$AdS_5 \times T^{1,1}$ and the Klebanov--Witten Spectrum

The last stop in this course is the first truly non-spherical example of AdS/CFT. The original $AdS_5\times S^5$ duality is maximally supersymmetric: the internal space is the round sphere, the field theory is four-dimensional $\mathcal N=4$ super-Yang—Mills, and the R-symmetry is $SO(6)\simeq SU(4)$. The next natural question is what survives when the internal space is not $S^5$ but another Einstein five-manifold.

A particularly elegant answer is obtained from the conifold. Its base is the five-dimensional Sasaki—Einstein space

$$T^{1,1}={SU(2)\times SU(2)\over U(1)},$$

and type IIB string theory on $AdS_5\times T^{1,1}$ is dual to the Klebanov—Witten $\mathcal N=1$ superconformal gauge theory,

$$\text{type IIB on }AdS_5\times T^{1,1} \quad\Longleftrightarrow\quad \mathcal N=1\text{ SCFT with gauge group }SU(N)\times SU(N).$$

This example teaches three lessons at once. First, the geometry of the internal space becomes the global symmetry and operator spectrum of the CFT. Second, less supersymmetry allows generic operator dimensions to be irrational at strong coupling, while chiral primaries remain protected. Third, D-branes at singularities naturally produce quiver gauge theories.

The conifold and its base

The conifold is the hypersurface in $\mathbb C^4$

$$\mathcal C=\{(z_1,z_2,z_3,z_4):z_1^2+z_2^2+z_3^2+z_4^2=0\}.$$

Equivalently, after a linear change of variables, arrange four complex coordinates into a $2\times2$ matrix

$$Z=\begin{pmatrix} z_{11}&z_{12}\\ z_{21}&z_{22} \end{pmatrix}, \qquad \det Z=0.$$

This second form makes the $SU(2)\times SU(2)$ symmetry transparent: one $SU(2)$ rotates the row index and the other rotates the column index.

The conifold is a cone,

$$ds^2_6=dr^2+r^2ds^2_{T^{1,1}},$$

where $T^{1,1}$ is the angular space at fixed radius. A useful coordinate system for the standard Einstein metric is

$$ds^2_{T^{1,1}} ={1\over9}\bigl(d\psi+\cos\theta_1d\phi_1+\cos\theta_2d\phi_2\bigr)^2 +{1\over6}\sum_{a=1}^2 \bigl(d\theta_a^2+\sin^2\theta_a d\phi_a^2\bigr),$$

with

$$0\leq \theta_a\leq \pi, \qquad 0\leq \phi_a<2\pi, \qquad 0\leq \psi<4\pi.$$

The two pairs $(\theta_1,\phi_1)$ and $(\theta_2,\phi_2)$ parametrize two copies of $S^2$, while $\psi$ is a circle fiber twisted over both spheres. Topologically,

$$T^{1,1}\simeq S^2\times S^3.$$

The Sasaki—Einstein space $T^{1,1}$ is a $U(1)$ fibration over $S^2\times S^2$. The fiber coordinate $\psi$ becomes the geometric origin of the $U(1)_R$ symmetry in the dual SCFT.

The metric is normalized so that

$$R_{mn}=4g_{mn}$$

on $T^{1,1}$. This is the same normalization used for a unit-radius $S^5$ in $AdS_5\times S^5$. The cone over an Einstein five-manifold with this normalization is Ricci-flat. More specifically, the cone over $T^{1,1}$ is Calabi—Yau: $T^{1,1}$ is Sasaki—Einstein.

