69 research outputs found

### Noninteger flux - why it does not work

We consider the Dirac operator on a 2-sphere without one point in the case of
non-integer magnetic flux. We show that the spectral problem for the
Hamiltonian (the square of Dirac operator) can always be well defined, if
including in the Hilbert space only nonsingular on 2-sphere wave functions.
However, this Hilbert space is not invariant under the action of the Dirac
operator; the action of the latter on some nonsingular states produces singular
functions. This breaks explicitly the supersymmetry of the spectrum. In the
integer flux case, the supersymmetry can be restored if extending the Hilbert
space to include locally regular sections of the corresponding fiber bundle.
For non-integer fluxes, such an extention is not possible.Comment: 10 pages. Eq.(20) correcte

### Dolbeault Complex on S^4\{.} and S^6\{.} through Supersymmetric Glasses

S^4 is not a complex manifold, but it is sufficient to remove one point to
make it complex. Using supersymmetry methods, we show that the Dolbeault
complex (involving the holomorphic exterior derivative and its Hermitian
conjugate) can be perfectly well defined in this case. We calculate the
spectrum of the Dolbeault Laplacian. It involves 3 bosonic zero modes such that
the Dolbeault index on S^4\{.} is equal to 3

### Self-duality and supersymmetry

We observe that the Hamiltonian H = D^2, where D is the flat 4d Dirac
operator in a self-dual gauge background, is supersymmetric, admitting 4
different real supercharges. A generalization of this model to the motion on a
curved conformally flat 4d manifold exists. For an Abelian self-dual
background, the corresponding Lagrangian can be derived from known harmonic
superspace expressions.Comment: 14 page

### Modified Korteweg-de Vries equation as a system with benign ghosts

We consider the modified Korteweg-de Vries equation, uxxx + 6u2ux + ut = 0, and explore its dynamics in spatial direction. Higher x derivatives bring about the ghosts. We argue that these ghosts are benign, i.e., the classical dynamics of this system does not involve a blow-up. This probably means that the associated quantum problem is also well defined

### Ultraviolet behavior of 6D supersymmetric Yang-Mills theories and harmonic superspace

We revisit the issue of higher-dimensional counterterms for the N=(1,1)
supersymmetric Yang-Mills (SYM) theory in six dimensions using the off-shell
N=(1,0) and on-shell N=(1,1) harmonic superspace approaches. The second
approach is developed in full generality and used to solve, for the first time,
the N=(1,1) SYM constraints in terms of N=(1,0) superfields. This provides a
convenient tool to write explicit expressions for the candidate counterterms
and other N=(1,1) invariants and may be conducive to proving
non-renormalization theorems needed to explain the absence of certain
logarithmic divergences in higher-loop contributions to scattering amplitudes
in N=(1,1) SYM.Comment: 55 pages, published version in JHE

### Supersymmetric Proof of the Hirzebruch-Riemann-Roch Theorem for Non-K\"ahler Manifolds

We present the proof of the HRR theorem for a generic complex compact
manifold by evaluating the functional integral for the Witten index of the
appropriate supersymmetric quantum mechanical system

### Monopole harmonics on $\mathbb{CP}^{n-1}$

We find the spectra and eigenfunctions of both ordinary and supersymmetric
quantum-mechanical models describing the motion of a charged particle over the
$\mathbb{CP}^{n-1}$ manifold in the presence of a background monopole-like
gauge field. The states form degenerate $SU(n)$ multiplets and their wave
functions acquire a very simple form being expressed via homogeneous
coordinates. Their relationship to multidimensional orthogonal polynomials of a
special kind is discussed. By the well-known isomorphism between the twisted
Dolbeault and Dirac complexes, our construction also gives the eigenfunctions
and eigenvalues of the Dirac operator on complex projective spaces in a
monopole background.Comment: 42 pages, 3 figures, v2: minor corrections, references adde

### Born--Oppenheimer corrections to the effective zero-mode Hamiltonian in SYM theory

We calculate the subleading terms in the Born--Oppenheimer expansion for the
effective zero-mode Hamiltonian of N = 1, d=4 supersymmetric Yang--Mills theory
with any gauge group. The Hamiltonian depends on 3r abelian gauge potentials
A_i, lying in the Cartan subalgebra, and their superpartners (r being the rank
of the group). The Hamiltonian belongs to the class of N = 2 supersymmetric QM
Hamiltonia constructed earlier by Ivanov and I. Its bosonic part describes the
motion over the 3r--dimensional manifold with a special metric. The corrections
explode when the root forms \alpha_j(A_i) vanish and the Born--Oppenheimer
approximation breaks down.Comment: typos correcte

- …