Holography for N = 1 on S 4 - PDF Free Download

Holography for N = 1 on S 4 Nikolay Bobev Instituut voor Theoretische Fysica, KU Leuven Benasque July 14 2015 1311.1508 + 15ΩΩ.ΩΩΩΩΩ + 15ΥΥ.ΥΥΥΥΥ with Henriette Elvang, Daniel Freedman, Silviu Pufu Uri Kol, Tim Olson

Motivation Powerful exact results for supersymmetric field theories on curved manifolds using equivariant localization [Witten], [Nekrasov], [Pestun],...

Motivation Powerful exact results for supersymmetric field theories on curved manifolds using equivariant localization [Witten], [Nekrasov], [Pestun],... Supersymmetric localization reduces the path integral of a gauge theory to a finite dimensional matrix integral. Still hard to evaluate explicitly in general! Make progress by taking the planar limit for specific 4d N = 2 (non-conformal) gauge theories. [Russo], [Russo-Zarembo], [Buchel-Russo-Zarembo] Evaluation of the partition function of planar SU (N ), N = 2 SYM on S 4. An infinite number of quantum phase transitions as a function of λ g 2 YMN. [Russo-Zarembo]

Motivation N = 1 SYM is a theory of an N = 1 vector multiplet and 3 massive chiral multiplets in the adjoint of the gauge group. It is a massive deformation of N = 4 SYM. There is a unique supersymmetric Lagrangian on S 4. [Pestun], [Festuccia-Seiberg] The result from localization for N, λ 1 for N = 2 is [Russo-Zarembo] F N =2 =2 S4 = log ZN S 4 = N 2 2 (1 + (mr)2 ) log λ(1 + (mr)2 )e 2γ+ 1 2, 16π 2 For N = 1 hard to calculate the partition function in the field theory The goal is to calculate F N =2 N =1 S4 and FS 4 holographically. Precision test of holography! In AdS 5/CFT 4 one typically compares numbers. Here we have a whole function to match.

Plan N = 1 SYM theory on S 4 The gravity dual of N = 2 on S 4 Holographic calculation of F N =2 S 4 Holography for N = 1 SYM theory on S 4 Outlook

N = 1 SYM theory on S 4

N = 1 SYM on R 4 The field content of N = 4 SYM is A µ, X 1,2,3,4,5,6, λ 1,2,3,4.

N = 1 SYM on R 4 The field content of N = 4 SYM is A µ, X 1,2,3,4,5,6, λ 1,2,3,4. Organize this into an N = 1 vector multiplet A µ, ψ 1 λ 4, and 3 chiral multiplets χ j = λ j, Z j = 1 ( ) Xj + ix j+3, 2 j = 1, 2, 3.

N = 1 SYM on R 4 The field content of N = 4 SYM is A µ, X 1,2,3,4,5,6, λ 1,2,3,4. Organize this into an N = 1 vector multiplet and 3 chiral multiplets A µ, ψ 1 λ 4, χ j = λ j, Z j = 1 2 ( Xj + ix j+3 ), j = 1, 2, 3. Only SU (3) U (1) R of the SO(6) R-symmetry is manifest. The N = 1 theory is obtained by giving (independent) mass terms for the chiral multiplets.

N = 1 SYM on S 4 The theory is no longer conformal so it is not obvious how to put it on S 4.

N = 1 SYM on S 4 The theory is no longer conformal so it is not obvious how to put it on S 4. When there is a will there is a way! [Pestun], [Festuccia-Seiberg],...

N = 1 SYM on S 4 The theory is no longer conformal so it is not obvious how to put it on S 4. When there is a will there is a way! [Pestun], [Festuccia-Seiberg],... L S4 N =1 = LS4 N =4 + 2 R 2 tr( Z 1 Z1 + Z 2 Z2 + Z 3 Z3 ) + tr ( m 1 m 1Z 1 Z1 + m 2 m 2Z 2 Z2 + m 3 m 3Z 3 Z3 ) 1 tr (m1χ1χ1 + m2χ2χ2 + m3χ3χ3 + m1 χ1 χ1 + m2 χ2 χ2 + m3 χ3 χ3) 2 1 tr [ ] m iɛ ijk Z i Zj Zk + m iɛ ijk ZiZ jz k 2 + i 2R tr ( m 1Z 2 1 + m 2Z 2 2 + m 3Z 2 3 + m 1 Z1 2 + m2 Z2 2 + m3 Z3 2 ). 15 (real) relevant couplings in the Lagrangian + 1 complex gaugino vev + 1 complexified gauge coupling. A 19 (real) parameter family of N = 1 deformations of N = 4 SYM. Only 18 of these parameters are visible as modes in IIB supergravity. For m 3 = m 3 = 0, m 1 = m 2 m and m 1 = m 2 m we get the N = 2 theory.

