Modularity in Argyres-Douglas Theories with a = c

: We consider a family of Argyres-Douglas theories, which are 4D N = 2 strongly coupled superconformal field theories (SCFTs) but share many features with 4D N = 4 super-Yang-Mills theories. In particular, the two central charges of these theories are the same, namely a = c . We derive a simple and illuminating formula for the Schur index of these theories, which factorizes into the product of a Casimir term and a term referred to as the Schur partition function. While the former is controlled by the anomaly, the latter is identified with the vacuum character of the corresponding chiral algebra and is expected to satisfy the modular linear differential equation. Our simple expression for the Schur partition function, which can be regarded as the generalization of MacMahon’s generalized sum-of-divisor function, allows one to numerically compute the series expansions efficiently, and furthermore find the corresponding modular linear differential equation. In a special case where the chiral algebra is known, we are able to derive the corresponding modular linear differential equation using Zhu’s recursion relation. We further study the solutions to the modular linear differential equations and discuss their modular transformations. As an application, we study the high temperature limit or the Cardy-like limit of the Schur index using its simple expression and modular properties, thus shedding light on the 1/4-BPS microstates of genuine N = 2 SCFTs with a = c and their dual quantum gravity via the AdS/CFT correspondence.


Introduction
Supersymmetric and superconformal field theories have been attracting intensive interest due to the feasibility of performing exact computations.Many protected sectors have also been discovered in supersymmetric and superconformal field theories, which turn out to enjoy rich mathematical structures.In the zoo of supersymmetric field theories, the 4D N = 2 superconformal field theories are of particular interest.On the one hand, the low energy limit on the Coulomb branch of 4D N = 2 superconformal field theories (SCFTs), which is a special Kähler geometry, can be effectively described using the famous Seiberg-Witten theory.On the other hand, the 4D N = 2 superconformal field theories also possess an interesting protected sector, known as the Schur sector, which encodes the information about Higgs branch.The Schur sector is particularly interesting because there is a nice correspondence between the Schur sector and the mathematical notion of vertex operator algebra (VOA), dubbed SCFT/VOA correspondence [1].One important item in validating this correspondence is given by the so-called Schur index, which just counts the Schur operators in the SCFTs.In general, the Schur index has the following structures: To distinguish the two quantities, we will refer to I as Schur index, and Z as Schur partition function.
While the Schur index I is more meaningful for counting and starts with 1 for the unique vacuum state, it is the Schur partition function Z that is identified with the vacuum character χ vac of the corresponding VOA following the SCFT/VOA correspondence [1].The Casimir factor e −c 2d /24 in Z is crucial to ensure the nice modular behavior in the chiral algebra. 1he superconformal index can be computed in many theories using various techniques.Once the Schur index is known, the Schur partition function can be obtained trivially by multiplying the Casimir factor.The multiplication is straightforward but appears somewhat artificial.It is natural to inquire whether the factorization of the Schur index (1.1)can be achieved in a more natural manner.We will demonstrate that the factorization of Schur index indeed naturally emerges in a family of Argyres-Douglas (AD) theories, which will be denoted as T (p,N ) with two coprime integers p = 2, 3, 4, 6 and N = 2, 3, 4, • • •.
The Argyres-Douglas theories T (p,N ) can be obtained by conformally gauging several copies of AD theories D p i (SU (N )) [2,3] which have flavor symmetry SU (N ), where the set of p i depends on the value of p.The Argyres-Douglas theories T (p,N ) share many features with N = 4 SU (N ) super-Yang-Mills (SYM) theories [4,5].In particular, the two central charges are exactly the same a = c.More surprisingly, the Schur index of T (p,N ) can be obtained from the Schur index of N = 4 SU (N ) SYM theory by specializing its fugacities to particular values.Meanwhile, the Schur index of N = 4 SU (N ) SYM admits a closed formula when the flavor fugacities are turned off [6].The generalization to flavored Schur index was established in [7].With the closed formula for the flavored Schur index of N = 4 SU (N ) SYM, we derived the Schur index of T (p,N ) in Eq. (2.62) (2.63).Surprisingly, the Schur index derived in this way is very simple and enjoys the obvious and natural factorization of the form in (1.1).We are then led to a remarkably simple formula for the Schur partition function of T (p,N ) AD theory.
Following the SCFT/VOA correspondence, the Schur partition function is identified with the vacuum character of the corresponding chiral algebra.In particular, this implies that Schur partition function satisfies the modular linear differential equation (MLDE) [8], D (k) q Z = 0 where D (k) q is the modular linear differential operator of weight 2k (see Appendix A for details).The modular linear differential equations transform covariantly with specific weight under modular transformations, significantly constraining the structure of modular linear differential equations.At a given weight, the modular linear differential equations are almost fully determined, up to several constants.Such a kind of simplicity allows one to numerically search for the modular linear differential equations which are satisfied by the Schur partition function.The numerical search becomes even simpler and tangible thanks to the simple closed form of the Schur partition function of T (p,N ) AD theory.To illustrate, we will find the MLDEs in several examples and study their solutions.
At a fundamental level, the modular linear differential equation is rooted in the existence of a specific kind of null state in the corresponding VOA.Once the null state is known, one can derive the modular linear differential equations systematically using Zhu's recursion relation [9] and the commutation relations in the chiral algebra.We will review the details in Appendix B. Unfortunately, the chiral algebra of T (p,N ) AD theory is generally complicated and not known explicitly, except for the simplest case of T (3,2) AD theory.The VOA of T (3,2) AD theory is known [10,11].With the explicit OPEs at hand, we manage to find the desired null state, and derive the corresponding MLDE of weight 10.Such a MLDE can be verified numerically to very high order.
As a physical application of MLDE, we further study the high temperature limit of the Schur index / Schur partition function.The high temperature limit of the Schur index of 4D N ≥ 2 SCFTs has been studied previously in [12,13] for N = 4 SYM theories and some other N = 2 SCFTs.The T (p,N ) AD theories of interest in this paper have central charges a = c and are honest 4D N = 2 SCFTs without enhanced supersymmetry.So understanding the high temperature limit of the Schur index of T (p,N ) AD theories is important and complementary to the examples studied before.The modularity of solutions to MLDE enables one to derive the high temperature limit systematically.
Based on examples studied in this paper and some results in the literature, we are motivated to propose some conjectures on MLDE and particularly a power law asymptotic behavior for the Schur partition function in (4.29), which in the special case of p = 2 can be proved using our simple formula for Schur partition functions.
The rest of the paper is organized as follows.In section 2, we will first introduce the T (p,N ) AD theories that will be studied in this paper.Then we will derive a simple and illuminating formula for the Schur index of T (p,N ) which admits an obvious factorization (1.1).In section 3, we will study the modular linear differential equations in T (p,N ) .We will also discuss the solutions to the MLDEs and their modular transformation behavior.In section 4, we will study the high temperature limit of the Schur index / partition function based on the MLDEs and modular properties.In section 5, we will summarize the main results of the paper and discuss several open questions for future explorations.We also include a few technical appendices.In appendix A, we will review the concepts of Eisenstein series and modular forms.We will also discuss the general structures of MLDEs and their solutions.In appendix B, we review the basic concepts of VOA, Zhu's recursion relation for torus one-point function, and discuss in detail how to derive MLDE from the null state of the VOA.In appendix C, we will present the explicit MLDEs in several families of AD theories, which are either found numerically or derived rigorously from the null state of the corresponding VOAs.