That last sentence is the geometric reason this background preserves four-dimensional $\mathcal N=1$ supersymmetry. For comparison,

$$S^5\quad\text{has cone }\mathbb C^3 \quad\text{and gives }\mathcal N=4,$$

while

$$T^{1,1}\quad\text{has cone the conifold} \quad\text{and gives }\mathcal N=1.$$

The near-horizon background

Place $N$ D3-branes at the conifold singularity. The near-horizon limit gives type IIB string theory on

$$AdS_5\times T^{1,1}$$

with $N$ units of self-dual five-form flux through $T^{1,1}$,

$${1\over (4\pi^2\alpha')^2}\int_{T^{1,1}}F_5=N.$$

The ten-dimensional metric takes the form

$$ds^2_{10}=R^2\left(ds^2_{AdS_5}+ds^2_{T^{1,1}}\right),$$

where the radius is fixed by flux quantization. Since

$$\operatorname{Vol}(T^{1,1})={16\pi^3\over27},$$

one finds

$$R^4={27\over4}\pi g_sN\alpha'^2.$$

This is the analogue of $R^4=4\pi g_sN\alpha'^2$ for $S^5$. The curvature is small in string units when $g_sN\gg1$, and string loops are suppressed when $g_s\ll1$, equivalently at large $N$ with large effective ‘t Hooft coupling.

For any Einstein space $X_5$ in an $AdS_5\times X_5$ solution, the leading large-$N$ central charge is governed by the inverse volume of $X_5$:

$$a=c={\pi^3\over4\operatorname{Vol}(X_5)}N^2.$$

Thus for $X_5=T^{1,1}$,

$$a=c={27\over64}N^2$$

at leading order in $N$. The smaller volume of $T^{1,1}$ compared with $S^5$ gives a larger central charge for the same five-form flux $N$.

The same number follows directly from the $\mathcal N=1$ anomaly formula. At large $N$, the two gauginos contribute $2N^2$ fermions of R-charge $1$. The four bifundamental chiral multiplets contain fermions of R-charge

$$R_\psi=R_\Phi-1=-{1\over2},$$

with $4N^2$ components. Hence

$$\operatorname{Tr}R=2N^2+4N^2\left(-{1\over2}\right)=0,$$

and

$$\operatorname{Tr}R^3=2N^2+4N^2\left(-{1\over2}\right)^3={3\over2}N^2.$$

Using

$$a={3\over32}\left(3\operatorname{Tr}R^3-\operatorname{Tr}R\right), \qquad c={1\over32}\left(9\operatorname{Tr}R^3-5\operatorname{Tr}R\right),$$

one obtains

$$a=c={27\over64}N^2.$$

Thus the inverse volume of $T^{1,1}$ computes the same protected anomaly as the strongly coupled quiver theory.

The Klebanov—Witten quiver

The dual field theory is a four-dimensional $\mathcal N=1$ quiver gauge theory. Up to decoupled Abelian factors, the gauge group is

$$SU(N)_1\times SU(N)_2.$$

There are four chiral superfields:

$$A_1,A_2: (\mathbf N,\overline{\mathbf N}), \qquad B_1,B_2: (\overline{\mathbf N},\mathbf N).$$

The $A_i$ form a doublet under a flavor symmetry $SU(2)_A$, and the $B_j$ form a doublet under another flavor symmetry $SU(2)_B$. The full continuous global symmetry is

$$SU(2)_A\times SU(2)_B\times U(1)_R\times U(1)_B.$$

The baryonic symmetry $U(1)_B$ assigns opposite charges to $A$ and $B$. In the gravity dual it comes from reducing the Ramond—Ramond four-form on the nontrivial three-cycle of $T^{1,1}$.

The Klebanov—Witten theory has two gauge nodes and two arrows in each direction. The quartic superpotential is forced by the $SU(2)_A\times SU(2)_B$ flavor symmetry and by $U(1)_R$.

The superpotential is

$$W=h\,\epsilon^{ik}\epsilon^{jl} \operatorname{Tr}(A_iB_jA_kB_l).$$

The $U(1)_R$ assignment at the superconformal fixed point is

$$R(A_i)=R(B_j)={1\over2}.$$

For a chiral primary operator in a four-dimensional $\mathcal N=1$ SCFT,

$$\Delta={3\over2}R.$$

Therefore the elementary chiral superfields have

$$\Delta(A_i)=\Delta(B_j)={3\over4}.$$

This is not a contradiction with the four-dimensional scalar unitarity bound, because $A_i$ and $B_j$ are not gauge-invariant local operators. The simplest gauge-invariant chiral fields are bilinears such as $\operatorname{Tr}(A_iB_j)$, with dimension $3/2$.

