From Killing spinor equations to Exceptional Sasaki Einstein conditions 

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The AdS/CFT Correspondence

In the previous section, we saw that the world-volume of a Dp-brane contains an abelian gauge field Aμ and 9 − p scalar fields i. In fact, the DBI action (1.5.2) provides also the dynamics for these fields. One can see this by expanding the square root and keep only the lowest order terms in 0. Then, one obtains a Maxwell term FμFμ from the expansion of the square root and a kinetic term for the scalars coming from the pullback of the ten-dimensional metric on the D-brane world-volume. There is also a constant term which expresses the rest mass of the D-brane which is irrelevant for the purpose of this section and therefore we drop it.
One can obtain more interesting gauge theories with a richer field content by considering more copies of D-branes. We will explore this set-up in more detail in section 2.1, but for the moment let us present some basic facts which are necessary to introduce the AdS/CFT Correspondence. We focus therefore on the case of N parallel  D3-branes sitting at the same point of the ten-dimensional space (coincident branes). The U(1) gauge symmetry in the case of a single brane gets now enhanced to a U(N) gauge symmetry. This can be explained by the fact that an open string can have endpoints lying on different branes. These are charged under different U(1) components of the gauge field corresponding to the different branes and moreover there should be transformations mixing them since they belong to the same string.
Therefore, there is now a U(N) gauge field (Aμ)a b with its corresponding non-abelian field strength. The scalar fields parametrizing the positions of the D3-branes now become also N ×N matrices (i)a b transforming in the adjoint of U(N). Although we have not been explicit with fermions, we should keep in mind that the D-brane action contains also a supersymmetric completion. It turns out that the fermionic content of the gauge theory living on the D3-branes is four Weyl fermions ()a b transforming in the adjoint of U(N).
The field content we just described is the field content of N = 4 super Yang-Mills theory. A complete analysis shows that the interactions derived from the D-brane action are the right ones for N = 4 SYM. More precisely, comparing the dilaton factor in the DBI action with the standard normalization of the Yang-Mills action, we get a relation for the couplings on the two sides of the correspondence. Moreover, in the CS action there is a term of the form C(0)F ^F providing the theta term for the gauge theory. The precise relations are g2Y M = 4gs, = 2C(0).

Moving towards the N = 0? theory

Our main goal for the rest of this chapter is to study the non-supersymmetric version of the Polchinski-Strassler story, and in particular to spell out a method to determine completely the D3-brane Coulomb branch potential (or the quadratic term in the polarization potential) for the N = 4 SYM theory deformed with a generic supersymmetrybreaking combination of fermion and boson masses. Many of the issues in the problem we are addressing have been touched upon in previous explorations, but when one tries to bring these pieces of the puzzle together one seems to run into contradictions. We will try to explain how these contradictions are resolved, and give a clear picture of what happens in the supergravity dual of the mass-deformed N = 4 theory. As explained in [6], a fermion mass deformation of the N = 4 SYM field theory, iMijj , corresponds in the bulk to a combination of R-R and NS-NS three-form field strengths with legs orthogonal to the directions of the field theory, that transforms in the 10 of the SU(4) R-symmetry group. The complex conjugate of the fermion mass, M†, corresponds to the complex conjugate combination transforming in the 10. Since the dimension of these fields is 3, the normalizable and non-normalizable modes dual to them behave asymptotically as r−3 and r−1.
The boson mass deformation in the field theory, aMabb, can be decomposed into a term proportional to the trace of M, which is a singlet under the SU(4) ‘ SO(6) R-symmetry, and a symmetric traceless mass operator, which has dimension 2 and transforms in the 200 of SO(6). The traceless mass operator in the 200 corresponds in the AdS5 × S5 bulk dual to a deformation of the metric, dilaton and the RR fourform potential that is an L = 2 mode on the five-sphere, and whose normalizable and non-normalizable asymptotic behaviors are r−2 and r−2 log r [41]. On the other hand, the dimension of the trace operator is not protected, and hence, according to the standard lore, turning on this operator in the boundary theory does not correspond to deforming AdS5 ×S5 with a supergravity field2, but rather with a stringy operator [42]. The anomalous dimension of this operator at strong coupling has consequently been argued to be of order (gsN)1/4.
On the other hand, there exist quite a few supergravity flows dual to field theories in which the sum of the squares of the masses of the bosons are not zero [43, 44, 45, 46, 47, 48, 49, 50, 51, 52], and none of these solutions has any stringy mode turned on, which seems to contradict the standard lore above. In the next sections of this chapter we would like to argue that the solution to this puzzle comes from the fact that the backreaction of the bulk fields dual to the fermions determines completely the singlet piece in the quadratic term of the Coulomb branch potential of a probe D3-brane. Therefore, the trace of the boson mass matrix that one reads off from the bulk will always be equal to the trace of the square of the fermion mass matrix.
This, in turn, indicates that in the presence of fermion masses, the stringy operator is not dual to the sum of the squares of the boson masses, but to the difference between it and the sum of the squares of the fermion masses. Mass deformations of the N = 4 theory where the supertrace of the square of the masses is zero can therefore be described holographically by asymptotically-AdS supergravity solutions [43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. However, to describe theories where this supertrace is nonzero, one has to turn on “stringy” non-normalizable modes that correspond to dimension-(gsN)1/4 operators, which will destroy the AdS asymptotics.