Schur partition function of T (p,N ) Argyres-Douglas theory
In this section, we will first review the properties of D p (SU (N )) AD theories, which are the building blocks of T (p,N ) AD theories.After discussing the construction and properties of T (p,N ) AD theories, we will then derive a simple formula for the Schur index of T (p,N ) AD theory.

D p (SU (N )) AD theory
We will start with a specific type of 4D N = 2 SCFTs denoted by D p (G), which was introduced in [2,3].In particular, in this paper we will be mainly focusing on the case of G = SU (N ), namely D p (SU (N )) where p, N > 1 are positive integers.Moreover, we will always impose the constraint that p and N are coprime.In this case, the flavor symmetry of D p (SU (N )) is SU (N ). 2he D p (SU (N )) theories are generalized AD theories [14,15] and have no direct N = 2 Lagrangian description.But they have class S realization: one can compactify the 6D (0,2) SCFT of type A N −1 on a Riemann sphere with an irregular puncture and a full regular puncture.To manifest this construction, one can equivalently use the following notation [15,16] where F means full puncture which is responsible for the SU (N ) flavor symmetry.The condition gcd(p, N ) = 1 ensures that the irregular puncture has no further contribution to flavor symmetry.The flavor central charge k F and c central charge are [3] while a central charge can be computed via It turns out for our theories with gcd(p, N ) = 1, there a simple formula for a central charge:

T (p,N ) AD theory
Suppose we take a collection of D p i (SU (N ))) theory and gauge the maximal diagonal flavor symmetry, the total one-loop where the first contribution comes from the vector multiplet and the second contribution comes from the individual D p i (SU (N )) theory [2].
In order to get a superconformal theory after gauging, we require that the beta function of gauge coupling should vanish, b = 0.This is equivalent to the requirement that i k Dp i (SU (N )) F = 4h ∨ (SU (N )) = 4N .Then one can easily show that the only possibilities are These conformally gauged AD theories are the four series of theories that we will consider in this paper.We will label them respectively as See figure 1 for the four series of theories.Therefore, the theories that we shall study is generally denoted as T (p,N ) where p = 2, 3, 4, 6 and N = 2, 3, 4 • • • subject to the condition that gcd(p, N ) = 1.
See [5] for further generalizations and an alternative notation for this family of theories that we show in table 1.
A remarkable feature of these theories is that the a and c central charges are exactly the same.
Then it is easy to see that a − c = b(N 2 − 1)/48 = 0.This property is reminiscent of the N = 4 SYM theories which have a = c due to maximally superconformal symmetry.
T (2,N ) : T (4,N ) : T (6,N ) : When p = 3, 4, 6, we can also realize T (p,N ) AD theory by considering IIB string compactified on isolated hypersurface singularity (IHS), which is a Calabi-Yau three-fold defined in terms of a quasi-homogeneous polynomial in C 4 [17].This geometric engineering way of constructing T (p,N ) AD theories is very useful.For example, one can easily compute the Coulomb branch spectrum from the deformation of singularity.The Higgs branch dimension can also be obtained from the resolution of the singularity [18].By considering small values of N , one can compute the Higgs branch dimension explicitly. 3The dependence on N turns out to be very simple and we find that the Higgs branch dimensions of T (p,N ) AD theories fit into the following simple formula where • is the floor function.Note that for p = 2, there is no known type IIB realization.However, it was argued in [19] that the T (2,2k+1) AD theories have Higgs branches of quaternionic dimension k, which is consistent with (2.10).
We list some properties of T (p,N ) SCFTs in table 1.In particular, we find that the central charges are uniformlly given by The notations and properties of T (p,N ) theories.Here N > 1 and k, N ∈ N.
Among the four infinite series of theories, there are several special theories which admit other constructions.For example, we have where the last notation comes from the class S realization by compactifying 6d (0,2) SCFT of type D 4 on a Riemann sphere with just one irregular puncture (and without any regular puncture) [15,20].The notation (A 2 , D 4 ) also comes from the type IIB realization, which is different but equivalent to the type IIB realization in table 1.Similarly, the other two special theories are