The superpotential has R-charge

$$R(W)=4\cdot {1\over2}=2,$$

so $\int d^2\theta\,W$ is classically compatible with superconformal symmetry. At the interacting fixed point the anomalous dimensions are such that the NSVZ beta functions vanish for both gauge groups.

The conifold as the moduli space of one brane

For one D3-brane, the non-Abelian structure disappears and $A_i,B_j$ become ordinary complex numbers. The gauge-invariant coordinates are

$$z_{ij}=A_iB_j.$$

The matrix $z_{ij}$ has rank one, so

$$\det z= z_{11}z_{22}-z_{12}z_{21}=0.$$

This is precisely the conifold. Thus a single D3-brane probes the space whose near-horizon geometry we used on the gravity side. For $N$ D3-branes, the mesonic branch is roughly the symmetric product

$$\operatorname{Sym}^N(\mathcal C),$$

up to the usual subtleties of non-Abelian gauge theory, coincident branes, and baryonic branches.

This matching is one of the cleanest ways to remember the quiver. The four bifundamental fields are not arbitrary matter: their gauge-invariant products are the coordinates of the conifold.

The chiral ring and mesonic operators

The most important single-trace mesonic chiral primaries are

$$\mathcal O_k= \operatorname{Tr}(A_{i_1}B_{j_1}A_{i_2}B_{j_2}\cdots A_{i_k}B_{j_k}),$$

with the $i$ indices symmetrized and the $j$ indices symmetrized. The F-term relations following from $W$ remove antisymmetric combinations and leave the representation

$$\left({k\over2},{k\over2}\right)$$

under $SU(2)_A\times SU(2)_B$. The R-charge and dimension are

$$R(\mathcal O_k)=k, \qquad \Delta(\mathcal O_k)={3k\over2}.$$

The first few cases are worth spelling out:

operator	$SU(2)_A\times SU(2)_B$ representation	$R$	$\Delta$
$\operatorname{Tr}(A_iB_j)$	$({1\over2},{1\over2})$	$1$	$3/2$
$\operatorname{Tr}(A_{(i_1}B_{(j_1}A_{i_2)}B_{j_2)})$	$(1,1)$	$2$	$3$
$\operatorname{Tr}(A_{(i_1}B_{(j_1}A_{i_2}B_{j_2}A_{i_3)}B_{j_3)})$	$({3\over2},{3\over2})$	$3$	$9/2$

At large $N$, single-trace operators correspond to single-particle supergravity states. Multi-trace products correspond to multi-particle states in $AdS_5$.

Harmonic analysis on $T^{1,1}$

The isometry group of $T^{1,1}$ is

$$SU(2)_A\times SU(2)_B\times U(1)_R.$$

Scalar harmonics are labeled by quantum numbers

$$(j,l,r),$$

where $j$ and $l$ are $SU(2)_A$ and $SU(2)_B$ spins, and $r$ is the $U(1)_R$ charge in the convention used for the dual CFT. A harmonic has angular dependence of the schematic form $e^{ir\psi/2}$ along the $U(1)$ fiber. The allowed values obey the usual weight constraints: $r/2$ must be an allowed magnetic quantum number inside both spin-$j$ and spin-$l$ representations, so in particular

$$|r|\leq 2j, \qquad |r|\leq 2l.$$

For the unit-radius metric above, the scalar Laplacian eigenvalue is

$$-\nabla^2_{T^{1,1}}Y_{jlr}=H_0(j,l,r)Y_{jlr},$$

with

$$H_0(j,l,r)=6\left[j(j+1)+l(l+1)-{r^2\over8}\right].$$

If a ten-dimensional minimally coupled scalar is expanded in such harmonics, the corresponding five-dimensional field has

$$m^2R^2=H_0(j,l,r),$$

and hence

$$\Delta=2+\sqrt{4+H_0(j,l,r)}.$$

This formula already displays an important difference between $T^{1,1}$ and $S^5$. On $S^5$, scalar harmonics have eigenvalues $k(k+4)$, so $\sqrt{4+k(k+4)}=k+2$ is simple. On $T^{1,1}$, generic quantum numbers give square roots of non-square numbers. Generic dimensions are irrational at strong coupling.

