Ω-background and equivariant integration

3.2 Instanton counting

3.2.2 Ω-background and equivariant integration

where the mappingsσandβare defined by

σ “

¨

˚

˚

˚

˝

z₁´B₁ z₂´B₂

J

˛

‹

‹

‹

‚

, β“

¨

˚

˚

˚

˝

0 Φr 0

´Φr 0 0

0 0 Φs

˛

‹

‹

‹

‚

. (3.23)

The ADHM equations ensure thatσ^˚β^˚σ “0, and the sequence (3.22) is another monad.

Analogously to the UpNq example, when varying the ADHM-parameters in the com- plex (3.22) we find the universal bundle E_SppNq. V is the k-dimensional solution space of theSppNqDirac operator onC², which carries a real structure, andW is the fiber ofEat infinity, and hence carries a symplectic structure.

Similarly, anySOpNqinstanton solutionEcan be represented as the cohomology bun- dle of the complex (3.22) as well, once we exchangeΦrwithΦsin the definition of the map β. The resulting complex is a short exact sequence, since according to the ADHM equa- tionsσ^˚β^˚σ “0. By varying the ADHM-parameters we find theSOpNquniversal bundle E_SOpNq. Note thatV is the 2k-dimensional solution space of theSOpNqDirac operator on C², which carries a symplectic structure, whereasWis the fiber ofEat infinity, and hence carries a real structure.

BPS equations turn into the monopole equations F_A,µν<sup>`</sup> ` i

2 q_αΓ_µν^α_βq^β “0, (3.25)

ÿ

µ

Γ^µ_αα9 D_A,µq^α“0.

Here, Γ^µ are the Clifford matrices and ř

µΓ^µDA,µ is the Dirac operator in the instanton background for the connectionA. Furthermore,q^αis the lowest component of the twisted hypermultiplet. The representation of the connection A in the connection D_A is deter- mined by the representation of the gauge group that the hypermultiplet is in. As is argued in section 3.4 of [68], it is possible to deform the action in a Q-exact way such that the first equation gets an extra factor

F_A,µν<sup>`</sup> ` i

2t q_αΓ_µν^α_βq^β “0, ,

for an arbitrary value oft. Taking the limittÑ 8reduces this BPS equation to the selfdual instanton equation. Given an instanton solution A, the remainder of the action is forced to localize onto solutions of the Dirac equation in the background ofA. The kernel of the Dirac operator thus forms a fiber over the instanton moduli spaceM^inst.

Let us start out by adding a single hypermultiplet in the fundamental representation of the gauge group. Remember that the corresponding kernel was already encoded in the vector bundleVbLin the original ADHM construction. N_f hypers are simply described by the tensor product of N_f vector bundles V. Each individual factor then carries the usual action of the dual group. Moreover, there is now also a natural action of the flavor symmetry group. By general arguments (see, for example, [74]) this partition function then localizes to the integral³

Z^inst“ ÿ

k

q^k

¿

M^inst_k

epV_bL_bMq (3.26)

of the Euler class of the vector bundle VbLof solutions to the Dirac equation over the

3Here (and elsewhere in the paper)Lis really just the fiber of the half-canonical bundleLat the origin of R⁴. More formally, we take the cup product of the bundleVbLbMoverM^inst_k ˆR⁴with the push-forward i˚of the fundamental classrM^inst_k s, whereiembeds the moduli spaceM^inst_k in the productM^inst_k ˆR⁴at the origin ofR⁴.

moduli spaceM^inst. The vector spaceM“C^Nf encodes the number of flavors.

Let us now discuss how to compute this partition function. First, however, note that (3.26) diverges. We will implicitly take care of the UV divergence in the next steps by computing a holomorphic character [53], but we also need to deal with IR divergences:

the instanton moduli space has flat directions where the instantons move off to infinity.

One way to regularize the instanton partition function is to introduce the so-called Ω- background, where we use equivariant integration with respect to the torus

T²_e1,e2 “Up1q_e1ˆUp1q_e2, (3.27) that acts on R⁴ “ C‘C by a rotation pz₁,z2q Ñ pe^ie1z₁,e^ie2z2q around the origin. This forces the instantons to be localized at the origin of R⁴. The resulting Ω-background is denoted byR⁴_e1,e2. The partition function in theΩ-background is defined by equivariantly integrating with respect to theT²_e1,e2–action.

