Advances in Theoretical and Mathematical Physics Articles (Project Euclid)
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The latest articles from Advances in Theoretical and Mathematical Physics on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTMon, 01 Nov 2010 09:46 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Equivariant cohomology of the chiral de Rham complex and the half-twisted gauged sigma model
http://projecteuclid.org/euclid.atmp/1278423128
<strong>Meng-Chwan Tan</strong><p><strong>Source: </strong>Adv. Theor. Math. Phys., Volume 13, Number 4, 897--946.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the perturbative aspects of the half-twisted
variant of Witten’s topological A-model coupled to a non-dynamical
gauge field with Kähler target space X being a G-manifold. Our main
objective is to furnish a purely physical interpretation of the equivariant
cohomology of the chiral de Rham complex, recently constructed
by Lian and Linshaw, called the “chiral equivariant cohomology.”
In doing so, one finds that key mathematical results such as the vanishing
in the chiral equivariant cohomology of positive weight classes,
lend themselves to straightforward physical explanations. In addition,
one can also construct topological invariants of X from the correlation
functions of the relevant physical operators corresponding to the nonvanishing
weight-zero classes. Via the topological invariance of these
correlation functions, one can verify, from a purely physical perspective,
the mathematical isomorphism between the weight-zero subspace of the
chiral equivariant cohomology and the classical equivariant cohomology
of X. Last but not least, one can also determine fully, the de Rham
cohomology ring of X/G, from the topological chiral ring generated by
the local ground operators of the physical model under study.
</p>projecteuclid.org/euclid.atmp/1278423128_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTCausal posets, loops and the construction of nets of local algebras for QFThttp://projecteuclid.org/euclid.atmp/1358950890<strong>Fabio Ciolli</strong>, <strong>Giuseppe Ruzzi</strong>, <strong>Ezio Vasselli</strong><p><strong>Source: </strong>Adv. Theor. Math. Phys., Volume 16, Number 2, 645--691.</p><p><strong>Abstract:</strong><br/>
We provide a model independent construction of a net of $C*$-algebras
satisfying the Haag–Kastler axioms over any spacetime manifold. Such a
net, called the net of causal loops , is constructed by selecting a suitable
base $K$ encoding causal and symmetry properties of the spacetime. Considering
$K$ as a partially ordered set (poset) with respect to the inclusion
order relation, we define groups of closed paths (loops) formed by the
elements of $K$. These groups come equipped with a causal disjointness
relation and an action of the symmetry group of the spacetime. In this
way, the local algebras of the net are the group $C*$-algebras of the groups
of loops, quotiented by the causal disjointness relation. We also provide a
geometric interpretation of a class of representations of this net in terms
of causal and covariant connections of the poset K. In the case of the
Minkowski spacetime, we prove the existence of Poincaré covariant representations
satisfying the spectrum condition. This is obtained by virtue
of a remarkable feature of our construction: any Hermitian scalar quantum
field defines causal and covariant connections of $K$. Similar results
hold for the chiral spacetime $S^1$ with conformal symmetry.
</p>projecteuclid.org/euclid.atmp/1358950890_Wed, 23 Jan 2013 09:21 ESTWed, 23 Jan 2013 09:21 ESTPersistence of gaps in the spectrum of certain almost periodic operatorshttp://projecteuclid.org/euclid.atmp/1358950891<strong>Norbert Riedel</strong><p><strong>Source: </strong>Adv. Theor. Math. Phys., Volume 16, Number 2, 693--712.</p><p><strong>Abstract:</strong><br/>
It is shown that for any irrational rotation number and any admissible
gap labelling number the almost Mathieu operator (also known as
Harper’s operator) has a gap in its spectrum with that labelling number.
This answers the strong version of the so-called "Ten Martini Problem".
When specialized to the particular case where the coupling constant is
equal to one, it follows that the "Hofstadter butterfly" has for any quantum
Hall conductance the exact number of components prescribed by the
recursive scheme to build this fractal structure.
