Loop Differential K-theory

In this paper we introduce an equivariant extension of the Chern-Simons form, associated to a path of connections on a bundle over a manifold M, to the free loop space LM, and show it determines an equivalence relation on the set of connections on a bundle. We use this to define a ring, loop differential K-theory of M, in much the same way that differential K-theory can be defined using the Chern-Simons form [SS]. We show loop differential K-theory yields a refinement of differential K-theory, and in particular incorporates holonomy information into its classes. Additionally, loop differential K-theory is shown to be strictly coarser than the Grothendieck group of bundles with connection up to gauge equivalence. Finally, we calculate loop differential K-theory of the circle.


Introduction
Much attention has been given recently to differential cohomology theories, as they play an increasingly important role in geometry, topology and mathematical physics. Intuitively these theories improve on classical (extra)-ordinary cohomology theories by including some additional cocycle information. Such differential cohomology theories have been shown abstractly to exist in [11], and perhaps equally as important, they are often given by some differential-geometric representatives. This illuminates not only the mathematical theory, but also helps give mathematical meaning to several discussions in physics. For example, differential ordinary cohomology (in degree 2) codifies solutions to Maxwell's equations satisfying a Dirac quantization condition, while differential K-theory (and twisted versions) aids in explaining the Ramond-Ramond field in Type-II string theories [6], [7]. Additionally, it is expected that several differential cohomology theories can be described in terms of low dimensional topological field theories, using an appropriate notion of geometric concordance [15].
Differential K-theory itself is a geometric enrichment of ordinary Ktheory, having several formulations, [11,3,13,14]. It is known that differential K-theory is determined uniquely by a set axioms, first given in [4]. For the purposes of this paper we focus on the even degree part of K-theory, denoted by K 0 , and the model of even differential K-theory presented by Simons-Sullivan in [14], which proceeds by defining an equivalence relation on the set of a connections on a bundle, by requiring that the Chern-Simons form, associated to a path of connections, is exact. Elements in this presentation of differential K-theory contain the additional co-cycle information of a representative for the Chern character.
In this paper, we show that a path of connections ∇ s in fact determines an odd differential form on the free loop space LM of the base manifold M , which we denote by BCS(∇ s ) and we call the Bismut-Chern-Simons form. When restricted to the base manifold along the constant loops, we obtain the ordinary Chern-Simons form, see Proposition 4.2. Furthermore, this form satisfies the following fundamental homotopy formula where ι is the contraction by the natural vector field on LM induced by the circle action, and BCh(∇) is the Bismut-Chern form on LM , see Theorem 4.3. Proceeding in much the same way as in [14], we prove that the condition of BCS(∇ s ) being exact defines an equivalence relation on the set of connections on a bundle, and use this to define a functor from manifolds to rings, which we call loop differential K-theory. Elements in this ring contain the additional information of the trace of holonomy of a connection, and in fact the entire extension of the trace of holonomy to a co-cycle on the free loop space known as the Bismut-Chern form, which is an equivariantly closed form on the free loop space that restricts to the classical Chern character [1], [8], [10], [16].
As we show, the loop differential K-theory functor, denoted by M → L K 0 (M ), maps naturally to K-theory by a forgetful map f , forgetting the connection, and to even (d + ι)-closed differential forms on LM , denoted Ω even (d+ι)−cl (LM ). This latter map, denoted BCh for Bismut-Chern character, gives the following commutative diagram of ring homomorphisms: K 0 (M ) [BCh] ( ( P P P P P P P P P P P P Ω even (d+ι)−cl (LM ) 7 7 n n n n n n n n n n n n Here H even S 1 (LM ) denotes (the even part) of the quotient of the kernel of (d + ι) by the image of (d + ι), where d + ι is restricted to differential forms on LM in the kernel of dι + ιd = (d + ι) 2 .
