Supersymmetric States in Large N Chern-Simons-Matter Theories

Supersymmetric States in Large Chern-Simons-Matter Theories

Abstract:

In this paper we study the spectrum of BPS operators/states in superconformal Chern-Simons-matter theories with adjoint chiral matter fields, with and without superpotential. The superconformal indices and conjectures on the full supersymmetric spectrum of the theories in the large limit with up to two adjoint matter fields are presented. Our results suggest that some of these theories may have supergravity duals at strong coupling, while some others may be dual to higher spin theories of gravity at strong coupling. For the theory with no superpotential, we study the renormalization of -charge at finite ’t Hooft coupling using “-minimization”. A particularly intriguing result is found in the case of one adjoint matter.

1 Introduction

The coupling constant of a four dimensional gauge theory coupled to matter generically runs under the renormalization group. While it is sometimes possible to choose the matter content and couplings of the theory so that the gauge function vanishes, such choices are very special. In three dimensions, on the other hand, gauge fields are naturally self coupled by a Chern-Simons type action. As the coefficient of the Chern-Simons term in the action is forced by gauge invariance to be integrally quantized, the low energy gauge coupling (inverse of coefficient of the Chern-Simons term) cannot be continuously renormalized and so does not run under the renormalization group. All these statements are for every choice of matter content and couplings of the theory. As a consequence CFTs are much easier to construct starting with Chern-Simons coupled gauge fields in than with Yang Mills coupled gauge fields in [1, 2, 3, 4, 5].

Precisely because the coefficient, , of the Chern Simons term is an integer, the Chern-Simons coupling cannot be varied continuously. The set of Chern-Simons CFTs obtained, by varying a given Lagrangian over the allowed values of , yields a sequence rather than a fixed line of CFTs. Consider, however an Chern Simons theory at level . Such a theory admits a natural ’t Hooft limit in which we take , with held fixed. As explained by ’t Hooft, is the true loop counting parameter or coupling constant in this limit. Several physical quantities - like the spectrum of operators with finite scaling dimension- are smooth functions of . Now a unit change in changes by , a quantity that is infinitesimal in the large limit. As a consequence, even though and are both integers, is an effectively continuous parameter in the large limit. Effectively, the discretum of Chern-Simon-matter CFTs at finite merges into an effective fixed line of Chern-Simon-matter theories at large , parameterized by the effectively continuous variable .

Lines of fixed points of large CFTs map to families of theories of quantum gravity, under the AdS/CFT correspondence [6]. CFTs at weak or finite ’t Hooft coupling are generically expected to map to relatively complicated higher spin theories of gravity [7, 8, 9] or string theories on AdS spaces of string scale radii. In many examples of explicit realization of AdS/CFT, the bulk description simplifies in some manner at strong . It is then natural to ask whether the large class of fixed lines of Chern-Simons-matter theories admit simple dual descriptions at large [5]. The first explicit realization of the gravity dual of a large Chern-Simons-matter theory, as a critical string theory, was achieved by ABJM [10]. At infinitely strong coupling the ABJM Chern-Simons-matter theory develops a supergravity dual description, which is a considerable simplification over the highly curved stringy dual description at finite coupling. A direct field theoretic hint for the nature of the dual of ABJM theory [10] at strong coupling comes from the observation that the set of single trace supersymmetric states in ABJM theory have spins (and in fact match the spectrum of supergravitons of IIA theory on ).

While many examples of gravity duals of supersymmetric Chern-Simons-matter theories have been proposed following the work of ABJM (see for instance [11, 12, 13, 14, 15, 16]), essentially all such proposals involve quiver type matter content in the field theory. The gravity duals of seemingly simpler Chern-Simons-matter fixed points, both with and without supersymmetry, remain unknown (and may well be most interesting in the non supersymmetric context). On the other hand, it is of significant interest to find the CFT duals of gravity theories in with as few four-dimensional bulk fields (apart from gravity itself) as possible, and one may hope that the Chern-Simons theories with simple matter content are good candidates.