Results from localization for N = 2 The SU (N ) gauge symmetry is generally broken to U (1) N 1 by a vev for Z 3. Z 3 = diag(a 1,..., a N ). After supersymmetric localization the path integral for the theory on S 4 reduces to a finite dimensional integral over the Coulomb moduli a i. [Pestun] N ( N ) Z = da i δ a i (a i a j) 2 Z 1-loop Z inst 2 e S cl, i<j where S cl = 8π2 N λ i=1 i=1 i=1 N N ai 2 H 2 (a i a j), Z 1-loop = H (a i a j + mr)h (a i a j mr), H (x) = n=1 i=1 ( ) n 1 + x2 e x n 2 2 /n. The function Z inst is Nekrasov s partition function with parameters ε 1 = ε 2 = 1/R.

Results from localization for N = 2 Russo and Zarembo solved (numerically) this matrix model at large N. They found an infinite number of (quantum) phase transitions as a function of λ.

Results from localization for N = 2 Russo and Zarembo solved (numerically) this matrix model at large N. They found an infinite number of (quantum) phase transitions as a function of λ. One caveat. They assume that Z inst = 1 for large N. This is important and will be checked holographically. The result for N, λ 1 is F S 4 = log Z = N 2 2 (1 + (mr)2 ) log λ(1 + (mr)2 )e 2γ+ 1 2. 16π 2 This answer depends on the regularization scheme. The scheme independent quantity is d 3 F S 4 d(mr) = 2N 2 mr((mr) 2 + 3) 3 ((mr) 2 + 1) 2 This is the unambiguous result one can aim to compute holographically.

The gravity dual

Supergravity setup Use 5d N = 8 gauged supergravity to construct the holographic dual.

Supergravity setup Use 5d N = 8 gauged supergravity to construct the holographic dual. Why is this justified?

Supergravity setup Use 5d N = 8 gauged supergravity to construct the holographic dual. Why is this justified? It is a consistent truncation of IIB supergravity on S 5 with fields dual to the lowest dimension operators in N = 4 SYM. [Lee-Strickland-Constable-Waldram] The gravity dual of N = 2 on R 4 was constructed first in 5d. [Pilch-Warner] From field theory we expect that the solutions we are after posses U (1) V U (1) H U (1) Y global symmetry. Impose this on the 5d N = 8 supergravity. The result is that only 3 real scalars, 2 Abelian gauge fields and the metric survive in the bosonic sector. The 3 scalars, {φ, ψ, χ}, are dual to 3 relevant operators O φ, Oφ = 2 ; O ψ, Oψ = 3 ; O χ, Oχ = 2.

Supergravity setup The Euclidean Lagrangian is [ ] L = 1 R + 12 µη µ η + 4 µz µ z 2κ 2 η 2 (1 z z) + V, 2 ( ) V 4 1 L 2 η + 1 + z z 4 2η2 1 z z + η8 (z z) 2. 4 (1 z z) 2 This is a non-linear sigma model with target R H 2 O(1, 1) SU(1,1) U(1). To preserve the isometries of S 4 take the domain-wall Ansatz ds 2 = L 2 e 2A(r) ds 2 S 4 + dr 2, η = η(r), z = z(r), z = z(r). The masses of the scalars around the AdS 5 vacuum are m 2 φl 2 = m 2 χl 2 = 4, m 2 ψl 2 = 3.

