Supersymmetry on three-manifoldshep.physics.uoc.gr/Slides/Fall_2012/CyrilClosset.pdf · Susy on curved space Example: Massless chiral supermultiplet (3d, N= 2 SUSY): L 0 = g @ ˚@

Supersymmetry on three-manifolds

Cyril Closset

Weizmann Institute of Sciences

CCTP, Heraklion, 01/11/2012

Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 1 / 28

Outline

Outline:I Introduction and motivationI Basics of rigid supersymmetry on curved spaceI Classification of supersymmetric three-manifolds (locally)I SUSY multiplets and LagrangiansI Comments on metric dependenceI Application: τrr from squashed sphereI Conclusions and outlook

Based on:I C.C., T. Dumitrescu, G.Festuccia, Z. Komargodski, [To appear]I C.C., T. Dumitrescu, G.Festuccia, Z. Komargodski, N. Seiberg,

1206.5218 and 1205.4142


Introduction

Rigid supersymmetry on curved manifolds

Given a d-dimensional supersymmetric field theory T in flat space anda Riemannian manifold (M, gµν), can we define a correspondingsupersymmetric theory on (M, gµν) ?

(T ,Rd, δµν) → (T ′,M, gµν)δT → δ′T ′

I Until recently, there was no systematic method. Case-by-casestudies forM simple enough. E.g. supersymmetric theories areknown on S4, S3 × S1, S3, S2 × S1,... [Sen, 1987; Romelsberger 2007; Pestun 2007; Kapustin,

Willett, Yaakov, 2010; Jafferis, 2010; Hama, Hosomichi, Lee, 2010, 2011; Imamura, Yokoyama, 2011; Benini, Cremonesi,

2012; Doroud, Gomis, Le Floch, Lee, 2012; ... ]

I General method has been proposed, based on backgroundsupergravity fields [Festuccia, Seiberg, 2011]


Introduction

Motivation: Exact results for supersymmetric theories

In recent literature, there has been some intense study ofsupersymmetric theories on spheres.

I Exact calculation of the partition functions Z(S3), Z(S2 × S1), Z(S2),· · · are known, using localisation. [See refs above]

I One can generalise such results to more general manifolds. Exactformulas for Z(M)? (I will not discuss this.) [Work in progress.]

I The more general approach allows to understand better previousresults on spheres, and extract more information from them.


Susy on curved space

Curved space rigid supersymmetry

Consider a supersymmetric quantum field theory described by someUV Lagrangian L0.

δ0L0 = ∂µ(· · · ) , δ0L0 = ∂µ(· · · ) .δ0, δ0 ∼ Pµ .

We can put this theory on a Riemannian manifold by the usualcovariantisation: δµν → gµν , etc.Recall that such a procedure is not unique: We can always add termsinvolving the curvature, that vanish in the flat space limit.



Example: Massless chiral supermultiplet (3d, N = 2 SUSY):

L0 = gµν∂µφ∂ν φ− iψγµ∇µψ − FF +αRφφ+ · · · .

This is not supersymmetric. We need additional corrections. Thesupersymmetry algebra itself is going to be modified.

Remark: We do not require nor use conformal invariance.

The procedure obviously relies on diffeomorphism invariance.Even though we fix the metric once and for all. We should think of themetric as a background field.

Possible because we consider theories with a conservedenergy-momentum operator Tµν .(It is like using U(1) gauge invariance to determine the correct couplingto a background magnetic field in a theory with a conserved current.)



Background supergravity fields

In any supersymmetric theory, we have a conserved supercurrent Sµα,which sits in the same supersymmetry multiplet as Tµν .

Sµ ∼ · · ·+ θSµ + θγν θ Tµν + · · ·

The detailed structure of the supercurrent multiplet can vary. Thegeneral supermultiplet S can often be improved to a simplersupercurrent. [Komargodski, Seiberg, 2010; Dumitrescu, Seiberg, 2011]

Festuccia-Seiberg proposal: To describe rigid supersymmetry in curvedspace, we should “weakly gauge” the supercurrent multiplet.⇒ Consider background supergravity [Festuccia, Seiberg, 2011]

Metric and its superpartners form a “background superfield”.



I We can think of rigid supersymmetry as some Mp →∞ of afull-fledged supergravity theory.

I For any given supercurrent there exists a correspondingsupergravity multiplet (gµν ,Ψµ,X). E.g. “old-minimal” or“new-minimal” in 4d. [Stelle, West, 1978; Ferrara, van Nieuwenhuizen, 1978; Sohnius, West, 1981]

I We should not impose any gravitational equation of motion. Needto consider off-shell formalism for the supergravity of interest.

