Yang-Mills gauge conditions from Witten’s open string field theory

Haidong Feng* and Warren Siegel†

C.N. Yang Institute for Theoretical Physics, State University of New York, Stony Brook, 11790-3840(Received 18 December 2006; published 15 February 2007)

We construct the Zinn-Justin-Batalin-Vilkovisky action for tachyons and gauge bosons from Witten’s 3-string vertex of the bosonic open string without gauge fixing. Through canonical transformations, we findthe off-shell, local, gauge-covariant action up to 3-point terms, satisfying the usual field theory gaugetransformations. Perturbatively, it can be extended to higher-point terms. It also gives a new gaugecondition in field theory which corresponds to the Feynman-Siegel gauge on the world-sheet.

DOI: 10.1103/PhysRevD.75.046006 PACS numbers: 11.25.Sq

I. INTRODUCTION

In the usual (super)string theory, the external states forgauge bosons are introduced by vertex operators with agauge condition b0 � 0 to have the right conformalweight. Although this constraint can be relaxed to findgauge-covariant unintegrated vertex operators, we stillneed the gauge-invariant equation of motion for the freevectors [1] and the effective action is valid only on-shell[2]. On the other hand, string field theory (SFT), thesecond-quantized approach to string theory, can be usedfor an off-shell analysis. A complete description of inter-acting strings and string fields was presented in the light-cone gauge [3] and generalized to the super case [4]. Acovariant, gauge-invariant formulation of the bosonic openstring field theory was given by Witten [5], based on therelation found between gauge transformations of the fieldsand first-quantized Becchi-Rouet-Stora-Tyutin transforma-tions in the free action [6]. It was made more concrete byseveral groups: The explicit operator construction of thestring field interaction was presented [7]; string field theorygeometry was formulated by writing each term in theaction as an expectation value in the 2D conformal fieldtheory on the world surface [8]; the tensor constructionswere analyzed from first principles [9]; etc.

To calculate with the action for Witten’s open string fieldtheory, it is helpful to fix the gauge. A particularly usefulgauge choice is the Feynman-Siegel gauge

b0j�i � 0: (1)

The antifields in the string field expansion, which areassociated with states that have a ghost zero-mode c0, aretaken to vanish. Then the action from the viewpoint ofquantum field theory is gauge fixed, while it is not clearwhat kind of gauge condition is applied. So we can onlyguess the action for these states (for example, the origin ofthe �A2 term is not clear for the lack of gauge covariance)but are not able to write it down gauge invariantly. The

simplest way to accomplish this is to find the Zinn-Justin-Batalin-Vilkovisky action [10,11] with all antifields. TheZJBV formalism was first developed to deal with therenormalization of gauge theories, but follows naturallyfrom any field theory action whose kinetic operator isexpressed as the first-quantized BRST operator [12]. Itallows the handling of very general gauge theories, includ-ing those with open or reducible symmetry algebras. TheZJBVaction includes both the usual gauge-invariant actionand the definition of the gauge (BRST) transformations.Here we will start from this ZJBV action for SFT and,through some canonical transformations (including fieldredefinitions and gauge transformations of both fields andantifields), get the explicit gauge-covariant action (andgauge transformations) for tachyons and massless vectorsup to 3-point terms. We will show for the first time that it isjust usual Yang-Mills coupled to scalars, plus F3 and �F2

interactions. These specific canonical transformations willtell us the gauge condition on the fields corresponding toFeynman-Siegel gauge on the world-sheet. Another advan-tage of this mechanism is that we pushed all nonlocalfactors in 3-point interactions to higher-point interactionsand make the 3-point interactions just the usual local YMform. But, as a price, there will be all possible higher-pointinteractions (nonrenormalizable in ordinary field theory),as shown in Sec. V.

The outline of this letter will be as follows: In Sec. II, wewill briefly review Witten’s open string field theory; inSec. III, we will give an introduction to the ZJBV formal-ism for Yang-Mills theory; in Sec. IV, we will calculate thefull ZJBV action for tachyons and massless vectors fromWitten’s open string field theory without the Feynman-Siegel gauge, and find the suitable canonical transforma-tions to get back the ‘‘original’’ gauge-invariant action tolowest order in the Regge slope for the nonlocal exponen-tial factors; in Sec. V, we will perform further transforma-tions to absorb the nonlocal factors in 3-point interactionsand push them to higher-point interactions, which will givethe usual local action up to 3-point terms; finally, we willgive some discussion and conclusions.

