators hW tr'J i can be extracted from the small g4-expansion of (3.34) W0 = = 1 + W = W0 g4 W1 + : : : W1 = 12 2YM I4 the number of diagrams with zero and one g4-vertices in gure 1. The insertion of tr'2 respect to . Indeed using (3.22) and (3.28) one nds htreC tr'2i = 2 2YM @ YM W0 = 2 YM I2 YM legs. These diagrams are counted by W0 = 8 YM I2 Subtracting this contribution from W1 one nds htreC tr'4i = YM) = localization and in [1, 4, 6] from perturbation theory. AGT duality: chiral correlators vs integrals of motion The CFT side with the insertion of the integrals of motion In introduced in [35]. The conformal eld theory algebra with commutation relations [Lm; Ln] = (m n) Lm+n + m) m+n;0 [am; an] = [Lm; an] = 0 The central charge c is parametrized by The primary elds V are de ned as c = 1 + 6 Q2 Q = b + V (z) = V vir a primary eld of the Virasoro algebra with dimension ( ) = (Q heis(z) = e2i( Q) Pn<0 ann z n e2i Pn>0 ann z n The commutation relations of the eld V and the generators Lm, an are [Lm; V (z)] = V [an; V (z)] = heis(z) zm+1@z + (m + 1) ( ) zm ) zn V (z) for Lmj0i = anj0i = 0 at zero and in nity respectively j i = V (0) j0i h j = lim z Consider the remaining Virasoro part of the four-point correlator Gvir ( i; jzi) = z1 hhVvi1r(z1) Vvi2r(z2) Vvi3r(z3) Vvi4r(z4) ii of conformal dimension and the factor z12 ( 1) is included to guarantee a nite limit at z1 ! 1. The fact that to z. For the choice above one nds [32] @z1 Gvir = 0 @z3 Gvir = @z Gvir @z2 Gvir = ( z@z + 2 1 @z4 Gvir = ((z 1)@z + = Pi4=1 i and sector one nds the conformal block i = ( i). Including also the contribution of the Heisenberg hh 1jV 2 (1) V 3 (z) j 4ii = (1 conformal block (4.12) G( ijz) = where C 1 2 are the Liouville structure constants [36, 37]. integrals are [35] I2 = L0 I3 = I4 = 2 X L k Lk + L k= 1;k6=0 k= 1;k6=0 a k Lk + 2 i Q X k a k ak + 3 i+j+k=0 k= 1;k6=0 i+j=k L k ai aj + 12 L0 jkja k Lk + 2(1 5 Q2) X k2 a k ak + 6iQ formal blocks. We de ne hh 1j[V 2 (1); a k] [ak; V 3 (z)] j 4ii + = z@z + Similarly for G3 one nds G3 = i X z the Virasoro part of the composite eld V . +(k +1) 2 3 +(k 1)(Q 5 G 3) z@z +(Q di erential operators acting on G. We write Gn( i; jz) = Ln G( i; jz) L2 = z@z + L3 = 3) z @z + (Q (see eq. (4.26)). The gauge/CFT dictionary to the partition function of the N characterizing the gravitational background parametrizes the central charge of the CFT. The full dictionary is given by [17] G( i; q) = ZSU4(2)(mi; q) 1 = 3 = (m3 + m4) 2 = 4 = 1 = b (m1 + m2) 2 = b 1 = 1 + 2 = Q z = q block G( i; jq) via with Zinst block is related to the SU(2) partition function via 4 for small q. On the other hand, the Virasoro conformal Gvir( ; i; q) = q 4 ZiSnUst(2)(a; miq) = q 3) ZiUns(t2)(a; miq) function on the sphere via valid at all-instanton orders for a -background. The results will be checked against Witten curve obtained in [25, 26]. Chiral relations: 1; 2 Using the AGT dictionary (4.20), (4.21) leads to 2 G2(q) htr'~2i = htr'~3i = 6 i G3(q) htr'~4i = 2h4 G4(q) = 1 + 2 h2 = 1 2 2 M3 + 2M4 + 2 q (M1 M3 M4) + h4 q(1 2q( M1 + M2) + q2( h2 + 2M2 htr'~2i = htr'~3i = htr'~4i = with Z = ZoUn(e2) loopZinst+tree and U(2) htr'~3i = htr'~4i = M2 = (1 + q) q@qhtr'~2i + M1 = M3 = mimj mk M4 = m1m2m3m4 We notice that the last two equations of (4.26) can be rewritten in the form q) + qM12 + M1 htr'~2i q) + qM1M3 + M3) which shows that in a nite -background, chiral correlators can be written in terms of instanton computation. Using (2.26){(2.39) one nds Zinst+tree = q h2 1 M1 + 2M2) M3 + 2 M4) + : : : : (4.30) The explicit form of Zone loop is irrelevant to our purposes since it is independent. It is to order q. We have checked this up to order q4. h2 ln Zinst nite. For the di erence equation [25, 26] ) y(z) y(z ) + (1 + q) P (z)y(z) 1 = 0 P (z) = z2 u1z + u2 Q(z) = 1 + X The chiral correlators can be extracted from the expansion at large z of tr '~2 = 2 a2 + tr '~4 = 2 a4 + tr '~3 = q 3 a2 M1 + 3 M3 + : : : M1 + 2 M2) M3 + 2 M4 + : : : a2 M1 +M3 @z log y(z) = y(z) = and using (4.33) one nds the chiral correlators as functions of the yi's. y34 + 4 y32 y4 3 y3 y4 + 3 y5 determines u1 while u2 can be solved in terms of htr'~2i. The results are u1 = u2 = ( 1 + q)htr'~2i + 2qM2 + 2q ( 2(1 + q) ring type. Explicitly htr'~3i = htr'~4i = (1 + q) q) + qM1M3 + M3) q) + qM12 + M1 htr'~2i for chiral correlators involving higher powers of the scalar eld. AGT duals of minimal models The minimal models is characterized by its Virasoro central charge There are (p 1g with conformal dimensions 1)=2 primary elds denoted as m;n, m 2 f1; 2; : : : ; p c = 1 m;n = where Q = b + 1=b, b = ipp=q and m;n = whose dimensions are Note the identi cation m;n p m;q n, which re ects the symmetry of the dimension (5.2) with respect to Since a primary eld m;n is degenerated at the level mn, any correlation function 2;1 satisfy a second order di erential equation and can be explicitly of squares of the corresponding conformal blocks. o = 1;1 = 1;2 " = 2;1 The non-trivial three-point structure constants are leading to the Ising fusion rules o = 0; " = Cooo = C o = C""o = 1 C " = [o] + ["] io = p h""""io = io = 2 z1=8(1 z)1=8 2z1=8 p i" = i = ""io = z1=8(1 z)1=8 one nds z)j1=4 h""""i = j z=2j2 jzj1=4j1 1 = 2 1 = b = i m3 + m4 = 3 = O3 = O3 = " O3 = O3 = " The masses, mi, are chosen such that the i's in (4.20) belong to the set ZiUns(t2;)+ = 2F1 ZiUns(t2;) = 2F1 1 B+A1;1 B+A2 q ; "g = Q; Q + ; Q + associated to the primary elds (5.4). Without any loss in generality we can discard expectation value a = a1 = a2. This integration can be formally evaluated with the a+ = Using (4.20), (5.13), (5.14) one nds + = O3 = O3 = " Young tableaux with a single row(column) at a+ and no boxes at a+ for O3 = Similarly, for a = a the relevant single-row(column) tableaux are centered at a . The whole instanton sum adds to a hypergeometric function [32] are the operators associated to O3 = : A1 = ( +)b = (a+ O3 = " : A2 = ( 1 + 2 + Q)b = (a+ B = 2 +b = (2a+ + ) 1 A1 = A2 = B = + = The partition function and the four-points conformal blocks model are listed in table 1. Using (5.16){(5.18) one nds the U(2) instanton partition ZhU"("2")";iionst = 2F1 ZhU("2)";iinst = 2F1 ZU(2);inst = ZhU"("2);inst = 2F1 h ""io io are reproduced from (5.20){(5.23) via (4.22) hO1O2O3O4iO = q 4 ZU(2);inst hO1O2O3O4iO = (1 = 1 O3 = O3 = " function vanishes at the corresponding critical values. The gauge partition function on S4 vs the four-point correlator hO1O2O3O4i constants C are de ned by = C+jq + 4 ZU(2) 4 ZU(2) 4 to rewrite c q a2 = q with (x) = (1(x)x) . Plugging the special values of i from table 1 into (5.28) one nds C =C+ get exchanged, so the involved in the corresponding conformal block The ratio between the two is given by [32] = Resa jZhoOne1Ol2oOop3O4ia j = CO1O2O C O O3O4 (A1) (A2) (B (B) (B A1) (B ZhU"("2")"i = j = j ZhU("2)"i = j q=2j2 jqj1=4 = j1 + p 2jqj1=4j1 hO1O2O3O4i = ZU(2) hO1O2O3O4i O3 = O3 = " with results in perfect agreement with (5.9). Acknowledgments mon projects in Fundamental Scienti c Research"-2013. Open Access. This article is distributed under the terms of the Creative Commons any medium, provided the original author(s) and source are credited. (1999) 125006 [hep-th/9904191] [INSPIRE]. [INSPIRE]. JHEP 05 (2008) 017 [arXiv:0711.3226] [INSPIRE]. 59 (1978) 35 [INSPIRE]. J. Math. Phys. 42 (2001) 2896 [hep-th/0010274] [INSPIRE]. Commun. Math. 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F. Fucito, J. F. Morales, R. Poghossian. Wilson loops and chiral correlators on squashed spheres, Journal of High Energy Physics, 2015, 64, DOI: 10.1007/JHEP11(2015)064