I In the rigid limit, Ψµ = 0, δΨµ = 0. (Much simpler than SUGRA.)I Given a set of background fields (gµν ,X), we have one rigid

supersymmetry for each spinor ζ solving the generalised Killingspinor equation

δζΨµ = D(g,X)ζ = 0 .



I For N = 1 supersymmetric theories in four dimensions, thisprogram has been recently completed. [Dumitrescu, Festuccia, Seiberg, 2012; Klare,

Tomassielo, Zaffaroni, 2012; Dumitrescu, Festuccia, 2012]

Complete classification of supersymmetric backgrounds.Rigid supersymmetry↔ hermitian structure onM.

We will apply the backround supergravity formalism to R-symmetricN = 2 supersymmetric theories in three dimensions.

A technical difficulty to tackle is that the corresponding N = 2 off-shellsupergravity has not been worked out, to date. (At least incomponents.)

Linearised supergravity is good enough. [C.C., Dumitrescu, Festuccia, Komargodski, Seiberg,

2012]


Killing spinor equation in 3d

3d R-multiplet and supergravity background fields

In a 3d N = 2 theory with an R-symmetry, we have an R-multiplet

Rµ = j(R)µ − iθSµ − iθSµ − (θγν θ)(2Tµν + iεµνρ∂ρJ(Z)

)−iθθ

(2j(Z)µ + iεµνρ∂ν j(R)ρ

)+ · · · .

There exists a metric multiplet

(gµν ,Ψµ, Aµ, Cµ, H) , Vµ ≡ −εµνρ∂νCρ .

The linearised coupling to the R-mutiplet operators is

−Tµνhµν + j(R)µ

(Aµ − 3

2Vµ)− ij(Z)µ Cµ + J(Z)H + ΨµSµ + c.c.

(here gµν = δµν + 2hµν)



Further remarks:I The linear submultiplet

J (Z) = J(Z) − 12θγµSµ +

12θγµSµ + iθθTµµ − (θγµθ)j(Z)µ + · · · ,

can be improved to zero in a superconformal theory.The background fields H and Cµ couple to redundant operators ina CFT.

I All the supergravity background fields would be real in a unitarytheory.

I We allow for complex H, Aµ, Vµ. Metric is real.



3d Killing spinor equations

For R-symmetric theories, 3d rigid supersymmetry is governed by:

(∇µ − iAµ)ζα = −12

H(γµζ)α −12εµνρVν(γρζ)α − iVµζα ,

(∇µ + iAµ)ζα = −12

H(γµζ)α +12εµνρVν(γρζ)α + iVµζα .

The spinors ζα, ζα are sections of S⊗ L, S⊗ L−1.Real part of Aµ is a U(1)R connection.

Note that these equations subsume all the Killing spinor equationsused in recent literature on 3d. In particular the round 3-spherecorresponds to H = −i, Aµ = Vµ = 0.



One supercharge: almost contact structure

An almost contact structure onM is the triplet (ξ, η,Φ) such that

η(ξ) = 1 , Φ Φ = −1 + ξ ⊗ η

It is metric-compatible if g(X,Y) = g(Φ(X),Φ(Y)) + η(X)η(Y) .

On a three-dimensional Riemannian manifold, any real (co)-vector fieldηµ of unit norm defines such a structure:

ξµ = ηµ , Φµν = εµ

νρηρ .

There always exists such a structure on (M, gµν).The frame bundle structure group is restricted to U(1).



Consider a solution ζ of the first Killing spinor equation on (M, gµν). Itis nowhere vanishing, and completely determined by its value at apoint.

Useful bilinear

ηµ =ζ†γµζ

ζ†ζ

Satisfies ηµηµ = 1.

Supersymmetry (one supercharge) onM3⇔ metric-compatible almost contact structure (M3, gµν , ηµ)

Using the Killing spinor equation, one can solve explicitly for thesupergravity background fields in term of the almost contact structure:

H =12∇µηµ +

i2

Φµν∇µην + iλ(η) ,

and similar expressions for Vµ, Aµ.Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 14 / 28


Two supercharges: Seifert manifold

If we have one ζ and one ζ, we can define the two almost contact struc-tures ηµ, ηµ, and also the Killing vector

Kµ = ζγµζ .

We restrict our attention to the case where Kµ is real. That impliesηµ = −ηµ = Ω−1Kµ.

We can introduce local coordinates (τ, z, z) and

ds2 = c(z, z)2dzdz + η2 , η = Ω(z, z)(dτ + b(z, z)dz + b(z, z)dz)

U(1) bundle over Riemann surface: Seifert manifold.



