5D partition functions, q-Virasoro systems and integrable spin-chains
Fabrizio Nieri
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Sara Pasquetti
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Filippo Passerini
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Alessandro Torrielli
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Field Theories
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Open Access
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c The Authors.
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Princeton
,
NJ 08544
,
U.S.A
1
Guildford
,
Surrey, GU2 7XH, U.K
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Department of Physics, Princeton University
3
Department of Mathematics, University of Surrey
We analyze N = 1 theories on S5 and S4 S1, showing how their partition functions can be written in terms of a set of fundamental 5d holomorphic blocks. demonstrate that, when the 5d mass parameters are analytically continued to suitable S1 partition functions degenerate to those for S3 and S2
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We explain this mechanism via the recently proposed correspondence between 5d partition
functions and correlators with underlying q-Virasoro symmetry. From the q-Virasoro
3point functions, we axiomatically derive a set of associated reflection coefficients, and show
that they can be geometrically interpreted in terms of Harish-Chandra c-functions for
quantum symmetric spaces. We link these particular c-functions to the types appearing
in the Jost functions encoding the asymptotics of the scattering in integrable spin-chains,
obtained taking different limits of the XYZ model to XXZ-type.
1 Introduction
2 5d holomorphic blocks
Squashed S5 partition functions and 5d holomorphic blocks
S4 S1 partition functions and 5d holomorphic blocks
Degeneration of 5d partition functions
Higher degenerations
3 5d partition functions as q-correlators
An example: Torus with one puncture
Degeneration of q-correlators
4 Reflection coefficients Liouville Field Theory id-CFT S-CFT
5 S-matrices A Special functions A.1 Bernoulli polynomials A.2
Multiple Gamma and Sine functions
A.4 Jacobi Theta and elliptic Gamma functions
B Instanton partition function degeneration
B.1 Classical term
C Transfer matrices and Baxter operators
D Relationships amongst the spin-chain parameters
E XXZ Baxter equation and 3d blocks
Introduction
The study of supersymmetric gauge theories on compact manifolds has attracted much
attention in recent years. After the seminal work by Pestun [1], the method of
supersymmetric localization has been applied to compute partition functions of theories formulated
on compact manifolds of various dimensions. A comprehensive approach to rigid
supersymmetry in curved backgrounds has been also proposed [2].
of N = 1 theories on S5 and S4
S1 were derived in [310] and [1113]. In the case of
the squashed S5, the partition function ZS5 was shown to localize to a matrix integral of
classical, 1-loop and instanton contributions. The latter in turn comprises of three copies
of the equivariant instanton partition function on R
S1 [14, 15] with an appropriate
south poles of S4.
identification of the equivariant parameters, each copy corresponding to the contribution
at a fixed point of the Hopf fibration of S5 over CP2. The S4
S1 case is similar with
the instanton partition function consisting of the product of two copies of the equivariant
instanton partition function on R4 S1, corresponding to the fixed points at the north and
Our first result is the observation that, by manipulating the classical and 1-loop part
which we name 5d holomorphic blocks B5d. In formulas:
to a form which respects the symmetry dictated by the gluing of the instanton factors, it
is possible to rewrite ZS5 and ZS4S1 in terms of the same fundamental building blocks,
ZS5 =
ZS4S1 =
pairings [16, 17]:1
where the brackets
glue respectively three and two 5d holomorphic
blocks, as described in details in the main text. This result is very reminiscent of the
3d case where S2
S1 and S3 partition functions were shown to factorize in terms of
ZS3 =
ZS2S1 =
Holomorphic blocks in three dimensions were identified with solid tori or Melving cigars
which was shown to be consistent with the decomposition of S2
Mq = D2
S1 partition functions, the subscripts id, S refer to the way blocks are fused
semiclassical R2
S1 theory but it also turns out to run over a basis of solutions to certain
difference operators which annihilates 3d blocks and 3d partition functions [17, 19, 20].
In fact, the similarity between the structure of 5d and 3d partition functions is not just
a coincidence, but it is due to a deep relation between the two theories. For example we
and show that, when the masses are analytically continued to certain values, 5d partition
functions degenerate to 3d partition functions of the SQED with U(1) gauge group and
four flavors respectively on S2 S1 and S3. Schematically:
ZSSC5QCD =
ZSSC4QSC1D =
S ZSSQ3ED =
id ZSSQ2ESD1 =
1For a proof of the factorization property of 3-sphere partition functions see [18].
where the 5d id and S gluings reduce to the corresponding pairing for 3d theories
introduced in [17]. The mechanisms that leads to this degeneration is the fact that, upon
analytic continuation of the masses, 1-loop terms develop poles which pinch the
integration contour. Partition functions then receive contribution from the (...truncated)