The explicit map between bosonic and fermionic mass matrices

In this section we will construct explicitly the maps (2.4.7) and (2.4.8), and the relationship between SU(4) and SO(6) representations. This will give the form of the possible terms in the supergravity fields that depend quadratically on fermion masses, which come from the backreaction of the fields dual to these masses. As shown in the previous section, the backreaction splits into two parts, corresponding to the 200 and 1 representations.
To build a map between SU(4) and SO(6) one identifies the 6a representation of SU(4) we have encountered above in (2.4.2) with the fundamental representation of SO(6). The former is given by a 4×4 antisymmetric matrix, ‘T = −’, that transforms as ‘ ! U’UT under U 2 SU(4). The complex 6 can be further decomposed into two real representations, 6 = 6+ + 6−, by imposing the duality condition:7

Constraints on the gauge theory from AdS/CFT

From the previous section we can arrive to another crucial observation. From (2.6.9) and the explicit form of the singlet (2.5.16) (or (2.5.13) for a generic mass matrix), we find Tr[boson masses2] = Tr[fermion masses2] (2.7.1) Tr(M2) = Tr(MM†) = Tr(mm†) + 2 ˆmI¯ˆmI + ˜m2 .
This result establishes that only theories where the supertrace of the mass squared is zero can be described holographically by asymptotically-AdS solutions. The sum of the squares of the boson masses, which is an unprotected operator (also known as the Konishi) and has been argued to be dual to a stringy mode of dimension (gsN)1/4, can be in fact turned on without turning on stringy corrections, as one could have anticipated from the solutions of [43, 45, 44]. In the presence of fermion masses, what is dual to a stringy mode is not therefore the sum of the squares of the boson masses, but rather the mass super-trace (the difference between the sums of the squares of the fermion masses and the boson masses). Theories where this supertrace is zero can be described without stringy modes, but to describe theories where this supertrace is nonzero, one has to turn on “stringy” non-normalizable modes which destroy the AdS asymptotics.

Supersymmetry, topology and geometry

The Killing spinor equations that we encountered in section 1.4 are actually statements about the internal manifold with or serving as the expansion coefficients of the ten-dimensional supersymmetry parameters in the solution under study (see (1.4.3) and (1.4.9)). Hence, when we write a Killing spinor equation for the internal manifold, it is always implied that such a mode expansion can be made for the ten-dimensional spinor which is a non-trivial requirement for the internal manifold M. Having that in mind, we distinguish the superysmmetry constraints in two classes:
• Topological constraints on the manifold. The internal space should have the right topological properties that allow for the existence of a spinor field (satisfying certain reality and chirality conditions) in a well-defined way. More technically, the spinor implies a reduction of the structure group2[79] of the tangent bundle of M.
• Differential constraints on the manifold. Equations (1.4.6) and (1.4.13) actually describe the parallel transport of the internal spinor. This is a statement about the connection defined on M and in technical terms leads to a reduction of the holonomy group3 of the internal space.
Let us now start discussing the simplest case of compactifications of type II supergravity in the absence of fluxes with (at least) a covariantly constant spinor (1.4.8) on the internal manifold. We will specialize to the case where the internal manifold is six-dimensional although most of the definitions and statements hold for all evendimensional manifolds which admit a covariantly constant spinor.