We have already introduced the other components of the equivariance group: The torus T^N_a of the gauge groupGacting on the fiberW with weights a_l, the torus T^k_φi of the dual group acting onVwith weightsφ_i, and lastly the torusT^Nm^f of the flavor symmetry group acting on the flavor vector spaceMwith weightsm_j. In total, we perform the equivariant integration with respect to the torus

T“T²_e1,e2 ˆT^N_a ˆT^k_φˆT^N_m^f . (3.28) The instanton partition function in theΩ-background is thus defined as the equivariant integral

Zpa,m,e₁,e₂q “ ÿ

k

q^k

¿

M_k

e_TpV_bL_bMq, (3.29)

wheree_Tis the equivariant Euler class with respect to the torusT. We will evaluate (3.29) in two steps, using the fact thatM_kis given by the solutionsµ^´1p0qto the ADHM equations quotiented by the dual groupG_k^D. We will thus first perform the equivariant integral over µ^´1p0q, and then take care of the quotient by integrating out G_k^D, which gives a multiple integral overφ_i.

To perform the first part of the above integral, we apply the famous equivariant local- ization theorem, which tells us that the integral only depends on the fixed points of the equivariant group and its weights at those points. This then leads to a rational function in all the weights.

More precisely, suppose that the action of the elementtPton the integration spaceM (which is represented by a vector fieldVt) has a discrete number of fixed points f. Then the equivariant localization theorem says that

ż

M

α“ ÿ

f

ι^˚α ś

kw_krtspfq, (3.30)

where ι embeds the fixed point locus in M _{and where} w_krtspfq are the weights of the action of the vector fieldV_t on the tangent space to the fixed point f P M. If we apply the localization theorem to the integral (3.29), the denominator of the resulting expression contains a product of weights of the torus action on the tangent bundle to the instanton moduli space. Its numerator is given by another product of weight of the torus action on the bundleVbLbMof Dirac zero modes.

Let us start with computing the weights in the numerator, and for convenience restrict the matter content to a single hypermultiplet in the fundamental representation of the gauge group. Since the bundle in the numerator is the kernel of the Dirac operator, we can equally well obtain these weights from the equivariant index Ind_T“ř

kn_ke^iwk of the Dirac operator. For the purpose of (3.30), the sum over weights can be translated into a product by the formula

ÿ

k

n_ke^iwk Ñź

i

pw_kq^nk. (3.31)

To compute the equivariant index Ind_T of the Dirac operator coupled to the instanton background, we make use of the equivariant version of Atiyah-Singer index theorem. It is given by

IndT “ ż

C²ChTpE_bL_qTdTpC²_{q “} ^ChTpE_bL_{q|z1“z2“0}

pe^ie1´1qpe^ie2´1q , (3.32) whereE is the universal bundle over the instanton moduli spaceM_kthat we constructed

in the previous section. Remember that the fiber ofE over an element Ain the instanton moduli space is given by the total space of the instanton bundleEwith connectionA. The second equality is obtained by applying the equivariant localization theorem and using the equivariant Todd class ofC²equals

Td_TpC²q “ e₁e₂

pe^ie1´1qpe^ie2´1q, (3.33) where the weights of the action ofT_e1,e2 onC²aree₁ande₂.

The purpose of all of this was to reduce everything to the equivariant Chern character of the universal bundle E, for which have found the simple expression (3.18) in terms of the Chern characters of the vector bundlesW, V, L andS. We can easily obtain the weights of the torus T on these bundles, so that we can compute the contribution of a fundamental hypermultiplet. We will write down explicit expressions in a moment, but let us first explain how to obtain the weights for other representations.

If we instead wish to extract the weights for an anti-fundamental hyper we just need re- place the equivariant character for universal bundleE by its complex conjugateE^˚. Other representations that are tensor products of fundamentals and anti-fundamentals (or sym- metric or antisymmetric combinations thereof) can be obtained similarly. For instance, the adjoint representation for a classical gauge group can be expressed as some product of the fundamental and the anti-fundamental representation. This product is the tensor product forUpNq, the anti-symmetric product for SOpNq and the symmetric product for SppNq.

Note that in those cases we also obtain the representations of the dual groups. We thus obtain the weights for an adjoint hypermultiplet by computing the character of the ap- propriate product of the universal bundle and its complex conjugate. The weights for the gauge multiplet are the same as for an adjoint hypermultiplet, but end up in the denomina- tor of the contour integral instead of the numerator. This is consistent with the localization formula (3.29), as the tangent space to the instanton moduli space can be expressed as the same product of the universal bundle and its dual.