The BPS equations Plug the Ansatz in the supersymmetry variations of the 5d N = 8 theory and use the conformal Killing spinors on S 4 to derive the BPS equations ˆ µζ = 1 2 γ5γµζ,

The BPS equations Plug the Ansatz in the supersymmetry variations of the 5d N = 8 theory and use the conformal Killing spinors on S 4 to derive the BPS equations ˆ µζ = 1 2 γ5γµζ, z = 3η (z z 1) [ 2(z + z) + η 6 (z z) ], 2η [η 6 ( z 2 1) + z 2 + 1] z = 3η (z z 1) [ 2(z + z) η 6 (z z) ], 2η [η 6 (z 2 1) + z 2 + 1] [ ( η 6 z 2 1 ) + z 2 + 1 ] [ η ( z 6 2 1 ) + z 2 + 1 ] (η ) 2 = 9L 2 η 2 (z z 1) 2, e 2A = (z z 1)2 [ η 6 ( z 2 1 ) + z 2 + 1 ] [ η 6 ( z 2 1 ) + z 2 + 1 ] η 8 (z 2 z 2 ) 2.

UV expansion The (constant curvature) metric on H 5 is ds 2 5 = dr 2 + L 2 sinh 2 ( r L ) ds 2 S 4.

UV expansion The (constant curvature) metric on H 5 is ds 2 5 = dr 2 + L 2 sinh 2 ( r L ) ds 2 S 4. Solving the BPS equations iteratively, order by order in the asymptotic expansion as r, we find (with L = 1) e 2A = e2r 4 + 1 6 (µ2 3) +..., [ ] η = 1 + e 2r 2µ 2 µ(µ + v) r + +..., 3 3 ] +..., [ 1 2 (z + z) = e 2r 2µ r + v 1 2 (z z) = µ e r + e 3r [ 4 3 µ( µ 2 3 ) r + 1 3 ( 2v(µ 2 3) + µ(4µ 2 3)) ] +....

IR expansion At r = r the S 4 shrinks smoothly to zero size. Solve the BPS equations close to r = r, and require that the solution is smooth to find

IR expansion At r = r the S 4 shrinks smoothly to zero size. Solve the BPS equations close to r = r, and require that the solution is smooth to find e 2A = (r r ) 2 + 7η012 + 20 81η 0 4 (r r ) 4 +..., ( ) [ ( ) ] 12 η0 1 η = η 0 (r r 27η 3 ) 2 85 + 12 131η0 1 (r r 0 810η 4 ) 2 +..., 0 [ ] 1 η 6 6 0 1 η 0 (z + z) = 2 η 06 + 1 η 06 + 2 2η0 8 (4η 6 0 + 5) (r r ) 2 +..., 15(η 06 + 2) 2 [ ] 1 η 6 0 1 2 (z z) = 2 η 06 + 1 η 06 + 2 + η0 2 (3η 12 0 10η 6 0 20) (r r ) 2 +.... 15(η 06 + 2) 2 Here, η 0 is the only free parameter since one can set r = 0 by the shift symmetry of the equations.

Numerical solutions One can find numerical solutions by shooting from the IR to the UV. There is a one (complex) parameter family parametrized by η 0, so v = v(η 0), and µ = µ(η 0). Hz+z é Lê2 Hz-z é Lê2 0.4 0.25 0.3 0.20 0.2 0.15 0.10 0.1 A -2 2 4 1.20 h e 2A ëe 2r 0.25 0.05 A -2 2 4 1.15 0.20 1.10 0.15 0.10 1.05 0.05-2 2 4 A 1 2 3 4 5 r

Numerical solutions For real η 0 one finds the following results for v(µ) 0.2 v h0>1 vêi h0<1 2 4 6 8 m/i 0.4 0.6 0.8 m -10-0.2-20 -0.4-30 -0.6-40 -0.8-50 From the numerical results one can extract the following dependence v(µ) = 2µ µ log(1 µ 2 )

Holographic calculation of F N =2 S 4

Calculating F from supergravity By the holographic dictionary the partition function of the field theory is mapped to the on-shell action of the supergravity dual. [Maldacena], [GKP], [Witten]

Calculating F from supergravity By the holographic dictionary the partition function of the field theory is mapped to the on-shell action of the supergravity dual. [Maldacena], [GKP], [Witten] The on-shell action diverges and one has to regulate it using holographic renormalization. [Skenderis],... There is a subtlety here. If we insist on using a supersymmetric regularization scheme there is a particular finite counterterm that has to be added. Only with it one can successfully compare d3 F with the field dµ 3 theory result.

Calculating F from supergravity The full renormalized 5d action is S ren = S 5D + S ct + S finite.