An analogous commutative diagram for (even) differential K-theory K 0 (M ) was established in [14], and in fact the commutative diagram above maps to this analogous square for differential K-theory, making the following commutative diagram of ring homomorphisms: K 0 (M ) [BCh] ( ( P P P P P P P P P P P P  Here H even (M ) denotes the deRham cohomology of M , ρ * is the restriction to constant loops, π is a well defined surjective restriction map by Proposition 4.2, g is the forgetful map, and Ch is the classical Chern character. In Corollary 7.2 we show the map π : L K 0 (M ) → K 0 (M ) is in general not one-to-one. In fact, elementary geometric examples are constructed over the circle to explain the lack of injectivity, showing loop differential K-theory of the circle contains strictly more information than differential K-theory of the circle. On the other hand, we also show in Corollary 7.3 that loop differential K-theory is strictly coarser than the ring induced by all bundles with connection up to gauge equivalence. The situation is clarified by a diagram of implications in section 6. In the final section of the paper, we calculate the ring L K 0 (S 1 ). In short, elements of L K 0 (S 1 ) are determined by the spectrum of holonomy.
Since the differential forms on LM do not have nice locality properties with respect to M , one does not expect L K 0 (M ) to have nice descent properties with respect to open sets of M . Giving prominence to such locality properties, Bunke, Nikolaus and Völkl construct in [2] a theory ku loop related to L K 0 (M ) through sheafification. Such a construction makes drastic changes in favor of locality with respect to open sets of M . It is shown in [2, Section 6.2], that there is a map L K 0 (M ) → ku loop (M ), which, in general, is not injective.
We close by emphasizing that the Bismut-Chern form, and many of the properties used herein, have been given a field theoretic interpretation by Han, Stolz and Teichner. Namely, they can be understood in terms of dimensional reduction from a 1|1 Euclidean field theory on M to a 0|1 Euclidean field theory on LM [10], [15]. We are optimistic that the extension of the Chern-Simons form to the free loop space, referred to here as the Bismut-Chern-Simons form, will also have a field theoretic interpretation, and may also be of interest in other mathematical discussions that begin with the Chern-Simons form, such as 3-dimensional TFT's, quantum computation, and knot invariants.

The Chern and Chern-Simons Forms on M
In this section we recall some basic facts about the Chern-Simons form on a manifold M , which is associated to a path of connections on a bundle over M . Definition 2.1. Given a connection ∇ on a complex vector bundle E → M , with curvature 2-form R, we define the Chern-Weil form by where ∇ s = ∂ ∂s ∇ s . Since connections are an affine space modeled over the vector space of 1forms with values in End(E), the derivative ∇ s lives in Ω 1 (M ; End(E)), so CS(∇ s ) is a well defined differential form on M . We note that the formula above agrees with another common presentation, where all the terms ∇ s are brought to the front. The fundamental homotopy formula involving CS(∇ s ) is the following [5,14]: For a path of connections ∇ s we have:

The Bismut-Chern Form on LM
Recall that the free loop space LM of a smooth manifold M is an infinite dimensional manifold, where the deRham complex is well defined [9]. In fact much of this theory is not needed here as the differential forms we construct can all be expressed locally as iterated integrals of differential forms on the finite dimensional manifold M .
The space LM has a natural vector field, given by the circle action, whose induced contraction operator on differential forms is denoted by ι.
Let Ω S 1 (LM ) = Ω even S 1 (LM )⊕Ω odd S 1 (LM ) denote the Z 2 -graded differential graded algebra of forms on LM in the kernel of (d + ι) 2 = dι + ιd, with differential given by (d + ι). We let H S 1 (LM ) = H even S 1 (LM ) ⊕ H odd S 1 (LM ) denote the cohomology of Ω S 1 (LM ) with respect to the differential (d+ι). Recall that this cohomology group can be computed completely in terms of the cohomology of M , see [12].
We remark that the results which follow can also be restated in terms of the periodic complex which is given by the operator (d + uι) on the Zgraded vector space Ω(LM )[u, u −1 ]], consisting of Laurent series in u −1 , where u has degree 2.
Associated to each connection ∇ on a complex vector bundle E → M , there is an even form on the free loopspace LM whose restriction to constant loops equals the Chern form Ch(∇) of the connection. This result is due to Bismut, and so we refer to this form as the Bismut-Chern form on LM , and denote it by BCh(E, ∇), or BCh(∇) if the context is clear.
In [16] we gave an alternative construction where BCh(E, ∇) = k≥0 T r(hol 2k ) and T r(hol 2k ) ∈ Ω 2k S 1 (LM ). We now recall a local description of this. On any single chart U of M , we can write a connection locally as a matrix A of 1-forms, with curvature R, and in this case the restriction T r(hol U 2k ) of T r(hol 2k ) to LU is given by Here R(t j ) is a 2-form taking in two vectors at γ(t j ) on a loop γ ∈ U , and ιA(t j ) = A(γ (t j )). This defines a differential form on LU since a tangent vector to a loop is a vector field along that loop, and we may evaluate the above expression by inserting the given vector fields at the prescribed times, and integrating.