In order to maintain a degree of technical control, however, in this paper we study only supersymmetric theories with at least four supercharges. We will consider large and Chern Simons theories with a single gauge group and adjoint chiral multiplets (for all integer ). Such theories have been studied perturbatively in [5, 17] (see also [18]). We will study theories both with and without superpotentials. We address and largely answer the following question: what is the spectrum of supersymmetric operators as a function of the ’t Hooft coupling ? In the rest of this introduction we elaborate on our motivation for asking this question. In the next section we briefly summarize our principal results.

In this paper we will compute (or present conjectures for) the supersymmetric spectrum of a large class of large Chern-Simons-matter CFTs. As we will now describe in some detail, the results we find for the supersymmetric spectrum of several fixed lines is quite simple, and has several intriguing features. As we describe in more detail below, an important difference between some theories with supersymmetry and theories with and higher supersymmetry is that the -charge of the chiral multiplets of the theories (and hence the charges and dimensions of supersymmetric operators) may sometimes be continuously renormalized as a function of [5] (and sometimes not [17]). Luckily, Jafferis [19] has presented a proposal that effectively allows the computation of the -charge as a function of in many of these theories. In the rest of this paper we assume the correctness of Jafferis’ proposal; we proceed to use a combination of analytic and numerical techniques to present a complete qualitative picture of this -charge as a function of and the number of chiral multiplets .

While we hope that our results will eventually inspire conjectures for relatively simple large descriptions of some of the theories we study, in no case that we have studied have our results proven familiar enough to already suggest a concrete conjecture for the dual description of large dynamics. Of the supersymmetric spectra we encounter in this paper, the one that appears most familiar is the spectrum of the theory with two chiral multiplets. As we will describe in more detail below, this spectrum includes only states of spins , and so might plausibly agree with the spectrum of some supergravity compactification: however a detailed study of the spectrum as a function of global charges reveals some unexpected features that has prevented us (as yet) from finding a supergravity compactification with precisely this spectrum. The spectrum of supersymmetric states in deformations of the theory also has similar features. We hope to return to an investigation into the possible meanings of these spectra in the future.

In the next section we will present a more detailed description of the theories we have studied and our results.


Note added in proof: Upon completion of this work, we received [20] which overlaps with section 4 of this paper.

2 Summary of results

2.1 Theories with a vanishing superpotential

-charge as a function of

The first class of theories studied in this paper consists of Chern Simons theories at level , coupled to adjoint chiral multiplets with vanishing superpotential. This class of theories was studied, and demonstrated to be superconformal (for all , and ) in [5]. In the free limit the -charge of the chiral fields in this theory equals . However, it was demonstrated in [5] that this -charge is renormalized as a function of ; indeed at first nontrivial order in perturbation theory [5] demonstrated that the -charge of a chiral field is given by

(1)

where . As the -charge of a supersymmetric operator plays a key role in determining its scaling dimension (via the BPS formula), the exact characterization of the spectrum of supersymmetric states in this theory at large clearly requires control over the function at large . Such control cannot be achieved by perturbative techniques, but is relatively easily obtained by an application of the extremely powerful recent results of Jafferis [19] to this problem. In [19] Jafferis used localization methods to derive a formula (in terms of an integral over variables, where is the rank of the gauge group) for the partition function on of the CFT in question, as a function of the -charges of all the chiral fields in the theory. He then demonstrated that the modulus of this partition function is extremized by the values of at the conformal fixed point, assuming the absence of accidental global symmetries. In the large limit of interest to this paper, Jafferis’ matrix integral is dominated by a saddle point. Using a combination of analytic and numerical techniques, it is not difficult to solve the relevant saddle point equations, extremize the action with respect to , and thereby evaluate . It turns out that is a monotonically decreasing function for all . In fact at large (but all values of )

(2)

Note that tends to a constant value at . At large this constant value is barely below the free value ; it is given by

Although there is no a priori reason for this formula to apply at of order unity, we have used numerical techniques to find that it appears to work at the 10-15 percent level even down to . More specifically, at , our numerics indicates , and at , (see Figs. 1 and 2 below); these do not compare badly with and as predicted by (2).

