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January 14, 2008

BV-Formalism, Part X: Symplectic Structures

Posted by Urs Schreiber

Last time in part IX (part I, II, III, IV, V, VI, VII, VIII) I finally started moving from discussion of the purely differential graded \infty-algebraic structure underlying BV-quantization towards those ingredients which make BV theory into BV theory: the BV-Laplacian, the antibracket, the master equation.

In the last installment I had reviewed Witten’s old (but not at all particularly wide spread, it seems) nice observation which indicated that all this new structure is just old familiar structure in unusual guise.

This time I want to add yet another facet to that. I had complained before at various places in our BV-discussions here that I am not entirely fond of the currently very popular perspective on BV-formalism in terms of supermanifolds. I said: if we are really talking about Lie \infty-algebroids, then it seems awkward to model all our internal imagery on supergeometry, just because the Chevalley-Eilenberg algebra of any Lie \infty-algebra happens to be that: a graded algebra. Instead, we should use Lie-algebraic imagery.

To add substance to this vague idea, I’ll here go through the standard constructions of the antifields- and antighosts- and anti-ghosts-of-ghosts-, etc.-parts, which is usually thought of as forming the cotangent bundle T *XT^* X of the supermanifold XX of physical configurations, by using instead the Lie \infty-algebraic point of view which we invoke in Lie \infty-connections and applications to String- and Chern-Simons nn-transport, combined with the Clifford-algebraic point of view that Witten highlighted.

In this spirit I will

- identify the configuration space XX as the action Lie nn-algebroid (g,V)(g,V) (here gg denotes an L L_\infty-algebra and VV a module for it) obtained from the L L_\infty-algebra gg of physical symmetries, symmetries of symmetries, etc., acting on the space VV of fields, whose dual algebra is the Chevalley-Eilenberg algebra CE(g,V)CE(g,V) (definition 2);

- identify the shifted tangent bundle TXT X with the inner automorphism Lie (n+1)(n+1)-algebroid inn(gg,V)inn(gg,V), corresponding to the tangent category of the groupoid integrating (g,V)(g,V), whose dual algebra is the Weil algebra W(g,V)=CE(inn(g,V)) W(g,V) = CE(inn(g,V)) (definition 5, section 4.1.1)

- identify the shifted cotangent bundle, dually with the Clifford algebra generated by CE(gg,V)CE(gg,V), which is like differential forms on XX together with the horizontal inner derivations on W(g)\mathrm{W}(g).

Here “horizontal” is with respect to the universal (g,V)(g,V)-bundle which dually reads CE(g,V) W(g,V) inv(g,V) \array{ CE(g,V) \\ \uparrow \\ \mathrm{W}(g,V) \\ \uparrow \\ inv(g,V) } (table 1)

- identify the inner pairing (often addressed as the graded symplectic pairing in the supermanifold imagery) of these (pairing of fields with anti-fields, ghosts with anti-ghosts, ghosts-of-ghosts with anti-ghosts-of-ghost, etc) with the co-adjoint action of horizontal vector fields on vertical vertor fields in the universal (g,V)(g,V)-bundle, which means in symbols that (ι X,ω):=L ι Xω=[[d W(g,V),ι X],ω]. (\iota_X, \omega) := L_{\iota_X} \omega = [[d_{\mathrm{W}(g,V)}, \iota_X], \omega] \,.

If you are an expert on BV-formalism in supermanifold language, you’ll find nothing new here after you unwrap my ideosyncratic terminology. Still I think this is worthwhile. The main change in perspective is:

instead of thinking of BV-formalism as living in the cotangent bundle of a supermanifold of physical configurations, we realize it as living in the horizontal derivations on the universal groupoid nn-bundle of the action nn-groupoid of gauge transformations acting on physical fields.

It seems that after this introduction I am too tired to spell out the details today. Maybe tomorrow… I could leave this as an exercise to the reader. Most everything one needs to know (apart from what was discussed in previous BV-installments here) is in Lie \infty-connections.