Four supercharges

Maximally supersymmetric background requires Aµ = Vµ, ∂µH = 0,∂µV2 = 0. Several cases:

I Vµ = 0,M3 = S3, T3 or H3.I H = 0,M = R× Σ.I H = ih, h ∈ R. M3 is a particular U(1)-fibration over a surface of

constant curvature.

The last case includes the “Imamura-Yokoyama three-sphere” [Imamura,

Yokoyama, 2011], which is a SU(2)× U(1)-isometric squashed sphere.

ds2 = (µ1)2 + (µ2)2 + h2(µ3)3

with H = ih and Vµdxµ = 2√

h2 − 1r2 e3.


Supersymmetric Lagrangians

Supersymmetry algebra and supermultiplets

Consider a supersymmetric manifold (M3, gµν ,Aµ,Vµ,H), with somesupersymmetries ζ and/or ζ.

One can work out the generalisation of the off-shell supersymmetrymultiplets from N = 2 flat-space supersymmetry to our case.

The supersymmetry algebra is

δ2ζϕ = 0 , δ2

ζϕ = 0 ,

δζ , δζϕ = −2iL(A−12 V)

K ϕ+ 2iζζ (Z −∆ϕH)ϕ

on a field ϕ of R-charge ∆ϕ. Here K = ζγζ.



Consider a ζ-sypersymmetry (ζ is similar). The real multiplettransformation rules become

δζ C = iζχ

δζ χα = ζαM

δζ χα = −(γµζ)α(∂µC − iaµ)− ζασδζ M = 0

δζ M = 2ζλ− 2iDµ(ζγµχ) + 4iHζχ

δζ aµ = −iζγµλ+ ∂µ(ζχ)

δζ σ = −ζλδζ λα = iζα(D + σH)− i(γµζ)α(εµνρ∇νaρ + iVµσ + ∂µσ)

δζ λα = 0

δζ D = ∇µ(ζγµλ)−iVµζγµλ− Hζλ

where Dµ = ∇µ − i∆ϕ(Aµ − 12 Vµ). Similarly we work out the rules for

chiral multiplets.Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 18 / 28



From a real multiplet, we have the D-term action

S =

∫d3x√

g(D− σH − aµVµ)

In particular, for a vector multiplet this is the FI term.

From this we can derive the vector multiplet kinetic term

LYM =14

f (V)µν f (V)µν+

12∂µσ∂

µσ− iλγµ(Dµ+i2

Vµ)λ− 12

(D+σH)2 +i2

Hλλ ,

with f (V)µν = fµν + iεµνρVρσ.

The Chern-Simons term is simply

LCS = iεµνρaµ∂νaρ + 2iλλ− 2σD .



We can similarly work out the matter Lagrangian (chiral multiplet ofR-charge ∆ coupled to vector multiplet):

L = DµφDµφ− iψγµDµψ − FF + φDφ+ φσ2φ− iψσψ

+i√

2(φλψ + φλψ) +H(∆− 12

)(2φσφ− iψψ)

+

(∆(∆− 1

2)H2 − ∆

4R +

∆− 12

2VµVµ

)φφ

with

Dµ = ∇µ−i∆(

Aµ −32

Vµ

)− i(∆−∆0)Vµ − iaµ .

I Note that the couplings depend heavily on the R-charge.I Superconformal value at ∆ = 1

2 (free field).I L reproduces expected R-multiplet operators around flat space.


Linearized analysis and Q-exactness

Metric dependence at first order

One can show that at first order around flat space, gµν = δµν + 2hµν ,

L = L0 + L1 +O(h2µν) , δ = δ0 + δ1 + · · ·

we must have

L1 = −hµνOµν = −hµν(Tµν + · · · ) , δ0Oµν = 0

to preserve one supercharge. Such δ0-closed operators in theR-multiplet are easily classified. In fact they are all δ0-exact.

I Matching L1 with the linearized SUGRA Lagrangian, we have anice check of our solution for the supergravity background fields.

I This δ0-exactness suggests that the partition function Z(M3, gµν)is “quasi-topological’ ’. This is indeed borne out by the knownexamples.