Note that T r(hol 0 ) is the trace of the usual holonomy, and heustically T r(hol U 2k ) is given by the same formula for the trace of holonomy except with exactly k copies of the function ιA replaced by the 2-form R, summed over all possible places. Since the terms X j are smooth they have bounded values and derivatives, so this series converges for the same reason that holonomy itself converges; it is comparable to an exponential series. This same argument is used to justify the convergence of related series below.
More generally, a global form on LM is defined as follows [16]. We first remark that if {U i } is a covering of M then there is an induced covering of LM in the following way. We fix a covering {U i } of M over which we have trivialized E| U i → U i , and write the connection locally as a matrix valued 1-form A i on U i , with curvature R i . For a given loop γ ∈ LM we can choose sets U = {U 1 , . . . , U p } that cover a subdivision of γ into p the subintervals [(k−1)/p, k/p], using a formula like (3.1) on the open sets U j together with the transition functions g i,j : U i ∩ U j → Gl(n, C) on overlaps. Concretely, we have where g i k−1 ,i k is evaluated at γ((k − 1)/p), and the second sum is a sum over all k-element index sets J ⊂ S of the sets S = {(i r , j) : r = 1, . . . , p, and 1 ≤ j ≤ n r }, and Note that T r(hol ) is precisely the trace of holonomy, and that heuristically T r(hol (p,U ) 2k ) is this same formula for the trace of holonomy but with k copies of R shuffled throughout.
In [16] it is shown that T r(hol 2k ) is independent of covering (p, U) and trivializations of E → M , and so defines a global form T r(hol 2k ) on LM . The techniques are repeated in Appendix A. Moreover, it is shown that these differential forms T r(hol 2k ) satisfy the fundamental property where d/dt is the canonical vector field on LM given by rotating the circle. The Bismut-Chern form is then given by and it follows from the above that (d + ι)BCh(∇) = 0 and (dι + ιd) × BCh(∇) = 0, where we abbreviate ι = ι d/dt . Therefore, BCh(∇) determines a class [BCh(∇)] in the equivariant cohomology H even S 1 (LM ), known as the Bismut-Chern class. It is shown in [17] that this class is in fact independent of the connection ∇ chosen. An independent proof of this fact will be given in the next section (Corollary 4.5), using a lifting of the Chern-Simons form on M to LM . Proof. Consider the restriction of formula (3.2) to M , for any p and U . Since the local forms ιA vanish on constant loops, the only non-zero integrands are those that contain only R. Now, R is globally defined on M , as a form with values in End(E), so we may take p = 1 and U = {M } for the definition of T r(hol 2k )(∇). In this case, the formula for T r(hol 2k ) (p,U ) agrees with the Chern form in (2.1) since 1/n! is the volume of the nsimplex.
The following proposition gives the fundamental properties of the Bismut-Chern form with respect to direct sums and tensor products. By restricting to constant loops, or instead to degree zero, one obtains the corresponding results which are known to hold for both the ordinary Chern form, and the trace of holonomy, respectively. In fact, we regard the proposition below as a hybridization of these two deducible facts. and Proof. We may assume that E andĒ are locally trivialized over a common covering {U i } with transition functions g ij and h ij , respectively. If ∇ and ∇ are locally represented by A i and B i on U i , then ∇ ⊕∇ is locally given by the block matrixes with blocks A i and B i . Similarly, this holds for transition functions and curvatures. The result now follows from Definition 3.1, since block matrices are a subalgebra, and trace is additive along blocks.
For the second statement, it suffices to show that for all k ≥ 0 Note for k = 0 this is just the well known fact that trace of holonomy is multiplicative. If we express ∇ and∇ locally by A i and B i on U i , then ∇ ⊗∇ is locally given by A i ⊗ Id + Id ⊗ B i . Similarly, the curvature is R i ⊗Id+Id⊗S i , if R i and S i are the curvatures of A i and B i , respectively.