Most interestingly, however, at , (see Fig. 4 below); i.e. the -charge of the chiral multiplet decreases without bound in this special (extreme) case (this was anticipated in [18]) , raising several interesting questions that we will come back to later.1

Spectrum of single trace operators

Given the function , it is not difficult to evaluate the superconformal index [21] of these theories as a function of . As was already noted in [21], this index demonstrates that the spectrum of supersymmetric single trace operators grows exponentially with energy for . In the absence of a superpotential one can actually compute a slightly more refined Witten index (adding in a chemical potential that couples to the global symmetry generators). Below we demonstrate that this refined index implies an exponentially growing density of states for the supersymmetric spectrum (in the theory without a superpotential) even for . This immediately suggests that the simplest possible dual description for all theories with (and the theory without a superpotential at ) is a string theory, with an exponential growth in supersymmetric string oscillator states.

However, the index indicates a sub exponential growth of supersymmetric states for all theories with and theories with a nontrivial superpotential when . This leaves open the possibility of a simpler dual (one with a field theory’s worth of degrees of freedom) in these cases. In this subsection we focus on theories without a superpotential, and so consider only the case . Making the assumption that the spectrum of supersymmetric states in this theory is isomorphic to the cohomology of the classical action of the susy operator, we have computed the full spectrum of single trace supersymmetric operators in this theory. Our explicit results are listed in Table (12). While the states listed in this table do grow in number with energy in a roughly Kaluza-Klein fashion, notice that the primaries listed in Table (12) include states of arbitrarily high spins, ruling out a possible dual supergravity dual description.

The supersymmetric states in Table (12) of course include the states in the chiral ring for all where is the scalar component of the chiral field. The scaling dimension of these chiral ring operators is given by ( was defined in the previous subsection). Unitarity, however, requires that every scalar operator in any 3 d CFT has scaling dimension , and that an operator with dimension is necessarily free (i.e. decoupled from the rest of the theory). Recall that decreases monotonically to zero as is increased. Let denote the unique solution to the equation

For , the scaling dimension of descends below its unitarity bound . (For later use we will also find it useful to define

is the value of the coupling at which a superpotential deformation by becomes marginal. Note that .)

Assuming Jafferis’ proposal, it follows from unitarity that our theory must either cease to exist 2 or must undergo a phase transition at a critical value, . While many possibilities are logically open, one attractive scenario (which is close to the scenario suggested in [18]) is the following. As is increased past then becomes free and decouples from the theory. As is further increased past then also becomes free and decouples. This process continues ad infinitum, leading to an infinite number of phase transitions.

The picture oulined above for the theory, namely that each of the decouple at successively larger values of can be subject to a consistency check. It was demonstrated in [18], using brane constructions, that the deformation of the zero superpotential system by the operator breaks supersymmetry precisely at (and in particular susy is not broken at smaller values of ).3 However the deformation of a superpotential by a free field always breaks supersymmetry. Consistency with the scenario outlined above therefore requires that . In other words

Our data (see Fig. 4) seems consistent with this bound, and moreover suggests (and we conjecture that) asymptotes to from above. It would be very interesting to establish this conjecture by analytic methods, but we leave that for future work.

If this picture outlined in the last two paragraphs is correct, then, in the limit we have an effective continuum of chiral primaries, with scaling dimension (all primaries with lower dimension have decoupled). The higher spin fields listed in Table (12) also reduce to a continuum at large . All this suggests that the large behavior of this theory is intriguing, and possibly singular.4

2.2 Theories with a superpotential

Let us now turn to the study of superconformal theories with a single gauge group, only adjoint fields (as above) but with appropriate superpotentials. For theories that reduce to free systems as , the superconformal index is independent of the details of the superpotential, other than the fact that the index cannot now be weighted with respect to a chemical potential for any global symmetry under which the superpotential is charged, and depends only on the -charge of matter fields which may be renormalized.5 So in particular, the index demonstrates the presence of an exponentially growing spectrum of supersymmetric states for , exactly as above. For this reason we focus our study on . Let us first start with .