Exemple of an application

Application: τrr from the squashed three-sphere

In any N = 2 superconformal theory, we have

〈Tµν(x)Tρσ(0)〉 = − τrr

64π2 (δµν∂2 − ∂µ∂ν)(δρσ∂

2 − ∂ρ∂σ)1x2

+τrr

64π2

((δµρ∂

2 − ∂µ∂ρ)((δνσ∂2 − ∂ν∂σ) + (µ↔ ν)) 1

x2 ,

〈j(R)µ (x)j(R)ν (0)〉 =τrr

16π2

(δµν∂

2 − ∂µ∂ν) 1

x2

determined by a unique parameter τrr, at separated points.(τrr = 1

4 for a free chiral multiplet.)We would like to compute τrr as

τrr ∼δ2

δgµνδgρσZ ∼ δ2

δAµδAνZ ,

using the exact results for Z(S3).Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 22 / 28


S3 is conformal to flat space.

Correlations functions in R3 and S3 are related by Weyl rescaling. Inparticular, for a conserved current

〈ja(x)jb(y)〉S3 = Ω(x)−2Ω(y)−2 〈ja(x)jb(y)〉R3

with Ω(x) = 2(1+x2)

.



To bring down j(R)µ on S3, we consider a one-parameter family ofsupersymmetric squashing. The most convenient one is the maximallysupersymmetric squashing of [Imamura, Yokoyama, 2011] we discussed before.

The supergravity background fields are

H = ih , Aµ = Vµ = v Kµ (K = e3), h =b + b−1

2, v = b− b−1

with b > 0 (b = 1 for the round sphere).

We have the coupling (Aµ − 32 Vµ) jµ(R) = −1

2 v Kµ jµ(R) .

All other couplings of the squashing to the CFT are through theparameter h. We can use that ∂bh|b=1 = 0 and ∂bv|b=1 = 2 to isolatethe R-symmetry current. One can see that (Fb = − ln Z(S3

b))

∂2bFb

∣∣∣b=1

= −∫

S3d3x√

g∫

S3d3y√

g Kµ(x)Kν(y)〈j(R)µ (x)j(R)ν (y)〉S3



One can evaluate that last integral, to arrive at

Re∂2

∂b2 Fb

∣∣∣b=1

=π2

2τrr

in term of the free energy Fb of the N = 2 SCFT on the squashedsphere.

I Since an exact formula is known for Fb, (at least) for any SCFTdescribed in the UV by a YM-CS-matter theory, the above is anexact and explicit formula for τrr.

I There can be contact term contributions to the integratedtwo-point functions. But one can show [C.C., Dumitrescu, Festuccia, Komargodski,

Seiberg] that contact terms contribute only to the imaginary part of Fb.That is why we must consider the real part in the above.



Some simple examples:I Free chiral multiplet. A very explicit form of Fb is

Fb = i∫ ∞

0

dxx

(sin(2x(z− ω)

sin(ω1x) sin(ω2x)− z− ωω1ω2x

)with z = i

2 h. One can compute ∂2bFb|b=1 = π2

8 .I Large N theories with a AdS4 × X7 dual. In this case, Fb simplifies

to [Imamura, Yokoyama, 2011; Martelli, Passias, Sparks, 2005]

Fb =(b + b−1)2

4F , F ≡ Fb=1 .

Moreover we have that F = π2

4 τrr at large N [Barnes, Gorbatov, Intriligator, Wright,

2005]. Our relation follows.


Conclusions and outlook

Conclusions

SummaryI Any orientable three-manifold preserve (at least) one supercharge

(we can put any N = 2 supersymmetric R-symmetric theory onM3 supersymmetrically).

I Supersymmetry onM3 is associated to an almost contactstructure.

I We developed a general formalism for curved space rigidsupersymmetry: supermultiplets, Lagrangians,... These resultscan be seen as a rigid limit of some as-yet unwritten off-shell“new-minimal” supergravity in 3d.

I As a first physical application of these technical results, wepresented an exact formula for the two-point function of Tµν in anyN = 2 SCFT in 3d.


Conclusions and outlook

Outlook

Further applications of our formalism :I Our results set the ground for a general discussion of localisation.

The next question is: Can we write down an exact formula forZ(M3, gµν , ηµ)? [Work in progress.]

The localisation locus on a general almost contact manifold is stillrelatively simple. The vector multiplet localises to solutions of theBogomolny equation (BPS monopole configurations).

I Possible to understand systematically the metric-dependence (orindependence) of Z(M3).

I It is easy to dimensionally reduce to 2d N = (2, 2) theories with anU(1)V R-symmetry. This makes the link with the recent work onthe round S2

[Benini, Cremonesi, 2012; Doroud, Gomis, Le Floch, Lee, 2012] and allows togeneralise it in various directions. [In progress]


Documents

Supersymmetry on three-manifoldshep.physics.uoc.gr/Slides/Fall_2012/CyrilClosset.pdf · Susy on curved space Example: Massless chiral supermultiplet (3d, N= 2 SUSY): L 0 = g @ ˚@