Schur partition function
In 4D N = 2 SCFTs, the Schur index is a useful quantity to count the Schur operators which are where the E is the energy, R is the SU (2) R weight, F is the fermion number, x j and F j are the fugacity and generators for flavor symmetries.The Schur index of D p (SU (N )) takes a very simple form [16]: where x is the fugacity for SU (N ) flavor symmetry, and χ is the character of SU (N ) in the adjoint representation.
The index of gauged theory can be obtained by taking the product of the individual matter components and vector multiplet contributions, and then projecting to gauge invariant sector.For our theory T (p,N ) , the index is given by where the Haar measure for SU (N ) is and the vector multiplet contribution is The notation PE stands for plethystic exponential defined by After combining all the contributions, the Schur index of T (p,N ) AD theory then takes the following explicit form On the other hand, the Schur index of N = 4 SYM, which can be thought as an N = 2 theory with one vector and one hyper transforming in the adjoint representation, is: where the hypermultiplet contribution is and y is the fugacity of the SU (2) F symmetry arising from the SU (4) R symmetry of N = 4 SYM.It is not obvious, but easy to verify that the indices of T (p,N ) and N = 4 SYM theories are related in a simple way .24)This index relation between AD theories and N = 4 SYM theories was first observed in [4].In the special case of p = 3, N = 2, it has been understood as a consequence of the operator map in VOAs of two theories.We will use this index relation to derive a simple formula for the Schur index of T (p,N ) theory.For this purpose, we need to first review the known closed formula for the Schur index of N = 4 SYM theory, which turns out to be simpler for U (N ) gauge group.
In [6], a closed form expression for the unflavored Schur index of N = 4 U (N ) SYM theory was given where and the q-Pochhammer symbol One useful property that we will use later is that in the limit x → 1, (1 − q j ) = (q, q) ∞ . (2.28) In [7], the generalization to the case of Schur index with flavor fugacity turned on was achieved using the fermi gas approach.For convenience, we introduce ξ via y = √ q/ξ, then the flavored Schur index of N = 4 U (N ) SYM theory can be written as where and Z N is defined via the generating function (2.31) Here u is an arbitrary parameter whose contribution to index cancels out finally.We will specialize u to different values in order to simplify the formula.Let us first consider the limit u → 1, but keep q, ξ generic.Using (2.28), we find (2.30) simplifies where Meanwhile, (2.31) becomes ) where and we have Combining (2.32) and (2.36), the N = 4 U (N )) SYM index (2.29)can be written as Taking N = 1, we get where we used (2.33) and (2.35).With this U (1) index, we can get the Schur index of N = 4 SU (N ) SYM theory: (2.39) where (2.41) Now we can use the index relation (2.24) to get the Schur index of T (p,N ) theory.To distinguish, we will use Q in place of q as the argument for the Schur index of T (p,N ) .Then (2.24) implies (2.42) Therefore we must set ξ = Q and (2.44) To proceed further, we take an alternative limit u → Q p .Using (2.28), we get (2.45) and where This implies (2.48) Combining (2.45) and (2.48) together, we get To further simplify the expression, we introduce λ = −µ/Q and I N via (2.50) It is easy to see that I N = (−Q) N −1 J N .Substituting it to (2.49), we get For convenience of notation, we change variables back to q and µ and get where 1 + µq j 1 − q jp .(2.54) Surprisingly, the formula can be further simplified for the T (p,N ) AD theories studied here.In particular, the values of p, N are restricted such that p = 2, 3, 4, 6 and gcd(p, N ) = 1.So in all cases we always have N = rp ± 1 for r ∈ Z.If N = rp + 1, we have (2.57)
We can then redefine Z T (p,N ) (q) = (−1) N p Z T (p,N ) (q) and get the final formula for the Schur index where Z T (p,N ) (q) is given by the generating function (2.63) Equivalently, we can write the generating function as Obviously, (2.62) has exactly the factorized form in (1.1).The factorization emerges naturally from our derivation. 4As we discussed in the introduction, the Schur partition function Z T (p,N ) (q) is identified with the vacuum character of the corresponding chiral algebra, and is expected to have nice modular properties.The Z T (p,N ) (q) defined via (2.63) is very simple, suggesting that it defines an elementary function with nice modular properties and deserves further studies.
The Schur partition function Z T (p,N ) (q) can be more explicitly written as (2.65) When p = 2, it is easy to see that m i and m j = −m i must appear in pair in each summand, otherwise, the two summands with m i and −m i would cancel.One can also see this point from the expression in (2.64).Therefore, we should have N = 2k + 1, and (2.66) Note that the sign factor (−1) N p = (−1) k precisely cancels the sign from the product, rendering a function with positive coefficients in q.In this case, Z T (2,2k+1) (q) turns out to be related to the known function via Z T (2,2k+1) (q) = A k (q 2 ), 5 where is known as the MacMahon's generalized 'sum-of-divisor' function [21].It satisfies the recursion relation [22] A and Here E 2k are Eisenstein series (A.6).This implies A k (q) is a quasi-modular form, and can be written as the polynomial of Eisenstein series E 2 , E 4 , E 6 .See appendix A for definitions of various concepts. 6or p = 3, N = 2, we have (2.70) = q 1 + q 2 + q 3 + 2q 6 + q 8 + q 11 + 2q 12 + q 15 + • • • . (2.71) For p = 4, N = 3, we have Note that (2.63) is derived from physics and the values of p and N are subject to various conditions.But at the level of mathematics, (2.63) applies to all p and N .Some may be trivial.For example, one can check that and more generally we expect Z T (p,jp) = 0 for j ∈ N. On the other hand for p = 4, N = 2, Z T (4,2) (q) is not zero and furthermore satisfies the MLDE of weight 18.It would be interesting to study these properties further.

