# Light Adjoint Quarks in the Instanton-Dyon Liquid Model IV

###### Abstract

We discuss the instanton-dyon liquid model with Majorana quark flavors in the adjoint representation of color at finite temperature. We briefly recall the index theorem on for twisted adjoint fermions in a BPS dyon background of arbitrary holonomy, and use the ADHM construction to explicit the adjoint anti-periodic zero modes. We use these results to derive the partition function of an interacting instanton-dyon ensemble with light and anti-periodic adjoint quarks. We develop the model in details by mapping the theory on a 3-dimensional quantum effective theory with adjoint quarks with manifest symmetry. Using a mean-field analysis at weak coupling and strong screening, we show that center symmetry requires the spontaneous breaking of chiral symmetry, which is shown to only take place for . For a sufficiently dense liquid, we find that the ground state is center symmetric and breaks spontaneously flavor symmetry through . As the liquid dilutes with increasing temperature, center symmetry and chiral symmetry are restored. We present numerical and analytical estimates for the transition temperatures.

###### pacs:

11.15.Kc, 11.30.Rd, 12.38.Lg## I Introduction

This work is a continuation of our earlier studies LIU1 ; LIU2 ; LIU3 of the gauge topology in the confining phase of a theory with the simplest gauge group . We suggested that the confining phase below the transition temperature is an “instanton dyon” (and anti-dyon) plasma which is dense enough to generate strong screening. The dense plasma is amenable to standard mean field methods.

The key ingredients in the instanton-dyon liquid model are the so called KVBLL instantons threaded by finite holonomies KVLL splitted into their constituents, the instanton-dyons. Diakonov and Petrov DP ; DPX have shown that the KvBLL instantons dissociate in the confined phase and recombine in the deconfined phase, using solely the BPS protected moduli space. The inclusion of the non-BPS induced interactions, through the so called streamline set of configuration, is important numerically, but it does not alter this observation LARSEN . The dissociation of instantons into constituents was advocated originally by Zhitnitsky and others ARIEL . Unsal and collaborators UNSAL proposed a specially tuned setting in which instanton constituents (they call instanton-monopoles) induced confinement even at exponentially small densities, at which the semi-classical approximations is parametrically accurate. Key feature of this setting is cancellation of the perturbative Gross-Pisarski-Yaffe holonomy potential.

The KvBLL instantons fractionate into constituents with fractional topological charge . Their fermionic zero modes do not fractionate but rather migrate between various constituents KRAAN . This interplay between the zero modes and the constituents is captured precisely by the Nye-Singer index theorem SINGER . For fundamental fermions, we have recently shown in the mean-field approximation that the center symmetry and chiral symmetry breaking are intertwined in this model LIU2 . The broken and restored chiral symmetry correspond to a center symmetric or center asymmetric phases, respectively. Similar studies were developed earlier in SHURYAK1 ; SHURYAK2 ; TIN .

In this work we would like to address this interplay between confinement and chiral symmetry breaking using massless quarks in the representation of color . We will detail the nature of the flavor symmetry group of the effective action induced by dissociated KvBLL calorons in the confined phase, and investigate its change into an asymmetric phase at increasing temperature. Lattice simulations with adjoint quarks LATTICEX have shown that the deconfinement and restoration of center symmetry occurs well before the restoration of chiral symmetry. These lattice results show that the ratio of the chiral to deconfinement temperatures is large and decrease with the number of adjoint flavors. More recent lattice simulations have suggested instead a rapid transition to a conformal phase LATTICEZ . Effective PNJL models with adjoint fermions have also been discussed recently OGILVIE ; KANA .

The organization of the paper is as follows: In section 2 we briefly review the index theorem on for an adjoint fermion with twisted boundary condition. In section 3, we detail the ADHM construction and use it to derive the anti-periodic adjoint fermion in self-dual BPS dyons. In section 4, we develop the partition function of an instanton-dyon ensemble with one light light quark in the adjoint representation of . By using a series of fermionization and bosonization techniques we construct the 3-dimensional effective action, accommodating light adjoint quarks with explicit flavor symmetry. In section 5, we discuss the nature of the confinement-deconfinement in the quenched sector () of the induced effective action. In section 6, we show that for a sufficiently dense instanton-dyon liquid with light adjoint quarks, the 3-dimensional ground state is still center symmetric and breaks spontaneously flavor symmetry. Center symmetry is broken and chiral symmetry is restored only in a more dilute instanton-dyon liquid, corresponding to higher temperatures. Our conclusions are summarized in section 6. In Appendix A we check that our ADHM construct reproduces the expected periodic zero modes for BPS dyons. In Appendix B we derive the pertinent equations for the anti-periodic adjoint fermions in a BPS monopole without using the ADHM method. In Appendix C we explicit the ADHM construction for the anti-periodic zero modes in a KvBLL caloron. In Appendix D we detail the Fock correction to the mean-field analysis. In Appendix E we briefly outline the 1-loop analysis for completeness. In Appendix F we quote the general result for the 1-loop contribution to the holonomy potential with adjoint massless quarks.

