The present paper shows that general relativity in the Arnowitt-Deser-Misner formalism admits a BV-BFV formulation. More precisely, for any d + 1 ≠ 2 (pseudo-) Riemannian manifold M with space-like or time-like boundary components, the BV data on the bulk induces compatible BFV data on the boundary. As a byproduct, the usual canonical formulation of general relativity is recovered in a straightforward way.

1.
A. S.
Cattaneo
,
P.
Mnëv
, and
N.
Reshetikhin
, “
Classical BV theories on manifolds with boundary
,”
Commun. Math. Phys.
332
(
2
),
535
-
603
(
2014
).
2.
I. A.
Batalin
and
G. A.
Vilkovisky
, “
Gauge algebra and quantization
,”
Phys. Lett. B
102
(
1
),
27
-
31
(
1981
).
3.
I. A.
Batalin
and
E. S.
Fradkin
, “
A generalized canonical formalism and quantization of reducible gauge theories
,”
Phys. Lett. B
122
(
2
),
157
-
164
(
1983
).
4.
I. A.
Batalin
and
G. A.
Vilkovisky
, “
Relativistic S-matrix of dynamical systems with boson and fermion costraints
,”
Phys. Lett. B
69
(
3
),
309
-
312
(
1977
).
5.
C.
Becchi
,
A.
Rouet
, and
R.
Stora
,
Phys. Lett. B
52
,
344
(
1974
);
C.
Becchi
,
A.
Rouet
, and
R.
Stora
,
Commun. Math. Phys.
42
,
127
(
1975
);
C.
Becchi
,
A.
Rouet
, and
R.
Stora
,
Ann. Phys.
98
,
2
(
1976
);
I. V.
Tyutin
,
Lebedev Phys. Inst.
39
, 22 pp. (
1975
); preprint arXiv:0812.0580.
6.
J.
Stasheff
, “
Homological reduction of constrained Poisson algebras
,”
J. Differential Geom.
45
,
221
-
240
(
1997
).
7.
A. S.
Cattaneo
,
P.
Mnëv
, and
N.
Reshetikhin
, “
Semiclassical quantization of Lagrangian field theories
,” in
Mathematical Aspects of Quantum Field Theories
,
Mathematical Physics Studies
(
Springer
,
2015
), pp.
275
-
324
.
8.
A. S.
Cattaneo
,
P.
Mnëv
, and
N.
Reshetikhin
, “
Perturbative quantum gauge theories on manifolds with boundary
,” e-print arXiv:1507.01221.
9.
M.
Atiyah
, “
Topological quantum field theories
,”
Inst. Hautes Etud. Sci. Publ. Math.
68
,
175
-
186
(
1988
).
10.
G.
Segal
, “
The definition of conformal field theory
,” in
Differential Geometrical Methods in Theoretical Physics
(
Springer
,
Netherlands
,
1988
), pp.
165
-
171
.
11.
P. A. M.
Dirac
, “
Generalized hamiltonian dynamics
,”
Can. J. Math.
2
,
129
-
148
(
1950
).
12.
B. S.
DeWitt
, “
Quantum theory of gravity. I. The canonical theory
,”
Phys. Rev.
160
,
1113
(
1967
).
13.
R.
Arnowitt
,
S.
Deser
, and
C.
Misner
, “
Dynamical structure and definition of energy in general relativity
,”
Phys. Rev.
116
(
5
),
1322
-
1330
(
1959
).
14.
C.
Blohmann
,
M. C.
Barbosa Fernandes
, and
A.
Weinstein
, “
Groupoid symmetry and constraints in general relativity
,”
Commun. Contemp. Math.
15
,
1250061
(
2013
).
15.
A. S.
Cattaneo
and
M.
Schiavina
, “
BV-BFV approach to general relativity. Part II: Palatini-Holst action
” (unpublished).
16.
A. S.
Cattaneo
and
M.
Schiavina
, “
On time
” (unpublished).
17.
P.
Mnev
, “
Discrete BF theory
,” e-print arXiv:0809.1160 (
2008
).
18.
G.
Felder
and
D.
Kazhdan
with an appendix by T. M. Schlank, “
The classical master equation
,” in
Contemporary Mathematics
(
AMS
,
2014
), Vol.
610
.
19.
J.
Stasheff
, “
Deformation theory and the Batalin-Vilkovisky master equation
,” in
Deformation Theory and Symplectic Geometry, Proceedings, Meeting
,
Ascona, Switzerland
,
June 16-22, 1996
; e-print arXiv:q-alg/9702012.
20.
D.
Roytenberg
, “
AKSZ-BV formalism and courant algebroid-induced topological field theories
,”
Lett. Math. Phys.
79
,
143
-
159
(
2007
).
21.
F.
Schaetz
, “
BFV-complex and higher homotopy structures
,”
Commun. Math. Phys.
286
(
2
),
399
-
443
(
2009
).
22.
F.
Schaetz
, “
Invariance of the BFV complex
,”
Pac. J. Math.
248
(
2
),
453
-
474
(
2010
).
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