Zonotopes and fourdimensional superconformal field theories
Abstract:
The maximization technique proposed by Intriligator and Wecht allows us to determine the exact charges and scaling dimensions of the chiral operators of fourdimensional superconformal field theories. The problem of existence and uniqueness of the solution, however, has not been addressed in general setting. In this paper, it is shown that the function always has a unique critical point which is also a global maximum for a large class of quiver gauge theories specified by toric diagrams. Our proof is based on the observation that the function is given by the volume of a three dimensional polytope called “zonotope”, and the uniqueness essentially follows from BrunnMinkowski inequality for the volume of convex bodies. We also show a universal upper bound for the exact charges, and the monotonicity of function in the sense that function decreases whenever the toric diagram shrinks. The relationship between maximization and volumeminimization is also discussed.
1 Introduction
One of the most important problems in quantum field theories is to understand the renormalization group (RG) flows and the universality classes.
In two dimensions, we have a fairly satisfactory global picture of the moduli space of quantum field theories. Zamolodchikov [1] introduced a real valued function and showed that the RG flow is a gradient flow of with respect to the metric defined by twopoint correlation functions. In particular, is monotonically decreasing along the RG flow. Each critical point of corresponds to a fixed point of the RG flow i.e. a conformal field theory, and the critical value is the central charge of the Virasoro algebra of the corresponding conformal field theory.
Considerable effort has been expended to generalize these ideas to to four dimensions. As the Zamolodchikov’s function is related the trace anomaly of the stress energy tensor, natural candidates for four dimensional theories are the coefficients and of trace anomaly [2]
where
where denotes the Riemann curvature of the background geometry. It is now believed [3] that function will play a similar role to Zamolodchikov’s function: decreases along any RG flow.
It is usually difficult to compute functions. The situation is much better if the field theories has supersymmetry. Any four dimensional superconformal fields theory (SCFT) has global symmetry supergroup ; is its bosonic subgroup. For the representation of the superconformal algebra on the chiral supermultiplet [4, 5], there is a simple relation between the charge and the conformal dimension of a operator
The scaling dimensions of chiral operators are protected from quantum corrections. Anselmi et al. [6, 7] have shown that the ’t Hooft anomalies completely determine the and central charges of the superconformal field theory:
Here denotes the generator of the symmetry and the traces are taken over all the fields in the field theory. Thus symmetry is extremely useful if correctly identified; it is in general, however, a nontrivial linear combination of all nonanomalous global symmetries.
The crucial observation by Intriligator and Wecht [8] is that the correct combination should be free of AdlerBellJackiw type anomalies i.e. the NSVZ exact beta functions [9] vanish for all gauge groups. Denoting by the global charges of nonanomalous symmetries, the conditions are
(1) 
where the second line is required by the unitarity of the conformal field theory. These conditions are succinctly stated as “exact charges maximize ”:
Theorem 1.1 ([8])
Among all possible combination of abelian currents
the correct current is given by the which attains a local maximum of the “trial” function
It is thus quite natural and important to investigate the existence and uniqueness of the solution to the maximization. By continuity of the trial function, a maximizer always exists on every closed set. But this may not be a critical point; the anomalyfree condition (1) requires that the point should be critical. On the other hand, if there are several local maxima, which one gives the “correct” charges? If there is a critical point which is not local maximum i.e. saddle point, what happens? How does the change of toric diagrams influence the maxima of trial functions? To the best of the authors knowledge, however, no general answer to these questions is known.
The purpose of this paper is to answer these questions. We prove that the function has always a unique critical point which is also a global maximum for a large class of quiver gauge theories specified by toric diagrams, i.e. two dimensional convex polygons. The monotonicity of function is also established in the sense that function decreases whenever the toric diagram shrinks. We derive these results purely mathematically, although the setting of the problem is substantially based on the conjectural AdS/CFT correspondence or gauge/gravity duality. Hopefully, our result will be useful toward the proof of these conjectures.