at

The superpotential deformation is marginal at , but is relevant at finite (this follows because for all finite ). It has been argued in [5] that the beta function for this superpotential term vanishes when its coefficient is of order (at small ) leading to a weakly coupled CFT with a superpotential turned on. The presence of the superpotential forces the -charge of the field to be fixed at at all values of in this new fixed line. While the superconformal index of this theory is blind to the presence of the superpotential, the spectrum of single trace supersymmetric operators is not. We present a conjecture for this spectrum in Table (13) below (this conjecture is based on the same assumption described above, namely that the supersymmetric spectrum is accurately captured by the classical supercharge cohomology at all .) As in the case of theories without a superpotential, our conjectured supersymmetric spectrum grows with energy in a manner expected of Kaluza-Klein compactification, but continues to include states of arbitrarily high spin.

at

is another relevant deformation of the weakly coupled theory at . This deformation leads to a term in the scalar potential of the theory. At least at weak coupling we would expect the RG flow seeded by this operator to end at the supersymmetric large analogue of a Wilson Fisher fixed point. At this fixed point the scaling dimension (and superconformal -charge) of is fixed to be . This fixed point can then be continued to large by varying the gauge coupling. For this reason the superconformal index of this theory cannot be calculated from a free path integral; however it can be computed using the techniques of localization using the results of [22] (following the original work of [23, 24]). We have performed this computation in section 10 below. While the computation of this index is exact at all and , the final result simplifies dramatically in the large ’t Hooft limit of interest to this paper, and in fact reduces to the index of the free theory with appropriate charge renormalizations (in order to account for the fact that the -charge of is rather than ). We have also computed the full spectrum of single trace supersymmetric primaries in this theory (using assumptions similar to those described above). Our rather simple final results are listed in Table (14) .

at

At small , operators of the form are irrelevant for . However if our description of the theory without a superpotential in the previous subsection is indeed correct then for each the operator is in fact relevant for . It seems likely that the RG flows seeded by these operators end in new lines of CFTs in which the scaling dimension of is fixed at . We expect the index of these theories (in the ’t Hooft limit) to once again be given by the formula for non-interacting theory but with renormalized charges for all fields. Using methods similar to those described above, it should be easy to compute the full spectrum of single trace superconformal primaries in this theory. We leave this computation to future work.

theory at

Let us now turn to theories with a superpotential. First consider superpotentials of the form . Like any quartic superpotential in this theory, this superpotential is marginal at , but is relevant at finite (regarded as a deformation about the theory with no superpotential). It was argued in [5] that the RG flow seeded by this operator ends with the coefficient of this superpotential stabilized at that finite value (of order ) that enhances the supersymmetry of the theory to . This theory enjoys invariance under an enhanced symmetry group, and also enjoys invariance under an flavour symmetry group. We have computed the spectrum of supersymmetric states in this theory; our results are presented in Table (4). Interestingly, it turns out that the spins of supersymmetric states in this theory grow roughly in the manner one would expect of a Kaluza-Klein compactification of a supergravity theory on . In particular the spins of supersymmetric states in this theory never exceed two. We have not, however, managed to identify a specific supergravity compactification that could give rise to this spectrum (there seems to be a qualitative difficulty in making such an identification, as we describe in more detail in section 8.2).

Superconformal deformations of the theory

There exists a manifold of exactly marginal deformations [17] of the theory described in the previous paragraph. This manifold can be characterized rather precisely in the neighbourhood of the fixed point using the recent results of [25]. We have computed the spectrum of supersymmetric states in these deformed theories. Our results are presented in Tables (17), (18) and (19). Qualitatively, our results for these deformed theories are similar to those described in the previous paragraph. The spins of all supersymmetric states are less than or equal to two, potentially describing the supersymmetric spectrum of a Kaluza-Klein compactification.

Other superpotentials at

Finally, as is increased superpotentials of up to order in the chiral fields eventually turn relevant (Here we use at explained above). Just as in the case of , this suggests the existence of new fixed lines with superpotentials of up to order in chiral fields turned on. In addition, at every value of there probably exist superconformal field theories with cubic superpotentials. We leave the investigation of these theories and their supersymmetric spectra to future work.