We calculate T r hol 2k (∇ ⊗∇) directly from Definition 3.1 using coordinate transition functions g ij ⊗ h ij : . . , p, and 1 ≤ j ≤ n r }, and On each neighborhood U i above, for each choice of m = n j and ≤ m, we can apply the fact that Now, integrating this expression over ∆ m and combining this integral with the sum over T m 1 ⊂ S m with |T m 1 | = m 1 , we see that this becomes an integral over Tm 1 ⊂Sm ) and we use the fact that these inclusions only intersect on lower dimensional faces. We therefore see that By multi-linearity, this shows Then (3.3) follows by taking trace of both sides, since T r(X ⊗ Y ) = T r(X)T r(Y ).

The Bismut-Chern-Simons Form on LM
Using a similar setup and collection of ideas as in the previous section, we construct for each path of connections on a complex vector bundle E → M , an odd form on LM which interpolates between the two Bismut-Chern forms of the endpoints of the path. Similarly to the presentation for BCh above, we begin with a local discussion. Let A s with s ∈ [0, 1] be a path of connections on a single chart U of M , with curvature R s . We let A s = ∂As ∂s and R s = ∂Rs ∂s . For each k ≥ 0, we define the following degree 2k + 1 differential form on LU , Here there is exactly one A s at t r , and there are exactly k wedge products of R s at positions t j 1 , . . . , t j k = t r , and the remaining factors are ιA s . Heuristically, (4.1) is similar to (3.1), except there is exactly one A s , summed over all possible times t r , and integrated over s = 0 to s = 1. This formula can be understood in terms of iterated integrals, just as BCh(∇) was understood in [8] and [16]. It is evident that the restriction of this form to U equals the degree 2k + 1 part of the Chern-Simons form on U since ιA vanishes on constant loops, and the volume of the n-simplex is 1/n!.
More generally, we define an odd form on LM as follows. Let {U i } be a covering of M over which we have trivialized E| U i → U i , with the connection given locally as a matrix valued 1-form ..,ip be the induced cover of LM , as in the previous section. For a given loop γ ∈ LM , we can choose sets U = {U 1 , . . . , U p } that cover a subdivision of γ into p subintervals, and then use a formula like (4.1) on the open sets U i , and multiply these together (in order) by the transition functions g i,j : U i ∩ U j → Gl(n, C). Concretely, we have

Loop differential K-theory
For each k ≥ 0, we define the following degree 2k + 1 differential form on LM , Furthermore, we define the Bismut-Chern-Simons form, associated to the choice (p, U), as ∈ Ω odd (LM ).
Heuristically, (4.2) is much like formula (3.2) for BCh(∇ s ), but with one copy of A s shuffled throughout, and integrated over s = 0 to s = 1.
In appendix A we show that BCS 2k+1 is independent of subdivision integer p, and covering U of local trivializations of E → M , and so it defines a global form BCS 2k+1 (∇ s ) on LM . Hence, the total form is also well defined. This form respects composition of paths of connections on E → M , in the sense that for two paths of connections ∇ s and∇ s with since the integral for BCS(∇ s • ∇ s ) breaks into a sum of two integrals. It furthermore satisfies the following property.
Proof. We'll first give the proof for the local expressions in (4.1) and (3.1), and then explain how the same argument applies to the general global expressions (4.2) and (3.2). Let Loop differential K-theory We first show that for each s we have The statement of the theorem (for the local case) will then follow from this by taking trace of both sides, integrating from s = 0 to s = 1, and using the fundamental theorem of calculus. Note also, that taking the bracket with A s (0) vanishes when taking the trace.
To prove (4.4) we evaluate ∂ ∂s on the right-hand side of (4.4) as Thus, we need to show that the left-hand side of (4.4) consists of exactly the two kinds of terms given in (4.5) and (4.6).
For the left-hand side of (4.4), we first apply ι to k≥0 I 2k+1 . Since ι acts as a derivation and ι 2 = 0, we have ιιA s = 0, so that we only obtain terms with exactly one ιR s or ιA s , i.e. we get the following integrands (suppressing the variables t i for better readability): In (4.7) the factor ιR s may appear anywhere in this product; in particular it may appear before the factor A s or after that factor. Since ιA s and R s are even, and A s is odd, the sign "±" in (4.7) is "+" if ιR s appears before A s , and "−" if ιR s appears after A s . Note, that (4.8) is precisely the term (4.5) on the right-hand side of (4.4).
Next, we apply the derivation d to k≥0 I 2k+1 . We now obtain terms containing exactly one dιA s , dR s , or dA s , i.e. (suppressing again the variables t i ): Again, the sign is "+" if the d term appears before A s , and "−" otherwise.
