4d SCFTs from negative-degree line bundles

We construct 4d N=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=1 $$\end{document} quantum field theories by compactifying the (2,0) theories on a Riemann surface with genus g and n punctures, where the normal bundle decomposes into a sum of two line bundles with possibly negative degrees p and q. Until recently, the only available field-theoretic constructions required the line bundle degrees to be nonnegative, although supergravity solutions were constructed in the literature for the zero-puncture case for all p and q. Here, we provide field-theoretic constructions and computations of the central charges of 4d N=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=1 $$\end{document} SCFTs that are the IR limit of M5-branes wrapping a surface with general p or q negative, for general genus g and number of maximal punctures n.


Background and motivations
One interesting way to obtain 4d quantum field theories is to compactify the 6d N = (2, 0) superconformal field theories on a Riemann surface. As the (2,0) theories are the lowenergy description of multiple M5-branes, the 4d theories that result from this procedure describe the low-energy limit of M5-branes that wrap the Riemann surface. The 4d SCFTs that have such a description are known as theories of class S. This strategy for obtaining 4d theories has been used in an attempt to study and classify theories with N = 2 supersymmetry in [1][2][3]. In [2], it was argued that strongly JHEP08(2018)199 coupled N = 2 SCFTs, denoted T N theories, arise as the low-energy effective description of N coincident M5-branes wrapping a 3-punctured sphere. The punctures (i.e. codimension 2 defects) each introduce an SU(N ) flavor symmetry in the field theory. The T N theories can be combined by gauging diagonal subgroups of these global symmetries, which geometrically corresponds to gluing the punctures to form more general Riemann surfaces with arbitrary genus and punctures [2]. These constructions provided an understanding of the 4d SCFTs discussed in [4] which arise for the case of branes wrapping a surface with no punctures, and which are holographically dual [5] to some solutions of 11d supergravity referred to in the literature as Maldacena-Nuñez solutions. The holographic duals of constructions for the more general system with punctures were considered in [5].
One can field-theoretically construct N = 1 theories of class S -including the duals of the N = 1 Maldacena-Nuñez solutions -via deformations of the T N theories, as discussed in [6]. This program was continued in [7] with constructions in which the T N theories are coupled with N = 1 vector multiplets. In the present work, we will be interested in generalizing an infinite class of 4d N = 1 SCFTs that were introduced in [8,9]. There, supergravity solutions that naturally generalize the Maldacena-Nuñez solutions (and which we'll refer to as BBBW solutions for authors Bah, Beem, Bobev, Wecht) were constructed and shown to be holographically dual to field-theoretic constructions that involve N = 1 gluing of T N building blocks to create a surface with no punctures.
We here consider M5-branes wrapping a genus g Riemann surface with n maximal punctures C g,n , where the surface is embedded in a Calabi-Yau 3-fold. 1 The normal bundle decomposes into a sum of two line bundles as The Calabi-Yau condition requires that the degrees of the line bundles p and q satisfy The BBBW supergravity solutions are valid for all p and q; however, there is only an explicit field theory construction for the case of p and q nonnegative. 2 Our main goal in the present work is to understand such a construction for the case where one of the line bundle degrees is negative. Our construction requires a more general building block than the T N theory. The necessary ingredient was provided in [11], which introduced a generalization of the T N theory denoted T (m) N , for m a positive integer (and whose features we will review in section 2.1). The field-theoretic constructions in [11] utilizing T (m) N building blocks provided the first generalized quiver field theories with p, q < 0.
1 Eight supercharges are preserved when the ambient geometry in which the Riemann surface is embedded in a Calabi-Yau 2-fold, while only four supercharges are preserved when the geometry is a Calabi-Yau 3-fold. For further discussion on the general structure of how N = 1 supersymmetry is preserved in the BBBW theories, see [9]. 2 See note added at the end of the Introduction; similar constructions were recently considered in [10]. JHEP08(2018)199

