2,870 research outputs found
Non-Renormalization Theorems in Non-Renormalizable Theories
A perturbative non-renormalization theorem is presented that applies to
general supersymmetric theories, including non-renormalizable theories in which
the integrand is an arbitrary gauge-invariant function
of the chiral superfields and gauge field-strength
superfields , and the -integrand is restricted only by gauge
invariance. In the Wilsonian Lagrangian, is unrenormalized except
for the one-loop renormalization of the gauge coupling parameter, and
Fayet-Iliopoulos terms can be renormalized only by one-loop graphs, which
cancel if the sum of the U(1) charges of the chiral superfields vanishes. One
consequence of this theorem is that in non-renormalizable as well as
renormalizable theories, in the absence of Fayet-Iliopoulos terms supersymmetry
will be unbroken to all orders if the bare superpotential has a stationary
point.Comment: 13 pages (including title page), no figures. Vanilla LaTe
Mathieu twining characters for K3
The analogue of the McKay-Thompson series for the proposed Mathieu group
action on the elliptic genus of K3 is analysed. The corresponding NS-sector
twining characters have good modular properties and satisfy remarkable
replication identities. These observations provide strong support for the
conjecture that the elliptic genus of K3 carries indeed an action of the
Mathieu group M24.Comment: 19 page
N=4 Superconformal Algebra and the Entropy of HyperKahler Manifolds
We study the elliptic genera of hyperKahler manifolds using the
representation theory of N=4 superconformal algebra. We consider the
decomposition of the elliptic genera in terms of N=4 irreducible characters,
and derive the rate of increase of the multiplicities of half-BPS
representations making use of Rademacher expansion. Exponential increase of the
multiplicity suggests that we can associate the notion of an entropy to the
geometry of hyperKahler manifolds. In the case of symmetric products of K3
surfaces our entropy agrees with the black hole entropy of D5-D1 system.Comment: 25 pages, 1 figur
Towards A Topological G_2 String
We define new topological theories related to sigma models whose target space
is a 7 dimensional manifold of G_2 holonomy. We show how to define the
topological twist and identify the BRST operator and the physical states.
Correlation functions at genus zero are computed and related to Hitchin's
topological action for three-forms. We conjecture that one can extend this
definition to all genus and construct a seven-dimensional topological string
theory. In contrast to the four-dimensional case, it does not seem to compute
terms in the low-energy effective action in three dimensions.Comment: 15 pages, To appear in the proceedings of Cargese 2004 summer schoo
Melting Crystal, Quantum Torus and Toda Hierarchy
Searching for the integrable structures of supersymmetric gauge theories and
topological strings, we study melting crystal, which is known as random plane
partition, from the viewpoint of integrable systems. We show that a series of
partition functions of melting crystals gives rise to a tau function of the
one-dimensional Toda hierarchy, where the models are defined by adding suitable
potentials, endowed with a series of coupling constants, to the standard
statistical weight. These potentials can be converted to a commutative
sub-algebra of quantum torus Lie algebra. This perspective reveals a remarkable
connection between random plane partition and quantum torus Lie algebra, and
substantially enables to prove the statement. Based on the result, we briefly
argue the integrable structures of five-dimensional
supersymmetric gauge theories and -model topological strings. The
aforementioned potentials correspond to gauge theory observables analogous to
the Wilson loops, and thereby the partition functions are translated in the
gauge theory to generating functions of their correlators. In topological
strings, we particularly comment on a possibility of topology change caused by
condensation of these observables, giving a simple example.Comment: Final version to be published in Commun. Math. Phys. . A new section
is added and devoted to Conclusion and discussion, where, in particular, a
possible relation with the generating function of the absolute Gromov-Witten
invariants on CP^1 is commented. Two references are added. Typos are
corrected. 32 pages. 4 figure
Comments on Non-holomorphic Modular Forms and Non-compact Superconformal Field Theories
We extend our previous work arXiv:1012.5721 [hep-th] on the non-compact N=2
SCFT_2 defined as the supersymmetric SL(2,R)/U(1)-gauged WZW model. Starting
from path-integral calculations of torus partition functions of both the
axial-type (`cigar') and the vector-type (`trumpet') models, we study general
models of the Z_M-orbifolds and M-fold covers with an arbitrary integer M. We
then extract contributions of the degenerate representations (`discrete
characters') in such a way that good modular properties are preserved. The
`modular completion' of the extended discrete characters introduced in
arXiv:1012.5721 [hep-th] are found to play a central role as suitable building
blocks in every model of orbifolds or covering spaces. We further examine a
large M-limit (the `continuum limit'), which `deconstructs' the spectral flow
orbits while keeping a suitable modular behavior. The discrete part of
partition function as well as the elliptic genus is then expanded by the