*Electronic address: hfeng@ic.sunysb.edu†Electronic address: siegel@insti.physics.sunysb.edu

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II. WITTEN’S 3-STRING VERTEX

In string field theory, the 3-string interaction can beinterpreted as

hh1�#A�h2�#b�h3�#c�i � hV123�jAi1 � jBi2 � jCi3� (2)

where #i is the vertex operator for each external state andhi�z� is the conformal mapping from each string state to thecomplex plane. In Witten’s bosonic open string field the-ory, strings interact by identifying the right half of eachstring with the left half of the next one. The conformalmapping for this interactive world-sheet geometry can beexpressed as

h1�z� � ei�2�=3�h�z�; h2 � h�z�; h3 � e�i�2�=3�h�z�

(3)

where

h�z� ��1� iz1� iz

�2=3: (4)

Then the action is

S � hV2j�; Q�i �g3hV3j�;�;�i (5)

where Q is the usual string theory BRST operator. Usingthe string oscillation modes �n of the matter sector and bn,cn of the ghost sector, the two-string ‘‘vertex’’ is

hV2j � �D�p1 � p2��h0;p1j � h0;p2j��c�1�0 � c

�2�0 �

exp�X1n�1

��1�n�1���1�n ��2�n � c

�1�n b

�2�n � c

�2�n b

�1�n �

�

(6)

and the 3-string vertex associated with the three-stringoverlap can be written as

hV3j �N �D�p1 � p2 � p3��h0jc�1c0��3���h0jc�1c0�

�2��

�h0jc�1c0��1�� exp

� X3

r;s�1

Xn;m1

1

2��r�m Nrs

mn��s�n

� p�r�Nrs0m�

�s�m �

1

2N00

X3

r�1

�p�r��2�

exp� X3

r;s�1

Xm0n1

b�r�m Xrsmnnc�s�n

�(7)

with the normalization factor N � 39=2=26 [13]. Becausewe will focus on the fields and antifields up to oscillationmodes 1, the only relevant Neumann coefficients are

N1111 � N22

11 � N3311 � �

5

27

N1211 � N23

11 � N3111 �

16

27

N1201 � �N

1301 � N23

01 � �N2101 � N31

01 � �N3201 � �

2���3p

9

N1100 � N22

00 � N3300 � �

1

2ln�27=16�

N1101 � N22

01 � N3301 � 0 (8)

for the matter sector and

X1111 � X22

11 � X3311 � �

11

27

X1211 � X23

11 � X3111 � X21

11 � X3211 � X13

11 � �8

27

X1201 � �X

1301 � X23

01 � �X2101 � X31

01 � �X3201 � �

4���3p

9

X1101 � X22

01 � X3301 � 0 (9)

for the ghost sector.Usually, the three-string interactions are calculated in

the Feynman-Siegel gauge

b0j�i � 0: (10)

Then what we get is the gauge-fixed action, and the gaugecondition for this action was never clear. Also we will getsome �A2 interactions whose origin was not obvious dueto the lack of gauge covariance. In the next section, we willconstruct the ZJBV action from string field theory to studythe gauge condition from the aspect of field theory.

III. ZJBV

In the usual Hamiltonian formalism for a phase space�q; p�, the Poisson bracket, which is useful for studyingsymmetry properties and relates to the commutator of thequantum theory, can be defined. In gauge field theory, thereis a similar interpretation where the fields (includingghosts) correspond to q and the antifields (with oppositestatistics) to p. In the YM case (including scalars), �, A�,C, ~C are fields and ��, A��, C�, ~C� are antifields. As ageneralization of the Poisson bracket, the ‘‘antibracket’’�f���; g���� � f � g is introduced [10]:

� �Zdx��1�I

��

���I

��I �

�

��I

���I

�: (11)

It has the following useful properties:

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�f; ga� � �f; g�a;

�af; g� � a�f; g�

�f; g� � ���1��f�1��g�1��g; f�

�f; gh� � �f; g�h� ��1��f�1�gg�f; h�

��1��f�1��h�1��f; �g; h�� � cyc � 0

(12)

The existence of a bracket with these properties allows thedefinition of a Lie derivative, LAB �A;B� and a unitarytransformation

S0 � eLGS � S�LGS�1

2!LGLGS� � � � (13)

For the example we are going to discuss, the antibracketsfor fields and antifields are:

�A��; A�� � ���; ���; �� � 1;

�C;C�� � 1; � ~C; ~C�� � 1(14)

The general Lagrangian path integral for BRST quanti-zation is

A �ZD Ie�iSgf ; Sgf � eL�SZJBVj (15)

where Sgf is evaluated at all antifields � � 0. Expandingthe ZJBVaction in antifields, using m and nm to indicateall minimal and nonminimal fields,

