View
5
Download
0
Category
Preview:
Citation preview
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
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 2 / 28
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]
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 3 / 28
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.
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 4 / 28
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.
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 5 / 28
Susy on curved space
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.)
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 6 / 28
Susy on curved space
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”.
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 7 / 28
Susy on curved space
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 .
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 8 / 28
Susy on curved space
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]
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 9 / 28
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)
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 10 / 28
Killing spinor equation in 3d
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.
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 11 / 28
Killing spinor equation in 3d
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.
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 12 / 28
Killing spinor equation in 3d
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).
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 13 / 28
Killing spinor equation in 3d
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
Killing spinor equation in 3d
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.
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 15 / 28
Killing spinor equation in 3d
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.
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 16 / 28
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.
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 17 / 28
Supersymmetric Lagrangians
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
Supersymmetric Lagrangians
Supersymmetric Lagrangians
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 .
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 19 / 28
Supersymmetric Lagrangians
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.
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 20 / 28
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.
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 21 / 28
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
Exemple of an application
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)
.
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 23 / 28
Exemple of an application
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
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 24 / 28
Exemple of an application
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.
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 25 / 28
Exemple of an application
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.
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 26 / 28
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.
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 27 / 28
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]
Cyril Closset (WIS) Supersymmetry on Three-Manifolds. CCTP, 01/11/12 28 / 28
Recommended