Sorry.

Posted at January 14, 2008 9:54 PM UTC

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3 Comments & 1 Trackback

Re: BV-Formalism, Part X: Symplectic Structures

I am in the process of writing this out in more detail, and explaining how, using the kind of internal homs in DGCAs which we were discussing over at Transgression, we can obtain BV-formalism from the charged nn-particle internalized into DGCAs.

It’s not much more than a sketch so far, plus one crucial consistency check computation (ordinary gauge theory), but maybe somebody wants to have a look while I go with Danny to the supermarket:

BV-formalism and the charged nn-particle

Abstract. We describe the general framework of the charged nn-particle in the differential realm (Lie \infty-algebroids instead of Lie \infty-groupoids) and show how it reproduces aspects of the standard BV formalism. We spell out the example of ordinary gauge theory in detail.

In particular, all comments from abstract nonsense experts concerning the construction in section 4 of DGCAs of maps maps(B,A) maps(B,A) between two given DGCAs AA and BB, as an approximation to the non-existent internal hom DGCAs(B,A) hom_{DGCAs}(B,A) would be very much appreciated.

Posted by: Urs Schreiber on January 15, 2008 6:17 PM | Permalink | Reply to this

Re: BV-Formalism, Part X: Symplectic Structures

Back from the supermarket. Somewhat more detailed discussion of maps(W(g),Ω (Y)) \mathrm{maps}(\mathrm{W}(g),\Omega^\bullet(Y)) now at the end of this. (But still pretty informal.)

The main statement is: the space maps(W(g),Ω (Y)) \mathrm{maps}(\mathrm{W}(g),\Omega^\bullet(Y)) of differential forms on the space of Lie-algebra gg valued differential forms has entirely the analogous structure as the Weil algebre W(g)\mathrm{W}(g) of gg, only that the generators of W(g)\mathrm{W}(g) are replaced by differential forms on YY. Roughly.

So maps(W(g),Ω (Y)) \mathrm{maps}(\mathrm{W}(g),\Omega^\bullet(Y)) is itself the Weil algebra of something, hence the the “algebra of functions on the shifted tangent bundle” of something. Passing to the cotangent bundle by looking at all horizontal inner derivation on maps(W(g),Ω (Y)) \mathrm{maps}(\mathrm{W}(g),\Omega^\bullet(Y)) yields the corresponding BV-setup, all ghosts and antifields included just as they should.

(I don’t discuss any action functionals yet, though, so no BV differential yet at this point.)

Posted by: Urs Schreiber on January 15, 2008 8:52 PM | Permalink | Reply to this

Re: BV-Formalism, Part X: Symplectic Structures

I observed:

So maps(W(g),Ω (Y))\mathrm{maps}(\mathrm{W}(g),\Omega^\bullet(Y)) is itself the Weil algebra of something,

Don’t know about you, but this I find pretty remarkable. It’s saying:

forming the configuration “space” of a gg-gauge theory internally yields automatically an inner automorphism (n+1)(n+1)-groupoid/tangent nn-category, which in turn supports the BV setup.

If you look at the derivation, this is a direct consequence of the fact that a non-flat G (n)G_{(n)}-transport is an INN(G (n))\mathrm{INN}(G_{(n)})-(n+1)(n+1)-transport, because that’s what is behind the fact (section 4.5 here) that gg-valued differential forms on YY are morphisms ω:W(g)Ω (Y) \omega : W(g) \to \Omega^\bullet(Y) from the Weil algebra to differential forms.

Posted by: Urs Schreiber on January 15, 2008 9:29 PM | Permalink | Reply to this
Read the post Frobenius algebras and the BV formalism
Weblog: The n-Category Café
Excerpt: Bruce Bartlett is looking at the latest article by Cattaneo and Mnev on BV-quantization of Chern-Simons theory.
Tracked: November 14, 2008 1:33 PM

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