Overview and summary of results
In the present work, we explicitly construct 4d N = 1 field theories that result from compactifying the (2,0) theories on a surface with negative p or q, thereby providing field theoretic constructions for the duals of the BBBW gravity solutions obtained by gluing T (m) N building blocks. We further generalize to the case of M5-branes wrapped on Riemann surfaces with maximal punctures, yielding formulae for the (trial) central charges of the resulting SCFTs that depend only on geometric data.
The organization of this paper is as follows. In section 2, we provide the field-theoretic construction of the 4d N = 1 SCFTs that are dual to the BBBW gravity solutions for negative p or q. We begin by reviewing the definition of the T (m) N theories as formulated in [11]. Then, we glue together (2g − 2) copies of the T (m) N theories, yielding a genus g > 1 surface with no punctures and possibly negative p or q, thus providing an inherently fieldtheoretic construction of 4d N = 1 field theories that arise as the IR limit of M5-branes wrapped on a surface with negative normal bundle degrees. We compute the central charges and operator dimensions for these theories, and find that they match precisely onto the BBBW formulae.
In section 3, we study the genus zero case, which requires closing punctures on chains of T (m) N theories. We consider the simplest case of a single T (m) N theory whose SU(N ) flavor groups are Higgsed, which can yield theories with twist |z| ≥ 2, and find that the trial central charges match onto the BBBW results. There will be corrections to these values from operators whose R-charges were shifted in the Higgsing procedure. We comment on these corrections, but leave their complete analysis to future work.
Having reviewed the machinery to close punctures in section 3, in section 4 we construct theories of genus g and n maximal punctures from T (m i ) N building blocks. With these general constructions, we are able to compute the central charges for the torus as well. We conclude with a brief discussion of future directions. Appendices review our conventions, relevant BBBW formulae we reference throughout, and list the 't Hooft anomalies needed in our constructions.
Note added. While this work was in the final stages of preparation, [10] appeared, which has some overlap with the present work.
2 Constructing the BBBW duals from T (m) N building blocks The T (m) N theories constructed in [11] are strongly coupled 4d N = 2 SCFTs. They have a natural description as being of class S, arising as the low-energy limit of wrapping N M5branes (the (2,0) theories of type A N −1 ) on a sphere with three punctures, C 0,3 . The sphere is embedded in a Calabi-Yau 3-fold, which decomposes into a sum of two line bundles as in (1.1). For our purposes, the novelty of this construction is that the degrees p and q of the line bundles are allowed to be negative, parameterized by a nonnegative integer m: (2.1) For m = 0, this construction reduces to the T N theory. These theories preserve a U(1) + × U(1) − global symmetry which derives from the parent N = (2, 0) SO(5) R symmetry, as well as an SU(N ) 3 global symmetry associated to the three punctures 3 which we denote SU(N ) A × SU(N ) B × SU(N ) C . Each puncture is labeled by a sign σ A,B,C = ±1; in the notation of [11], +1 is blue-colored and -1 is red-colored, and in diagrams we'll take +1 to be unshaded and −1 to be shaded. Denoting the generators of U(1) + × U(1) − by J ± , the exact superconformal R-symmetry is given by the linear combination (see appendix A for more on our conventions)

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The 't Hooft anomalies for the T that arises from the construction of the T (m) N theories by gluing m + 1 copies of the T N theory -see [11] for more details. We summarize the resulting UV generalized quiver description for the T   trivial in the ring. As checked in [11], none of the gauge-invariant chiral operators that are nontrivial in the ring decouple. 4

Gluing procedure
The gluing procedure corresponds to decomposing the geometry into pairs of pants, where each can be associated with T (m) N theories, and gauging subgroups of the flavor symmetries associated to the punctures. We will label the ith block by T Before we do the general case, let us first illustrate the procedure of gluing two T (m i ) N , i = 1, 2 theories with either an N = 1 or N = 2 vector multiplet, as in [11]. Label the degrees of the blocks as since p i and q i switch roles for an unshaded versus shaded block. We gauge an SU(N ) flavor symmetry of the two T (m i ) N theories, leading to a superpotential for the moment-map operators of the gauged block, For instance, µ + could be chosen to derive from µ A in table 1 for one block, and µ − to come from µ A for the other block. In order to write a superpotential of this form, when gluing with an N = 1 vector the (J + , J − ) charge assignment of one of the T (m i ) N blocks must be flipped such that µ − has J ± charges given by (J + , J − ) = (0, 2). In general, two JHEP08(2018)199 blocks of the same color/shading should be glued by an N = 2 vector, while two blocks of differing colors/shadings should be glued by an N = 1 vector.
The result of this procedure is a four-punctured sphere C 0,4 , where the total degrees p and q of the embedding space satisfy p + q = 2g − 2 + n = 2.

