A convenient way to extract the volume of the moduli space, which does not require the geometrical technologies involving the Reeb vector and individual 5-cycles, is related to counting the mesonic operators of the field theory. The counting can be performed by the Hilbert series, which is the partition function for the mesons on the M2 moduli space, see e.g. [152, 159, 166].
As a crucial fact, the pole of the series gives the volume of the Sasaki-Einstein space, while keeping track of the dependency on the global symmetries [167]. We want to elab- orate a bit on how exactly this track-keeping works in our context.
In the toric case the counting becomes particularly easy, since we we can systematically solve the F -terms through perfect matchings, which results in the quotient description of the moduli space
M =Cd/ (C∗)d−4 . (9.12)
Here, d denotes the number of perfect matchings assigned to external points of the toric diagram and the charge matrix of the quotient is given by the linear relations amongst the corresponding vectors in the fan. This quotient construction makes manifest the dependency on all of the global symmetries, which the moduli space inherits from the natural isometries of the ambient space Cd. Generically, there are more external perfect matchings than global symmetries. This is because we had to introduce extra fields together with spurious symmetries, which are not seen by the physical fields. It is the prize we have to pay when solving the F -terms via perfect matchings. Upon parameterising the symmetries by the perfect matchings we might encounter a redundancy.
Nevertheless, it is a prize we are happy to pay; we will find it convenient to parameterise the set of global symmetries of the field theory in terms of coordinates onCd, as this will
make the comparison with the geometry most straightforward.
The Hilbert series for the flat ambient space reduces to the geometrical series Hil ∼ 1/(1 − t)d, and the quotient can be realised by projecting on the singlets under the (C∗)d−4
action, Hil(ti; M) = I d−4 Y k=1 dzk 2πizk 1 Qd i=1(1 − tiZi) , (9.13)
where Zi = Zi(zk) is the monomial weight of the i-th homogeneous coordinate under the
(C∗)d−4 action in (9.12). If we further set ti = e−2ai and take the → 0 limit, we have
[152, 167]
Hil(ti; M) ∼
Volai(Y )
which gives us an expression for the volume of the base Y in terms of the parameters ai.
The equation (9.14) is a geometric formula that knows about the flavour symmetries of the field theory, precisely through the ai’s which appear on the CFT side via the perfect
matchings. We will see in section10.2how we can reproduce this geometric formula from a purely field theoretical computation.
Chapter 10
The free energy in the large N limit
and holography
10.1
The large N free energy of vector-like quivers
In this section we discuss the computation of the leading order term of the free energy of vector-like field theories in a large N expansion. Localisation leads to a matrix model for the partition function on the three-sphere [11,14, 15],
Z[∆] = Z Y i,a dλ(a)i exp − F λ(a)i , ∆ (10.1) where the integral is over the Cartan of the gauge group Qg
a=1U (N )ka, parameterised by the g × N many variables λ(a)i , i = 1 . . . N and
F λ(a)i , ∆ = lnY a " eiP kaλ (a) i 2 4π −P ∆ (a) mλ(a)i Y i<j sinh2 λ (a) i − λ (a) j 2 ! Y ρ el(1−∆+iρ(2πλ)) # (10.2) Furthermore, ∆(a)m is a monopole charge associated with the ath gauge group [14] and `(z)
is the one loop determinant of the matter fields computed in [14, 15] `(z) = −z ln 1 − e2πiz + i 2 πz2+ 1 πLi2e 2πiz − iπ 12 (10.3) with derivative `0(z) = −πz cot (πz) . (10.4)
Finally, ρ refers to the weights of the representation of each matter field with R-charge ∆, the product is over all chiral fields. Note that the full matrix model is a functional of the R-charges.
We want to solve this matrix model in a large N approximation, following closely the procedure of [18]. The integral at large N and finite ka is dominated by the minimum
of the free energy F λ(a)i , ∆ and can be approximated by configurations that obey the saddle point equations ∂λ(a)
i
F = 0. For vector-like theories, it turns out that a sensible ansatz for the eigenvalues λ is given by
λ(a)i = N1/2xi+ iy (a)
i (10.5)
For large enough N , one can replace the discrete set (10.5) with g continuous variables. The real part of the eigenvalues becomes a dense set with density ρ(x) = ds/dx and the imaginary parts yi(a) become functions ya(x). The free energy can be split in two parts, a
piece from the Chern-Simons and monopole terms FCS = N3/2 2π Z dx ρ(x) x X a kaya+ 2π∆(a)m (10.6) and a second piece from the one loop determinant of the vector and the matter fields. For a pair of bifundamental and anti-bifundamental fields (Xab, Xba) in a vector-like theory,
charged under the ath and the bth gauge group, with R-charges (∆ab, ∆ba), we have
F1−loop= −N3/2 (2 − ∆+ab) 2 Z dxρ(x)2 δy2ab−π 2 3 ∆ + ab(4 − ∆ + ab) (10.7) where δyab ≡ ya(x) − yb(x) + π∆−ab and ∆
±
ab = ∆ab ± ∆ba. For an adjoint field we use
(10.7) with a = b and divide by a factor two. Equation (10.7) is only valid in the range |δyab| ≤ π∆+ab, which will indeed be satisfied by all our solutions.
In the continuum limit, the resulting free energy is extremised as a functional of ρ and the y’s at the saddle point. The eigenvalue density is subjected to the consistency constraints
Z
dx ρ(x) = 1 ρ(x) ≥ 0 point − wise
One can impose the first constraint through a Lagrange multiplier µ. This set of rules is enough to compute the free energy of the vector like theories as a function of the R charges, F (∆) [18].
As observed in [18] the expressions (10.6) and (10.7) possess flat directions which parameterise the symmetries on the eigenvalues and on the R charges. By defining the real parameters η(a) they are
ya → ya− 2πη(a)
∆ab → ∆ab+ η(a)− η(b) (10.8)
∆(a)m → ∆(a)