## Ii Index theorem for twisted quarks

In this section we revisit the general Nye-Singer index theorem for fermions on a finite temperature Euclidean manifold for a general fermion representation. For periodic fermions a very transparent analysis was provided by Popitz and Unsal POP . We will extend it to fermions with arbitrary “twist” (phase), which is the used for our case of anti-periodic fermions in the adjoint representation.

### ii.1 Index

Consider chiral Dirac fermions on interacting with an anti-self-dual gauge field through

(1) |

with twisted fermion boundary conditions ()

(2) |

Here satisfies

(3) |

For monopoles, the difference between the zero modes of different chiralities in arbitrary R-representation is captured by Calias index CAL

(4) |

with the Trace carried over spin-color-flavor and space-time. Using the local chiral anomaly condition for the iso-singlet axial current in Euclidean 4-dimensional space

(5) |

we can re-write the index in the following form

### ii.2 L, M Dyons

To evaluate (6) for twisted adjoint fermions in the background of an anti-self-dual or dyon, we follow Popitz and Unsal POP and write

(7) | |||

In the anti-dyon background, we have at asymptotic spatial infinity

(8) |

The compact character of on breaks . After expanding the ratio with in (7), only the first term carries a non-vanishing net flux in (6) on thanks to the asymptotic in (II.2). If we recall that the Trace now carries a summation over the windings along labeled by , and using the identity

(9) |

we have

(10) |

For color , in the fundamental representation and in the adjoint representation. For the latter,

(11) |

it follows that

(12) |

For the L-dyon we note that the surface contributions satisfy since the asymptotics at spatial infinity have the same with of opposite sign. Therefore we obtain

(13) |

whatever the twist as expected. For anti-periodic fermions with , we find that for , the L-dyon carries 4 anti-periodic zero modes and the M-dyon carries 0 zero mode. For , the M-dyon carries 4 zero modes and the L-dyon carries 0 zero mode. The confining holonomy with is special as the zero modes are shared equally between the - and -dyon, 2 on each. These observations are in agreement with those made recently in KANA .

## Iii ADHM construction of Adjoint zero modes

In this section we first remind the general framework for the ADHM ADHM ; MATTIS ; KVLL ; DPX construction for adjoint fermions, and then apply it to to the special case of adjoint fermions in the background field of BPS dyons. A concise presentation of this approach can be found in MATTIS ; DPX whose notations we will use below. Throughout this section we will set the circle circumference , unless specified otherwise. We note that our construction is similar in spirit to the one presented in ADHMX for adjoint fermions in calorons, but is different in some details. In particular, it does not rely on the replica trick and therefore does not double the size of the ADHM data.

### iii.1 ADHM construction

The basic building block in the ADHM construction is the asymmetric matrix of data of dimension for an gauge configuration of topological charge . The null vectors of can be assembled into a matrix-valued complex matrix of dimension , satisfying or specifically

(14) |

with the ADHM label running over , and referring to the Weyl-Dirac indices which are raised by . They are orthonormalized by . In terms of (14) the classical ADHM gauge field with reads

(15) |

For it is a pure gauge transformation with a field strength that satisfies the self-duality condition . For it still satisfies the self-duality condition provided that MATTIS

(16) |

with a positive matrix of dimension . The matrix of data is taken to be linear in the space-time variable

(17) |

with the quaternionic notation and .

### iii.2 Anti-periodic adjoint fermion in general

Given the matrix of ADHM data as detailed above, the adjoint fermion zero mode in a self-dual gauge configuration of topological charge reads MATTIS

(18) |

which can be checked to satisfy the Weyl-Dirac equation provided that the Gassmanian matrix of dimension satisfies the constraint condition

(19) |

To unravel the constraints (16) and (19) it is convenient to re-write the ADHM matrix of data in quaternionic blocks through a pertinent choice of the complex matrices , i.e.

(20) |

with

(21) |

In quaternionic blocks, the null vectors (14) are

with the normalization . To solve the constraint condition (19) we also define

(23) |

and its conjugate . Therefore the solution to (19) satisfies and the new constraint between the Grassmanians

(24) |

finally, for periodic gauge configurations on such as the KvBLL calorons or BPS dyons, the index is extended to all charges in . It is then more convenient to use the Fourier representations

(25) |

which are z-periodic of period 1.

### iii.3 Anti-periodic adjoint fermion in a BPS Dyon

For BPS dyons the previous arguments apply NAHM . In particular, for the M-dyon on , the preceding construct simplifies. In particular, the quaternion blocks in the ADHM matrix of data in (20) are simply

(26) |

The normalized null vector is readily found in the form

(27) |

with

(28) |

with the vev .

The constraint (16) following from the self-duality condition translates to the equation for the resolvent

The solution is

We have explicitly checked that (III.3) satisfies the identities used in the ADHM construction as noted in MATTIS . In our case these identities read

with the definition , and

(32) | |||||

We note that the periodicity on translates to the quasi-periodicities

(33) |

For the adjoint fermion zero-mode, the Grassmanian matrix also simplifies

(35) |

with normalized constant spinors . This allows to re-write (34) in the explicit form

With the above in mind, the adjoint zero-mode solution (18) in the BPS M-dyon simplifies to

For the integrations can be undone. For that we translate the vectors in (III.3) to spinors using the quaternionic form , and make (III.3) more explicit. The result is

(38) | |||||

with defined as

where we have set and . Asymptotically, the zero modes (III.3) simplify to

(40) |

and therefore (38) is asymptotically ()

(41) |

We will use this approximation to carry explicitly the analysis below. The 4 zero modes (41) are localized on the M-dyon for , and by duality on the L-dyon for , in agreement with the index theorem reviewed above.For the integration vanishes with .

