Baryonic Generating Functions

Alberto Zaffaroni; Amihay Hanany; Davide Forcella

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Baryonic Generating Functions

Davide Forcella, Amihay Hanany, Alberto Zaffaroni

High Energy Physics - Theory · Submitted 25 Jan 2007 · 35 references

Pith's verdict: A solid, concrete paper: the plethystic exponential really does extend to baryonic operators, with three independent derivations of the conifold one-brane result and honest checks at N=2 and N=3.

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CitationAlberto Zaffaroni, Amihay Hanany, Davide Forcella. "Baryonic Generating Functions." arXiv:hep-th/0701236 (2007). https://pith.science/paper/GANMAQKX

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Editor's note

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The thing worth knowing: this paper takes the plethystic program, which until then only counted mesonic operators, and makes it work for baryonic ones. The prescription is Eq. (2.1) — the plethystic exponential of the one-brane, fixed-baryon generating function gives the all-N fixed-baryon count. That is the real content, and it is backed by more than assertion. The conifold g_{1,B} is derived three ways (direct Laurent expansion, Chebyshev identities, localization) and the three agree. The N=2 closed form (Eq. 3.47) is checked against hand operator counting at dimension four and six, 48 down to 45 and 56 down to 52, and the plethystic log predicts exactly the right number of relations. That is a genuine, non-trivial check, not a consistency tautology. The extension to C^2/Z_2 x C is worked through N=3. The paper is careful about what it can and cannot do, and the writing is unusually explicit about the mechanics — you can actually follow the residue computations and the localization weights without reconstructing them yourself. That matters for a paper whose value is in the machinery. The stress-test worry about the symmetric-product premise is real but weaker than stated. The epsilon contraction in Eq. (3.50) makes the symmetrization manifest by construction, so the plethystic exponential is the natural counting device rather than a guess. And the N=3 case is a better test than the note allows: the plethystic log predicts the two non-factorizable baryons at t1^4 t2 and t1 t2^4 (Eq. 3.79), which is exactly the regime where the symmetric-product picture should be least reliable, and it comes out right. So the premise is tested where it is most at risk, not just at low N. The soft spots are elsewhere and are minor. The Kähler-modulus identification (abstract:s4) is an interpretation of the lattice picture, not a derivation, and the word 'surprisingly' is doing more work than the argument behind it. The general multi-charge formula (2.3) is called schematic by the authors themselves. The orbifold geometric-to-field-theory map fails at B <= -2 and is patched by hand (Sec. 4.2) — acknowledged openly, and the conifold results do not depend on it. The abstract's 'solves a long standing problem' is a bit grand for two geometries plus a prescription, but the prescription is the point and it holds. Who should read it: anyone counting chiral ring operators in quiver theories, or working on the geometry-to-gauge-theory map for toric cones. It is a methods paper and will be cited as one. Send it to referees.

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Major objections (3)