The organization of the paper is as follows. In section 2, we briefly summarize the rule how a toric diagram determines the trial function of a quiver gauge theory. Section 3 is devoted to set up a mathematical framework of maximization and state our main theorems. In section 4, we observe that the function is given by the volume of a three dimensional polytope called “zonotope”; the uniqueness of the critical point then follows from BrunnMinkowski inequality as we discuss in section 5. In Section 6 we show the existence of the critical point, i.e. the solution to the maximization. In Section 7, we derive a universal upper bound on charges using the interpretation as a volume. The monotonicity of function is established in Section 8. In section 9, the relationship between maximization and volume minimization proposed by Martelli, Sparks and Yau [10, 11] is discussed. In particular, the Reeb vector is shown to be pointing to the zonotope center and the results of Butti and Zaffaroni [12] is rederived. In the final section the results are summarized and a short outlook is given.
2 Toric diagrams and functions
There is a general formula for functions based only on toric diagrams, which we summarize below. For more details we refer the reader to [13, 14, 12, 15, 16] and references therein.
A toric diagram is a two dimensional convex integral polygon embedded into height one; the coordinates of each vertex is of the form (Figure 1). Let denote the set of toric diagrams. For an gon , denote its vertices^{1}^{1}1As usual, we assume that all the vertices are extremal points of . A point is called an extremal point of a polytope if cannot be expressed as where are distinct points in and are positive numbers such that . by in counterclockwise order, so that
Here and throughout this paper, denotes the determinant of the matrix whose columns are . We adopt the convention that the indices are defined modulo , ; thus for all . The cone over the base defined by
will be important for the relation with volume minimization (see Section 9).
With each toric diagram there is associated a quiver gauge theory. A quiver is a directed graph encoding a gauge theory which gives rise to a SCFT. For our purpose, however, a specific form of the quiver is not needed. Let and denote the number of the vertices and the edges of the quiver, respectively. Each vertex is in onetoone correspondence with a factor of the total gauge group ; each edge represents a chiral bifundamental field. The numbers and can be extracted directly from the toric diagram :
(2) 
Here denotes the usual absolute value and is the unit vector .
The charges of chiral bifundamental fields are given as follows. Let be the set of all the unordered pairs of edges of . An element of will be simply denoted by , with the convention that the oriented edge can be rotated to in the counterclockwise direction with an angle . For each , we introduce a chiral field of charge
with the multiplicity
By our orientation convention, are nonnegative. The charges are constrained as
(3) 
The function of the quiver gauge theory is then given by
(4) 
Figure 2 is an example of a toric diagram and the chiral field content of the corresponding quiver gauge theory.
Benvenuti, Zayas, Tachikawa [17] and Lee, Rey [18] have shown that under the constraint (3), the function (4) can be neatly rewritten as
(5) 
where
(6) 
is proportional to the area of the triangle with vertices sitting inside (see Figure 3). The formulas (5) and (6) make the starting point our investigation.
3 Mathematical setup and main results
In this section, we discuss the mathematical formulation of maximization and state our main theorems.
Let be a toric diagram and the vertices of in counterclockwise order, as described in Section 2. Define a homogeneous cubic polynomial in by
(7) 
Choose a real constant and fix it. Let denote the linear function
and set
which is an dimensional simplex. Its relative interior i.e. the interior as a topological subspace of its affine hull will be denoted by
The function is defined to be the restriction of to . Obviously, is a model for function (5), and (or ) represents a physically allowed region of charges. The choice of is not important for maximization; the homogeneity allows us to choose any positive real number. Usually we set to match the convention (3).
For each toric diagram , define its modulus by
The modulus is independent of and is normalized so that the smallest toric diagram has unit modulus. This is the quantity of our primary interest.
Let be the subgroup of which leave invariant the set of lattice points on hyperplane . induces integral affine transformations on this hyperplane: . acts naturally on the set of toric diagrams : for and a polygon with vertices , is the polygon with vertices . The action defines an equivalence relation on :
We denote by the equivalence class of . The functions are invariant, , because depends on only through the areas of triangles inscribed in . The modulus is thus welldefined on . In the physical context, two equivalent toric diagrams and are associated with the identical quiver gauge theory and the same dual SasakiEinstein geometry, so there is no reason to distinguish the two.
In connection with RG flow, it is interesting to compare and for toric diagrams and which are not necessarily equivalent. The inclusion relation on naturally induces a partial order on , namely,
(8) 
The basic question we shall be concerned with is the existence and uniqueness of the critical point of ; we want to establish this as a mathematical fact independent of duality conjectures. This problem is not so simple as it may appear at a first glance. For example, consider the function
which corresponds to the toric diagram . This has a unique critical point in . However,
has no critical points in ; it is maximized at
has two critical points in ; one is a local maximum and the other is a saddle point. Consequently the conditions such as

are nonnegative integers,

is invariant under any permutation of indices , and

unless are distinct,
are not enough to guarantee the existence and uniqueness of the critical point in . The fact that the coefficients are given by the areas of triangles will be heavily used in this paper.