That is not a pathology. In an interacting CFT, unprotected operator dimensions need not be rational. The rational dimensions are selected by shortening conditions.

Scalar harmonics on $T^{1,1}$ are labeled by $(j,l,r)$. The protected mesonic chiral primaries live on the special line $j=l=r/2$.

The chiral-primary line is

$$j=l={r\over2}.$$

Set $r=k$. Then

$$\begin{aligned} H_0\left({k\over2},{k\over2},k\right) &=6\left[2\,{k\over2}\left({k\over2}+1\right)-{k^2\over8}\right]\\ &={9k^2\over4}+6k. \end{aligned}$$

Therefore

$$H_0+4=\left({3k\over2}+2\right)^2.$$

In the Kaluza—Klein reduction, the scalar harmonics occur not only in minimally coupled scalars but also in coupled systems involving metric fluctuations and the Ramond—Ramond four-form. Diagonalizing those coupled modes gives a short-multiplet branch with

$$\Delta=-2+\sqrt{H_0+4}={3k\over2}.$$

This precisely matches the chiral-ring result

$$\Delta(\mathcal O_k)={3k\over2}.$$

The appearance of the shifted branch is not an ad hoc trick; it is the gravitational expression of superconformal shortening.

The full Kaluza—Klein spectrum in words

The full spectrum on $AdS_5\times T^{1,1}$ is much richer than the scalar Laplacian formula. Type IIB supergravity contains the metric, dilaton, axion, two two-forms, and the self-dual four-form. Expanding all of these fields on $T^{1,1}$ gives towers of five-dimensional fields of different spins. Because the internal space has only $\mathcal N=1$ supersymmetry, these fields assemble into four-dimensional $\mathcal N=1$ superconformal multiplets.

The practical dictionary is:

gravity side	CFT side
isometry $SU(2)_A\times SU(2)_B$	flavor symmetry acting on $A_i$ and $B_j$
Reeb $U(1)$ fiber	superconformal $U(1)_R$
scalar harmonics $Y_{jlr}$	operators in representation $(j,l)_r$
shortened KK multiplets	protected chiral or conserved multiplets
generic KK multiplets	unprotected operators with strong-coupling dimensions
wrapped D3-branes on $S^3\subset T^{1,1}$	baryonic determinant operators

A useful mental model is this: the spherical harmonics on $S^5$ are replaced by coset harmonics on $T^{1,1}$. The representation labels are no longer just $SO(6)$ labels; they are labels of the smaller symmetry group $SU(2)_A\times SU(2)_B\times U(1)_R$. The price of reduced symmetry is a less rigid spectrum. The reward is a much more general mechanism for constructing holographic SCFTs.

Baryons and wrapped D3-branes

The topology

$$T^{1,1}\simeq S^2\times S^3$$

has a nontrivial three-cycle. A D3-brane wrapped on this $S^3$ becomes a particle in $AdS_5$, and its dual is a baryonic operator. In the gauge theory, schematic baryons are determinant-like operators such as

$$\mathcal B_A\sim \epsilon_{\alpha_1\cdots\alpha_N} \epsilon^{\beta_1\cdots\beta_N} (A_i)^{\alpha_1}{}_{\beta_1} \cdots (A_i)^{\alpha_N}{}_{\beta_N},$$

and similarly for $B_j$. Since each $A$ or $B$ has dimension $3/4$, these baryons have

$$\Delta(\mathcal B_A)=\Delta(\mathcal B_B)={3N\over4}.$$

The same scaling is obtained from the mass of a D3-brane wrapped on the minimal $S^3$ inside $T^{1,1}$,

$$\Delta=mR.$$

This is one of the most beautiful non-mesonic checks of the correspondence: topology of the internal space becomes baryonic operator physics.