[Abstract (sentence s4); §3.1.3, Fig. 4; Conclusions] The abstract states as a result that 'the baryonic charge turns out to be the quantized Kähler modulus of the geometry.' The evidence offered in §3.1.3 is that the boundary of the conical-pyramid lattice of charges coincides with the (p,q) web and that the flop between the two resolutions occurs at B=0. This is a suggestive correspondence, but no quantitative relation is derived: there is no equation of the form B = (1/2 pi) integral over S^2 of J, no comparison of B with a Kähler modulus computed independently from the toric data, and no test beyond the conifold and C^2/Z_2. Since this sentence is in the abstract as a headline claim, it should either be derived (even at the level of a tree-level relation for the resolved cone) or explicitly labelled a conjecture. As written, the abstract claims more than §3.1.3 establishes.
[§3.3, Eqs. (3.50)-(3.53); §3.4, Eqs. (3.67)-(3.71)] The load-bearing step for Eq. (2.1) is the claim that baryonic operators at finite N are symmetric products of N one-brane blocks, so that the plethystic exponential is the correct counting device. The argument given is that the epsilon contraction in Eq. (3.50) is manifestly symmetric in the A_I's; this establishes that such operators are symmetric, but not that the map from symmetric products to gauge invariants is injective modulo the relations that the plethystic logarithm counts. The only independent checks are at N=2 and N=3. At N=2 the check is that 48 operators at t1^4 t2^2 are rewritten in terms of 45 (Eqs. 3.67-3.69), which establishes 'at least 3 relations', after which the text asserts 'one can check that the relations are exactly three'; the same pattern occurs at t1^3 t2^3 (56 reduced to 52, 'exactly four'). The word 'exactly' is what makes the check conclusive, and it is asserted rather than demonstrated. Please supply the argument (for instance a rank computation for the relevant tensor spaces, or an explicit basis of the 45) or state the check as a lower bound. A check at N=4, where the paper already reports non-factorizable baryons at (5,1) and (1,5) in Eq. (3.83), would also materially strengthen the case.
[§4.2, Eqs. (4.8)-(4.9)] The authors acknowledge that for C^2/Z_2 x C the geometric generating function of Eq. (4.8) reproduces the field-theory g_{1,B} only for B >= -1 and fails for B <= -2, because the homogeneous coordinate x_3 is not associated with an elementary field, and they repair the field-theory answer by hand using a Z_2 symmetry (Eq. 4.9). The acknowledgment is clear and the conifold results do not depend on it. What remains unresolved is the general criterion: §2 presents g_{1,B} as the character of H^0(C, O(B)) (Eq. 3.25) and cites [2] for the localization computation, which reads as an algorithm, but the orbifold shows that the geometric character and the field-theory count can differ. Please state explicitly the conditions under which the geometric computation is valid (for example, a condition on the toric diagram ensuring that every homogeneous coordinate corresponds to a field), and test the criterion on one further cone such as dP_1 or L^{1,2,1}. Without such a criterion the 'general prescription' of §2 is a collection of worked examples.

Minor objections (7)

[§3.1, Eq. (3.10)] Each of the two terms is singular at x = ±1 (and, for the B <= 0 line, at y = ±1); only the sum is regular. A reader evaluating term-by-term at x = 1 will get a divergent expression. Please add a sentence that the expression is to be read as a sum in which the singularities cancel, as indeed happens in passing to Eq. (3.11).
[§3.2.1, Eq. (3.46)] The integrand of Eq. (3.45) has eight poles, at s = 1/(t1 x), x/t1, t2 y, t2/y, t1 x, t1/x, 1/(t2 y), and y/t2. The text lists four (s = t1 x, t1/x, t2 y, t2/y), which mixes poles from the two factors. Please state the contour convention that selects these four.
[§3.1.3, Eq. (3.32)] The matrix displayed for B = 1 appears to have eleven columns and nine rows. For B = 1 the x-spin exceeds the y-spin by 1/2, so the number of x-values should be even and the number of y-values odd, consistent with the stated 2x1, 3x2, ... rectangles. Please check the array.
[Abstract] 'This paper solves a long standing problem' is stronger than the explicit results support, since closed forms are given only for the conifold, its two truncations, and C^2/Z_2 x C. A phrasing such as 'provides a general method and solves the problem for ...' would be more accurate.
[§4.3, Eq. (4.12)] The passage from f_2(t1,t2,x) to the expression in t and b uses t for the diagonal t1 = t2 = t while t1 and t2 are still in play two lines above. Please disambiguate the notation.
[§3.4, Eqs. (3.65), (3.79), (3.80)] The ellipses hide precisely the relations that the section is about. A short table listing generators and relations for N = 2 and N = 3, with the operator identifications of Eqs. (3.66) and (3.81), would make the section much easier to check.
[§2, Eq. (2.3)] The text itself calls this formula 'very schematic' and notes that the general case would accommodate multiple baryonic charges. Since Eq. (2.3) is the only place where the sum over B is written, a more careful statement, with the b^{NB} factor and the B-dependence of the exponent made explicit, would help the reader.

