UUITP08/07
HIP200728/TH
NORDITA200715
YITPSB0718
New vector multiplets
[12mm] Ulf Lindström, Martin Roček, Itai Ryb,
Rikard von Unge, and Maxim Zabzine
[8mm] Department of Theoretical Physics Uppsala University,
Box 803, SE751 08 Uppsala, Sweden
HIPHelsinki Institute of Physics, University of Helsinki,
P.O. Box 64 FIN00014 SuomiFinland
NORDITA, Roslagstullsbacken 23,
SE10691 Stockholm, Sweden
C.N.Yang Institute for Theoretical Physics, Stony Brook University,
Stony Brook, NY 117943840,USA
Institute for Theoretical Physics, Masaryk University,
61137 Brno, Czech Republic
Abstract
We introduce two new vector multiplets that couple naturally to generalized Kähler geometries. We describe their kinetic actions as well as their matter couplings both in and superspace.
1 Introduction
Generalized Kähler geometry has aroused considerable interest both among string theorists and mathematicians, e.g., [2, 3, 4]. Recently, several groups have tried to construct quotients [5, 6, 7, 8]; however, it is unclear how general or useful the various proposals are. Experience has shown that supersymmetric models are often a helpful guide to finding the correct geometric concepts and framework for quotient constructions [9, 10]. In this paper, we take the first step in this direction; further results will be presented in [11].
The basic inspiration for our work is the interesting duality found in [12, 13]. As was shown in [10, 14], Tdualities arise when one gauges an isometry, and then constrains the fieldstrength of the corresponding gauge multiplet to vanish. Here we address the question: what are the gauge multiplets corresponding to the duality introduced in [12, 13]?
In section 2, we analyze the types of isometries that arise on generalized Kähler geometries which are suitable for gauging, and describe the corresponding multiplets in superspace. In addition to the usual multiplets with chiral or twisted chiral gauge parameters, we find two new multiplets: one with semichiral gauge parameters, which we call the semichiral gauge multiplet, and one with a pair of gauge parameters, one chiral and one twisted chiral; the last has more gaugeinvariant components than other multiplets, and hence we call it the large vector multiplet.
In section 3, we describe the superspace content of these mulitplets; this exposes their physical content. We describe both multiplets and their couplings to matter, and discuss possible gauge actions for them. The component content of the various multiplets that arise is well known and can be found in [15].
Throughout this paper we follow the conventions of [16].
2 Generalized Kähler geometry: superspace
Generalized Kähler geometry (GKG) arises naturally as the target space of supersymmetric models. As shown in [16], such models always admit a local description in superspace in terms of complex chiral superfields , twisted chiral superfields and semichiral superfields [17]. These models have also been considered in superspace [18, 19].
These geometries may admit a variety of holomorphic isometries that can be gauged by different kinds of vector multiplets. We now itemize the basic types of isometries.
2.1 Isometries
The simplest isometries act on purely Kähler submanifolds of the generalized Kähler geometry, that is only on the chiral superfields or the twisted chiral superfields ; for a single isometry away from a fixed point, we may choose coordinates so that the Killing vectors take the form:
(2.1) 
In [12, 13], new isometries that mix chiral and twisted chiral superfields or act on semichiral superfields were discovered; we may take them to act as
(2.2)  
(2.3) 
where , etc. One might imagine more general isometries that act along an arbitrary vector field; however, compatibility with the constraints on the superfields (chiral and twisted chiral superfields are automatically semichiral but not viceversa) allows us to restrict to the cases above; in particular, if the vector field has a component along or , we can (locally) redefine to eliminate any component along .
A general Lagrange density in superspace has the form:
(2.4) 
For the four isometries listed above the corresponding invariant Lagrange densities are^{1}^{1}1Generally, isometries may leave the Lagrange density invariant only up to a (generalized) Kähler transformation [21, 16], but as our interest here is the structure of the vector multiplet, we are free to choose the simplest situation.:
(2.5)  
(2.6)  
(2.7)  
(2.8) 
In general, the isometries act on the coordinates with some constant parameter :
(2.9) 
where is any of the coordinates , etc.
2.2 Gauging and Vector Multiplets
We now promote the isometries to local gauge symmetries: the constant transformation parameter of (2.9) becomes a local parameter that obeys the appropriate constraints.
(2.10) 
To ensure the invariance of the Lagrange densities (2.52.8) under the local transformations (2.2), we introduce the appropriate vector multiplets. For the isometries (2.5,2.6) these give the well known transformation properties for the usual (un)twisted vector multiplets:
(2.11) 
whereas for generalized Kähler transformations we need to add triplets of vector multiplets.
For the the semichiral isometry , we introduce the vector multiplets:
(2.12) 
We refer to this multiplet as the semichiral vector multiplet.
For the isometry we introduce the vector multiplets
(2.13) 
and refer to this multiplet as the large vector multiplet due to the large number of gaugeinvariant components that comprise it.
2.3 fieldstrengths
We now construct the gauge invariant fieldstrengths for the various multiplets introduced above.
2.3.1 The known fieldstrengths
The fieldstrengths for the usual vector multiplets are well known:
(2.14) 
Note that , the fieldstrength for the chiral isometry is twisted chiral whereas , the fieldstrength for the twisted chiral isometry, is chiral.