To evaluate (4.9), we use the relation By the fundamental theorem of calculus, the integral over ∂ ∂t A s is given by evaluation at the endpoints of integration, i.e.
. Thus the variable t i has been removed, and either A s is being multiplied to its adjacent term on the right, or (−A s ) is being multiplied to its adjacent term on the left. This can be further analyzed by considering the following four cases.
(1) If dιA s is the first or last factor in a summand of I 2k+1 , we obtain terms −A s (0) and −A s (1) from the evaluation at the endpoints. These two terms are precisely −A s (0) . Thus, this cancels with the bracket [A s (0), −] in (4.4).
(2) If dιA s is adjacent to ιA s , we obtain −ιA s A s +A s ιA s = −ι(A s ∧A s ) which, when combined with −ι(dA s ) from (4.12) above, equals −ι(dA s + A s ∧ A s ) = −ιR s . Each such term appearing in dI 2k+1 cancels with the corresponding term (4.7) coming from ιI 2k+3 .
Thus, we have shown identity (4.4), and with this the claim of the theorem in the local case. For the general case, using multi-linearity and a similar calculation shows that The only new feature comes from the apparent terms g ij in (4.2), which are not in (4.1). For these, note that all the terms g ij A j and A i g ij which appear from the fundamental theorem of calculus applied to ∂ ∂t A s , cancel Proof. For any path ∇ s from ∇ 1 to ∇ 0 , the degree zero part of (d + i)BCS(∇ s ) is i(BCS 1 )(∇ s ), which is the difference of the traces of the holonomies. We remark that this corollary, and also Corollary 4.6 below, were first proven by Zamboni using completely different methods in [17].
Proof. First, for any path of connections ∇ s from ∇ 1 to ∇ 0 , BCS(∇ s ) is in the kernel of dι + ιd since BCH is (d + i)-closed: The corollary now follows from Theorem 4.3 since the space of connections is path connected.
Let K 0 (M ) be the even K-theory of complex vector bundles over M , i.e. the Grothendieck group associated to the semi-group of all complex vector bundles under direct sum. Elements in K 0 (M ) are given by pairs (E, E ), thought of as the formal difference E − E . This is a ring under tensor product. Using Corollary 4.5, Proposition 3.3, and Proposition 3.2, we have the following: is the ordinary Chern character to deRham cohomology, and ρ * is the restriction to constant loops.

Further properties of the Bismut-Chern-Simons Form
We now show that, up to (d + ι)-exactness, BCS(∇ s ) depends only on the endpoints of the path ∇ s .
i.e. there is an even form H ∈ Ω even S 1 (LM ) such that  F (s, r). The idea is to define an even form on LM using the formula similar to that for BCS(∇ r s ), expect with an additional term ∂ ∂r ∇ r s shuffled in, and integrated from r = 0 to r = 1.
Here A r s,i is the local expression of ∇ r s in U i , with curvature R r s,i . It is shown in Proposition A.6 that H(∇ s r ) is independent of the local trivialization chosen in the above expression for (5.1), and thus defines a well defined global form on LM .
Using the same techniques as in Theorem 4.3 to calculate (d+ι)BCS(∇), and the equality of mixed partial derivatives, we can calculate that where g i k−1 ,i k is evaluated at γ((k − 1)/p), and the second sum is a sum over all k-element index sets J ⊂ S of the sets S = {(i α , j) : α = 1, . . . , p, and 1 ≤ j ≤ n α }, and singletons (i q 1 , m 1 ) ∈ S − J, and Now, using the fundamental theorem of calculus with respect to s, we see that Z 2 (∇ r s ) = 0, because ∂ ∂r A r 0,i = ∂ ∂r A r 1,i = 0, as ∇ r 0 and ∇ r 1 are constant. On the other hand, using the fundamental theorem of calculus with respect to r we have and completes the proof.

Definition 5.2 (BCS-equivalence)
. Let E → M be a complex vector bundle. We say two connections ∇ 0 and ∇ 1 on E are BCS-equivalent if BCS(∇ s ) is (d + ι)-exact for some path of connections ∇ s from ∇ 0 to ∇ 1 .