3 superconformal algebras and their unitary representations 6

In order to lay out the background (and notation) for our analysis of supersymmetric states, in this section we present a brief review of the structure of the superconformal algebras, their unitary representations and their Witten indices. We also explicitly decompose every representation of the relevant superconformal algebras into irreducible representations of the conformal algebra. The paper [21] is useful background material for this section. The reader who is familiar with the superconformal algebras and their representation theory may wish to skip to the next section.

3.1 The superconformal algebras and their Witten indices

In this section we briefly review the representation theory of the and superconformal algebras. The bosonic subalgebras of these Lie super algebras is given by (for ) and (for ). Primary states of these algebras are labeled by where is the scaling dimension of the primaries, is its spin and is its -charge (or symmetry highest weight). can be any positive or negative real number for , but is a positive half integer for .

The labels of unitary representations of the superconformal algebra obey the inequalities forced by unitarity. When the condition

is necessary and sufficient for unitarity. When unitary representations occur when

or when

The isolated representations, and those that saturate this bound, are all short.

The Witten index vanishes on all long representations of the supersymmetry algebra but is nonzero on short representations. This index captures information about the state content of a conformal field theory. The only way that the Witten index of a CFT can change under continuous variations of parameters (like the parameter in our theory), is for the -charge to be renormalized as a function of that parameter. Note that the -charge is fixed to be a half integer at , but can in principle be continuously renormalized at .

In the case of theories, we have glossed over a detail. At the purely algebraic level, in this case, there are really two independent Witten indices; and . These are defined as

and

respectively. We used the notation for the dilatation operator, the third component of the spin, and the symmetry generator. The indices above are distinct, even though they both evaluate to quantities independent of . The first index receives contributions only from states with ; all such states are annihilated by and lie in the cohomology of the supercharge with charges . The second index receives contributions only from states with ; all such states are annihilated by and lie in the cohomology of the supercharge with charges . The existence of two algebraically independent Witten Indices is less useful than it might, at first seem, in the study of quantum field theories, as the two indices are closely linked by the requirement of CPT invariance.

3.2 State content of all unitary representations of the superconformal algebra

In the rest of this section we will list the conformal representation content of all unitary representations of the superconformal algebra, and use our listing to compute the index of all short representations of this algebra.

To start with, we present a group theoretic listing of the state content of an antisymmetrized product of supersymmetries. This is given in Table (1).

Operator States
,
, , ,
,
Table 1: A decomposition of the antisymmetrized products of supersymmetries into irreducible representations of the maximal compact bosonic subgroup of the relevant superalgebras. Representations are labeled by where is the scaling dimension, the angular momentum (a positive half integer) and the -charge of the representation. The same labeling convention is used in all tables in this section.

The conformal primary content of a long representation of the superconformal algebra is given by the Clebsh Gordon product of the state content of the product of susy generators above with that of the primary. We list the conformal primary content of an arbitrary long representation in Table (2).

Primary Conformal content Index
,
, ,
, ,
, , ,
, ,
, ,
, ,
,
, ,
, , ,
, ,
,
Table 2: Conformal primary content of long representations. Representations are labeled by where is the scaling dimension, the angular momentum (a positive half integer) and the -charge of the representation.

Note that a long representation of the susy algebra decomposes into long representations of the conformal algebra when ; when we must delete the representations with negative values for the highest weight ( and ) from the generic result leaving us with a total of conformal representations. The Witten index of all long representations automatically vanishes.

Let us now turn to the short representations. To start with consider representations with , and . These representations are short because they include a family of null states. These null states themselves transform in a short representation of the superconformal algebra, with quantum numbers (when is positive) and (when is negative). It is not too difficult to verify that the conformal primary content of such a short representation (represented by ) and the Witten index of these representations is as given in Table (3) (we list the result for positive ; the result for negative follows from symmetry).7 Note that all conformal primaries that occur in this decomposition are long (recall we have assumed ).

