Half-hypermultiplets

3.6 Instanton counting for half-hypermultiplets

3.6.1 Half-hypermultiplets

There are two types ofN “2 supersymmetry multiplets: theN “2 vector multiplet and the hypermultiplet. The former consists of a vector field A_µ, two Weyl fermions λ_α and ψα, and one complex scalarB. All of them transform in the adjoint representation of the gauge group. InN “ 1 language such anN “ 2 vector multiplet consists of oneN “1 vector multiplet with component fieldspAµ,λ_αqand one chiral multiplet with components pB,ψ_αq. A hypermultiplet requires a choice of representation R of the gauge group. In N “ 1 language it consists of two chiral multiplets Qand ˜Q, the former transforming in

the representationRand the latter in its complex conjugateR^˚. The chiral multipletQhas component fieldspφ,χαq, and the anti-chiral multiplet ˜Qhas componentspφ, ˜˜ χαq. We call bothQand ˜Qhalf-hypermultiplets.

The half-hypermultiplets Qand ˜Qform massless representations of theN “ 2 SUSY algebra. However, even though the helicities of the states in a half-hypermultiplet form a CPT complete distribution, the half-hypermultiplet does not transform as a real repre- sentation of the SUSY algebra. Indeed, notice that the masslessN _“ 2 SUSY algebra is equal to the Clifford algebra Cl_4,0with invariance groupSOp4q. The four-dimensional rep- resentation of the Clifford algebra Cl_4,0, under which the half-hypermultiplet transforms, is pseudo-real instead of (strictly) real. A generic half-hypermultiplet will therefore not be invariant under CPT.

It is possible though to circumvent this constraint. More precisely, anN “2 multiplet is CPT invariant if it transforms under a real representation of the product of the SUSY algebra, the gauge group and possible flavor groups. So, apart from the obvious possibility of combining a chiral and an anti-chiral multiplet into a full hypermultiplet, we can also consider a single half-hypermultiplet in a pseudo-real representation of the gauge group G.

Nevertheless, there is an additional requirement. Even when a single half-hyper-multiplet transforms in a real representation of Cl_4,0ˆG, such a theory may still be anomalous due to Witten’s anomaly argument [93]. According to this argument, for example, a single half-hypermultiplet in the fundamental representation ofSUp2qis anomalous (its partition function vanishes) since it contains an odd number of chiral fermions. On the contrary, a single half-hypermultiplet in the fundamental representation ofSUp2q³contains four chi- ral fermions in each SUp2q-representation. Therefore, quiver gauge theories with SUp2q trifundamental half-hypermultiplets are free of Witten’s SUp2q anomaly as well as CPT invariant.

Other examples of consistent theories with half-hypermultiplets occur when we con- sider massless bifundamental couplings betweenSOandSpgauge groups. A half-hypermultiplet transforming under the bifundamental ofSOˆSpis in a pseudo-real representation of the gauge groupG“SOˆSpand is free of the Witten anomaly as well.

Half-hypermultiplet as a constrained hypermultiplet

Since working with full hypermultiplets is often much more efficient than with half-hypermultiplets, in what follows we find an alternative method to deal with half-hypers. Instead of consid-

ering a half-hypermultiplet by itself, we start with a full hypermultiplet (consisting of two half-hypermultiplets) and impose a constraint on it which only leaves a half-hypermultiplet.

A full hypermultiplet can be thought of as a multiplet formed out of twoN _“1 chiral multipletsQand ˜Q. The chiral multipletQtransforms in representation R and the anti- chiral multiplet ˜Qin its complex conjugate R^˚. By the remarks above, for the theory of a single half-hyper to make sense, R needs to be a pseudoreal. Letσ_G be the anti-linear involution that maps the representation R to its complex conjugate R^˚. Since the repre- sentationRis pseudo-real, it obeysσ_G² “ ´1. For example, in the case of the fundamental representation ofSUp2q, the involution σ_Gis given by the e-tensor iσ₂. For the trifunda- mental representation ofSUp2qit is given by the product of threee-tensors, one for each SUp2qgauge group.