Modular linear differential equation
In this section, we will study the modular linear differential equations in T (p,N ) AD theories.In the simplest case of T (3,2) whose VOA is known, we will derive the corresponding MLDE using the formalism in Appendix A and B. In other cases, we will present the modular linear differential equations satisfied by Schur partition function, which are found through numerics.We will also discuss the solutions to the MLDEs and their modular transformation properties.
The OPEs among them are given by where we ignore the coordinate of operators on the RHS, which is 0. Actually the full OPE can be easily bootstrapped using the associativity of OPEs and the information about the conformal dimensions of these operators.
The mode expansion of these operators are Using the previous OPEs, we manage to derive the commutation relations of these modes ) where 7 7 The Λ, Λ, Υ constructed in this way have simple commutation relations with stress tensor.E.g. [Lm, Λn] = With the explicit OPEs, we can try to find the null operators.See appendix C.3 for detailed discussions.In particular, we find a null operator at dimension 10: In terms of modes, the corresponding null state is This null state has the form of (B.13), namely The presence of such a kind of null operator enables us to derive the MLDE using Zhu's recursion relation (B.9)(B.10).We defer the detailed discussions and derivations to the appendix C.3, and only provide the final result here.
The resulting MLDE we find takes the following simple form where E 2k are Eisenstein series (A.6), and are modular covariant differential operators (A.20).See appendix A for review and discussion on the notations and properties.
The explicit and simple expression of Z T (3,2) is given in (2.70).One can then numerically verify that the above MLDE is indeed true.
We would like to find the full set of solutions to MLDE, which correspond to the characters of some modules in the corresponding A(6) chiral algebra.We can use the following ansatz Then we get the following indicial equation Due to the degeneracy and integral spacing of the roots, the ansatz (3.15) is not valid in general.instead we should use the following ansatz: where N b depends on the structure of roots to the indicial equation.See appendix A.3 for more discussions.
We then find the following set of solution to MLDE (3.14) b = 0 : b = 0 : χ log 0 = log q 1 + 6q + 6q 3 + 6q 4 + 12q 7 + 6q 9 + 6q 12 + 12q 13 + 6q 16 = q 1 3 1 + q + 2q 2 + 2q 4 + q 5 + 2q 6 + q 8 + 2q 9 + 2q 10 + 2q 12 + 2q 14 + 3q 16 b = 1 : = q + q 3 + q 4 + 2q 7 + q 9 + q 12 + 2q 13 + q 16 + 2q 19 Note that since there is no constant term in MLDE (3.14), namely the coefficient of q is zero, χ 0 = 1 is also a solution.One can also see that The five linearly independent solutions above are the full solutions to the MLDE.They are also the characters of some modules in the corresponding chiral algebra.And we expect that they form a (weakly holomorphic logarithmic) vector-valued modular under modular transformation.More precisely, we can consider the vector of solutions Then, under modular S-and T -transformation, we expect where S, T are modular S and T matrices.The modular T matrix can be easily derived.In the absence of log term, χ b transforms to e 2πib χ b .In the presence of log term, χ b gets mixed with other items in χ, and the modular T matrix is not diagonal anymore.For example, it is easy to see that As a result, we find the modular T matrix The modular S matrix is generally more complicated to compute.Fortunately, we find that some entries in χ reduce to known functions whose modular transformation is understood.By numerically computing the solution χ 1 3 to very high order 8 , we find that it actually can be written as where η(τ ) = q ) is the Dedekind eta function.Under modular transformation, the Dedekind eta function transforms as With the help of these formulae, we can easily find that under S-transformation, the solution (3.28) becomes where we write the S-transformed character as the linear combination of the original untransformed characters in light of (3.25); this can be easily achieved by comparing the coefficients of their qexpansions.
Similarly, χ log transforms as The modular S matrix has to satisfy the condition S 2 = 1.Imposing such a constraint, we are able to determine S matrix completely: Note that the S matrix is not symmetric.It is easy to verify that S 2 = (ST ) 3 = 1, as required by modular S and T matrices. 10 As a result, we find Using (3.23), we then find ) which agrees with (3.32).This thus provides a strong consistency check of our results.

T (2,3)
For the T (2,3) AD theory, the corresponding VOA is not known.So we will resort to numerics to find the corresponding MLDE.The Schur partition function is given by (2.66): This simple expression allows one to numerically compute the series expansion to very high order efficiently.Furthermore, the SCFT/VOA correspondence indicates that Z T (2,3) satisfies the MLDE of specific weight, say 2k: where is the modular linear differential operator and transforms covariantly under modular transformations.More explicitly, it takes the following form (A.22) (3.37) 9 One can rescale the character by replacing log q with log q/(6 √ 3π) and get the modular S matrix with rational coefficients.But this would introduce the factor √ 3 to T matrix. 10More generally, the condition is S 2 = (ST ) 3 = C where C is the charge conjugation matrix.
Here M 2r (Γ) denotes the modular form of weight 2r, and is freely generated by E 4 and E 6 .So f r can be written as the polynomial of E 4 and E 6 with finitely many undetermined coefficients.Therefore, the MLDO D (k) q is almost fixed completely by modular covariance, up to a finite number of coefficients.To fix these coefficients, we can consider the series expansion of both D (k) q and Z T (2,3) in q, and check MLDE in (3.36).Starting with k = 1 in (3.36), one can check whether it has is a solution.If yes, we then find the desired MLDE, otherwise we increase k and repeat the same procedure.This then offers an efficient way to find MLDE numerically.
After implementing the algorithm above, we find that up to the order q 100 , there is indeed a MLDO of weight 24 which annihilates the Schur partition function Z T (2,3) , namely where D (12)   q = D (12)   q − 1510E 4 D (10)   q q . (3.39) Note that as in the case of T (3,2) AD theory, there is also no constant term in D here.The MLDE above is a differential equation of order 12, so it should have other 11 solutions corresponding to other modules of the VOA, in addition to the vacuum character Z T (2,3) .To find the rest of solutions, we can use the ansatz χ i = q b (1 + a 1 q + • • •) and substitute it into D (12) q χ i = 0.At leading order in q, this gives the indicial equation (3.40) The solutions can be easily found to be (ignoring multiplicity) In particular, note that b = 0 is the minimal root and has degeneracy 3.In principle, one can proceed further and find the full set of solutions order by order in q.In practice, this computation is tedious as the MLDO is a differential operator of very high order being 12.The computation is further complicated by the degeneracies of the roots which means logarithmic term in the solutions.Given these complications, we will not study the explicit solutions here, and just be content with the indicial equation.As we will discuss later, the indicial equation is already useful enough and can be used to understand the high temperature limit of the Schur index / partition function.

T (4,3)
We can use the same numerical algorithm as before to find the MLDE in T (4,3) AD theory.It turns out that up to the order of q 100 , the Schur partition function (2.72) satisfies the following MLDE of weight 34 where the explicit form of MLDO D (17) q is given in (C.73).
Similarly, we find the indicial equation where The set of solutions is given by (ignoring multiplicity) where β s are the four roots of g(b) = 0.In particular, note that b = 0 is the minimal root and has degeneracy 3.