Primary Conformal content Indices (; )
, ,
(, ) , , ;
, , ,
, ,
Table 3: Conformal primary content and index of generic short representation. Representations are labeled by where is the scaling dimension, the angular momentum (a positive half integer) and the -charge of the representation.

It is not difficult to verify that . This expresses the fact that the state content of a long representation just above unitarity is equal to the sum of the state content of the short representation it descends to plus the state content of the short representation of null states.

Let us now turn to the special case of short representation . The null states of this representation consist of a sum of two irreducible representations with quantum numbers and . It is not too difficult to convince oneself that the primary content of such a short representation is as given in Table (4). Note that all conformal representations that appear in this split are short.8 It may be verified9 that .

Primary Conformal content Index
, = =
, ,
Table 4: Conformal primary content and index for representation. Representations are labeled by where is the scaling dimension, the angular momentum (a positive half integer) and the -charge of the representation.

Let us next turn to the special case of a short representation with and with quantum numbers . Such representations are often referred to as semi short, to distinguish them from the ‘short’ representations we will deal with next. We will deal with the case (the results for can then be deduced from symmetry). The primary for the null states of this representation has quantum numbers . Note that the null states transform in an isolated short representation. The state content and Witten index of a semishort representation are listed in Table (5). Note that ; this formula captures the split of a long representation into the short representation and null states.

Primary Conformal content Index(; )
, , ;
, ,
, , , ,
Table 5: Conformal content and index of representation . Representations are labeled by where is the scaling dimension, the angular momentum (a positive half integer) and the -charge of the representation.

The short representation with primary labels is a bit special; its null states have primaries with quantum numbers and (these are isolated short representations, see below). The state content and Witten index of this representation are given in Table (6). Of the conformal primaries that appear in this split, only the representation with quantum numbers is short.10 Using the results we present below, it is possible to verify the character decomposition rule:11

Primary Conformal content Index
,
,
Table 6: Conformal content and index for representation . Representations are labeled by where is the scaling dimension, the angular momentum (a positive half integer) and the -charge of the representation.

Now let us turn to the isolated short representations for . The primaries for the null states of these representations have quantum numbers . The null states transform in a (short) non-unitary representation, reflecting the fact that the isolated representations cannot be regarded as a limit of unitary long representations but can be regarded as the limit of non-unitary long reps. The conformal primary content and Witten indices for these representations are given in Table (7). Here we have written conformal content for positive; the result for negative is given by symmetry.

Primary Conformal content Index(; )
, , ;
, ,
Table 7: Conformal content and index for representation .

Recall that for the representations we have just discussed. The lower bound of this inequality, , is a special case. The conformal decomposition and index for the and cases are given in Table (8).

Primary Conformal content Index
, ,
,
Table 8: Conformal content and index for representations and

3.3 Decomposition of all unitary representations of the algebra into representations

In this subsection we record the decomposition of all representations into representations. Representations of the algebra are labeled as , where is the highest weight under the Cartan of the symmetry (normalized to be a half integer). A generic long representation with breaks as follows

(3)

where denotes a direct sum and denotes the direct sum of representations.

Here the subscript denotes of the algebra; i.e. denotes a representation of the algebra, while denotes a representation of the algebra.

The summation outside the brackets on the RHS of (3) reflects the fact that a primary that transforms in a given irreducible ( symmetry in algebra) representation consists of several different primaries (with distinct -charges). The four terms in the bracket on the RHS of (3) represent the states obtained by acting on the primary with supercharges that belong to the algebra, but are absent in the algebra.

The decomposition (3) may be rewritten as follows:

(4)

In this equation we have grouped together terms on the RHS for the following reason. Recall that the decomposition of a long representation - with - at the unitarity bound, into short unitary representations of the superconformal algebra, is given both at and at by

(5)

This formula should apply to (3.3). Comparing (5) and (3.3), it is plausible (and correct) that the generic short representation decomposes into representations according to the formula

(6)

where all representations that saturate the unitarity bound, on the RHS of (6), are short.