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The paper

The paper claims

The paper argues that one generating function — the count of baryonic BPS operators on a single D-brane at fixed baryon charge — exponentiates, through the plethystic exponential, into the exact count for any number of D-branes, and that in

The reading

Counting gauge-invariant operators in a quiver gauge theory is normally a hard, case-by-case problem, and it becomes harder still when the operators carry baryon number, because baryons are determinants rather than traces. The paper's claim is that this difficulty collapses to a single input: the generating function $g_{1,B}$ that counts BPS operators of fixed baryon charge $B$ on one D-brane. Once that is known, the plethystic exponential $g_B(ν) = PE_ν[g_{1,B}] = Σ_N ν^N g_{N,NB}$ produces the exact generating function for $N$ D-branes at baryon charge $NB$, for every $N$ at once. The authors make this concrete for the conifold and for the orbifold $C^2/ℤ_2 × ℂ$, where they obtain closed forms by three independent routes — direct Laurent expansion, Chebyshev-polynomial identities, and a toric localization formula — and check them against explicit tensor-algebra counting at $N=2$. Along the way they find that the baryonic charge behaves as the quantized Kähler modulus of the geometry, with a flop transition sitting at $B=0$. What follows, if the construction holds, is that the generators and relations of the chiral ring of a quiver theory on its baryonic branch can be read off order by order from a plethystic logarithm of a single known function.

What carries the argument

Two ingredients carry the argument. The first is the plethystic exponential, $PE_ν[f(t)] = exp(Σ_{k≥1} ν^k f(t^k)/k)$, which converts a generating function for a set of one-brane objects into the generating function for all symmetric products of those objects; its inverse, the plethystic logarithm, extracts generators (positive terms) and relations (negative terms) of the chiral ring. The second is the geometric identification of $g_{1,B}$ with the character of the space of holomorphic sections of a line bundle $O(B)$ over the cone, computed either by expanding the homogeneous-coordinate generating function in a Laurent series in the baryonic fugacity or by the toric index theorem, which writes the character as a sum over fixed points of the torus action with weights read off the $(p,q)$ web. The paper shows these three computations — direct expansion, Chebyshev identities built on $U_n(cos θ)$, and localization — agree for the conifold, which is what makes the one-brane input trustworthy enough to exponentiate.

Extended reading notes

Core claim

The central claim is equation (2.1): for a Calabi-Yau cone with a single baryonic charge, the generating function for $N$ D-branes at fixed baryon number factorises through the plethystic exponential of the one-brane, fixed-charge generating function, $g_B(ν;{t_i}) = PE_ν[g_{1,B}({t_i})] = Σ_N ν^N g_{N,NB}({t_i})$. The physical justification is that a baryonic operator on $N$ branes is a fully symmetrised product of $N$ one-brane blocks, exactly as a mesonic operator is a symmetrised product of one-brane fields, so the plethystic exponential — the machine that counts symmetric products — is the right passage from $N=1$ to general $N$. The paper verifies this on the conifold, where it obtains $g_{1,B}$ in closed form for all $B$ and then derives $g_{N,NB}$ for arbitrary $N$ and $B$, and on $C^2/ℤ_2 × ℂ$, where the same procedure yields explicit expressions at $N=1$ and $N=2$. A plethystic-logarithm analysis at $N=2$ for the conifold predicts three relations of weight $(4,2)$, four of weight $(3,3)$ and three of weight $(2,4)$ among ten basic generators, and the authors confirm these numbers by explicitly reducing 48 operators to 45 and 56 to 52 using the epsilon tensor identity. The paper further identifies the baryonic charge with the quantized Kähler modulus of the resolved cone, reading the two phases of the conifold as the two signs of $B$ separated by a flop at $B=0$.

Load-bearing premise

The load-bearing premise is that the geometric count of holomorphic sections of the line bundle $O(B)$ over the cone equals the gauge-theory count of baryonic operators at charge $B$ on one brane; the paper itself shows this identification needs hand repair for $C^2/ℤ_2 × ℂ$ at $B ≤ -2$, so the general prescription is not yet an automatic algorithm.