High temperature limit
One virtue of modularity is that it relates states in the UV to that in the IR, which allows one to infer the high energy or high temperature behavior.For example, modular invariance of 2D CFT gives rise to the Cardy formula which characterises the high energy density of states universally in terms of central charge [25].The same philosophy applies here for the Schur partition function of 4D N = 2 SCFTs, as we will discuss below.
We would like to understand the behavior of the Schur index I(q), or equivalently the Schur partition function Z(q), in the limit τ → 0, which will be referred to as the high temperature limit or Cardy limit.
In many cases, the leading asymptotic of the Schur index / partition function is [26,27] It has been observed that this is valid in many example where a − c < 0, but violated in a few examples when a − c > 0. See [13] for more discussion on this point.The theories studied here have exactly a = c, so the formula (4.1), if correct, would predict a finite constant leading term in the limit τ → 0 or equivalently q → 1.
For N = 4 SYM theories which have a = c, the Cardy limit of the index has been studied a lot.In particular, the Schur partition function of N = 4 SU (2) SYM with central charge a = c = 3/4 has the leading asymptotic [28] which is obviously different from (4.1).This indicates that (4.1) may be not valid in theories with equal central charges a = c.However, the N = 4 SYM theories are special as they have enhanced SUSY.
Instead the T (p,N ) AD theories studied here are genuine 4D N = 2 SCFTs with a = c.Of course, one can try to infer the high temperature behavior of Schur index of T (p,N ) theories based on their index relation with N = 4 SYM.However, except for the special case of p = 2, this requires the knowledge of high temperature limit of flavored Schur index of N = 4 SYM, which is generally not known. 11Given this fact, we will study the high temperature asymptotic behavior of Schur index of T (p,N ) theories directly using MLDE and modular property.In the case of p = 2, it turns out that we can derive the high temperature limit for all T (2,2k+1) using recursion relation and the defining generating function.These results motivate us to make some conjectures about MLDE and high temperature limit of Schur partition function.

T (3,2)
Let us first consider the T (3,2) AD theory, whose modular properties have been discussed extensively in subsection 3.1.We wan to understand the high temperature limit τ → 0 of Schur partition function Z T (3,2) (τ ), which is identified with the vacuum character χ 1 .For our purpose of application, we rewrite (3.33) as This can be used to study the behavior of the Schur partition function in the high temperature limit, namely τ → 0, τ = −1/τ → ∞, q → 1, q = e 2πiτ = e −2πi/τ → 0. In this limit, it is easy to see that χ 1 (τ ), χ 1 3 (τ ) → 0 up to exponentially small corrections.Therefore in the high temperature limit τ → 0, we have up to exponentially suppressed corrections.

T (2,2k+1)
We now derive the high temperature asymptotic behavior of the Schur partition function of T (2,2k+1) AD theories.We will provide two ways to derive it.The first way to derive is to use Z T (2,2k+1) (q) = A k (q 2 ), where A k (q) is MacMahon's generalized 'sum-of-divisor' function defined in (2.67) and satisfies the recursion relation (2.68) and (2.69).It turns out these formulae are useful enough to derive the asymptotic behavior of A k (q).
Let us first derive the asymptotic behavior of A 1 (q).Using the modular behavior of E 2 in (A.7), we find Meanwhile, we have where τ = − 1 τ → ∞.Combining them together, we get up to exponential corrections.Further using (2.69), we get which gives the asymptotic behavior of A 1 (q).We can find similar formula for other A k (q) by using the recursion relation (2.68) Obviously, we can insert (4.8) into this recursion relation and get the asymptotic behavior of A 2 (q).Repeating the procedure in a recursive way, we can find the asymptotic behavior of all A k (q).For simplicity, let us focus on the most singular terms of A k (q) in the limit τ → 0. By noticing that A 1 ∼ − 1 24τ 2 and q d dq = τ d dτ , we easily see that the most singular term in (4.9) is With this relation, we can easily derive the following asymptotic behavior where we also show the large k limit of the coefficients using Stirling's formula.This also gives the high temperature asymptotic behavior of the Schur partition function 12) It is very straightforward to generalize the above derivation and computing all the subleading corrections.For example, at sub-leading order, we have We now give another derivation of the above asymptotic formula based on the defining generating function (2.63), which will be denoted as Ω p (µ, τ ): (4.14)In the particular case of p = 2, we have ) where we used Z T (2,2k) = 0 and Z T (2,1) = 1.We would like to use this formula to derive the high temperature limit τ → 0 of Z T (2,2k+1) .It turns out that in the generating function Ω 2 , we need to consider the double scaling limit by taking µ, τ → 0 but keeping the ratio µ/τ fixed.In such a limit, the multiplicand in the infinite product of (4.15) reduces to Substituting it back to the infinite product, we get12 where we used the formula (4.18) One can then perform series expansion in µ in order to get the asymptotic behavior of Z T (2,2k+1) in (4.12).Equivalently, we will show that (4.12) gives the same generating function (4.17).
Indeed, from (4.12) we have where # and dots represent less singular terms and exponentially suppressed terms.Substituting it back to the generating function (4.15), we get ) which exactly recovers (4.16).Note that in the above formula, we also need to take the double scaling limit µ, τ → 0 with µ/τ fixed, in order to suppress the contribution from the less singular terms in (4.19).
In the case of k = 1, (4.12) gives which has also been verified numerically.