Construction of C g>1,n=0 and computation of a and c
Here, we glue together (2g − 2) copies of the T with no punctures and possibly negative degrees p and q.
(The sphere and torus cases are constructed separately in later sections.) We consider the general case of 1 (shaded) blocks with σ i = −1, and 2 (unshaded) blocks with σ i = +1, glued together with n 1 (shaded) N = 1 vector multiplets and n 2 N = 2 (unshaded) vector multiplets. Given the geometries we wish to construct, these parameters satisfy Label the degrees of the blocks as in (2.4). Then, the total degrees p and q of the space in which the genus g surface is embedded are given by The sum over {σ i = +1} runs over the 2 unshaded nodes, while the sum over σ i = −1 runs over the 1 shaded nodes. As an example, the options for forming a genus 2 surface in this manner are shown in figure 2. We now compute the central charges a and c for these configurations. For a general 4d N = 1 SCFT, the central charges a and c are determined by the 't Hooft anomalies [17], building blocks, in the absence of accidental symmetries, the N = 1 superconformal R-symmetry R = R( ) takes the form (see appendix A for conventions)

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Then, the exact superconformal R-symmetry at an IR fixed point is determined by amaximization with respect to [18]. The contributions to a and c of the various components of our constructions can be computed using the 't Hooft anomalies given in appendix C, substituted into equations (2.8) and (2.9). The contribution of the i'th T (m i ) N block is given by [11] a T (m i ) (2.10) For convenience, we've defined A 0 ( ) as the piece of a T (m i ) N ( ) that's independent of m i , and A 1 ( ) as the piece proportional to m i . Our convention is that these formulae as written correspond to an unshaded (σ i = +1) block, while taking → − (equivalently, swapping J + and J − ) yields the formulae for a shaded block. The contributions of an N = 2 and N = 1 vector multiplet are Here, the convention for a N =2 ( ) is that as written we're gluing two unshaded flavor groups; 5 gluing two shaded flavor groups with an N = 2 vector corresponds to taking → − . Then, a( ) for this class of theories is given by a sum over these pieces, (2.13) Maximizing with respect to yields where we've written the answer in terms of the twist parameter z, in (2.14) matches the value computed in [9], as expected. The argument of the square root is always positive for the valid ranges of the parameters, N ≥ 2, g ≥ 2. is singular only for q = p = g − 1, and re-maximizing with respect to for this special point yields = 0. Substituting into (2.13), and performing the similar computation for c, we find (4+3N (1+N )) . (2.17) We plot a as a function of z for various values of N in figure 6. These results precisely match the values of a and c that were computed by BBBW in [9] via integrating the anomaly eight-form of the M5-brane theory over C g,0 . We emphasize that we've found this same result with a field theoretic construction. It is worth highlighting that it is nontrivial that the dependence on the m i in (2.13) cancels to yield central charges (2.16) and (2.17) that depend only on the topological data z and g (and choice of N ).

Operator dimensions and large-N
With fixed in (2.14), the dimensions of chiral operators can be determined by ∆(O) = We can construct gauge-invariant operators out of Q and Q that correspond to M2brane operators wrapping the surface C g,0 , as described in [9]. Schematically, these are From (2.18), the dimensions of these operators are for given in (2.14), and where 0 ≤ | 1 − 2 | ≤ 2g − 2. None of these operators decouple. | | in (2.14) ranges from 0 at z = 0, to 1 √ 3 at large z and N . In particular, the new range of accessible for negative p and q versus the previously studied case of p, q ≥ 0 [9] is the range 1 3 ≤ | | ≤ 1 √ 3 . In this range of , no operators violate the unitarity bound; thus, assuming no accidental IR symmetries, a and c are given by (2.16) and (2.17). We note that a and c given in (2.16), (2.17) are always positive and nonimaginary, and always (for g > 1) satisfy the Hofman-Maldacena bounds for N = 1 SCFTs [19], At large N , the leading-order term of a = c in (2.16) scales as N 3 , and is given by theory. The Higgsing procedure we review below was detailed in [11], and also utilized in [20] (where the |z| = 2 case was first studied) and [10]. First, switch the color of the punctures to be opposite the color of the background T  . . , j (e.g. see relevant discussion in [13] and [11]). The only components that don't decouple are those with m = −j, which we'll denote as M (A,B,C) j below. After decoupling operators, the remaining superpotential is (3. 3) The UV quiver is depicted in 4. The Higgsing shifts the R-charges for m (i) the weights of the SU (2) representations. The resulting R-charges of operators are given in table 2. Note that the trifundamental Q's have decomposed into N 3 singlets, with R-charges shifted due to (3.4). Geometrically, closing the punctures reduces the degrees of the normal bundle; starting with p = m + 1, q = −m as in (2.1), we flow to a theory with Note that this construction only yields 4d theories with |z| ≥ 2, since starting with a T

Computation of a trial and c trial
Next, we compute a and c for the Higgsed T (3.6) With these ingredients, we find that is given by and the central charges a and c are given by (4+3N (1+N ))) . However, as we discuss in the next section, this is not the whole story, and there will be field theory corrections from operators in the theory hitting the unitarity bound. For this reason, we explicitly include the label a trial , c trial .