Once we obtain the index, we can extract the equivariant weights from it by using the rule (3.31). Finally, we need to integrate out the dual groupG_k^D. This leads to a multiple integral overdφ_ialong the real axis. We will absorb factors appearing from this integration such as the Vandermonde determinant of the Haar measure and the volume of the dual

group into the contribution of the gauge multipletz^k_gauge. The resulting integrand actually has poles on the real axis, which we cure by giving small imaginary parts to the equivari- ance parameters. We will describe this in more detail once we turn to the actual evaluation of such integrals. Since the integrand obtained is a rational function in the parameters of G_k^D, we can convert the integral into a contour integral around the poles of the integrand.

In total the equivariant integral (3.29) over the moduli space thus reduces to Z_kpa,m,e₁,e₂q “

¿ ź

i

dφ_iź

R

z^k_Rpφ_i,a,m,e₁,e₂q, (3.34)

where z^k_R are the integrands that represent the matter content of the gauge theory and where theφ_i’s parametrize the dual group. We will discuss which poles (3.34) is integrated around shortly.

Equivariant index forUpNqtheories

Let us see how this works out explicitly for gauge group UpNq. We first have a look at the weights of the torusT²_e1,e2 at the fibers of the half-canonical bundle Land the spinor bundles S^˘ at the origin ofR⁴. Remember that the torus T²_e1,e2 acts on the coordinates z₁ andz₂ with weights e₁ande₂, respectively. It thus acts on local sectionssof the half- canonical bundle as

sPL: sÞÑe^ie`s,

withe_˘ “ ^e1˘e₂ ². Local sections of the four-dimensional spinor bundlesS^˘can be written in terms of those of the two-dimensional spinor bundles onR². Since the weights of the torusT¹_ej on the local sections of the two spinor bundles onR²are˘^e₂^j, the torusT²_e1,e2 acts on local sectionsψ˘of the four-dimensional spinor bundle as

ψ˘PS^˘: ψ˘ÞÑdiagpe^ie˘,e^´ie˘qψ˘.

Let us continue with the weights of the equivariant torus T^N_a ˆT^k_φ. The equivariant

torus then acts on the linear ADHM data as

vPV : vÞÑdiagpe^iφ1,¨ ¨ ¨ ,e^iφkqv, wPW : wÞÑdiagpe^ia1,¨ ¨ ¨,e^iaNqw.

Combining all weights and using the formula (3.18), we find that the equivariant Chern character of the universal bundleE_UpNqis given by

Ch_TpE_UpNqq|z₁“z₂“0 “ ÿn

l“1

e^ial´ pe^ie1´1qpe^ie2´1q ÿk

i“1

e^iφi´ie`. (3.35)

Using the index formula (3.32) we have now computed the contribution for a fundamental massless hypermultiplet. As we explained before we can easily generalize this to other representations. In particular, we can give the hypermultiplet a mass by introducing a weightmfor the flavor torusT¹_m. This will act on the linear ADHM data as

vPV : vÞÑdiagpe^im,¨ ¨ ¨,e^imqv, wPW : wÞÑdiagpe^im,¨ ¨ ¨ ,e^imqw.

For gauge groupUpNqthe poles of the resulting contour integral (3.34) can be labeled by a set of colored Young diagramsY“ pY₁,Y₂,¨ ¨ ¨Y_Nq[28, 55, 56]. Therefore, the partition function can be written as

Zpa,m,e₁,e₂q “ÿ

Y

q^|Y|ź

R

z_R,|Y|pY;a,m,e₁,e₂q. (3.36)

When the gauge group is a product of M factors, the instanton partition function can be written as a sum over M colored Young diagramsY. ForSO{Spgauge groups, we will see that the structure of the contour integral is similar. However, the poles are no longer labeled by a simple set of colored Young diagrams.

Equivariant index forSO{Spgauge theories

For SppNq the weights of the equivariant torus action on the vector spacesV andW are given by

vPV : v ÞÑdiagpe^iφ1,¨ ¨ ¨,e^iφn,p1q,e^´iφ1,¨ ¨ ¨,e^´iφnqv wPW : w ÞÑdiagpe^ia1,¨ ¨ ¨ ,e^iaN,e^´ia1,¨ ¨ ¨,e^´iaNqw,

wherek “2n`χ, withn “ rk{2sandχ”kpmod 2q. Thep1qis inserted whenχ“1 and omitted whenχ“0. The equivariant character of the universal bundle is thus given by

Ch_TpE_Spq|_z1“z2“0 “

N

ÿ

l“1

´

e^ial `e^´ial

¯

(3.37)

´ pe^ie1´1qpe^ie2´1q

˜ _n ÿ

i“1

´

e<sup>iφi´ie`</sup>`e^´iφi´ie`

¯

`χe<sup>´ie`</sup>

¸ .