modular completions of irreducible discrete characters, which are parameterized
by both continuous and discrete quantum numbers modular transformed in a mixed
way. This limit is naturally identified with the universal cover of trumpet
model. We finally discuss a classification of general modular invariants based
on the modular completions of irreducible characters constructed above.Comment: 1+40 pages, no figure; v2 some points are clarified with respect to
the `continuum limit', typos corrected, to appear in JHEP; v3 footnotes added
in pages 18, 23 for the relation with arXiv:1407.7721[hep-th
The Whitham Deformation of the Dijkgraaf-Vafa Theory
We discuss the Whitham deformation of the effective superpotential in the
Dijkgraaf-Vafa (DV) theory. It amounts to discussing the Whitham deformation of
an underlying (hyper)elliptic curve. Taking the elliptic case for simplicity we
derive the Whitham equation for the period, which governs flowings of branch
points on the Riemann surface. By studying the hodograph solution to the
Whitham equation it is shown that the effective superpotential in the DV theory
is realized by many different meromorphic differentials. Depending on which
meromorphic differential to take, the effective superpotential undergoes
different deformations. This aspect of the DV theory is discussed in detail by
taking the N=1^* theory. We give a physical interpretation of the deformation
parameters.Comment: 35pages, 1 figure; v2: one section added to give a physical
interpretation of the deformation parameters, one reference added, minor
corrections; v4: minor correction
The non-compact elliptic genus: mock or modular
We analyze various perspectives on the elliptic genus of non-compact
supersymmetric coset conformal field theories with central charge larger than
three. We calculate the holomorphic part of the elliptic genus via a free field
description of the model, and show that it agrees with algebraic expectations.
The holomorphic part of the elliptic genus is directly related to an
Appell-Lerch sum and behaves anomalously under modular transformation
properties. We analyze the origin of the anomaly by calculating the elliptic
genus through a path integral in a coset conformal field theory. The path
integral codes both the holomorphic part of the elliptic genus, and a
non-holomorphic remainder that finds its origin in the continuous spectrum of
the non-compact model. The remainder term can be shown to agree with a function
that mathematicians introduced to parameterize the difference between mock
theta functions and Jacobi forms. The holomorphic part of the elliptic genus
thus has a path integral completion which renders it non-holomorphic and
modular.Comment: 13 page
Bounds for State Degeneracies in 2D Conformal Field Theory
In this note we explore the application of modular invariance in
2-dimensional CFT to derive universal bounds for quantities describing certain
state degeneracies, such as the thermodynamic entropy, or the number of
marginal operators. We show that the entropy at inverse temperature 2 pi
satisfies a universal lower bound, and we enumerate the principal obstacles to
deriving upper bounds on entropies or quantum mechanical degeneracies for fully
general CFTs. We then restrict our attention to infrared stable CFT with
moderately low central charge, in addition to the usual assumptions of modular
invariance, unitarity and discrete operator spectrum. For CFT in the range
c_left + c_right < 48 with no relevant operators, we are able to prove an upper
bound on the thermodynamic entropy at inverse temperature 2 pi. Under the same
conditions we also prove that a CFT can have a number of marginal deformations
no greater than ((c_left + c_right) / (48 - c_left - c_right)) e^(4 Pi) - 2.Comment: 23 pages, LaTeX, minor change
Non-holomorphic Modular Forms and SL(2,R)/U(1) Superconformal Field Theory
We study the torus partition function of the SL(2,R)/U(1) SUSY gauged WZW
model coupled to N=2 U(1) current. Starting from the path-integral formulation
of the theory, we introduce an infra-red regularization which preserves good
modular properties and discuss the decomposition of the partition function in
terms of the N=2 characters of discrete (BPS) and continuous (non-BPS)
representations. Contrary to our naive expectation, we find a non-holomorphic
dependence (dependence on \bar{\tau}) in the expansion coefficients of
continuous representations. This non-holomorphicity appears in such a way that
the anomalous modular behaviors of the discrete (BPS) characters are
compensated by the transformation law of the non-holomorphic coefficients of
the continuous (non-BPS) characters. Discrete characters together with the
non-holomorphic continuous characters combine into real analytic Jacobi forms
and these combinations exactly agree with the "modular completion" of discrete
characters known in the theory of Mock theta functions \cite{Zwegers}.
We consider this to be a general phenomenon: we expect to encounter
"holomorphic anomaly" (\bar{\tau}-dependence) in string partition function on
non-compact target manifolds. The anomaly occurs due to the incompatibility of
holomorphy and modular invariance of the theory. Appearance of
non-holomorphicity in SL(2,R)/U(1) elliptic genus has recently been observed by
Troost \cite{Troost}.Comment: 39+1 pages, no figure; v2 a reference added, some points are
clarified, typos corrected, version to appear in JHE
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