Comments on ruling out g = 0 SCFTs
For the Higgsed T (m) N theory, the chiral operators are summarized in table 2. | | runs from 1 3 19 7 for N = 2 and large-z, to 1 6 (1 + √ 13) for z = 2 and large-N . For instance, the (3.9) We generically find that some operators decouple, and at finite-N there will be field theory corrections to the central charges (3.8), and thus corrections to the BBBW results. We will not discuss these corrections in general, and instead briefly point out some features in the |z| = 2, 3 cases. Our construction does not reach |z| = 0, 1. The case of |z| = 2, i.e. m = 0, is studied field-theoretically in [20], where they find that the central charges violate the Hofman-Maldacena bounds due to accidental symmetries from the decoupled operators along the flow to the IR SCFT.
For z = 3, i.e. m = 1, and N > 2, the operator Trµ 2 decouples. Q, Q operators with violate the unitarity bound. For instance, for N = 2 and z = 3, the three operators Q (s)(t)(u) with s + t + u = 1/2, and one with 3/2 would have R-charges that unitarity. In general, the decoupling of these operators could lead to violations of the Hofman-Maldacena bounds, but we do not pursue this direction here. 7 4 General C g,n from T Now we consider the most general case of constructing a genus g surface with n maximal punctures from T (m i ) N building blocks, and computing a trial and c trial . One useful way to arrange this computation is to glue in chains of T (m i ) N theories to the n = 0 cases we constructed in section 2.3. The result will be a genus g surface where the number of punctures depends on how many chains we add. Then, we can close arbitrarily many of these punctures via the Higgsing procedure discussed in section 3.
Let us begin with the class of theories we considered in section 2.3: start with 1 T (m i ) N (σ i = −1) blocks, and 2 T (m i ) N (σ i = +1) blocks, glued with n 1 N = 1 vectors, and n 2 N = 2 vectors, where 1 + 2 = 2g − 2 and n 1 + n 2 = 3(g − 1). 7 A discussion of the chiral operators in the ring of these theories on a sphere and their decoupling is given in [10], as well as some discussion on which geometries do not flow to SCFTs in the IR.
Next, glue in some number of additional T (m i ) N blocks such that we do not change the genus of the surface. In particular, introduce 1 T blocks. This will require that we introduce 1 + 2 vectors to glue in these chains, where the number of additional N = 2 vectors n 2 is given by 2 − 1 . These 1 + 2 blocks also introduce 1 + 2 punctures to the surface, of various colors. As in section 3, we can close each of these punctures by flipping their color, coupling in an adjoint chiral field, and letting the adjoint take a nilpotent expectation value that breaks the SU(N ) flavor group of the puncture. In particular, let us close h 1 of the σ i = −1 punctures, and h 2 of the σ i = +1 punctures. This will leave us with n (−) = 1 − h 1 minus punctures, and n (+) = 2 − h 2 plus punctures. An example of this construction applied to a genus 2 surface that results in one plus puncture is given in figure 5.
The total degrees p and q for the surface will be given by (4.1) The sum over {σ i = +1} runs over the 2 + 2 plus nodes, while the sum over {σ i = −1} runs over the 1 + 1 minus nodes. Below, we write out answers in terms of z = (p − q)/(p + q), the combinations and the Euler characteristic of the surface, χ = −2g + 2 − n tot . Summing the contributions to the trial central charges, we find a( ) is given by and c( ) is given by
At large-N , computed in (4.5) matches the BBBW result listed in (B.6). Then, the leading order piece of a and c in a large-N expansion is given by This is simply our result (2.22) with (2g − 2) → (2g − 2 + n tot ). Thus at large N , the central charges depend on the Riemann surface only through the Euler characteristic χ = −2g + 2 − n tot . We highlight that these answers depend only on topological data: the total number of punctures n tot , the difference between the number of plus and minus punctures n dif , JHEP08(2018)199 the twist z defined in terms of the normal line bundle degrees p and q, and the genus g of the surface (or equivalently, the Euler characteristic). It is satisfying that the computation organized such that the other parameters dropped out, leaving the (relatively) nice expressions (4.3)-(4.5).

