The ω-deformed gauging has a nice geometric description in terms of Lie algebroid gauge theories.6 The geometric description of the gauging is as follows. We have a
Lie algebroid π:E →M, together with a choice of local frame {ea}. The image of this frame under the anchor map ρ(ea) =va=vai∂i are the vector fields with which we will gauge, and they have associated to them the structure functions
[ea, eb] =fcabec. (5.15)
The gauge fields, A, are one-forms on the worldsheet with values in the pullback bundle X∗E. The fields X are simply scalars on the worldsheet, and the Lagrange multipliers are scalars on the worldsheet with values in the dual pullback bundle
X∗E∗. Under a change of frame ea → e0a = ebΛba, the fields A, X and η are tensorial, so their components change as
Xi →Xi
Aa→(Λ−1)abAb
ηa→ηbΛba.
Lemma 5.1.1. Under the change of frame ea→e0a =ebΛba, the structure functions
change as
fabc→f0abc = (Λ−1)adΛebΛfcfdef −(Λ−1)adΛebΛdc,ivie+ (Λ−1)adΛecΛdb,ivie (5.16)
Proof. This follows from the defining property of the fundamental vector fields (5.15). After a change of frame, the new structure functions are defined by [e0b, e0c] =
f0a bce
0
a. Expanding the bracket [e
0
b, e
0
c] = [emΛmb , enΛnc] by applying the Leibniz prop- erty
[X, f Y] = (ρ(X)f)Y +f[X, Y]
5.1. ω-DEFORMED GAUGING 127
and antisymmetry of the bracket, and then re-expressing in terms of e0 gives us (5.16).
Lemma 5.1.2. Under the change of frame ea→e0a =ebΛba, the matrix of one-forms
ωa
b changes as
ωab →ω0ba= (Λ−1)amωmnΛnb + (Λ−1)amdΛmb .
It follows from Appendix C that ω determines the connection coefficients for a con- nection ∇ on E:
∇ea =ebωab
Proof. This follows from insisting that the form of the gauge transformations (5.2) does not change under the change of frame.
The connection ∇determined by the connection formωba pulls back naturally to a connection on X∗E, which we will denote by the same symbol to avoid cluttering notation. We can also define an E-connection,E∇,7 by
E∇(s, t) =∇
ρ(s)(t). (5.17)
We will refer toE∇asthe E-connection associated to ∇, with the tacit understand- ing that there may be many possible E-connections we can construct from a given connection∇ . We now note that we can write the field strength for the gauge field in the following way:8
Fa= dAa+ωa b ∧ Ab+ 1 2 f a bc+ωbiavci −ωacivib Ab∧ Ac = (∇A)a− 1 2T a bcA b∧ Ac, where Ta
bc =−(fabc+ωabivci −ωacivib) is simply the torsion of the E-connection E∇. Note that although the field strength is written as a sum of terms which are covariant in E, the same is not true of the variation of the gauge fields:
δAa = (∇)a− TbcaA b
c+ωabivicAbc.
The requirements that the field strength transform homogeneously, that is the van- ishing of (5.9a) and (5.9b), now take on a geometric interpretation. The expression
7See AppendixCfor details
(5.9a) for Rab is precisely the expression for the components of the curvature of the connection ∇, or the associated E-connection E∇. These vanish if and only if the connection is flat. The expression (5.9b) forSa
bccan be written in terms of geometric quantities as Sa bc=∇T a bc+ιvbR a c−ιvcR a b (5.18) Since we require Ra
b = 0, the conditionSbca = 0 reduces to
∇Ta
bc = 0. (5.19)
From the flat connection we deduce (5.11), and then the ‘field redefinitions’ (5.12) are simply the change in the components of the fields induced by the change of frame:
ea→eb(K−1)ba.
Once we know the connection must be flat, we can simply choose a frame in which the connection formωa
b is zero. Note that in this frame, the condition (5.19) reduces to dfa
bc = 0, and so the structure functions are simply structure constants. We thus recover the standard non-abelian gauging.
