Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 47 (2014)

Unitarity bounds from generalised Käbler identities
J. Schmude∗1
A textbook result in Kähler geometry relates thede Rham with the Dolbeault Laplacian, ∆ = 2∆∂¯.
The topic of this note is a similar identity in the caseof Sasaki-Einstein manifolds and its application in to
the unitarity bounds in superconformal gauge theories(SCFTs):∆ = 2∆∂¯b − £2η − 2ı(n − d0 )£η + 2LΛ
+ 2(n − d0 )Lη Λη + 2ı(Lη ∂¯b∗ − ∂¯b Λη ).

(1)The right hand side features the tangential CauchyRiemann operator, the Lefschetz operator, and the action of the Reeb vector. The equation ∆ = 2∆∂¯ canbe derived from the Kähler identities, commutators between the Dolbeault and Lefschetz operators and theiradjoints. The proof of equation 1 follows a similarroute by obtaining Kähler-like identities that hold on
Sasaki-Einstein manifolds. Those identities as well thedetails of the proof were worked out in1) .
Equation 1 ﬁnds application in the AdS/CFT correspondence. Freund-Rubin compactiﬁcation on SasakiEinstein manifolds yields supergravity duals of superconformal ﬁeld theories. The AdS/CFT dictionary
links the conformal energy of SCFT operators to thespectrum of ∆, their R-charge to that of the Liederivative along the Reeb vector, £η . The conformalenergy, R-charge, and spin of any SCFT operator haveto satisfy the unitarity bounds4,5) , which should be reﬂected on the supergravity side in the spectrum of ∆.
Indeed, it is possible to re-derive the unitarity bounds
from supergravity when using equation 1 in conjunction with the calculations in2,3) .
This leads us to the spectral problem for ∆. Decompose the cotangent bundle as T ∗ S = D∗ ⊕ η =Ω1,0 ⊕ Ω0,1 ⊕ η and consider a k-form ω with £η ω = ıq,q ≥ 0, and d0 ≤ n. Clearly all terms on the right handside of 1 are positive deﬁnite except for the mixedterm M = ı(Lη ∂¯b∗ − ∂¯b Λη ) = N + N ∗ . M is self2
adjoint ∧and its spectrum∧∗ ∗ is real. Moreover,∧∗ ∗ N = 0∗∗and N ( D ) ⊂D ∧ η and N ( D ∧ η) = 0.
That is, N maps horizontal to vertical forms and annihilates the latter. N ∗ behaves accordingly and it
follows that ⟨ω, M ω⟩ vanishes if ω is horizontal or vertical. This is also the case if ω is neither horizontal
nor vertical yet holomorphic in the ∂¯b -sense. As longas we restrict to one of these cases, 1 takes the form of
a bound on the spectrum of ∆.
This was conjectured and partially shown in the context of the calculations of the superconformal index
in2,3) . Here, the spectrum was constructed from primitive elements of Ωp,q . For such forms, 1 clearly implies∗1∆ ≥ q 2 + 2q(n − d0 )(2)with equality if and only if ∂¯b ω = ∂¯b∗ ω = 0. In theKähler case, the latter of these is implied by transversality — d∗ ω = 0. Here however, d∗ ω = 0 leads only
to the vanishing of the horizontal component of ∂¯b∗ ω.
Indeed,∂b∗ ω = ıLη Λω,∂¯b∗ ω = −ıLη Λω,(3)which vanishes since ω was assumed to be primitive.
Assuming that every element of H∂p,q¯b (S) has a repre∗¯sentative closed under ∂b , the bound 2 is saturated
on the elements of H∂p,q¯b (S). These are the forms that
correspond to the short multiplets in the SCFT, and2 together with the expressions for the derived eigenmodes of ∆ given in2,3) allows to recover the unitarity
bounds from supergravity.
Since we found Sasaki-Einstein equivalents of both
∆ = 2∆∂¯ and the Kähler identities, it is tempting to
ask how much more of Kähler geometry can be generalized. For example, since ∆∂¯ is self-adjoint and elliptic, one can show that ΩkC = Hk ⊕ ∆∂¯(ΩkC ) which implies Hodge’s theorem. Similarly, the relation between
the de Rham and Hodge Laplacians allows for an isomorphism between the respective spaces of harmonicforms. However, ∆∂¯b is not elliptic. Recall that ∆∂¯b iselliptic if the symbol σ∆∂¯ : Hom(ΩkC , ΩkC ) ⊗ S 2 (T ∗ S)
bmaps any non-zero ω ∈ T ∗ S to an automorphism onΩkC . When calculating the symbol one essentially keeps
only those terms of ∆∂¯b that are of highest order inderivatives. In the context of the tangential CauchyRiemann operator, this means that ∂b and ∂¯b can betaken to be anticommuting and that the overall result
is essentially the same as for the symbol of the Dolbeault Laplacian on a Kähler manifold. It turns out,that σ∆∂¯ (η) = 0 and ∆∂¯b is not elliptic, yet transverb
sally elliptic.
An obvious problem of interest is the extension of theresults presented here beyond the Sasaki-Einstein case.
As long as there is a dual SCFT, there is a unitarity
bound meaning that there should be some equivalentof 1.
References
1) J. Schmude, arXiv:1308.1027 [hep-th].
2) R. Eager and J. Schmude, arXiv:1305.3547 [hep-th].
3) R. Eager, J. Schmude and Y. Tachikawa, arXiv:1207.0573[hep-th].
4) J. Kinney, J. M. Maldacena, S. Minwalla andS. Raju, Commun. Math. Phys. 275, 209 (2007) [hepth/0510251].
5) J. Bhattacharya, S. Bhattacharyya, S. Minwalla andS. Raju, JHEP 0802, 064 (2008) [arXiv:0801.1435 [hepth]].

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