By Proposition 5.1, if BCS(∇ s ) is (d + ι)-exact
for some path of connections ∇ s from ∇ 0 to ∇ 1 , then BCS(∇ s ) is (d + ι)-exact for any path of connections ∇ s from ∇ 0 to ∇ 1 . Moreover, given two connections ∇ 0 and ∇ 1 on E, there is a well defined element which is independent of the path ∇ s between ∇ 0 and ∇ 1 . Two connections since we may choose a path ∇ s from ∇ 0 to ∇ 2 that passes through ∇ 1 , and then the integral over s defining BCS(∇ s ) breaks into a sum. The Bismut-Chern-Simons forms satisfy the following relations regarding direct sum and tensor product, which will be used to define loop differential K-theory.
and so where in the last step we have used that (d + ι) is a derivation of ∧, and BCh is (d + ι)-closed.

Gauge Equivalence, BCS equivalence, CS equivalence
In this section we clarify how the condition of BCS-equivalence, defined in the previous section, is related to the notions of gauge equivalence, and to Chern-Simons equivalence, the latter defined in [14]. The definitions we need are as follows. ( In general, gauge equivalence does not imply gauge path equivalence, but if the gauge group consisting of bundle automorphisms f : E → E covering id : M → M is path connected, then gauge-path equivalence and gauge equivalence coincide. It is shown in [14] that CS-equivalence is independent of path ∇ s . This also follows from Propositions 5.1 and 4.2. All three of these are equivalence relations, and Figure 6.1 describes how these are related to BCS-equivalence. The entries in the diagram are each conditions on a pair of connections ∇ 0 and ∇ 1 on a fixed bundle. Note the four entries labeled BCS or CS, is exact or closed, mean that BCS(∇ s ), CS(∇ s ) is exact or closed for some path of connections ∇ s from ∇ 0 to ∇ 1 , and this is well defined independent of path ∇ s , by Propositions 5.1 and 4.2.

Counterexamples to converses
We give a single counterexample to the converses of implications 1 , 5 , and 12 by constructing a bundle with a pair of connections that are BCS-equivalent, but do not have conjugate holonomy, as follows.
Consider the trivial complex 2-plane bundle C 2 × S 1 → S 1 over the circle. There is a path of flat connections given by Finally, we remark that the endpoint connections A 1 and A 0 are not gauge equivalent since the holonomies are not conjugate.

Loop differential K-theory
In this section we gather the previous results to define loop differential K-theory, and give some useful properties. This definition given here is similar to the definition of (even) differential K-theory given in [14], which uses CS-equivalence classes. The Grothendieck functor L can be constructed by considering equivalence classes of pairs (w, x) ∈ N × N , where (w, x) ∼ = (y, z) if and only if w + z + k = y + x + k for some k ∈ N , and defining addition by In this case, the identity element is represented by (x, x) for any x ∈ N , and the monoid map N → LN is given by x → (x, 0). A sufficient though not necessary condition that the map N → LN is injective is that the monoid satisfies the cancellation law (w + k = y + k =⇒ w = y).

Relation to (differential) K-theory
We have the following commutative diagram of ring homomorphisms K 0 (M ) [BCh] ( ( P P P P P P P P P P P P is well defined by Theorem 4.3, the map [BCh] comes from Corollary 4.6, and the map Ω even (d+ι)−cl (LM ) → H even S 1 (LM ) is the natural map from the space of (d + ι)-closed even forms given by the quotient by the image of (d + ι).
The analogous commutative diagram for differential K-theory was established in [14], and in fact the commutative diagram above maps to this analogous square for differential K-theory, making the following commute.
Here ρ * is the restriction to constant loops, π is well defined by Proposition 4.2, and g is the forgetful map. On the other hand, we have the following. Let G(M ) denote the Grothendieck group of the monoid of complex vector bundles with connection over M , up to gauge equivalence, under direct sum. This is a ring under tensor product, and although this ring is often difficult to compute, we do have by Corollary A.5 a well defined ring homomorphism κ : G(M ) → L K 0 (M ). Proof. Surjectivity follows again from the definition. The first example in subsection 6.1 provides two connections ∇ and∇ on a trivial bundle E over S 1 that do not have conjugate holonomy, and so are not gauge equivalent, but are BCS-equivalent. Therefore, the induced element ((E, ∇), (E,∇)) in G(S 1 ) maps to zero in L K 0 (S 1 ). It remains to show that ((E, ∇), (E,∇)) is non-zero in G(S 1 ), i.e. that ((E, ∇), 0) and ((E,∇), 0) are not equal.