Comparison with $S^5$

It is helpful to put the two most important $AdS_5$ examples side by side.

feature	$AdS_5\times S^5$	$AdS_5\times T^{1,1}$
cone	$\mathbb C^3$	conifold $\det z=0$
dual CFT	$\mathcal N=4$ SYM	$\mathcal N=1$ Klebanov—Witten theory
gauge group	$SU(N)$	$SU(N)\times SU(N)$
matter	three adjoint chirals	bifundamentals $A_i,B_j$
global symmetry	$SU(4)_R$	$SU(2)_A\times SU(2)_B\times U(1)_R\times U(1)_B$
internal volume	$\pi^3$	$16\pi^3/27$
leading central charge	$N^2/4$	$27N^2/64$
protected mesons	$\operatorname{Tr}\Phi^{(i_1}\cdots\Phi^{i_k)}$	$\operatorname{Tr}(A_{(i_1}B_{(j_1}\cdots A_{i_k)}B_{j_k)})$
generic dimensions	highly constrained by maximal SUSY	often irrational at strong coupling

The conceptual move from $S^5$ to $T^{1,1}$ is enormous. The internal space is no longer a round sphere, but the logic of holography remains the same: geometry organizes the spectrum, flux sets $N$, the AdS radius sets the strong-coupling expansion, and Kaluza—Klein modes become local operators.

Final perspective

The Klebanov—Witten example is the doorway to a large world of holographic quiver gauge theories. Replacing $T^{1,1}$ by more general Sasaki—Einstein spaces $X_5$ gives new $\mathcal N=1$ SCFTs. Adding fractional branes leads to cascading gauge theories and warped deformed conifolds. Adding flavor branes introduces fundamental matter. Studying wrapped branes reveals baryons, domain walls, defects, and confinement mechanisms.

The essential lesson is already visible here:

$$\boxed{ \text{Calabi--Yau cone geometry} \quad\Longleftrightarrow\quad \text{quiver SCFT data} }$$

For $T^{1,1}$, that dictionary is unusually explicit. The metric knows the global symmetry, the Laplacian knows the operator representations, the conifold equation knows the moduli space, and the chiral ring knows the protected Kaluza—Klein spectrum.

Exercises

Exercise 1. The contact one-form of $T^{1,1}$

Define

$$\eta={1\over3}\bigl(d\psi+ \cos\theta_1d\phi_1+\cos\theta_2d\phi_2\bigr).$$

Compute $d\eta$ and explain why the metric can be viewed as a circle fibration over $S^2\times S^2$.

Solution

Taking the exterior derivative gives

$$d\eta=-{1\over3}\left( \sin\theta_1d\theta_1\wedge d\phi_1+ \sin\theta_2d\theta_2\wedge d\phi_2 \right).$$

The two terms are the volume forms on the two unit two-spheres. Thus the $\psi$ circle is not a trivial product circle; it is twisted over both $S^2$ factors. The metric separates into a fiber contribution plus a base contribution,

$$ds^2_{T^{1,1}}=\eta^2+{1\over6}\sum_{a=1}^{2} \bigl(d\theta_a^2+\sin^2\theta_a d\phi_a^2\bigr),$$

up to the normalization already included in $\eta$. This is the geometric origin of viewing $T^{1,1}$ as a $U(1)$ bundle over $S^2\times S^2$.

Exercise 2. The chiral line in the scalar spectrum

Starting from

$$H_0(j,l,r)=6\left[j(j+1)+l(l+1)-{r^2\over8}\right],$$

set $j=l=k/2$ and $r=k$. Show that

$$H_0+4=\left({3k\over2}+2\right)^2.$$

Then explain why the short-multiplet branch gives $\Delta=3k/2$.

Solution

Substitution gives

$$\begin{aligned} H_0 &=6\left[2\cdot {k\over2}\left({k\over2}+1\right)-{k^2\over8}\right]\\ &=6\left[{k^2\over2}+k-{k^2\over8}\right]\\ &=6\left[{3k^2\over8}+k\right]\\ &={9k^2\over4}+6k. \end{aligned}$$

Thus

$$H_0+4={9k^2\over4}+6k+4 =\left({3k\over2}+2\right)^2.$$

The relevant shortened supergravity branch has

$$\Delta=-2+\sqrt{H_0+4}.$$

Therefore

$$\Delta=-2+\left({3k\over2}+2\right)={3k\over2}.$$

This matches the chiral primary $\operatorname{Tr}(AB)^k$, whose R-charge is $k$ and whose dimension is $3k/2$.