General case
To study the high temperature limit of Schur partition function, the key point is to use the modular transformation of characters: where 1 corresponds to the vacuum, and χ j are the characters of some modules of the corresponding VOA, which are also the solutions to MLDE.Once we understand the behavior of χ j (τ ) in the limit of τ → ∞ or equivalently q = e 2πiτ → 0, and the modular S-matrix, we can establish the high temperature limit of Schur partition function via (4.22).
Using the modular transformation properties, the authors in [8] proposed the following high temperature limit From our previous examples of T (3,2) and T (2,2k+1) , we know that the asymptotic behaviors in both (4.1) and (4.23) are not valid.However, (4.24) seems to be still valid.
In AD T (p,N ) theories considered here, the two central charges are the same, so c − a = 0. On the other hand, in the case of T (3,2) , T (2,3) , T (4,3) , by analyzing the indicial equation of MLDE, we do find that min i b i = 0.
In [13], a more careful analysis was done and found that the following high temperature limit for Schur partition function where d i is the degeneracy of the root b i (counting also all the roots that are less than b i by an integer), and q = e 2πiτ , τ = −1/τ .Focusing on the most singular term which gives the leading contribution, we get where b j is the minimal b i .Depending on the sign of min i b i , we can have either exponentially enhanced / suppressed leading contribution, or power law leading asymptotic if min i b i = 0.
If min i b i > 0, we see Z(q) → 0 in the limit τ → 0, which looks very unlikely as it means an almost perfect cancellation between bosonic and fermionic states.If min i b i = −α < 0, α > 0, τ = iβ, then we have exponentially growing contribution of the form e 2πα/β , namely This is reminiscent of the index for counting black hole entropy. 13Since our theories have a = c which means that they are supposed to have nice holographic dual description, such a kind of exponential contribution to Schur partition function would indicate the presence of 1/4-BPS black holes.However, in all our explicit examples, we did not see any case with min i b i < 0.
On the other hand, if min i b i = 0, we have where d 0 is the degeneracy of the root b = 0, namely in the indicial equation we have the factor b d 0 .Interestingly, we find that all the examples we studied fall into this class.For T (3,2) , the asymptotic growth is given by (4.4), while for T (2,2k+1) , the asymptotic growth (4.12).For T (2,3) and T (4,3) , their MLDEs are shown explicitly in the previous section, and their indicial relation are given by (3.40) (3.43).As a result, we do see that the min i b i = 0 and the degeneracy is 3, namely d 0 = 3 in (4.28).
Based on these discussions and the results in the literature, we are then motivated to make the following set of conjectures: 1) There is no constant term in MLDE, so χ = 1 is a solution to MLDE and b = 0 is a root to the indicial equation; 2) b = 0 is the minimal root to indicial equation; 3) The degeneracy of root b = 0 is N , so the high temperature asymptotic behavior of the Schur partition function of T (p,N ) with gcd(p, N ) = 1 is Note that this kind of asymptotic behavior has been proved for p = 2. Moreover conjecture 1) is a consequence of (4.24).
To further understand the case of other p, let us attempt to generalize the previous generating function techniques.As in (4.16), we can take the double scaling limit of the multiplicands in (4.14) where the linear term in µ is absent due to the double scaling limit.Taking infinite product, we get where we used (4.18) again.This is an even function in µ, so it would give zero to Z T (p,N ) for even N according to (4.14).But (4.4) has shown that 3τ .This discrepancy is supposed to arise from the order of taking the double scaling limits and infinite product.To exemplify this point, let us consider the µ-linear term in (4.14): This would give Z T (3,2) ∼ i 6πτ , which is slightly different from but close to (4.4); the difference in the coefficients is due to our approximation in (4.33).Nevertheless, we get the right scaling behavior consistent with conjecture (4.29) for N = 2.A very careful and systematic analysis is needed to study the general case and get the exact coefficient, and we leave this important question to the future.

Conclusion
In this paper, we studied the Schur sector of a family of AD theories denoted as T (p,N ) .The theories we have studied are interesting as they share many features with N = 4 SYM theory.In particular, the two central charges are the same a = c, indicating that they have holographic dual descriptions in terms of supergravity in AdS with some special features.
We derived an enlightening formula (2.62) for the Schur index of this family of AD theories, which naturally factors out the Casimir term.The remaining Schur partition function takes a simple form (2.63) and is expected to satisfy the modular linear differential equation.We study the MLDEs numerically based on our simple formula.We also derive the MLDE analytically and investigate the modular properties of its solution for the theory T (3,2) , whose VOA is known.Combining the modularity of MLDE with the explicit simple formula for Schur partition function, we discuss their high temperature limits.All of our explicit results suggest that the high temperature limit of Schur index / partition function of T (p,N ) AD theories diverges following a power law, rather than exponentially.This motivates us to propose a set of conjectures about MLDEs and the high temperature behavior (4.29) for general T (p,N ) AD theories.In the case of p = 2, we prove the conjecture and show the asymptotic growth explicitly (4.29).In general, the exponential growth of index in the high temperature limit is closely related to the black hole entropy in the dual AdS quantum gravity. 14ur conjecture on power law divergence (4.29) indicates the absence of 1/4-BPS black hole.
In the appendix, we also review many important concepts and useful techniques which are used for developing the results in the main body.In addition, we also present some new results there, including computing the torus one-point function to higher weight (B.27)(B.28),and deriving the explicit MLDEs for some AD theories in the family of (A k−1 , A n−1 ) and D p (SU (N )) in appendix C.1 and C.2, whose chiral algebras are W-algebras and Kac-Moody algebras.
There are some open questions.First of all, it would be interesting to study the properties of the functions (2.63) from a mathematical perspective.In particular, these functions are well-defined for general p and N (some may be trivial), rather than just for coprime integers p = 2, 3, 4, 6 and N = 2, 3, 4 • • •.We expect they all enjoy some nice modular properties.For example, one can take p = 4 and N = 2 which does not correspond to the SCFT studied here, and check that Z T (4,2) defined by (2.63) satisfies a MLDE of weight 18.It is also important to study the asymptotic behavior of these functions in the limit τ → 0. This characterises the high energy/temperature growth of 1/4-BPS states in SCFTs.We have made the conjecture in subsection 4.3, and it would be interesting to prove or disprove it.Another interesting and related limit is τ → Q.
Secondly, it would be fascinating to find the chiral algebra of T (p,N ) AD theory, and use it to derive the MLDE analytically.So far, only the chiral algebra of T (3,2) AD theory is known.The intriguing operator map between T (p,N ) AD theory and N = 4 SYM theory proposed in [4] may offer some insights and help to find or bootstrap the chiral algebra.
Thirdly, our Schur partition function is derived from the index of N = 4 SYM theory based on the index relation (2.24).Then a natural question is whether there is a direct connection at the level of MLDE.It would be amazing if one could establish an explicit map between the (flavored) MLDE of N = 4 SYM theory and the MLDE of T (p,N ) AD theory.For N = 4 SU (N ) SYM with odd N ≤ 7 , it has been observed in [8] that Schur partition functions satisfy monic MLDEs of weight (N + 1) 2 /2.It is natural to ask about the dependence of the weights of MLDEs on the parameters p and N for T (p,N ) AD theory.
Finally, it would be interesting to find the holographic dual of T (p,N ) AD theories and study various properties using supergravity techniques.The important feature of a = c implies some remarkable cancellations of higher derivative corrections in the supergravity Lagrangian.Once the holographic dual is known, one could then study the "giant graviton expansion" of the Schur index of T (p,N ) AD theories and try to reproduce it from supergravity. 15 We leave these interesting questions to the future.and remains finite in the limit Im τ → +∞.Due to the invariance under T transformation, any modular form has a convergent Fourier expansion in q and is finite in the limit q → 0: a n q n , q = e 2πiτ .(A.5) One can relax the finiteness condition at infinity and allow a finite number of terms with negative q exponents in the above Fourier expansion, which defines a weakly holomorphic modular form.
A particular set of modular forms is given by the Eisenstein series which is defined by16 where B 2k is the 2k-th Bernoulli number.When the integer k > 1, E 2k (τ ) is a modular form for Γ of weight 2k.The case of E 2 (τ ) is special as it is not a modular form but a quasi-modular form since it transforms anomalously under modular transformations: The space of modular forms of weight k is denoted by M k (Γ).The ring of the modular forms for modular group Γ = PSL(2, Z) is freely generated by E 4 (τ ) and E 6 (τ ): In particular, this implies that E 2k with k ≥ 4 can be written as the polynomial of E 4 and E 6 .For example, The space M k (Γ) is finite dimensional: We can further genearlize and define the vector-valued modular form similarly.Consider a homomorphism ρ : Γ → GL(n, C) , (A.11) which gives an n-dim representation of Γ.A vector-valued modular form of weight k ∈ Z and and all the component functions are finite in the limit Im τ → +∞.If the component functions have exponential growth in this limit, then χ(τ ) is referred to as a weakly holomorphic vector-valued modular form.
where N i depends on the structure of the roots to indicial equation.See e.g.[32] for detailed discussions.In particular N i ≤ k.
The set of linearly independent solutions of MLDE transform as a weakly holomorphic logarithmic vector-valued modular form under the modular group Γ, and ρ(T ) is not a diagonal matrix any more; the off-diagonal entries arise from the logarithmic term due to the fact that log q → log q + 2πi when we perform T -transformation τ → τ + 1.