This follows from a more general fact: if for some point x ∈ M , the holonomies of ∇ and∇ for loops based as x are not related by conjugation by any automorphism of the fiber of E over x, then ∇⊕∇ and∇⊕∇ are not gauge equivalent for any (Ẽ,∇). To see this, we verify the contrapositive. Suppose ∇ ⊕∇ and∇ ⊕∇ are gauge equivalent for some (Ẽ,∇). Then ∇ ⊕∇ and∇ ⊕∇ have conjugate holonomy for loops based at any point. But hol(∇ ⊕∇) = hol(∇) ⊕ hol(∇) and similarly hol(∇ ⊕∇) = hol(∇) ⊕ hol(∇). By appealing to the Jordan form, we see that hol(∇) and hol(∇) are conjugate.
The general fact implies the desired result, since for the given example, the holonomies of (E, ∇) and (E,∇) are not conjugate at any point of x ∈ S 1 .
. . . , λ n ρ m ], which satisfies the distributive law. The map i is a homomorphism with respect to these structures, and an inclusion. Moreover, BCh maps onto the image of i, since we can construct a bundle over S 1 of any desired holonomy, and therefore any desired eigenvalues.
The Grothendieck functor L takes surjections to surjections, and injections to injections if the target monoid satisifies the cancellation law. In particular, groups are monoids satifying the cancellation law, and the Grothendieck functor is the identity on groups. Therefore we can apply the Grothendieck functor, and obtain the following commutative diagram of rings This shows that L K 0 (S 1 ) maps surjectively onto the ring L( n∈N (C * ) n /Σ n ), which imbeds into Ω 0 (d+ι)−cl (LS 1 ). Since the diagram above commutes, this calculates the image of BCh.
We now calculate the ring L K 0 (S 1 ) directly from the definition. We need the following Lemma 8.1. Every C n -bundle with connection (E → S 1 , ∇) over S 1 is isomorphic, as a bundle with connection, to one of the form (C n ×S 1 → S 1 , ∇ = Jdt), where J is a constant matrix in Jordan form.
Proof. A C n -bundle with connection over S 1 is uniquely determined up to isomorphism by its holonomy along the fundamental loop, which is a well defined element [g] ∈ GL(n, C)/ ∼, where the latter denotes conjugacy classes of GL(n, C).
The exponential map from all complex matrices M (n, C) respects conjugacy classes, it is surjective, so that it is surjective on conjugacy classes, and every conjugacy class is represented by a Jordan form.
Given a bundle with connection over S 1 , let [g] be the conjugacy class that determines it up to isomorphism. We can choose J in Jordan form so that [e J ] = [g] ∈ GL(n, C)/ ∼. Regard Jdt as a connection on the trivial bundle over S 1 . Since the connection is constant, e J is the holonomy of this connection along the fundamental loop, which completes the proof.
By the Lemma, an element in the monoid M(S 1 ) which defines L K 0 (S 1 ) can always be represented by a bundle with connection of the form (C n × S 1 → S 1 , ∇ = Jdt). The next lemma gives a sufficient condition for when two such are BCS-equivalent.  (1) hol γ (Adt) and hol γ (Bdt) have the same eigenvalues, where γ is the fundamental loop.
(2) Adt and Bdt are isomorphic. Proof. The map sends (a 1 , . . . , a n ) to the equivalence class of the trivial bundle with connection given by the constant diagonal matrix with entries a 1 , . . . , a n . It is well defined since reordering the a i , or changing some a i by an element in Z produces an isomorphic bundle with connection. It is straightforward to check it is a semi-ring homomorphism. It is surjective by Corollary 8.3 above, and injective by the Proposition 8.5 above.
More intuitively, the elements of M(S 1 ) are determined uniquely by log of the spectrum of holonomy. It follows that the group L K 0 (S 1 ) is simply the Grothendieck group L ( n (C/Z) n /Σ n ) of this monoid n (C/Z) n /Σ n . Finally we have: Proof. Via the isomorphisms the map is induced from the map on monoids given by (a 1 , . . . , a n ) → (e a 1 , . . . , e an ).