SZJBV � Sgi � �Q m� �m �

�nm

�nm; (16)

then

Sgf � eL�SZJBVj

� Sgi � ���=� m��Q m� � ���=� nm�2 (17)

where Sgi and � depend only on coordinates I. Also, theBRST transformations can be written as �Q I ��SZJBV;

I�. Gauge independence requires

��1�I�2SZJBV

� �I � I � i

1

2�SZJBV; SZJBV� � 0 (18)

which is called the ‘‘quantum master equation’’. It is theapproach to BRST of Zinn-Justin, Batalin, and Vilkovisky(ZJBV).

To see the equivalence of the ZJBV combination of thegauge-invariant action with the BRST operator to ordinaryBRST, here is an example, pure Yang-Mills theory. TheZJBV action in YM can be written as

SZJBV � �F��F�� � 2� ~C��2 � 2i�5�; C�A

�� � C2C�:

We have the usual BRST transformations of fields fromQ � �S; �:

QA� � �2i�5�; C�; QC � �C2;

Q ~C � 4 ~C�; Q ~C� � 0:(19)

Taking

� � trZ 1

4~Cf�A�; (20)

we find the usual gauge-fixed action

Sgf � Sgi �1

4f�A�2 �

i2

~C@f@A� �5; C� (21)

as from the usual BRST formalism.

IV. THE GAUGE-COVARIANT ACTION

In this section, we will use Witten’s 3-string vertex to getthe interactions for tachyons and vectors without theFeynman-Siegel gauge. The action will be in the ZJBVformalism including fields and antifields. From this ZJBVaction, through some canonical transformations, we canget the gauge-invariant action back. Observing the forms ofthese transformations, we will be able to tell which gaugecondition in field theory corresponds to the Feynman-Siegel gauge in Witten’s string field theory.

In string field theory, the general external state (withoutb0 � 0) is

j i � �C��c1 � A � a�1c1 � ~Cc�1c1 � ~C�c0

���c0c1 � A� � a�1c0c1 � C�c�1c0c1

� � � ��j0; ki: (22)

It gives the free terms and 3-point interactions for tachy-ons, YM gauge bosons, ghosts, antighosts, and their anti-fields. The free part is

SZJBV2 � hV2j�; Q�i

� �1

2���� 2���

1

2A��A� � ~C�C

� 2i�@�C�A�� � 2� ~C��2 � 2i�@ � A� ~C� (23)

and the interaction part is

SZJBV3 �

g3hV3j�;�;�i � S�0�3 � S

�1�3 � S

�2�3 � S

�3�3 (24)

where (to lowest order in Regge slope for those nonlocalfactors e�1=2�Nrr

00�P2i�m

2i �; we will discuss them in the next

section)

S�0�3 �1

3�3 ��A2 � � ~C��2��

1

2� ~C;C���

1

2f ~C;Cg ~C�

� fC;��g�� C2C� � ���; C� ~C� � �A�; C�A��

(25)

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S�1�3 �i2@���A�;�� �

i4�@�A� � @�A���A�; A��

�i2� ~C�; @� ~C��A� �

i4

~C�A�; @�C�

�i4@� ~C�A�; C� �

i2���f@�C; A�g � @�fC;A�g�

�i2A����C; @

� ~C�� � �@�C; ~C���

�i2A���fC; @

��� � �@�C;��� (26)

S�2�3 � ��@�A���@�A�� �

1

2�f@��@ � A�; A

�g

�1

4��@ � A�2 �

�1

2�@�C; @�A�� �

1

2�@�@�C; A��

�1

4�C; @��@ � A�� �

1

4�@�C; �@ � A��

�A�� (27)

S�3�3 �i6�@�@�A���A�; @

�A�� �i

24@��@ � A��@�A�; A��:

(28)

This gives the gauge-fixed action after setting antifieldsto zero. Before setting them to zero, it is related to the usualZJBV action by a canonical (with respect to the anti-bracket) transformation. Since such transformations canmix fields and antifields, the transformation itself (fol-lowed by setting antifields to zero) is one way to definethe gauge-fixing procedure in this formalism. So, one wayto find the gauge-invariant action is to undo thistransformation.

Another way is to take this action with antifields, dropall fields with nonvanishing ghost number, and theneliminate the remaining zero-ghost-number antifields(Nakanishi-Lautrup fields) by their equations of motion.However, the resulting action is kind of messy and hasunusual gauge transformations.

The advantage of working with the entire ZJBVaction isthat it contains both the gauge-invariant action and thegauge (BRST) transformations. Furthermore, canonicaltransformations perform field redefinitions (including anti-field redefinitions that define the gauge fixing) in a way thatpreserves the (anti)bracket (as in ordinary quantum me-chanics). Thus, we look for canonical transformations thatproduce the standard form for gauge transformations of thefields, a well as eliminate terms in the action that couldnormally be ignored ‘‘on shell’’.

Notice there are antifield-independent terms from gaugefixing in the ZJBV action of (23) and (24). So we have tofind transformations to ‘‘undo’’ the gauge fixing. For ex-ample, the gauge transformation generated by� i

2 �@ � A�~C

will cancel the gauge-fixing term ~C�C because �� i2

�@ � A� ~C;�2i�@�C�A��� � � ~C�C. Also notice that the

ZJBV actions of (23) and (24) do not give the usual gauge

transformations (from terms linear in antifields), so we alsolook for transformations to give them the usual form. Forinstance, the term ���; C� ~C� will give unusual contribu-tions for gauge transformations of � and ~C, but it can becanceled through the field redefinition generated by 1

4

���; C� ~C. We also look for terms that generate field re-definitions that cancel cubic antifield-independent termsthat are proportional to the linearized field equations. Forexample, 1

2A2�� will generate the counter term ��A2 �

12 ����A

2, which converts �A2 into � 12 ����A

2, whichwill be part of the covariant interaction �F��F��.

The calculation is straightforward, but to find the com-plete transformation we need more steps, because sometransformations applied to cancel terms we do not wantwill have byproducts to be canceled by further transforma-tions. The complete transformation is given as follows:First, make the transformation generated by

Gg � �i2�@ � A� ~C�

1

16~C��A� � @��@ � A�; A��

�1

8~C2C�

1

16�@� ~C�2C�

i8

~Cf@ � A;�g (29)

to undo the gauge fixing. It is independent of antifields, andso can be identified with gauge fixing. Then we make thetransformation

G0 �1

4���; C� ~C�

1

2A2�� �

1

4fC;�gC� �

1

8f�; ~C�g ~C

�1

2f�;A��gA

� �i4A���@

�A�; A��

�i8fC;A��g�@� ~C� �

i8� ~C�; A���@� ~C�: (30)

This generator is linear in antifields, and so can be identi-fied with a field redefinition. (However, there is somesubtlety in that the Nakanishi-Lautrup fields in this formof ZJBVappear as antifields ~C�.) As the result of the abovetransformations, the action (up to 3-point terms and lowestorder in Regge slope) can be written as

S � S2 � S3

�1

2�r�;���r�;�� ��2 � F��F�� � 2i�5�; C�A��

� 2� ~C��2 � fC;��g�� C2C� �1

3�3 � 2�F��F

��

�4

3F��F��F

�� (31)

with5� � @� �i2A�. Now it is explicitly gauge covariant

(to this order) even off-shell! Thus the F3 interactionappears explicitly (which was done only on shell before),and a new gauge-invariant interaction term �F2 is found.Furthermore, the YM gauge condition corresponding to theworld-sheet Feynman-Siegel gauge is now known: Theusual gauge-fixing function @ � A of the Fermi-Feynman

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gauge is modified to

@ � A�i8��A� � @��@ � A�; A

�� �1

4f@ � A;�g

�i8f ~C;Cg �

i16f� ~C;Cg �

i16f@� ~C; @�Cg: (32)

The additional gauge-fixing terms simplify the F3 and�F2

interactions, and make the gauge-fixed action symmetric inghosts and antighosts [14].

V. HIGH ORDERS OF REGGE SLOPE

This is not the end of the story, because we only madethe action manifestly gauge invariant to lowest order in theRegge slope expanded from the nonlocal factors.Remember, in the 3-string vertex in (7), the Neumanncoefficients 1

2Nrr00 � �� will contribute nonlocal factors

to interactions. That means the full interaction will havethe form of replacing each (anti)field i in (24) bye���p

2i�m

2i � i. But the above canonical transformations

can be performed in the same way except that the (anti)-fields i in Gg and G0 are replaced by e���i�m2

i � i. Thenwe will get the full action as in (31) while attaching thefactor e���i�m2

i � to each (anti)field i in the interaction part:

Sfull2 � �

1

2���� 2���

1

4@��A��@��A��

� 2i�@�C�A�� � 2� ~C��2 (33)

Sfull3 ��� �

i2@���A

�; �� �i4F���A

�; A�� �1

3�3

� �A�; C�A�� � fC; ��g�� C2C�

� 2�F��F�� �

4

3F��F

��F

�� (34)

where

F �� � @��A�� (35)

and

� � e����2��; �� � e����2���;

A � e��A; A� � e��A�

C � e��C; C� � e��C�;

~C � e�� ~C; ~C�� e�� ~C�:

(36)

We now perform more field redefinitions to push thesenonlocal factors into higher-point interactions and restorethe usual gauge-invariant action up to 3-point terms. Let usfirst expand the exponential factor e���i�m2

i � to the firstorder. Then there are extra terms like �2����� 2���from 1

3�3 to be absorbed. The naive guess is making the

field redefinition through G � ��2��, which will give acounter term through the antibracket:

�S3 � �G; S2� �

���2��;�

1

2���� 2��

�

� ���2���� 2��� (37)

where we use S2 to represent the free part and S3 theinteraction part in (31) (to lowest order in Regge slope).

Fortunately, it turns out this is almost the right guess. Tothe first order in Regge slope, the redefinition should comethrough

G � ����; S3��� � ��A��; S3�A��

� ��C�; S3�

��i2

��@ � A� � ��@ � A; S3�

�i2C��: (38)

Then

�����; S3���; S2� � ����; S3����; S2�

� �������; S3�; S2�

� �����; S3���� 2��

� ����S2; ���; S3��: (39)

Using the properties of antibrackets in (12) and the gauge-invariant condition �S3; S2� � 0,

� �S2; ���; S3�� � ���; �S3; S2�� � �S3; �S2; ���� � 0

) �S2; ���; S3�� � �S3; �S2; ���� � �S3; ��� 2���

� ���C;����; ��� 2��� � ��C;����� 2�: (40)

Thus (39) gives

�����; S3���; S2� � �����; S3���� 2��

� �fC; ��� 2���g� (41)

which will cancel the additional terms from the first-orderexpansions of e����m

2� for �’s and ��’s in the 3-pointinteractions. Similar calculations show that G in (38) doescancel all additional terms from the first-order expansionsof e���i�m2

i � for all (anti)fields:�,��, A�, A��, C,C�, ~C, ~C�

in Sfull3 .

Basically, we can do it order by order, and here is thefield redefinition for all orders:

G ����;

Z �

0d�Sfull

3 ������ �

�A��;

Z �

0d�Sfull

3 �����A���

�

�C�;

Z �

0d�Sfull

3 ������i2

��@ � A�

�

�@ � A;

Z �

0d�Sfull

3 �����i2C��: (42)

The integral is easy to perform:

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Z �

0d�Sfull

3 ��� �1

��1 �m21� � ��2 �m2

2� � ��3 �m23�

�e���1�m21�����2�m2

2�����3�m23� � 1�S3

(43)

where the indices 1, 2, 3 indicate the three fields in eachterm of S3. The proof is very similar to the first-order caseand we will not bother to give the details here.

Then we will have N-point interactions for any big Njust from a 3-string interaction in SFT. This is because inthe above calculation we only accounted for corrections upto 3-point, while the full transformed action should be

eLGS � S� �G; S� �1

2!�G; �G; S�� � � � � : (44)

Essentially, we can perform this mechanism perturbativelyin higher-point interactions. We have not studied whetherthe nonlocal interactions can be eliminated at any finiteorder of perturbation, or whether this procedure is consis-tent nonperturbatively.

VI. DISCUSSION AND CONCLUSIONS

In this paper, we computed the ZJBV action for Witten’sopen string field theory for tachyons and massless vectors,

including all ghosts and antifields. We find after somecanonical transformations that the action up to 3-pointterms is just the usual Yang-Mills one plus �F2 and F3

interactions as in (31), which is explicitly gauge invariantnow. The gauge condition in field theory which corre-sponds to the Feynman-Siegel gauge on the world-sheetis also known. Furthermore, there are no nonlocal inter-actions in the action up to 3-point terms. (A higher-pointanalysis would require analyzing the massive fields, sinceredefinitions of massive fields appearing in propagators, in4-point and higher diagrams, will produce new local termsfor massless fields on external lines.) We pushed thesenonlocal factors in 3-point interactions to higher-pointinteractions. It may be possible that all such explicit factorscan be eliminated in the complete action, so that all ‘‘non-locality’’ can be attributed to the presence of higher-spinfields.

ACKNOWLEDGMENTS

It is a great pleasure to thank Leonardo Rastelli for hishelpful discussion. This work is supported in part byNational Science Foundation Grant No. PHY-0354776.

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