Geometry of Localized Effective Theories, Exact Semi-classical Approximation and the Algebraic Index

Gui, Zhengping; Li, Si; Xu, Kai

doi:10.1007/s00220-021-03944-z

Cited by 5 publications

(8 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let us first describe the case Y = R 2n with the standard symplectic form ω = n ∑ i=1 dp i ∧ dq i . The BV quantization in the sense of Section 1.1 is worked out in [24] and is summarized as follows.…”

Section: One Dimensional Topological Theory and The Algebraic Indexmentioning

confidence: 99%

“…For any Lie subalgebra of the inner derivations of V βγ−bc , we construct a cocycle in the Lie algebra cochain complex valued in the linear dual of the elliptic chiral chain complex. This construction can be viewed as a chiral algebra analogue of the construction in the algebraic index theory (see [39,5,16,24] and reference therein), where a cocycle in the Lie algebra cochain complex valued in the linear dual of the Hochschild complex of the Weyl algebra was constructed. This cocycle computes the algebraic index.…”

Section: Applicationsmentioning

confidence: 99%

“…Our construction is motivated by the Gelfand-Fuks construction on Lie algebra cohomology of formal vector fields [20]. This type of construction appears in mathematical formulations of non-linear sigma models, see [17,29,30] and later developments [22,24]. In particular, [22] shows that the one loop part of the partition function is the formal Witten genus (see [8,7] for the original geometric approach) in the relative Lie algebra cohomology of formal vector fields.…”

Section: Trmentioning

confidence: 99%

“…In [25], Gwilliam and Grady studied the one-dimensional Chern-Simons theory (or topological quantum mechanics) on the total space of a cotangent bundle and showed that the one-loop effective theory encodes the Todd genus. Later [23,24] established the exact connection between the algebraic index theorem and topological quantum mechanics via a trace map constructed in the Batalin-Vilkovisky(BV) formalism. This sets up a mathematical understanding of the physics approach to index theorem in terms of an exact low-energy effective quantum field theory [24].…”

mentioning

confidence: 99%

“…Later [23,24] established the exact connection between the algebraic index theorem and topological quantum mechanics via a trace map constructed in the Batalin-Vilkovisky(BV) formalism. This sets up a mathematical understanding of the physics approach to index theorem in terms of an exact low-energy effective quantum field theory [24].…”

mentioning

confidence: 99%

See 4 more Smart Citations

Elliptic Trace Map on Chiral Algebras

Gui¹,

Li²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Trace map on deformation quantized algebra leads to the algebraic index theorem. In this paper, we investigate a two-dimensional chiral analogue of the algebraic index theorem via the theory of chiral algebras developed by Beilinson and Drinfeld. We construct a trace map on the elliptic chiral homology of the free βγ − bc system using the BV quantization framework. As an example, we compute the trace evaluated on the unit constant chiral chain and obtain the formal Witten genus in the Lie algebra cohomology. CONTENTS ZHENGPING GUI AND SI LI 4. Applications 44 4.1. Trace map valued in Lie algebra cohomology 45 4.2. Trace map and the formal Witten genus 48 4.3. Trace map for the cosets 53 5. Appendix 55 5.1. The Wick theorem 55 5.2. Proof of the identies (3.3)-(3.8) 56 5.3. Regularized integrals 57 5.4. Feynman diagram computations 60References 63 * M for each finite set I. Here ∆ (I) : X ֒→ X I is the diagonal embedding. A chiral algebra A can be viewed as a Lie algebra object in (M(X S ), ⊗ ch ) via the embedding ∆ (S ) * : we have a binary operationinduced by the chiral bracket µ on A (see Definition 2.2). Then µ (S ) is a (graded) Lie bracket since µ is (graded) skew-symmetric and satisfies the Jacobi identity.Let C(A) be the reduced Chevalley-Eilenberg complex of the Lie algebra object corresponds to the chiral algebra Awhich is an admissible complex in M(X S ) in the sense of [4]. This means that we can view the complex C(A) as a complex of D-modules on the Ran space R(X). The chiral homology is defined as followsAny actual functorial complex represents C ch (X, A) is called chiral chain complex. We will use the one in [4, Section 4.2.15.] which involves Dolbeault and Spencer resolutions. See Section 2.2 for details, there we use the same notation Cch (X, A) Q as in [4]. Now let X be the elliptic curve C/Z + Zτ, and we focus on the free βγ − bc system. The general set up in Section 1.1 becomes (see Section 1.4 for a more detailed introduction) * A = A ⊗ D X D X→X 3 , ∆ X→X 3 is the diagonal embedding.

show abstract

Section: One Dimensional Topological Theory and The Algebraic Indexmentioning

confidence: 99%