B Zhu's recursion relation
We will review Zhu's recursion relation [9] and its application in deriving modular linear differential equation.A detailed discussion can be found in the original reference [9] and [8,33].

B.1 Vertex operator algebra
We now introduce the vertex operator algebras.We will only introduce the basic concepts and refer the reader to [8,33] for details.For simplicity, we will only consider Z ≥0 -graded conformal vertex operator algebra, meaning that the vertex operator algebra has a subalgebra which is isomorphic to Virasoro vertex operator algebra, and the conformal dimension of all operators are non-negative integers.As a result, the vertex operator algebra V can be decomposed as and L 0 operator from Virasoro algebra acts as L 0 a = na for a ∈ V n .The operator-state correspondence dictates that for each state a with integer weight h a , the corresponding mode expansion of the corresponding vertex operator is given by where the modes act as endomorphism a n : V k → V k+n .We further introduce the following notation for the zero mode of a o(a) := a 0 .
For the convenience of formulating recursion relations on the torus, we consider an alternative expansion of the same vertex operator where we have introduced the "square-bracket" modes.They are related to the usual modes in (B.2) via a where the coefficients are defined via For Virasoro primary state a, the commutation relations for a [n] are identical to those of a n .For stress tensor, which is not Virasoro primary, we instead define These square-bracket Virasoro modes then satisfy the same commutation relation as the original Virasoro modes L n .We also need the vacuum state which is defined such that17 The two definitions of vacuum are consistent as a Since the vacuum state and the commutation relations defined via the square-bracket modes are the same as those of usual modes, a null vector in the vacuum Verma module of a VOA formulated in terms of the usual mode will still be null after replacing the usual modes with square bracket modes.In another word, once we have a null state acting on vacuum, we can replace all the a n in the null state with a [n] , and the resulting state is still null.

B.2 Zhu's recursion relation for torus one-point function
The virtue of the square-bracket modes is that one can elegantly formulate the recursion relation for torus one-point functions in the VOA.We will call such kinds of recursion relation as Zhu's recursion relation, as they were originally derived by Zhu [9].The simplest Zhu's recursion relation is18 (B.9) More generally, for n ≥ 1 we have and obtain the zero mode of state with strictly smaller conformal dimension.This then gives a way to compute the torus one-point function recursively.

B.3 MLDE from null state
Given a VOA V, one can define a subspace C 2 (V) ⊂ V as By operator-state correspondence, the state a −ha−1 Ω is associated to the vertex operator ∂ n a.The vector space C 2 (V) can be understood as the space of those normally-ordered composite operators that contain at least one derivative, including those operators without explicit derivatives apparently but can be rewritten in terms of operators with derivatives after using appropriate null relations in the VOA.We can further define the quotient space The corresponding algebra is known as the C 2 -algebra of V, which is a commutative, associative Poisson algebra.In many interesting cases, the VOA contains some null state of the form In other words, we have The existence of such a null state is closely related to the existence of MLDE for the vacuum character of V.
Due to the isomorphism between square modes and usual modes that we discussed before, a null state of the form in (B.13) indicates another null state in the Verma module of VOA where C [2] (V) is the square mode analogue of (B.11).Because N [T ] is a null state, correlation functions with insertions of N [T ] must vanish.In particular, the torus one-point function of N [T ] must be zero The torus one-point function with insertions above can be evaluated using Zhu's recursion relation in (B.9) and (B.10).Consequently and in ideal cases, 19 we get a modular covariant differential operator acting on the vacuum character, namely we obtain the MLDE for the vacuum character of VOA (A.23) From these discussions, we see that the MLDE for the vacuum character arises as the consequence of a null state of the form (B.13).