2.3.2 Semichiral fieldstrengths
To find the gaugeinvariant fieldstrengths for the vector multiplet that gauges the semichiral isometry it is useful to introduce the complex combinations:
(2.15) 
Then the following complex fieldstrengths are gauge invariant:
(2.16) 
where is chiral and is twisted chiral.
2.3.3 Large Vector Multiplet fieldstrengths
As above it is useful to introduce the complex potentials:
(2.17) 
Because are (twisted)chiral respectively, the following complex spinor fieldstrengths are gauge invariant:
(2.18) 
The higher dimension fieldstrengths can all be constructed from these spinor fieldstrengths:
(2.19) 
the chirality properties of these fieldstrengths are summarized below:
(2.20) 
3 Gauge multiplets in superspace
To reveal the physical content of the gauge multiplets, we could go to components, but it is simpler and more informative to go to superspace. We expect to find spinor gauge connections and unconstrained superfields. As mentioned in the introduction, the component content of various multiplets can be found in [15].
The procedure for going to components is wellknown; for a convenient review, see [16]. We write the derivatives and their complex conjugates in terms of real derivatives and the generators of the nonmanifest supersymmetries,
(3.1) 
and components of an unconstrained superfield as , , and .
3.1 The semichiral vector multiplet
We first identify the components of the semichiral vector multiplet, and then describe various couplings to matter.
3.1.1 components of the gauge multiplet
We can find all the components of the semichiral gauge multiplet from the field strengths (2.3.2) except for the spinor connections . The only linear combination of the gauge parameters that does not enter algebraically in (2.2) is , and hence the connections must transform as:
(3.2) 
This allows us to determine the connections as:
(3.3) 
where the terms vanish in WessZumino gauge. The gaugeinvariant component fields are just the projections of the fieldstrengths (2.3.2) and the fieldstrength of the connection :
(3.4) 
These are not all independent–they obey the Bianchi identity:
(3.5) 
Thus this gauge multiplet is described by an gauge multiplet and three real unconstrained scalar superfields:
(3.6) 
Though not essential, the simplest way to find the reduction of various quantities is to go to a WessZumino gauge, that is reducing the gauge parameters to a single gauge parameter by gauging away all components with algebraic gauge transformations. Here this means imposing

(3.7) 
on the gauge multiplet and
(3.8) 
on the gauge parameters. This leads directly to:
(3.9) 
3.1.2 Coupling to matter
We start from the gauged Lagrange density:
(3.10) 
In the WessZumino gauge defined above, we have
(3.11) 
and spinor components:
(3.12) 
Then for the tuple and the isometry vector defined as
(3.13) 
we write the gauge covariant derivative as it appears in [10]
(3.14) 
We can compute
(3.15) 
Using
(3.16)  
we obtain the gauged Lagrange density
(3.17) 
with:
(3.18) 
Here in the reduced Lagrange density is that same as for the ungauged model [16, 20].
3.1.3 The vector multiplet action
Introducing the notation
(3.19) 
and using the (twisted)chirality properties
(3.20) 
we find
(3.21) 
with
(3.22) 
Starting from an action:
(3.23) 
we write the reduction to in terms of the gaugeinvariant components :
(3.24) 
where
(3.25) 
To obtain real and positive definite we require which yields one gauge multiplet and three scalar multiplets. In particular, when , we find the usual diagonal action.
Other gaugeinvariant terms are possible; these are general superpotentials and have the form
(3.26) 
where are holomorphic functions. These terms reduce trivially to give:
(3.27) 
Particular examples of such superpotentials include mass and FayetIliopoulos terms.
3.1.4 Linear terms
To perform Tduality transformations, one gauges an isometry, and then constrains the fieldstrength to vanish [10, 14]. We will discuss Tduality for generalized Kähler geometry in detail in [11]; it was introduced (without exploring the gauge aspects) in [12, 13]. Here we describe the superspace coupling and its reduction to . We constrain the fieldstrengths to vanish using unconstrained complex Lagrange multiplier superfields
(3.28) 
integrating by parts, we can reexpress this in terms of chiral and twisted chiral Lagrange multipliers , to obtain
(3.29) 
This reduces to an superspace Lagrange density (up to total derivative terms)
(3.30)  
where are the obvious projections of the corresponding Lagrange multipliers. When we perform a Tduality transformation, we add this to the Lagrange density (3.17).
3.2 The Large Vector Multiplet
We now study the components of the large vector multiplet.
3.2.1 gauge invariants
Starting with the eight secondorder gauge invariants (2.3.3), we descend to superspace and identify the gauge fieldstrength.
Imposing the condition that the gauge connection transforms as
(3.31) 
we find the quantities
(3.32) 
of course, any gaugeinvariant spinor may be added to . It is useful to introduce the real and imaginary parts of :
(3.33) 
These form a basis for the gaugeinvariant spinors. The fieldstrength of the connection
(3.34) 
is manifestly gauge invariant. The remaining gaugeinvariant scalars are:
(3.35) 
The decomposition of the invariants is
(3.36) 
3.2.2 Matter couplings in superspace
We start from the gauged Lagrange density:
(3.37) 
We reduce to superfields, which in the WessZumino gauge
(3.38) 
are simply
(3.39) 
It is useful to introduce the notation
(3.40) 
and the covariant derivatives
(3.41) 
This gives
(3.42) 
and
(3.43)  
where and