Comments on operators
Deferring a detailed analysis of the chiral operators of this class of theories to the future, we point out one interesting feature in the chain operators O M 2 , O M 2 . For illustration, consider the theories in figure 5. There will be N 2 fundamentals (and anti-fundamental) operators of the form (QQQQQ) label the N 2 operators and yield a shift in the J − charges of these operators as in table 2, and runs from 1, . . . , N is an index for the SU(N ) flavor symmetry of the remaining puncture. If we close the remaining puncture, then there will be N 3 degenerate chain operators that should correspond to M2-branes wrapping the surface.
More generally, for a class of theories constructed from gluing h 1 Higgsed shaded flavor groups and h 2 Higgsed unshaded flavor groups such that all the punctures are closed, there will be gauge-invariant chain operators as well as the corresponding operators constructed from the Q's. The R-charges of these operators will be given by (4.8) Thus, this field-theoretic analysis suggests a degeneracy of possible M2-brane operators. 8

Computing a trial and c trial for the torus
Up to this point, we've considered g = 1. The case of M5-branes compactified on the 2torus is special because the torus admits a flat metric, implying that the maximal amount of supersymmetry can be preserved by fixing the normal bundle to the M5-brane worldvolume to be trivial. The singular behavior at g = 1 in the computation of a and c is related to the fact that the M5-brane tension causes the volume of the torus to shrink. In our constructions, this means that we should get a 4d N = 4 field theory in the IR when g = 1 and z = 0. One can formulate a nonsingular construction that preserves only N = 1 supersymmetry by taking the torus to have line bundles of equal and opposite degrees fibered over JHEP08(2018)199 it, i.e. taking p = −q. Letting g = 1 and p = −q in (4.1) means that we should require the total number of punctures be zero. In the construction detailed in section 4.1, the simplest generalized quiver that this could correspond to is closing one of the punctures on a T (m) N block, and then gluing the other two.
Redoing the computation of section 4.1 for the torus, we find and (4.10) Indeed, (4.10) matches the BBBW result that we've written in (B.7), where our definition of p matches their |z|.

Future directions
There are many directions one can think about based on the present work, some of which we will list below. First, we note that the authors of [10] study the chiral rings of these theories (much as [21] and [22] studied the chiral rings of the T N theory in detail). It would be further interesting to study the moduli space of vacua of the 4d N = 1 SCFTs realized by our constructions.
In the present work we don't discuss possible confinement of the gauge theories that result from the gluing procedure. However, as shown in [11], when two T (m) N blocks with the same m and opposite shading are glued with an N = 1 vector multiplet, the gauge node confines. It would be interesting to study the structure of confinement for our general constructions, as well as to understand how various duality maps of T (m) N theories (discussed in [11], and [10]) act on our constructions.
It would be interesting to obtain a field-theoretic construction of the N = 2, g = n = 0 theories with |z| = 0, 1. While these theories do not have a conformal phase, it could be useful to study the IR dynamics from the field theory side. 9 Also, it is shown in [23] that the BBBW solutions with rational central charges allow probe M5-branes to break into multiple M5-branes at special points; in particular, when a and c at large-N (given in (2.22)) are rational. It would be interesting to understand field-theoretically what happens at these special points in our constructions.

A Conventions and main T N formulae
The T N theory is N = 2 supersymmetric with global symmetries SU(2) R ×U(1) R ×SU(N ) 3 . We use a basis for the Cartan subalgebra of the N = 2 R-symmetry SU(2) R ×U(1) R labeled by (I 3 , R N =2 ). The R-symmetry of an N = 1 subalgebra is given by We can rewrite these in terms of the generators J + , J − of the U(1) + ×U(1) − symmetry preserved by the N = 1 theories of class S, using R N =2 = J − , and I 3 = 1 2 J + . With these conventions, for example, the adjoint field in the N = 2 vector multiplet has (J + , J − ) = (0, 2).
The (J + , J − ) charges of chiral operators of the T N theory are The The IR superconformal R-charge for operators of the T N theory of color σ i = ±1 is given by maximizing the following combination of R-charges with respect to :

B Relevant BBBW results
In [9], Bah, Beem, Bobev, and Wecht (BBBW) compute a and c of the IR N = 1 SCFTs obtained from compactifying the 6d (2,0) theories on a Riemann surface C g , where the surface is embedded in a Calabi-Yau three-fold that decomposes into a sum of line bundles as in (1.1). These are computed by integrating the anomaly eight-form of the M5-brane theory over the surface C g , and matching with the anomaly six-form, which is related to the anomalous divergence of the 4d N = 1 R-current by the descent procedure and given by F is the S 1 bundle which couples to the R-symmetry, T 4 is the tangent bundle to the 4d spacetime manifold, c 1 is the first Chern class, and p 1 is the first Pontryagin class. Due to the presence of an additional global symmetry U(1) F , the superconformal R-symmetry takes the form