Editorial extensions

If this is right

For the conifold, the chiral ring's generators and relations are fixed for every $N$ and $B$ by a single known function, so the ring's structure can be read off order by order from the plethystic logarithm rather than computed by hand.
Non-factorisable baryons, which cannot be written as a product of mesons and basic determinants, first appear at $N=3$ on the conifold, with two operators of weight $(4,1)$ and two of weight $(1,4)$, and their number grows systematically with $N$.
The baryonic charge is the quantized Kähler modulus of the cone, so the two resolutions of the conifold correspond to the two signs of $B$ and the flop transition sits at $B=0$.
The same plethystic exponential is the geometric quantisation of the phase space of D3-branes wrapped on three-cycles, so the gauge-theory count and the geometric count of sections agree, at least in the cases checked.
A statistical analysis of quiver theories on their baryonic branches — entropies of BPS states at large charge — becomes a well-posed calculation rather than an open combinatorial problem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the baryon number really is the quantized Kähler modulus, the generating function is a discrete sum over two-cycle areas, which suggests one could read off a discrete spectrum of volumes directly from the coefficients and, at large $B$, extract a BPS entropy for wrapped branes.
The paper's own orbifold computation shows the geometric character of $O(B)$ matching the field-theory count only for $B ≥ -1$ and failing for $B ≤ -2$, because the homogeneous coordinate $x_3$ has no elementary-field partner; the authors repair this by hand with a $ℤ_2$ symmetry. A systematic rule based on the dimer/tiling rather than the toric cone would remove that patching and turn the prescri
Because the plethystic exponential depends only on the one-brane spectrum, the whole scheme is cheaply testable: a Molien–Weyl contour integral at small $N$ gives an independent count, and the paper performs that check only for half-the-conifold at $N=2$. Doing it for the full conifold at $N=3$ would test the non-factorisable baryon predictions directly.
The construction counts only bosonic BPS operators; including fermionic ones would replace symmetric products with a graded symmetric algebra and would likely alter which terms in the plethystic logarithm count as generators versus relations.

Original abstract

We show how it is possible to use the plethystic program in order to compute baryonic generating functions that count BPS operators in the chiral ring of quiver gauge theories living on the world volume of D branes probing a non compact CY manifold. Special attention is given to the conifold theory and the orbifold C^2/Z_2 times C, where exact expressions for generating functions are given in detail. This paper solves a long standing problem for the combinatorics of quiver gauge theories with baryonic moduli spaces. It opens the way to a statistical analysis of quiver theories on baryonic branches. Surprisingly, the baryonic charge turns out to be the quantized Kahler modulus of the geometry.

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We show how it is possible to use the plethystic program in order to compute baryonic generating functions that count BPS operators in the chiral ring of quiver gauge theories living on the world volume of D branes probing a non compact CY manifold. Special attention is given to the conifold theory and the orbifold C^2/Z_2 times C, where exact expressions for generating functions are given in detail. This paper solves a long standing problem for the combinatorics of quiver gauge theories with baryonic moduli spaces. It opens the way to a statistical analysis of quiver theories on baryonic branches. Surprisingly, the baryonic charge turns out to be the quantized Kahler modulus of the geometry.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show how it is possible to use the plethystic program in order to compute baryonic generating functions that count BPS operators in the chiral ring of quiver gauge theories living on the world volume of D branes probing a non compact CY manifold.	contribution 0.90	0.94	0.38	asserted 0.75
s1	Special attention is given to the conifold theory and the orbifold C^2/Z_2 times C, where exact expressions for generating functions are given in detail.	contribution 0.70	0.91	0.08	asserted 0.99
s2	This paper solves a long standing problem for the combinatorics of quiver gauge theories with baryonic moduli spaces.	contribution 0.41	0.90	0.41	asserted 1.00
s3	It opens the way to a statistical analysis of quiver theories on baryonic branches.	interpretation 0.56	0.40	0.09	asserted 0.77
s4	Surprisingly, the baryonic charge turns out to be the quantized Kahler modulus of the geometry.	result 0.82	0.88	0.26	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.52 · evidence from proof or derivation 1.00.

