## June 21, 2004

### Nonabelian 2-form connection from SCFT deformation

#### Posted by urs

Over at sci.physics.strings Charlie Stromeyer made me think about $p$-gerbes a little bit. A literature search revealed that apparently the SCFT deformation formalism that I am playing with could be useful for understanding nonabelian Kalb-Ramond fields.

In particular, in

Ch. Hofman: Nonabelian 2-forms (2002)

a connection on loop space induced by a nonabelian 2-form $B$ field on target space is proposed.

I would like to show how something very similar can be derived from the string theory perspective by simply deforming the SCFT algebra of the open string appropriately. (A detailed version of the following can be found LaTeXified in this draft.)

The key observation is the following: As discussed before we can combine the left- and rightmoving supercharge $G$ and $\overline{G}$ on the worldsheet to obtain the deformed exterior derivative on loop space

(1)${d}_{K}:=G+i\overline{G}={ℰ}^{†\left(\mu ,\sigma \right)}{\partial }_{\left(\mu ,\sigma \right)}+iT{ℰ}_{\left(\mu ,\sigma \right)}{X}^{\prime \left(\mu ,\sigma \right)}\phantom{\rule{thinmathspace}{0ex}},$

where ${K}^{\left(\mu ,\sigma \right)}={X}^{\prime \mu }\left(\sigma \right)$ is the reparameterization Killing vector on loop space.

It is also easy to check that sending

(2)${d}_{K}↦{e}^{-W}{d}_{K}{e}^{W}$

with ${W}^{\left(A\right)}=i\oint {A}_{\mu }{X}^{\prime \mu }$ or ${W}^{\left(B\right)}=\oint \frac{1}{2}{B}_{\mu \nu }{ℰ}^{†\mu }{ℰ}^{†\nu }$ turns on a gauge field and a $B$ field, respectively, for the closed string. This can straightforwardly be generalized to the open string by using an appropriate boundary state and by setting $B=\mathrm{dA}/T$ all the worldsheet bulk terms cancel and the above deformation gives just the covariant exterior derivative for the Chan-Paton endpoints of the string:

(3)${e}^{-{W}^{\left(A\right)}-{W}^{\left(B\right)}}{d}_{K}\left(\sigma \right){e}^{{W}^{\left(A\right)}+{W}^{\left(B\right)}}{=}_{\left(\mathrm{open}\mathrm{string}\right)}{d}_{K}+i{ℰ}^{†\mu }{A}_{\mu }\left(\sigma \right)\left(\delta \left(\sigma -\pi \right)-\delta \left(\sigma \right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

This has a totally obvious generalization to the non-abelian case. Simply replace the exponentiated integral with the corresponding path ordered expression and set $B\to {d}_{A}A/T+B$. (Don’t take a trace, since the matrices must act on the CP factors.)

[Update 06/23/04: So I am thinking here of the ‘bare’ boundary state that all the operator $R$ below acts on as a module of the algebra ${M}_{N}\left(H\right)$, namely the algebra of $N×N$ matrices with values in the Heisenberg algebra $H$ of worldsheet oscillators. The matrix entries of course correspond to the strings stretching between two of the $N$ branes in the stack.

The relation between boundary states and modules of (non-commutative) algebras and how this relates to various brane configurations is discussed nicely in

Yonatan Zunger: Constructing exotic D-branes with infinite matrices in type IIA string theory (2002) .

Even though it is not explicitly stated there, the fact that the above matrices ${M}_{N}\left(H\right)$ take non-commutative values is related, of course, to the noncommutativity introduced by the spatial extension of the string.]

Hence I am saying that the connection on loop space induced by a nonabelian 2-form that follows from string theory considerations is the 1-form component of

(4)${R}^{-1}\circ {d}_{K}\circ R$

where

(5)$R=\mathrm{P}\mathrm{exp}\left({\int }_{0}^{\pi }d\sigma \phantom{\rule{thinmathspace}{0ex}}\left(i{A}_{\mu }{X}^{\prime \mu }+\frac{1}{2}{\left(\frac{1}{T}{d}_{A}A+B\right)}_{\mu \nu }{ℰ}^{†\mu }{ℰ}^{†\nu }\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

Performing some loop-space gymnastics this can be evaluated and yields

(6)${R}^{-1}\circ {d}_{K}\circ R=$
(7)${d}_{K}+$
(8)$+{i{ℰ}^{†\mu }{U}_{A}\left(0,\sigma \right){A}_{\mu }{U}_{A}\left(\sigma ,0\right)\mid }_{\sigma =0}^{\sigma =\pi }+$
(9)$+{\int }_{0}^{\pi }d\sigma \phantom{\rule{thinmathspace}{0ex}}{U}_{A}\left(0,\sigma \right)\left({ℰ}^{†\mu }{B}_{\mu \nu }{X}^{\prime \nu }\right){U}_{A}\left(\sigma ,0\right)+\cdots \phantom{\rule{thinmathspace}{0ex}}$

(up to terms of higher form degree, which don’t have an interpretation as connection terms)

where ${U}_{A}\left(\sigma ,\kappa \right)$ is the holonomy of $A$ along the string from $\kappa$ to $\sigma$.

The first term is the unperturbed loop space $K$-deformed exterior derivative. The second is the gauge connection of $A$ on the Chan-Paton factors. The third is obviously the connection associated with the nonablian $B$-field.

It has a nice and plausible heuristic interpretation: The CP factor is parallel-transported, using the gauge field $A$, along the string from the endpoint to the point $\sigma$ in the worldsheet bulk. There it is multiplied with the B-field density at that point and then it is parallel-transported by $A$ back to the string’s endpoint.

This is pretty much as expected from general considerations on 2-form gauge theory. For instance see the text related to figure 1) of

Amitabha Lahiri: Parallel transport on non-Abelian flux tubes (2003) .

It is almost precisely the same action as that of the $B$ field connection in the above paper by Hofman, the only difference being that here the restoring parallel transport is also present, which looks very plausible.

The correct gauge invariance of the above construction is manifest, it reduces to known constructions in the appropriate special cases and is the only obvious natural generalization of these.

(As before, the above considerations don’t take quantum divergencies into account. But Hashimoto has shown that demanding Wilson lines of the above form to have a well defined action is equivalent to demanding the background field’s equations of motion.)

Posted at June 21, 2004 8:06 PM UTC

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