For we note that the 4 adjoint zero modes are normalizable as they fall asymptotically with . (III.3) can be explicitly checked to be normalized as

(42) |

We note that at the normalization is upset. This is precisely where the zero modes re-organize equally between the L- and M-instanton-dyon, a pair on each.

### iii.4 Anti-periodic adjoint fermion in a BPS Dyon with

The case for the adjoint zero mode is more subtle. The preceding arguments show that the exponent in it asymptotic decay disapperin this case: . In this limit, the index theorem states that 2 zero modes are localized on the M-dyon and 2 zero-modes on the L-dyon. In this section, we show that the reduction of the result (III.3) for simplifies. Specifically,

(43) | |||

with

(44) |

(43) can be written in a more concise form by translating the vectors to spinors using the quaternionic form

(45) |

with

(46) |

Using , the transposed of the second term in (III.4) can be reduced. The result is

(47) |

with the identified spinor . The (color) invariant group norm of (III.4) is finite. Specifically, if we set , then

(48) |

which is convergent in . Note the difference between the Matsubara arrangements in (III.4) and (III.3). For completeness, we note that (III.4) is the analogue of the gluino condensate using the anti-periodic zero modes. The periodic zero modes are briefly discussed in Appendix A using the same ADHM construct. In Appendix B, we verify explicitly that the ADHM zero modes are consistent with a direct reduction of the Dirac equation. For completeness, we detail in Appendix C the ADHM construct for the zero modes around KvBLL instantons.

## Iv Partition function with adjoint fermions

In this section we will use the adjoint zero modes made explicit in (III.3-41), to construct the partition function for an ensemble of interacting dyons and anti-dyons with adjoint fermions. We will show that the partition function is amenable to a 3-dimensional effective theory. The derivation will be for the non-symmetric case with , where all the 4 adjoint zero modes are localized on the -dyon (anti-dyon). The non-symmetric case with with the adjoint zero modes localized on the -dyon (anti-dyon) is equivalent and follow by duality and . The symmetric case with each and dyons carrying 2 of the 4 adjoint zero modes, will be understood in the limit .

### iv.1 Partition function

In the semi-classical approximation, the Yang-Mills partition function is assumed to be dominated by an interacting ensemble of instanton-dyons (anti-dyons). They are constituents of KvBLL instantons (anti-instantons) with fixed holonomy KVLL . The grand-partition function with adjoint Majorana quarks is

Here and are the 3-dimensional coordinate of the i-dyon of m-kind and j-anti-dyon of n-kind. Here a matrix and a matrix whose explicit form are given in DP ; DPX . The fugacities are related to the overall dyon plus anti-dyon density CALO-LATTICE .

is the streamline interaction between dyons and anti-dyons as numerically discussed in LARSEN ; LIU1 . For the case it is Coulombic asymptotically LIU1

The strength of the Coulomb interaction in (LABEL:DDXX) is . Following SHURYAK1 , we define the core interactions between and respectively, which we assume to be step functions of height and range

(51) |

with normalized to the dimensionless unit volume

(52) |

We recall that the and channels are repulsive. A sketch of the interaction potentials is given in Fig. 1. Below the core value of , the streamline configuration annihilates into perturbative gluons.

### iv.2 The determinant of the adjoint fermions

The fermionic determinant in (LABEL:SU2) is composed of all the hoppings between the dyons and anti-dyons through the adjoint fermionic zero modes. To explicit the hopping, we consider in details the case , and only quote at the end the generalization to arbitrary . To explicit the hopping for , we define

(53) |

with the sum running over all dyons and anti-dyons, and the 2 Matsubara frequencies subsumed in the zero modes. The adjoint dyon and anti-dyon zero modes are labelled by

(54) |

Here is a 2-component Grassmanian spinor and a valued matrix, both of which refer to a D-dyon (anti-dyon). From (III.3-41) the Fourier transforms of read

with

(56) |

Here and .

with . We note that the matrix entries in (IV.2) are valued or quaternionic, and that the matrix overall is anti-symmetric under transposition. This observation is consistent with the observations made in ADJOINT . The matrix entries in (IV.2) satisfy

(58) | |||||

(59) |

Using (IV.2) we can rewrite (LABEL:SU2) for massive fermions in the basis as follows

with dimensionality . Each of the quaternionic entry in is a “hopping amplitude” for a fermion between an instanton-dyon and an instanton-anti-dyon. Each of the quaternion entry in is an overlap between two instanton-dyons or two instanton-anti-anti-dyons.

### iv.3 Hopping amplitudes

In momentum space the quaternionic entries are given by

(61) |

with again . Since

(62) |

we also have the identities