A different E-connection
There are many ways to lift a connection ∇ on a Lie algebroid to anE-connection. Our definition (5.17) has the advantage of being rather simple. A different E- connection which could have been useful to use is the so-called dual connection
([13]),9 defined by
E
e
∇(s, t) =E∇(t, s) + [s, t]
=∇ρ(t)(s) + [s, t].
The E-torsion of the dual connection is (minus) the E-torsion of the E-connection we have defined, and theE-curvature of the dual connection can be written in terms of our E-connection as:
(Remn)ab = (∇bTmn)
a
+ (Rmb)an−(Rnb)am.
9Dual is an unfortunately loaded word in this thesis. Here it is used to point out that the
5.1. ω-DEFORMED GAUGING 129
The quantity appearing in (5.9b) now has a much simpler geometric interpretation. We have
(Remn)ab =ιρ(eb)Smna .
The gauging constraints enforcing the vanishing of Sa
bc can therefore be interpreted as the requirement that the dual connection has a vanishing E-curvature.
It is useful to have an expression for connection coefficients of an E-connection. For the E-connectionE∇, we have
E∇(e a, eb) = ∇ρ(ea)eb =∇vi a∂ieb =vai∇ieb = (vaiωbic)ec.
For the dual connection E
e ∇, we have E e ∇(ea, eb) =∇ρ(eb)ea+ [ea, eb] =∇vi b∂iea+f c abec =vib∇ieb+fcabec = vibωaic +fcab ec.
That is, the connection coefficients for the E-connection and the dual connection are Eωc ab =v i aω c bi E e ωabc =vbiωaic +fcab.
It is an interesting fact that theE-torsion of theE-connection is simply the difference between the connection coefficients of the E-connection and its dual:
Closure of the gauge algebroid
We can also see the constraints which arise from gauge invariance of the action from a different perspective, and that is by considering the closure of the gauge algebroid on the space of fields. In order for the gauge algebroid to have a representation on the space of fields, we require
δ1δ2 −δ2δ1 −δ[1,2]
Aa = 0.
We need to be careful here, however, since ∈ C∞(Σ, X∗E), and we are not guar- anteed to have a bracket structure on the pullback bundle, since the Lie algebroid structure doesn’t naturally pull back. We can, however, always pull back a Lie alge- broid to a Lie algebroid with trivial anchor map. To see how, note that a basis for sections of E pulls back to a basis of sections of X∗E, so any i can be expressed as
i =ai (X
∗ ea).
We can now define a C∞(Σ)-linear bracket [·,·]∗ on the pullback bundle by simply
taking the bracket [·,·]E on the corresponding sections in E, and then pulling it back: [1, 2]∗ = [b1X ∗ (eb), c2X∗(ec)]∗ :=b1c2X∗([eb, ec]E) =fabcb1c2X∗(ea). Note that this bracket satisfies
[X, hY]∗ =h[X, Y]
for functionsh∈C∞(Σ), and therefore defines a Lie algebroid bracket on the vector bundle X∗E →Σ where the anchor map is zero. In other words, X∗E is a bundle of Lie algebras over Σ. With this bracket, we now have a well-defined way of asking if the gauge algebroid closes on the space of fields. That is, does the quantity
δ1δ2 −δ2δ1 −δ[1,2]∗
Aa
vanish? This calculation was done in [96], where the authors found that the gauge algebra closes only up to a term proportional to DXi:
δ1δ2 −δ2δ1 −δ[1,2]
5.1. ω-DEFORMED GAUGING 131
whereSa
bcis the quantity given in (5.9b), or equivalently (5.18). Thus from a gauging perspective we only require the vanishing ofSa
bc. That is, we could in principle have a non-flat connection∇, although the dual connection∇e must still be flat. Of course,
if we want to do T-duality we need a field strength transforming homogeneously, and therefore also require a flat connection.