B.4 Stress tensor one-point function
To derive the MLDE, one important ingredient is to derive the torus one-point function with insertion o((L [−2] ) k Ω).Let us start with the simplest case of k = 1.Using Zhu's recursion relation (B.9), we get (B.17) where we used that For higher k, we can similarly derive the torus one-point function using (B.9) and (B.10). 19After using the recursion relation, one may encounter the zero mode of the form o(a 1 −h 1 • • • a j −h j ), which generally gives an obstruction for further evaluation.See [8] for discussions.We don't encounter such kinds of obstructions in this paper.
where A k,r is defined by Explicitly, one can compute A k,r using Virasoro commutation relations repeatedly and the property of vacuum L [m] Ω = 0 for m > −2.In particular, one can easily show that A 1,r = 2r, namely As a result, we find the following trace formula where q , (B.22) P 10 = D (5)  q + 5 ((16 + c)E 4 ) D (3)  q + 100(8 + c)E P 16 = D (8)  q + 14 ((32 + c)E 4 ) D (6)  q + 560 ((20 + c)E 6 ) D (5)   In the special case of k = 2, n = 2r + 1, we get (A 1 , A 2r ) AD theory, whose chiral algebra is simply given by (2, 2r + 3) minimal model VOA, whose only strong generator is the stress tensor.The SCFT/VOA correspondence then implies the vacuum character of (2, 2r + 3) VOA coincides with the Schur index of (A 1 , A 2r ) up to the Casimir factor, namely The VOA of (A 1 , A 2 ) AD theory is given by the (2,5) Lee-Yang VOA.It has a null operator of the form Equivalently, there is a null state of the form Following the previous general discussions, the torus one-point function with the insertion of the zero mode this null state is Equivalently, there is a null state of the form Following the general discussions in the previous subsections, the insertion of zero mode of this null state in the torus partition function leads to a MLDE.In particular, the insertion of the first and last mode in the bracket of (C. One can also verify this equation numerically as the character is known (C.3).
The OPEs are given by where Λ is given by (3.10).
In terms of modes, the commutation relations are The VOA is W 3 algebra with c = −114/7.At dimension 5, we find a null operator At dimension 6, we find a null operator Then we can consider the normal order product T N 6 , where T W 2 becomes W ′′ W using the null operator N 5 .Consequently, we find a null operator at dimension 8: 6) . (C.28) The corresponding state is Obviously, we have L 4 −2 ∈ C 2 (V).We can then use Zhu's recursion relation to derive the MLDE.The only non-trivial computation is for W generators, which can be computed using (C.24): As a result, we find the following MLDE D (4)  q − 370 7 The VOA is W 3 algebra with c = −23.The MLDE can be derived similarly and turns out to be given by D (5)  q − 125E 4 D A monic MLDE has been found to be D (10)   q Meanwhile, a non-monic MLDE is also found We now consider the family of D p (SU (N )) AD theory reviewed in (2.1).Here we again impose the condition gcd(p, N ) = 1.
The central charge of the corresponding chiral algebra is The Schur index is given in (2.15).For simplicity, we turn off all the flavor fugacities.Then the corresponding Schur partition function is For D p (SU (N )) theory, the chiral algebra is given by the Kac-Moody algebra su(N ) − N (p−1) p , and the only strong generators are Kac-Moody currents J a with a = 1, 2, • • • , N 2 − 1.In particular, the stress tensor is given via Sugawara construction: But this kind of null relation does not lead to a MLDE as L −2 Ω ̸ ∈ C 2 (V).
In case N = 2, we have AD theories which can be denoted alternatively as D 2k−1 (SU (2)) = (I 2,2k−3 , F ) = (A 1 , D 2k−1 ).In [8], the authors found the null state of the form (for small k explicitly) )) AD theory.For p = 2 and N = 2k + 1 > 3, the above examples suggest that the corresponding MLDEs always have weight 6.Assuming this and using 20 However, this seems to be not true for k = 8.
In this case, we find a MLDE of weight 12 satisfied by the Schur partition function D (6) q − 3493 5 E4D (4) q + 7028 5 E6D (3) q The chiral algebra of T (3,2) AD theory is given by the A(6) algebra [4,23,24].It contains 3 strong generators, denoted by T, Φ, Φ, whose conformal dimensions are 2, 4, 4, respectively.While the first generator T is the stress tensor, the latter two Φ, Φ are fermionic Virasoro primary operators.The OPEs among them are given by + −48 (T ′ ) 2 − 84T ′′ T + 36T 3 + 7T (4)  420z 2 + 60 −5T ′′ T ′ + 6T ′ T 2 − 2T (3) T + 7T (5)  2800z , (C.50) where we ignore the argument of operators on the RHS, which is 0. Actually the full OPE can be easily bootstrapped using the associativity of OPEs and the information of conformal dimensions of these operators. 21his algebra admits a free field realization in terms of chiral boson φ satisfying the OPE where P 5 is an operator with conformal dimension 5 built as a polynomial in ∂ j φ.Since it is quite complicated, we don't write it down explicitly here.One can verify that the free field realization is consistent with the OPEs above.The mode expansion of these operators are T (6) .(C.62) Due to the presence of Ψ Ψ, the corresponding state does not fit into the form of (B.13), namely (L −2 ) 4 Ω ̸ ∈ C 2 (V).However, we can consider the normal order product T N 8 , then the fermion bilinear terms can be rewritten T Ψ Ψ → Ψ ′′ Ψ using the null operator (C.61).Consequently, we get a null operator at level 10: The Schur partition function of T (4,3) AD theory is annihilated by the following MLDO of weight 34:

Figure 1 .
Figure 1.T (p,N ) from gauging of several copies of D pi (SU (N )).Yellow circle denotes SU (N ) gauge node.

24 , 24 = 0 .
(B.10) where the prime indicates that the term with 2k = n + 1 should be removed because o(a [−ha+1] b) is a commutator and hence STr o(a [−ha+1] b)q L 0 − c Note that since on the RHS, we have −h a − n + 2k ≥ −h a + 1, so a [−ha−n+2k] annihilates the vacuum (B.8).After using the commutation relation of the algebra, we can move the left-most oscillator a [−ha−n+2k] to the right inside o(a [−ha−n+2k] b)

(A 1 , A 4 )
used (B.30) and (B.21).Using (B.27), we then find the following MLDE D AD theory.The VOA of (A 1 , A 4 ) AD theory is given by the (2,7) minimal model VOA.It has a null operator of the form