To impose our constraint, we need a mapτthat relatesQto ˜Qand vice versa. Since ˜Q appears in the complex conjugate,τneeds to be anti-linear. Moreover, it needs to preserve the representation, which means that it must involveσ_G. Let us write a full hypermultiplet asQ^awhose two half-hypermultiplet components are given byQ¹“QandQ² “Q˜^˚. The involutionτis then defined by

Q^a ÞÑτpQ^aq “σ_Gbσ_IpQ^aq^˚, (3.109) where

σ_I

¨

˝ Q¹ Q²

˛

‚“

¨

˝

´Q² Q¹

˛

‚. (3.110)

It is straightforward to check that indeedτ² “ 1. We can describe a half-hypermultiplet as a hypermultiplet that stays (anti-)invariant underτ, i.e., that is an eigenvector ofτof eigenvalue˘1. More explicitly, such a hyper is given bypQ,˘σGQ^˚q.

This description of a half-hypermultiplet is, for instance, convenient to find the La- grangian for a half-hyper Qstarting from the Lagrangian of a full hyper. Recall that the

Lagrangian for the hypermultipletQ^a “ pQ, ˜Q^˚q coupled to a vector multiplet pV,Φq is given by

L_fh“ ż

d²θd²θ¯

´

Q^:e^VQ`Q˜^te^´VQ˜^˚

¯

`2? 2 Re

ż

d²θ`Q˜^tΦQ˘ .

For pseudo-real representations we can apply the constraint ˜Q “ ˘σ_GQ to recover the Lagrangian

L_hh “ ż

d²θd²θ¯

´ Q^:e^VQ

¯

˘

?2 Re ż

d²θ`

Q^tσ_G^t ΦQ˘

for a single half-hypermultiplet. Here we rescaledQÑ ^?1

2Qto give the kinetic term in the Lagrangian a canonical coefficient. We also used

Q˜^te^´VQ˜^˚ “Q^tσ_G^te^´Vσ_GQ^˚“Q^te^VtQ^˚ “Q^:e^VQ,

sinceσ_G^´1Tσ_G“ ´T^tforTPg. Since we found the LagrangianL_hhby starting out with the LagrangianL_fhfor a full hypermultiplet and then applying the constraint ˜Q“ ˘σ_GQ, it is automatically invariant underN “2 supersymmetry.

Let us spell this out in some more detail. Substituting the constraint ˜Q “ ˘σ_GQin theN “ 2 supersymmetry equations yields two identical copies of the supersymmetry variations for the components ofQ, which depend on all of the N “ 2 supersymmetry parameters. The LagrangianL_hhis obviously invariant under these variations. TheSUp2qR

symmetry now acts on the vector of complex scalarspq,˘σ_Gq^˚q^t.

As an example, the Lagrangian for the trifundamentalSUp2qhalf-hypermultiplet reads in components

L_trif“ ż

d²θd²θ¯

´

Q^˚_abce^pV1qaa1Q^a1bc`Q<sup>˚</sup><sub>abc</sub>e<sup>pV2qbb1</sup>Q<sup>ab1c</sup>`Q^˚_abce^pV3qcc1Q^abc1

¯

(3.111)

˘

?2 Re ż

d²θ

´ e^bb

1

e^cc

1Q_abcΦ^aa1Q_a¹_b¹_c¹`e^aa

1

e^cc

1Q_abcΦ^bb1Q_a¹_b¹_c¹`e^aa

1

e^bb

1Q_abcΦ^cc1Q_a¹_b¹_c¹

¯ . We can obtain the bifundamental hyper by demoting one of gauge groups to a flavor group. From this perspective it is clear that the bifundamental has an enhanced SUp2q flavor symmetry, as we already knew from general principles. We discuss these and other aspects of theSUp2qtrifundamental in detail in appendix D.