Our main results are as follows^{2}^{2}2The integrality of vertices will be important for constructing quiver gauge theories or CalabiYau cones. But the integrality is not needed to establish the analytic properties of obtained in this paper. In fact, can be any convex polygon on a hyperplane not passing through the origin.:
Theorem 3.1 (Theorem 6.4)
The function has a unique critical point in and is also the unique global maximum of .
Theorem 3.2 (Theorem 7.1)
The critical point satisfies the universal bound
Here, the equality holds for some if and only if .
Theorem 3.3 (Theorem 8.5)
The maximum value of is monotone in the following sense: Suppose and are toric diagrams satisfying . Then . The equality holds if and only if .
Some comments are in order here. The unitarity of the representation of superconformal algebra requires that all gauge invariant chiral operators must have charge [4, 5]. Theorem 3.2, however, yields opposite inequalities in the conventional normalization (see (3)). This is not a contradiction because are charges of gauge noninvariant bifundamental fields.
Theorem 3.3 can be regarded as a combinatorial analogue of “theorem”: the function always decreases whenever the toric diagram shrinks.
There are two key ingredients in the proof of the main results. First, function is identified with the volume of a three dimensional polytope called “zonotope” (Proposition 4.2); BrunnMinkowski inequality asserts that (cubic root of) volume function is a concave function on the space of polytopes. This concavity guarantees the uniqueness of the critical point (Proposition 5.5). Second key point is to show the monotonicity of modulus under simple change of the toric diagrams, e.g. deleting a vertex. This property is also used to prove the existence of the critical point.
Here is an application of our results. The uniqueness of the maximizer implies that there is no spontaneous symmetry break down in maximization:
Corollary 3.4
If a nontrivial element of fixes a toric diagram , then the critical point of is also fixed by .
4 Polytopes and Zonotopes
Let denote a dimensional real vector space. A subset is called convex if whenever and . For any set , its convex hull is, by definition, the smallest convex set containing :
A polytope is the convex hull of a finite set in .
The Minkowski sum, or vector sum, of two subsets and in is (see Figure 4)
whereas the dilatation by the factor is
If and are polytopes, then , are also polytopes. Let denote the family of all convex polytopes in . Two basic operations, Minkowski sum and dilatation, make the family a convex set: for any and nonnegative numbers such that one has .
Let be line segments, each of nonzero length, in . The polytope defined as the Minkowski sum
is called a zonotope and are called its generators (Figure 5). For a finite collection of vectors , we put , and write the corresponding zonotope. Equivalently,
The zonotope is the image of an dimensional cube under a linear projection defined by the matrix . may also be defined as the convex hull of points
is centrally symmetric, and its center is located at .
The zonotope can be decomposed into dimensional parallelepipeds called cubes, each of which is a translation of
The crucial fact is that, although such decomposition is not unique, all tuples appear exactly once in any decomposition. Since the volume of each cube is simply given by , this leads to the following volume formula for zonotopes, which will play a crucial role in this paper.
Theorem 4.1 (Shephard [19], attributed to McMullen)
(9) 
In the rest of this paper, we will specialize to case, i.e. three dimensional zonotopes. If no three of the vectors are coplanar, all the facets (i.e. two dimensional faces) of are parallelograms. For a given generator , the faces which has a edge parallel to form a zone going around a zonotope. Each zone consists of pairs of opposite faces, there are altogether pairs of opposite faces, pairs of opposite edges, and therefore pairs of opposite vertices.
5 Uniqueness of the critical point
In this section, the uniqueness of the critical point of is proved; the existence is shown in the next section.
A realvalued function on a convex set is concave if
for all and . If the above inequality can be replaced by
then is strictly concave. We will use the following well known properties of concave functions:
Theorem 5.1
Any local maximizer of a concave function defined on a convex set of is also a global maximizer of . If in addition is differentiable, then any stationary point is a global maximizer of . Any local maximizer of a strictly concave function defined on a convex set of is the unique strict global maximizer of on .