ForSOpNqthe weights are given by

vPV : v ÞÑdiagpe^iφ1,¨ ¨ ¨,e^iφk,e^´iφ1,¨ ¨ ¨ ,e^´iφkqv wPW : w ÞÑdiagpe^ia1,¨ ¨ ¨,e^ian,p1q,e^´ia1,¨ ¨ ¨ ,e^´ianqw,

whereN “2n`χ, such thatn “ rN{2sandχ ” Npmod 2q. Again,p1qis inserted when χ “ 1 and omitted when χ “ 0. The equivariant character of the universal bundle is therefore equal to

ChTpE_SOq|z₁“z2“0 “ ÿn

l“1

´

e^ial `e^´ial

¯

`χ (3.38)

´ pe^ie1´1qpe^ie2´1q

k

ÿ

i“1

´

e<sup>iφi´ie`</sup>`e^´iφi´ie`

¯ .

Building from these expressions we can obtain the instanton partition functions of quiver gauge theories containing matter fields in various representations.

3.2.3 Contour integrals for various matter fields with different gauge groups Let us collect various contour integrands for the instanton counting.

Fundamental ofUpNq

The equivariant index for a fundamentalUpNqhypermultiplet of massmis given by IndT “

ż

C²ChTpE_UbLbMqTd_TpC²q

“ ř_N

l“1e^ipal`e``mq pe^ie1´1qpe^ie2´1q ´

ÿk

i“1

e^ipφi`mq. (3.39)

The first term corresponds to the 1-loop factor. We read the matter contribution to the contour integral as

z^N_k “

k

ź

i“1

pφ_i`mq. (3.40)

The anti-fundamental matter can be obtained similarly:

Ind_T “ ż

C²Ch_TpE_U^˚_bL_bMqTd_TpC²q

“ ř_N

l“1e^{´ipal´e`´mq} pe^ie1´1qpe^ie2´1q ´

ÿk

i“1

e^{´ipφi`e`´mq} (3.41)

so that

z^N_k^¯ “ źk

i“1

pφ_i´mq. (3.42)

Note thatz^N_k pφ_i,´m,e₁,e₂q “z_k^N¯pφ,m,e₁,e₂q.

Adjoint ofUpNq

The equivariant index is Ind_T “

ż

C²Ch_TpEbE^˚bLbMqTd_TpC²q

“ ř_N

l,m“1e^{ipal´am`e``mq} pe^ie1´1qpe^ie2´1q ´

k,N

ÿ

i,l“1

e^ipφi´al`mq´

k,N

ÿ

i,l“1

e^{´ipφi´al´mq}

` pe^ie1´1qpe^ie2´1q ÿk

i,j“1

e^{ipφi´φj´e``mq}, (3.43)

and the integrand is given by

z^adj_k “ źk,N

i,l“1

pφ_i´a_l`mq źk,N

i,l“1

pφ_i´a_l´mq źk

i,j“1

pφ_ij`m´e´qpφ<sub>ij</sub>`m`e´q

pφ_ij`m`e`qpφ<sub>ij</sub>`m´e`q, (3.44) whereφ_ij “φ_i´φ_jande_˘“ ^e1˘e₂ ².

UpNqgauge multiplet

In general, the equivariant index of a gauge multiplet is given by the character of the tangent space of the moduli space [30],

IndTpT_pMq “ ´IndTpEbE^˚q. (3.45) Therefore,

Ind_T “ ´ ż

C²Ch_TpEbE^˚qTd_TpC²q

“ ´ ř_N,N

l,m“1e^ipal´amq pe^ie1´1qpe^ie2´1q`

k,N

ÿ

i,m“1

e<sup>ipφi´e`´amq</sup>`

k,N

ÿ

j,l“1

e^{´ipφj`e`´alq}

´ pe^ie1´1qpe^ie2´1q

k,k

ÿ

i,j“1

e^{ipφi´φj´eq}. (3.46)

The resulting contour integral is given by

Z_k “ 1 k!