We may deduce the split of a generic short representation into representations of the algebra using identical reasoning. To start with we note that long representation at splits up into long representations as follows

(7)

We next note that, both in the and the algebras,

(8)

These two equations allow us to deduce that, for short representations,

(9)

In the equation above, representations that saturate the BPS bound are short.

The breakup of the isolated short representations needs slightly more indirect reasoning to deduce; we simply present the final result:

(10)

The complete decomposition of the algebra into representations is given in Table (9) for long representations and Table (10) for short representations.

Spin primary primaries
Table 9: Decomposition of long representations into representations.
Spin primary primaries
Generic short
Isolated short
Table 10: Decomposition of short representations into representations.

4 The -charge as a function of in the absence of a superpotential

It was explained in [5] (see the introduction) that the Chern Simons theory with chiral multiplets and no superpotential, is superconformally invariant at every value of and , and so at every value of , in the large limit. In the free limit the -charge of each of the chiral multiplets in this theory is equal to half. As was explained in [5], however, this -charge is renormalized as a function of . As the -charge of an operator appears in the BPS formula that determines its scaling dimension, the determination of the charge of a chiral field, as a function of , is perhaps the most elementary characteristic of the supersymmetric spectrum of the theory. In this section we will adopt a proposal by Jafferis [19] to perform this determination.

4.1 The large saddle point equations

According to the prescription of [19], the superconformal -charge of the theories we study is determined by extremizing the magnitude of their partition function on with respect to the trial -charge,12 , assigned to a chiral multiplet. The partition function itself is determined by the method of supersymmetric localization to be given by the finite dimensional integral

(11)

where is the ’t Hooft coupling, () are real numbers (and the integration range is from to ), , and the function is given by

(12)

where is the dilogarithm function. While the function is complicated looking, its derivative is elementary

and is all we will need in this paper.

According to [19], once the partition function is obtained by performing the integral in (11), the -charge of the chiral fields is determined (up to caveats we will revisit below) by solving the equation . This gives the exact superconformal -charge.

In the large limit the integral in (11) may be determined by saddle point techniques. The saddle point equations, together with the equation (which determines , given the saddle point) are given by

(14)

4.2 Perturbative solution at small

While we have been unable to solve the equations (4.1) in general even at large , it is not difficult to solve these equations either at small (at all ) or at large (for all ). In this subsection we describe the perturbative solution to these equations at small (for all ). In the next subsection we will outline the perturbative procedure that determines at all but large .

It is apparent from a cursory inspection of (4.1) that the eigenvalues must become small in magnitude (in fact must scale like ) at small . It follows that complicated functions of simplify to their Taylor series expansion in a power expansion in . This is the basis of the perturbative technique described in this subsection.

More quantitatively, at small we expand and as

(15)
(16)

and attempt to solve our equations order by order in . At leading nontrivial order, , equation (14) reduces to

(17)

which tells us that at leading order in . On the other hand, equation (4.1) at its leading nontrivial order, namely , reduces to

(18)

Apart from an unusual factor of , this is precisely the large saddle point equations of the Wigner model. The extra factor of may be dealt with by working with the rescaled variable

in terms of which

(19)

The solution to this equation is well known in the large limit. The eigenvalues cluster themselves into a “cut” along the interval with

The density of eigenvalues, , in this interval is given by

(20)

Using the fact that , we see that, at leading order in , the saddle point is given by the eigenvalues clustering along a straight line of length of order , oriented at degrees in the complex plane.

Note that the distribution of eigenvalues has symmetry and in particular the average value of eigenvalues is zero. The symmetry is an exact symmetry of the saddle point equations, and we will assume that it is preserved in the solution (i.e. not spontaneously broken) in the rest of this paper.

Let us now proceed beyond the leading order. (14) is automatically satisfied at (this is because ). At order the same equation reduces to

(21)

Now recall that the phase of is . As a consequence the real part vanishes and hence, from (21), .

In order to compute the correction to at we need to find the first correction to the eigenvalue distribution. We now turn to that task. At order , (4.1) reduces to