Recall that the family of polytopes in is a convex set under the operations Minkowski sum and dilatation. Thus it makes sense to talk about the concavity of a function defined on , such as volume function. The following is a fundamental result in the theory of convex bodies (for extensive survey, see [20, 21]).
Theorem 5.2 (BrunnMinkowski inequality)
The th root of volume is a concave function on the family of convex bodies in . More precisely, for convex bodies and for ,
Equality for some holds if and only if and either lie in parallel hyperplanes or are homothetic. ^{3}^{3}3Two sets are called homothetic if for some and , or one of them is a single point.
In Proposition 4.2, is identified with the volume of a three dimensional zonotope. Actually, we are interested in the “family” of zonotopes parametrized by . In order to apply BrunnMinkowski inequality to this family, let us investigate under what conditions two zonotopes are homothetic to each other.
Lemma 5.3
For , two zonotopes , of nonzero volume are homothetic if and only if for some . In particular, two zonotopes , with are homothetic if and only if .
Proof. Suppose and are homothetic. By assumption, they have nonzero volume and cannot be in a hyperplane. Thus there exists and such that . In fact because both zonotopes have as the bottom vertex. Each of them have a unique edge starting from and parallel to for all . Homothethy implies , so holds for all . In particular, if , then , so .
Here we come to the key point of our analysis.
Proposition 5.4
The function
(11) 
is concave. Moreover, its restriction, is strictly concave.
Proof. Let us denote the function (11) by . It suffices to show that for any , ,
and the equality holds if and only if for some . One can easily check that
holds as an equality in . Using the notation (10), this is written as
Then the claim immediately follows from Theorem 5.2 and Lemma 5.3.
Since the function is a strictly increasing function, is maximal (resp. critical) at if and only if is maximal (resp. critical) at . Combining Theorem 5.1 and Proposition 5.4, we have established the uniqueness of the solution to maximization:
Proposition 5.5
Suppose is a critical point or a local maximum of . Then is the unique critical point and is also the global maximum over .
A remark is in order here: is not necessarily concave although the cubic root is. The conifold ,
is already a counterexample; the Hessian of at
6 Existence of the critical point
This section is devoted to the proof of Theorem 3.1 (Theorem 6.4). The key idea is as follows. The continuous function always has a global maximum on the closed set . If the maximum point is in , then from Proposition 5.5 it is also a critical point and there is no other local maxima. But if the maximum point is on the boundary , it is not necessarily a critical point — physical SCFT. Therefore to establish Theorem 3.1, it suffices to show that a point on the boundary can never be a local maximum of .
For this purpose, we investigate the behavior of the maximum values under the change of toric diagrams. More precisely we will prove the following
Proposition 6.1
Let be a toric diagram with vertices in counterclockwise order. Let be a toric diagram obtained by deleting from , i.e. the convex hull of vertices as in Figure 6. Then, for any , there exits such that .
Note that under the natural inclusion
the point corresponds to a point on the boundary facet of , and is none other than the restriction of to this facet. Clearly, any boundary point of is obtained in this manner. Thus Proposition 6.1 immediately implies
Corollary 6.2
No boundary point of can be a local maximum of .
Corollary 6.3
Suppose a toric diagram is obtained from another toric diagram by removing one vertex, then .
By the argument given in the first paragraph of this section, we deduce from Corollary 6.2 the following
Theorem 6.4
Suppose is a toric diagram with vertices . Then has a unique critical point in and is also the unique global maximum of .
Let us turn to the proof of Proposition 6.1. Our strategy is to show that for any there is at least one “inward” direction in which is strictly increasing. Consider two straight paths in emanating from the boundary point , defined by
for . If either or is proved, then we are done.
Note that for three vectors , the relation holds, where denotes the cross product and is the standard inner product. Let be a vector defined by
It is easy to see
Thus it suffices to show either or holds.
Let us choose three vectors as a basis of and express other ’s as
In the affine coordinates , the toric diagram and looks like polygons sitting in the first quadrant of , as depicted in Figure 7. Note that for all .
A straightforward calculation shows
Since , the claim follows from the next lemma.
Lemma 6.5
Let be positive numbers and
be points in satisfying
(12) 
Then, at least one of the following inequalities holds:
(13) 
Proof. Define by
Clearly and , and the sequence
(14)  