ˆ e e₁e₂

˙_k¿ ˜ źk

i“1

dφ_i 2πi

¸

(3.47)

ˆ 1

ś_k,N

i,m“1pφ_i´a_mqś_k,N

j,l“1pφ_j´a_l`e<sub>1</sub>`e₂q ź

1ďiăjďk

φ²_ijpφ²_ij´e²q pφ_ij²´e₁²qpφ²_ij´e²₂q.

where φ_ij “ φ_i´φ_j. The poles of the contour integral can be classified in terms of N- colored Young diagrams with k boxes. In the case of UpNq linear quiver theory, there is no additional pole besides coming from the gauge multiplet. Therefore, the instanton partition function for theUpNqtheories can be simply written in terms of sum over colored Young diagrams~_{Y“ pY}

1,¨ ¨ ¨ ,Y_Nq.

BifundamentalpN₁, ¯N₂qofUpN₁q ˆUpN₂q The index is

IndT “ ż

C²ChTpE₁bE₂^˚bLbMqTd_TpC²q

“

ř_N1,N2

l,m“1e^{ipal´bm`m`e`q} pe^ie1´1qpe^ie2´1q ´

k1,N2

ÿ

i,l“1

e^{ipφ1,i´bm`mq}´

k2,N1

ÿ

i,l“1

e^{´ipφ2,j´al´mq}

` pe^ie1´1qpe^ie2´1q

k1,k2

ÿ

i,j“1

e^{ipφ1,i´φ2,j´e``mq}. (3.48)

The corresponding integral is

z^pN_k ^{1, ¯N2q}

1,k2 “

k1,N2

ź

i,l“1

pφ_1,i´b_l`mq

k2,N1

ź

i,l“1

pφ_2,i´a_l´mq

ˆ

k₁,k₂

ź

i,j“1

pφ_1,i´φ_2,j`m´e´qpφ<sub>1,i</sub>´φ<sub>2,j</sub>`m`e´q

pφ_1,i´φ_2,j`m`e`qpφ<sub>1,i</sub>´φ<sub>2,j</sub>`m´e`q. (3.49) We can obtainN₂ fundamental matter contributions (3.40) with massesm´b_l by setting k₁“k,k2“0. Also, by settingφ_1,i “φ_2,i“φ_i anda_i“b_i, we obtain adjoint matter (3.43).

Fundamental ofSppNq

The equivariant index for a fund.SppNqhypermultiplet of massmis given by Ind_T“

ż

C²Ch_TpE_SpbLbMqTd_TpC²q (3.50)

“ 1

pe^ie1´1qpe^ie2´1q ÿN

l“1

´

e<sup>ial`im`ie`</sup>`e^´ial`im`ie`

¯

´ ÿn

i“1

´

e<sup>iφi`im</sup>`e<sup>´iφi`im</sup>`χe^im

¯ ,

where k “ 2n`χ and e` “ ^e1`e₂ ². Here, we tensored the universal bundle E_Sp by the vector space M – C on whose elements the flavor symmetry Up1q_m acts by v ÞÑ e^imv.

Since the first term in the index computes perturbative terms in the free energy, the contour

integrand for the instanton contribution to the free energy is

z_k^N “m^χ źn

i“1

pφ_i`mqpφ_i´mq. (3.51)

SppNqgauge multiplet

The equivariant index for anSppNqgauge multiplet is given by Ind_T“ ´

ż

C²Ch_TpSym²E_SppNqqTd_TpC²q (3.52) The resulting contour integral is

Z_k “ p´1qⁿ 2ⁿn!

ˆ e e₁e₂

˙_n„

´1 2e₁e₂Ppe`q

χ

(3.53) ˆ

¿ ˜ _n ź

i“1

dφ_i 2πi

¸ ∆p0q∆peq

∆pe1q∆pe2q źn

i“1

1

p2φ²_i ´e²₁qp2φ_i²´e₂²qPpφ_i`e`qPpφ_i´e`q withe“e<sub>1</sub>`e₂and

Ppxq “ źN

l“1

px²´a²_lq,

∆pxq “

« _n ź

i“1

pφ²_i ´x²q ff_χ

ź

iăj

`pφ<sub>i</sub>`φ_jq²´x²˘ `

pφ_i´φ_jq²´x²˘ .