Geometrizing the NS-NS degrees of freedom

The starting point of generalized geometry is the extension of the tangent bundle TM of the internal manifold to a generalized tangent bundle E in such a way that the elements of this bundle generate all of the bosonic symmetries of the theory (diffeomorphisms and gauge transformations). The generalized tangent bundle transforms in a given representation of the corresponding duality group acting on the symmetries.
Following the historical path, we start by discussing the O(d,d) generalized geometry, relevant to the NS-NS sector of type II theories compactified on d-dimensional manifolds.
In section 3.3, we introduce Ed(d) generalized geometry which encodes the full bosonic sector of type II theories compactified on a (d − 1)-dimensional manifold, or M-theory on a d-dimensional geometry.
The NS-NS sector of type II supergravity contains the metric g(mn), the Kalb- Ramond field B[mn] and the dilaton . The symmetries of this theory are diffeomorphisms generated by vectors k and gauge transformations of the B-field which leave the H = dB invariant and which are parametrized by one-forms . The latter corresponds to the restriction of the first transformation in (1.2.5) restricted on the d-dimensionalinternal manifold. The combined action of these symmetries can be thought to be generated by a single object V = (k, ) , k 2 TM , 2 TM.

Table of contents :

1 Strings, fields and branes 
1.1 Supersymmetric relativistic strings
1.2 Type II supergravities
1.3 Supergravity in D=
1.4 Supersymmetric vacuum solutions
1.5 D-branes
1.6 The AdS/CFT Correspondence
2 Mass deformations of N = 4 SYM and their supergravity duals 
2.1 Myers effect
2.2 The N = 1? theory
2.3 Moving towards the N = 0? theory
2.4 Group theory for generic mass deformations
2.4.1 Fermionic masses
2.4.2 Bosonic Masses
2.5 The explicit map between bosonic and fermionic mass matrices
2.6 Mass deformations from supergravity
2.7 The trace of the bosonic and fermionic mass matrices
2.7.1 Constraints on the gauge theory from AdS/CFT
2.7.2 Quantum corrections in the gauge theory
3 Supersymmetry and (Generalized) Geometry 
3.1 Supersymmetry, topology and geometry
3.2 O(d,d) Generalized Geometry
3.2.1 Geometrizing the NS-NS degrees of freedom
3.2.2 Supersymmetry in O(d,d) Generalized Geometry
3.3 Exceptional Generalized Geometry
4 Generalized Geometric vacua with eight supercharges 
4.1 Supersymmetry in Exceptional Generlaized Geometry
4.1.1 Backgrounds with eight supercharges
4.1.2 Supersymmetry conditions
4.2 From Killing spinor equations to Exceptional Sasaki Einstein conditions
4.2.1 The Reeb vector
4.2.2 The H and V structures as bispinors
4.2.3 Proof of the generalized integrability conditions
4.3 The M-theory analogue
4.4 Some constraints from supersymmetry
4.4.1 Type IIB
4.4.2 M-theory
4.5 The moment map for Ja
4.5.1 Type IIB
4.5.2 M-theory
4.6 The Dorfman derivative along K
4.6.1 Type IIB
4.6.2 M-theory
A ’t Hooft symbols 
B Spinor conventions 
C E6 representation theory 
C.1 SL(6) × SL(2) decomposition
C.2 USp(8) decomposition
C.3 Transformation between SL(6) × SL(2) and USp(8)


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