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arXiv listing abstract v2

not examined

full text
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Figures

**Figure 1.** The dimer representation for the conifold. The fundamen

**Figure 2.** The (p, q) webs for the two possible resolution of the conifold. The left figure corresponds to positive baryon number, B > 0, while the right figure corresponds to negative baryon number, B < 0. The vertices correspond to the fixed points of the torus action on the resolution. The equations for g1,B take the general form of equation 6.8 in [2]. It is therefore suggestive that the formulas for g1,B will always be of this form and will come from localization, giving rise to contributions from the vertices of the (p, q) web which are the fixed points of the torus action

**Figure 3.** Localization data for the N = 1 baryonic partition functions. The vertices Vi are in correspondence with homogeneous coordinates xi and with a basis of divisors Di . Two different resolutions, related by a flop, should be used for positive and negative B, respectively. Each resolution has two fixed points, corresponding to the vertices of the (p, q) webs; the weights m (I) i , i = 1, 2, 3 and m (I) B at the fixed points are indicated in black and red, respectively. g1,B(q; C) = q nB X PI q m (I) B Q3 i=1(1 − q m (I) i ) , (3.26) where the index I denotes the set of isolated fixed…

**Figure 4.** Examples of the “integral conical pyramid”s for the conifo

**Figure 5.** Quiver for half the conifold. relations and all baryons are factorized. The gauge theory data can be encoded in the quiver in

**Figure 6.** The quiver (1) and the toric diagram (2) for the

**Figure 7.** The generating functions g1,B can be computed using the index theorem and written as sum over fixed points corresponding to the (p, q) web vertices of C2 Z2 × C. As in the conifold case, localization needs to be performed in a correctly normalized basis of charges for T 3 , which are related to (t3, x, t) by equation (3.27). The vertical legs contribute the factorized factor 1/(1 − t3) in equation (4.3). By comparing Equation (4.4) with Equation (3.19) we find that g1 admits a Chebyshev polynomial expansion, with the assignments x = e iθ, b = e iβ , g1(t, b, x; C 2 Z2 ) = X∞ n=0 t…

Figures reproduced from arXiv:hep-th/0701236 by the authors.

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L. Grant and K. Narayan, “Mesonic Chiral Rings in Calabi-Yau Con es from Field Theory,” arXiv:hep-th/0701189

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[35]

Y. Nakayama, “Finite N Index and Angular Momentum Bound from Gravity,” arXiv:hep-th/0701208. 44

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Cited by 18 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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How the literature uses it

43 arXiv papers cite this in 109 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 60 sentences without a confident label

Builds on it 1 paper

[This assumes a conjecture of [ref] that baryonic generating functions of theories on multiple D3-branes are obtained by symmetrizing the baryonic generating functions of theories on a single D3-brane.] The proof was detailed for ABJM [ref] , but it is a straightforward exercise to generalise it to all flavoured ABJM theories.

Semiclassics of three-dimensional SCFTs from holography · Chiral operators · 0.80

Uses its method or tool 10 papers

Using the formalism which is developed to count gauge invariant operators for the conifold [ref] we find that the $N=2$ generating function does indeed decompose into characters of irreducible representations of $SU(4)$.

The Master Space of N=1 Gauge Theories · Hidden Symmetries for Higher Plethystics · 0.87
The baryonic Hilbert series for quiver quivTsigrhoSUN is given by the Molien-Weyl integral [ref] g[T^σ⃗_ρ⃗(SU(N))](t; w⃗_1, …, w⃗_ ℓ̂ ; B_1, …, B_ℓ'-1) = ∫∏_i=1^L ( 1/N_i!∏_1≤k ≤N_i 1/2πids_i,k/ s_i,k^1+B_i ) ∏_i=1^L ∏_ k ≠p^N_i (1- s_i,k/s_i,p) ∏_k , p^N_i (1- t^2 s_i,k/s_i,p)/∏_ k=1^N_i ∏_p=1^N_i+1 ∏_q=1^M_i (1- t s_i+1,p / s_i,k)(1- t s_i,k / s_i+1,p)(1- t w_i,q / s_i,k)(1- t s_i,k / w_i,q) where $L= ℓ'-1$ as before. $w⃗_j = (w_j,1, …, w_j, M_j)$ with $1 ≤j ≤ℓ̂$ are fugacities for the flavor symmetry S ( ∏_i U(M_i) ) , and the integration variables $s_i,k$ with $1 ≤k ≤N_i$ parameterise the Cartan of the gauge groups $U(N_i)$.