We describe a way to enumerate the poles of the above contour integral in appendix B, using what we call generalized Young diagrams.⁴

Bifundamental ofSppN₁q ˆSppN₂q

The equivariant index of a bifund.SppN₁q ˆSppN₂qhyper with massmis given by Ind_T“

ż

C²Ch_TpE_Sp¹ bE_Sp² bLbMqTd_TpC²q. (3.54) whereM –Cis acted upon by the flavor symmetry groupUp1q_m. Here we extended the universal bundlesE_Sp¹ andE_Sp² over the product M_SppN1q,k1ˆM_SppN2q,k2ˆR⁴ by pulling

4The cases wheree1 “ ´e₂were derived and discussed in [70, 71]. But their derivation can not be easily generalized to the generale₁,e₂.

them back using the respective projection mapsπ_i :M_{SppN1q,k1 ˆ}M_{SppN2q,k2 Ñ}M_SppNiq,ki. Define

P₁px,aq “

N₁

ź

l

px²´a²_lq,

P₂px,bq “

N2

ź

m

px²´b_m²q,

∆pxq “

n₁,n₂

ź

i,j“1

ppφ_i`φ˜_jq²´x²qppφ_i´φ˜_jq²´x²q,

∆1pxq “ź

i

pφ_i²´x²q,

∆2pxq “ź

i

pφ˜_i²´x²q,

wherek_i “2n_i`χ_i. Then the instanton contour integrand is given by

z_k^N1,N2

1,k₂ “

n₁

ź

i

P2pφ_i`mqP2pφ<sub>i</sub>`m`eq

n₂

ź

j

P₁pφ˜_j`mqP<sub>1</sub>pφ˜<sub>j</sub>`m`eq (3.55)

ˆ

N1

ź

l

pa²_l ´m²q^χ2

N2

ź

k

pb²_k´m²q^χ1

ˆ

ˆ∆pm´e´q∆pm`e´q

∆pm´e`q∆pm`e`q

˙ ˆ∆pm´e´q∆pm`e´q

∆pm´e`q∆pm`e`q

˙χ₂

ˆ

ˆ∆pm´e´q∆pm`e´q

∆pm´e`q∆pm`e`q

˙_χ1ˆ

pm´e´qpm`e´q pm´e`qpm`e`q

˙_χ1χ2 ,

wheree“e₁`e₂. Note that there are additional poles that involve the mass parameterm.

The contour prescription is to assumee₃ “ ´m´e_{`</sub>ande<sub>4</sub> “ m´e<sub>`} to have a positive imaginary value. This is the same prescription as for the massive adjoint hypermultiplet in theN _“2^˚theory [75].

Fundamental ofSOpNq

The equivariant index of a fund.SOpNqhypermultiplet of massmis given by IndT “

ż

C²ChTpE_SObLbMqTdTpC²q (3.56)

“ 1

pe^ie1´1qpe^ie2´1q

˜

χe<sup>ie`</sup>` ÿn

l“1

´

e<sup>ial`im`ie`</sup>`e^´ial`im`ie`

¯

¸

´ ÿk

i“1

´

e<sup>iψi`im</sup>`e^´iψi`im

¯ ,

whereN“2n`χ. The corresponding instanton integrand is

z_k^N “ źk

i“1

pψi`mqpψ_i´mq. (3.57)

SOpNqgauge multiplet

The equivariant index for anSOpNqgauge multiplet is given by Ind_T“ ´

ż

C²Ch_Tp^²E_SOpNqqTd_TpC²q (3.58) The resulting contour integral is

Z_k“ p´1q^kpN`1q 2^kk!

ˆ e e₁e₂

˙_k¿ ˜ źk

i“1

dψ_i 2πi

¸ ∆p0q∆peq

∆pe1q∆pe2q

ψ²_ipψ²_i ´e²_`q

Ppψ_i`e<sub>`</sub>qPpψ_i´e<sub>`</sub>q (3.59) wheree“e<sub>1</sub>`e₂and

Ppxq “ x^χ źn

l“1

`x²´a²_l˘ ,

∆pxq “ ź

iăj

`pψi´ψ<sub>j</sub>q<sup>2</sup>´x<sup>2</sup>˘ `

pψi`ψ_jq²´x²˘ .

Whene₁“ ´e₂the pole structure is simplified and described by a set ofN-colored Young diagrams like for the gauge groupUpNq.⁵

5The cases wheree₁ “ ´e₂were derived and discussed in [70, 71]. But their derivation can not be easily generalized to the generale1,e2.