Calculating F from supergravity The full renormalized 5d action is S ren = S 5D + S ct + S finite. Differentiate the renormalized action w.r.t. µ to find df SUGRA = N 2 ( ) ( ) dµ 2π vol(s 4 ) 4µ 12v(µ) = N 2 1 2 3 µ v(µ). Finally we arrive at the supergravity result d 3 F SUGRA dµ 3 = N 2 v (µ) = 2N 2 µ (3 µ 2 ) (1 µ 2 ) 2. Set µ = imr and compare this to field theory d 3 F S 4 d(mr) 3 = 2N 2 mr((mr) 2 + 3) ((mr) 2 + 1) 2.

N = 1

N = 1 The dual of the 18-parameter family of deformations of N = 4 SYM is captured by a 5d N = 2 gauged supergravity with a scalar coset O(1, 1) O(1, 1) SO(4, 4) SO(4) SO(4). This model is a consistent truncation of IIB supergravity. There are special cases which allow for a very explicit analysis. m i = m i and m 1 = m 2 = m 3 - in flat space this is the GPPZ/PS flow. [Girardello-Petrini-Porrati-Zaffaroni], [Polchinski-Strassler]. It has SO(3) global symmetry. On S 4 we need 4 supergravity scalars. m 1 = m 1 and m 2 = m 3 = 0 - in flat space this is the Leigh-Strassler flow. [Freedman-Gubser-Pilch-Warner] On S 4 we need 3 supergravity scalars.

LS on S 4 From holographic renormalization one again finds d 3 F SUGRA dµ 3 = N 2 v (µ).

LS on S 4 From holographic renormalization one again finds d 3 F SUGRA dµ 3 = N 2 v (µ). On the whole complex µ-plane v (µ) vanishes linearly around µ 0. It is an odd function of µ and falls off for µ 1 as v (µ) 2 µ. v'' 100 v''/i 4 50 2-1.0-0.5 0.5 1.0 μ -3-2 -1 1 2 3 μ -50-2 -100-4 There is a pole at µ = ±1 with v (µ) 1 (1 ± µ) 3.

Summary We found a 5d supegravity dual of N = 2 SYM on S 4. After careful holographic renormalization we computed the universal part of the free energy of this theory. The result is in exact agreement with the supersymmetric localization calculation in field theory. This is a precision test of holography in a non-conformal Euclidean setting. Extension of these results to N = 1 SYM where our results can be viewed as supergravity lessons for the dynamics of the gauge theory.

Outlook Uplift of the N = 2 solution to IIB supergravity. Will allow for a holographic calculation of Wilson or t Hooft line vevs. In addition one can study probe D3-branes. [Chen-Lin-Zarembo] Holography for N = 1 on other 4-manifolds. [Cassani-Martelli] Extensions to other N = 2 theories in 4d with holographic duals, e.g. pure N = 2 SYM? [Gauntlett-Kim-Martelli-Waldram] Extensions to other dimensions. Can we see some of the large N phase transitions argued to exist by Russo-Zarembo in IIB string theory? Revisit supersymmetric localization for N = 1 theories on S 4. Can one find the exact partition function (modulo ambiguities)? [Gerchkovitz-Gomis-Komargodski] For 4d N = 2 conformal theories Z S 4 gives the Zamolodchikov metric. What is the meaning of Z S 4 for non-conformal theories? Broader lessons for holography from localization?

GRACIAS!

Comments For a CFT on S 4 of radius R with cutoff ɛ 0 ( ) R 4 ( ) R 2 ( ) R F S 4 = α 4 + α 2 + α 0 a anom log + O(ɛ/R). ɛ ɛ ɛ For supersymmetric theories with a supersymmetric regularization scheme α 4 = 0. For massive theories there is an extra scale, m, so α 2 = α 2(mɛ) and α 0 = α 0(mɛ). Expand this for small mɛ α 2 = α 2 + m 2 ɛ 2 β 2 + O(m 4 ɛ 4 ), α 0 = α 0 + O(m 2 ɛ 2 ). The nonuniversal contribution to the free energy is α 2 ( R ɛ ) 2 + α 0 + β 2(mR) 2. Thus 3 derivatives w.r.t. mr eliminate the ambiguity. For nonsupersymmetric theories the 5th derivative of F S 4 universal. w.r.t. mr is