T^{\sigma}_{\rho}(G) Theories and Their Hilbert Series · The baryonic generating function for the Higgs branch of $T^σ⃗_ρ⃗(SU(N))$ · 0.86
For the bosonic part of the spectrum we have the usual Plethystic function [ref] , that from now on we will call it the bosonic Plethystic function ( $PE^ℬ$ ): [equation] This function takes a certain generating function $g_1(q)$ and generates new partition functions $g_N(q)$ counting all the possible $N$ times symmetric products of the constituents of $g_1(q)$, implementing in this way the bosonic statistic, as it is possible to see from the left hand side of ([ref]).

BPS Partition Functions for Quiver Gauge Theories: Counting Fermionic Operators · The general approach to the complete generating functions for quiver gauge theories · 0.82

Uses its data or values 2 papers

An explicit expression is (for simplicity, we set the fugacities $x$ and $y$ to unity) g_2(t, x=1, y=1; B) = ∮_|b|=1 d b/2πi b^B+1 b^2 t^8+t^8/b^2-3 b^2 t^6-3 t^6/b^2+4 t^8-3 t^6+t^4+t^2+1/(1-t^2)^3 (1-t^2/b^2)^3 (1-b^2 t^2)^3 where the integrand is given by (3.47) of [ref] with $t_1 =t b$ and $t_2 = t b^-1$.

The moduli spaces of $3d$ ${\cal N} \ge 2$ Chern-Simons gauge theories and their Hilbert series · The Hilbert series for $N=2$ · 0.85
Hence equations ([ref],[ref]) become: [equation] The $g_1,B^0(q)$ generating functions for the scalar part of the chiral ring were given in [ref] and they are: [equation] hence [equation] Let us start as usual with the $N=1$ generating function.

BPS Partition Functions for Quiver Gauge Theories: Counting Fermionic Operators · The complete generating function for the conifold · 0.88

Compares with it 3 papers

Now, using the formula for the dimension of a generic irreducible representation of $SU(4)$, [equation] we find that the unrefined generating function for $N=2$ sums to [equation] as expected and in agreement with Equation (3.47) of [ref] .

The Master Space of N=1 Gauge Theories · Hidden Symmetries for Higher Plethystics · 0.98
Indeed, in many examples we consider, we compute the generating function of the Coulomb branch Hilbert series and match it with the baryonic generating function [ref] on the Higgs branch of the mirror theory.

Coulomb branch Hilbert series and Hall-Littlewood polynomials · Introduction · 0.95
The results of equation ([ref]) agree with the computations in equation 3.83 of [ref] but are still missing two essential operators that transform as $χ[2]_(2)_tχ[1]_(2)_γ/β$ and $χ[1]_(2)_tχ[2]_(2)_γ/β$ – the so called non factorizable baryons.

4d N=1 from 6d N=(1,0) on a torus with fluxes · $N=3, k=2$ · 0.96

Cites it as support 2 papers

The beautiful fact is that the full counting [ref] , including baryons as well, is achieved with the Hilbert series of $$ for one brane!

The Master Space of N=1 Gauge Theories · Introduction · 0.97
The beautiful fact is that the full counting [ref] , including baryons as well, is achieved with the Hilbert series of $$ for one brane!

Master Space and Hilbert Series for N=1 Field Theories · Generalities · 0.97

Cites it as background 24 papers

Recently a lot of papers studied generating functions counting $1/2 BPS$ operators in $𝒩=1$ quiver $CFT$ [ref] .

BPS Partition Functions for Quiver Gauge Theories: Counting Fermionic Operators · Introduction · 0.94
All these issues and the determination of the partition function depending on all the $d$ charges are discussed in [ref] , and will be review them in the following two chapters.

Master Space and Hilbert Series for N=1 Field Theories · Setting the Counting Problem · 0.81
These have been recently analyzed, in the context of D3 branes at singularities, using the concepts of Master space, Hilbert series and Plethystic exponential in [ref] .

Master Space, Hilbert Series and Seiberg Duality · Introduction · 0.95

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