We can also calculate the following maps: The Grothendieck group G(S 1 ) of all bundles with connection over S 1 , up to isomorphism, is isomorphic to the Grothendieck group of the monoid of conjugacy classes in GL(n, C) under block sum and tensor product. The isomorphism is given by holonomy. The group K 0 (S 1 ) is isomorphic to Z ⊕ C/Z, as can be computed by the character diagram in [14], or directly using a variation of the argument above used to calculate L K 0 (S 1 ). With respect to these isomorphisms, the first map G(S 1 ) → L K 0 (S 1 ) is given by taking log of the eigenvalues of a conjugacy class, while the second map L K 0 (S 1 ) → K 0 (S 1 ) is induced by (a 1 , . . . , a n ) → (n, a 1 + · · · + a n ), where the sum is reduced modulo Z = {2kπi}.
Appendix A.
In this appendix we prove that the Bismut-Chern-Simons form BCS(∇ s ), associated to a path of connections ∇ s on a vector bundle E → M , is a well defined global differential form on LM . We also gather some useful corollaries.
Let {U i } be a covering of M over which the bundle is locally trivialized, and write the connections ∇ s locally as A s,i on U i , with curvature R s,i . For a given loop γ ∈ LM we can choose sets U 1 , . . . , U p that cover a subdivision of γ into p subintervals [(k − 1)/p, k/p], and for this choice we interpreted as above with Similarly, for any p ≥ 0 and collection U = (U i 1 , . . . , U ip ), subdivide each of the p intervals of [0, 1] into its r subintervals, and let U be the cover using the same open set U i j for all of the r subintervals of the j th interval, The proposition follows from these two facts, since for γ ∈ N (p, U i 1 , . . . ) ∩ N (p , U j 1 , . . . ), we may assume by (1) that p = p , and then by (2) that the forms agree on the overlap. Note that (1) follows from Lemma A.1, which is the analogous statement for the integrand I (p,U ) 2k+1 . We now prove (2). Since trace is invariant under conjugation, it suffices to show that for each fixed s the integrand I (p,U ) 2k+1 changes by conjugation if we perform a collection of local gauge transformations on each U i ∈ U. In fact, it suffices to prove this for the integral expression ∆ n j on the j th subinterval, since the sum defining BCS (U ,p) 2k+1 (∇ s ) can be re-ordered as a sum first over all 1 ≤ j ≤ p and n j ≥ 0, where the forms A s and R s vary on this interval for each possible arrangement on the remaining intervals. To this end, we'll drop the s dependence and it suffices to show for each k ≥ 0 that where ∆ n [0,1/p] = {0 ≤ t 1 ≤ · · · ≤ t k ≤ 1/p}, g = g i,j : U i ∩ U j → Gl(n, C) is the coordinate transition function, and A j = g −1 A i g + g −1 dg. We first prove this for k = 0, i.e. that there are no R's in the above expression. The general case will follow by similar arguments.
We use the following multiplicative version of the fundamental theorem of calculus for the iterated integral. For r < s, Here the k-simplex used in the integral is ∆ k [r,s] = {r ≤ t 1 ≤ · · · ≤ t k ≤ s}. One proof of this is given by observing that the function satisfies f (r) = Id, and also f (s) = 0, since the left hand side of (A.2) is the formula for parallel transport for the connection A = g −1 dg and so is the solution to the ordinary differential equation x (t) = x(t)ι d/dt (g −1 dg). The latter equation can also be checked by direct calculation using the fundamental theorem of calculus.
Remark A.3. In [16], we have used similar techniques to show that BCh(∇) is a well defined global form on LM , independent of the choice of local trivialization charts U, and the subdivision integer p ∈ N. In particular this shows that if two connections ∇ 0 and ∇ 1 are gauge equivalent, then BCh(∇ 0 ) = BCh(∇ 1 ). Proof. The first statement follows from the theorem since BCS is well defined independent of local trivializations, and a global gauge transformation induces local gauge transformations.
By exactly the same argument as in Proposition A.2, we can also prove that the form H appearing in Proposition 5.1 is a well-defined form on LM . Proof. The proof is the same of the one in Proposition A.2, since the only difference between BCS(∇ s ) and H(∇ r s ) is that BCS(∇ s ) contains precisely one factor ∂As ∂s whereas H(∇ r s ) contains one factor ∂A r s ∂r and one factor ∂A r s ∂s . We denote the dependence on the local data by H (U ,p) (∇ r s ). Then, proceeding as in Proposition A.2 as well as using the same notation, it suffices to check: