Nonlinear gauge invariance and WZW-like action for NS-NS superstring field theory
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c The Authors. 0 1
0 Komaba , Meguro-ku, Tokyo 153-8902 , Japan
1 Institute of Physics, University of Tokyo
We complete the construction of a gauge-invariant action for NS-NS superstring field theory in the large Hilbert space begun in arXiv:1305.3893 by giving a closedform expression for the action and nonlinear gauge transformations. The action has the WZW-like form and vertices are given by a pure-gauge solution of NS heterotic string field theory in the small Hilbert space of right movers.
String Field Theory; Superstrings and Heterotic Strings; Gauge Symmetry
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Nonlinear gauge invariance and
WZW-like action for
NS-NS superstring field theory
1 Introduction
Nonlinear gauge invariance
Quartic vertex
WZW-like expression
Coalgebraic description of vertices
Gauge-invariant insertions
NS string products
Wess-Zumino-Witten-like action
Nonlinear gauge invariance
Gauge-invariant insertions of picture-changing operators
B Some identities
A Heterotic theory in the small Hilbert space
Introduction
While bosonic string field theories have been well-understood [1–8], superstring field
theories remain mysterious. A formulation of supersymmetric theories in the early days [9],
which is a natural extension of bosonic theory, has some disadvantages caused by
picturechanging operators inserted into string products: singularities and broken gauge
invariances [10]. To remedy these, various approaches have been proposed within the same
There exists an alternative formulation of superstring field theory: large space
theory [18–26]. Large space theories are formulated by utilizing the extended Hilbert space of
operators. One can check the variation of the action, the equation of motion, and gauge
invariance without taking account of these operators. Of course, the action implicitly
includes picture-changing operators, which appear when we concretely compute
scattering amplitudes after gauge fixing. The singular behaivor of them is, however, completely
regulated and there is no divergence [28, 29].
The cancellation of singularities can also occur in the small Hilbert space. Recently,
by the brilliant works of [30, 31], it is revealed how to obtain gauge-invariant insertions
of picture-changing operators into (super-) string products in the small Hilbert space: the
NS and NS-NS sectors of superstring field theories in the small Hilbert space is completely
formulated. In this paper, we find that using the elegant technique of [31], one can construct
the WZW-like action for NS-NS superstring field theory in the large Hilbert space.
A pure-gauge solution of small-space theory is the key concept of WZW-like
formulation of NS superstring field theory in the large Hilbert space, which determines the vertices
of theory, and we expect that it goes in the case of the NS-NS sector. There is an attempt
to construct non-vanishing interaction terms of NS-NS string fields utilizing a pure-gauge
solution GB of bosonic closed string field theory [22]. However, the construction is not
complete: the nonlinear gauge invariance is not clear and the defining equation of GB is
ambiguous. To obtain nonlinear gauge invariances, we have to add appropriate terms to
these interaction terms defined by GB at each order. Then, the ambiguities of vertices are
removed and we obtain the defining equation of a suitable pure-gauge solution GL, which
we explain in the following sections.
In this paper, we complete this construction begun in [22] by determining these
additional terms which are necessitated for the nonlinear gauge invariance and by giving
closed-form expressions for the action and nonlinear gauge transformations in the NS-NS
sector of closed superstring field theory. We propose the action
S =
the NS heterotic string equation of motion in the small Hilbert space of right movers. The
action has the WZW-like form and the almost same algebraic properties as the large-space
action for NS open and NS closed (heterotic) string field theory [20, 21].
This paper is organized as follows. In section 2 we show that cubic and quartic actions
can be determined by adding appropriate terms and imposing gauge invariance. In section
3, we briefly review the method of gauge-invariant insertions of picture-changing
operaIn section 4, first, we give the defining equation of GL and associated fields which are
necessitated to construct the NS-NS action. Then, we derive the WZW-like expression for the
string equation of motion just as other large-space theories. We end with some conclusions.
Nonlinear gauge invariance
string field theory in powers of κ: S = α2′ P
number 0, and right-mover picture number 0 state. The free action S2 is given by
S2 =
use the symbol (G|p, p˜) which denotes that the total ghost number is G, the left-mover
picture number is p, and the right-mover picture number is p˜. Then, ghost-and-picture
that the inner product hA, Bi gives a nonzero value if and only if the sum of A’s and B’s
S2 is invariant under the (...truncated)