Exercise 3. The conifold from one D3-brane

For $N=1$, take $A_i$ and $B_j$ to be complex numbers. Define $z_{ij}=A_iB_j$. Prove that $z_{ij}$ obeys the conifold equation $\det z=0$.

Solution

The matrix

$$z=\begin{pmatrix} A_1B_1&A_1B_2\\ A_2B_1&A_2B_2 \end{pmatrix}$$

is the outer product of the column vector $(A_1,A_2)^T$ and the row vector $(B_1,B_2)$. It therefore has rank at most one. A $2\times2$ matrix has rank at most one exactly when its determinant vanishes:

$$\det z=(A_1B_1)(A_2B_2)-(A_1B_2)(A_2B_1)=0.$$

Thus the gauge-invariant coordinates of one probe brane satisfy the conifold equation.

Exercise 4. Marginality of the quartic superpotential

Use $R(A_i)=R(B_j)=1/2$ and the $\mathcal N=1$ chiral-primary relation $\Delta=3R/2$ to show that the quartic superpotential is marginal.

Solution

Each chiral superfield has

$$\Delta(A_i)=\Delta(B_j)={3\over2}\cdot {1\over2}={3\over4}.$$

The superpotential term contains four chiral superfields, so its operator dimension is

$$4\cdot {3\over4}=3.$$

In four-dimensional $\mathcal N=1$ superspace, $\int d^2\theta$ has dimension $1$, so a superpotential operator of dimension $3$ gives a four-dimensional marginal interaction. Equivalently, the superpotential must carry R-charge $2$, and indeed

$$R(W)=4\cdot {1\over2}=2.$$

Exercise 5. The baryon dimension

A baryonic operator made from $N$ copies of one $A$ field has the schematic form $\det A$. Using $\Delta(A)=3/4$, compute its dimension. Why is this naturally of order $N$?

Solution

The determinant operator contains $N$ elementary chiral fields. Its dimension is therefore

$$\Delta(\det A)=N\Delta(A)={3N\over4}.$$

This is of order $N$, not order one, so it is too heavy to be a perturbative supergravity Kaluza—Klein mode. In the dual string theory it is a wrapped D3-brane state. The D3-brane tension scales as $1/g_s$, while the AdS radius and flux relation convert the wrapped-brane mass into a conformal dimension proportional to $N$.

Exercise 6. Central charge from the volume of $T^{1,1}$

Use

$$a=c={\pi^3\over4\operatorname{Vol}(X_5)}N^2$$

and $\operatorname{Vol}(T^{1,1})=16\pi^3/27$ to compute the leading large-$N$ central charge of the Klebanov—Witten theory.

Solution

Substituting the volume gives

$$a=c={\pi^3\over4\cdot 16\pi^3/27}N^2.$$

Canceling $\pi^3$,

$$a=c={27\over64}N^2.$$

This agrees with the large-$N$ field-theory anomaly calculation using two vector multiplets and four bifundamental chiral multiplets with $R(A)=R(B)=1/2$.

Exercise 7. Why irrational dimensions are expected

Consider a minimally coupled scalar mode with quantum numbers $(j,l,r)=(1,0,0)$. Compute $H_0$ and the corresponding $\Delta=2+\sqrt{4+H_0}$. Is the answer rational?

Solution

For $(j,l,r)=(1,0,0)$,

$$H_0=6\left[1(1+1)+0-{0\over8}\right]=12.$$

The minimally coupled scalar formula gives

$$\Delta=2+\sqrt{4+12}=2+4=6.$$

This particular example happens to be rational. But this is accidental. For example, with $(j,l,r)=(1,1,0)$,

$$H_0=6(2+2)=24,$$

so

$$\Delta=2+\sqrt{28}=2+2\sqrt7,$$

which is irrational. Generic nonprotected dimensions in an interacting CFT are not expected to be rational. Protected chiral operators are special because superconformal shortening fixes their dimensions by their R-charges.