</p>projecteuclid.org/euclid.atmp/1358950891_Wed, 23 Jan 2013 09:21 ESTWed, 23 Jan 2013 09:21 ESTVortex equation and reflexive sheaveshttp://projecteuclid.org/euclid.atmp/1358950892<strong>Indranil Biswas</strong>, <strong>Matthias Stemmler</strong><p><strong>Source: </strong>Adv. Theor. Math. Phys., Volume 16, Number 2, 713--723.</p><p><strong>Abstract:</strong><br/>
It is known that given a stable holomorphic pair $(E,\phi)$, where $E$ is a
holomorphic vector bundle on a compact Kähler manifold $X$ and $\phi$ is a
holomorphic section of $E$, the vector bundle $E$ admits a Hermitian metric
solving the vortex equation. We generalize this to pairs $(E,\phi)$, where $E$
is a reflexive sheaf on $X$.
</p>projecteuclid.org/euclid.atmp/1358950892_Wed, 23 Jan 2013 09:21 ESTWed, 23 Jan 2013 09:21 ESTLocalization for Wilson Loops in Chern–Simons Theoryhttp://projecteuclid.org/euclid.atmp/1375124722<strong>Chris Beasely</strong><p><strong>Source: </strong>Adv. Theor. Math. Phys., Volume 17, Number 1, 1--240.</p><p><strong>Abstract:</strong><br/>
We reconsider Chern–Simons gauge theory on a Seifert manifold $M$,
which is the total space of a non-trivial circle bundle over a Riemann
surface $\Sigma$, possibly with orbifold points. As shown in previous work with
Witten, the path integral technique of non-abelian localization can be
used to express the partition function of Chern–Simons theory in terms
of the equivariant cohomology of the moduli space of flat connections
on $M$. Here we extend this result to apply to the expectation values of
Wilson loop operators that wrap the circle fibers of $M$ over $\Sigma$. Under
localization, such a Wilson loop operator reduces naturally to the Chern
character of an associated universal bundle over the moduli space. Along
the way, we demonstrate that the stationary-phase approximation to the
Wilson loop path integral is exact for torus knots in $S^3$, an observation
made empirically by Lawrence and Rozansky prior to this work.
</p>projecteuclid.org/euclid.atmp/1375124722_Mon, 29 Jul 2013 15:05 EDTMon, 29 Jul 2013 15:05 EDTFramed BPS stateshttp://projecteuclid.org/euclid.atmp/1408562451<strong>Davide Gaiotto</strong>, <strong>Gregory W. Moore</strong>, <strong>Andrew Neitzke</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 2, 241--397.</p><p><strong>Abstract:</strong><br/>
We consider a class of line operators in $d = 4, \mathcal{N} = 2$ supersymmetric field theories, which leave four supersymmetries unbroken. Such line operators support a new class of BPS states which we call "framed BPS states."
These include halo bound states similar to those of $d = 4, \mathcal{N} = 2$ supergravity, where (ordinary) BPS particles are loosely bound to the line operator. Using this construction, we give a new proof of the Kontsevich-Soibelman
wall-crossing formula (WCF) for the ordinary BPS particles, by reducing it to the semiprimitive WCF. After reducing on $S^1$, the expansion of the vevs of the line operators in the IR provides a new physical interpretation of the
"Darboux coordinates" on the moduli space M of the theory. Moreover, we introduce a "protected spin character" (PSC) that keeps track of the spin degrees of freedom of the framed BPS states. We show that the generating
functions of PSCs admit a multiplication, which defines a deformation of the algebra of holomorphic functions on $\mathcal{M}$. As an illustration of these ideas, we consider the sixdimensional (2, 0) field theory of $A_1$ type
compactified on a Riemann surface $\mathcal{C}$. Here, we show (extending previous results) that line operators are classified by certain laminations on a suitably decorated version of $\mathcal{C}$, and we compute the
spectrum of framed BPS states in several explicit examples. Finally, we indicate some interesting connections to the theory of cluster algebras.
</p>projecteuclid.org/euclid.atmp/1408562451_20140820152051Wed, 20 Aug 2014 15:20 EDTOn the mechanics of crystalline solids with a continuous distribution of dislocationshttp://projecteuclid.org/euclid.atmp/1408562452<strong>Demetrios Christodoulou</strong>, <strong>Ivo Kaelin</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 2, 399--477.</p><p><strong>Abstract:</strong><br/>
We formulate the laws governing the dynamics of a crystalline solid in which a continuous distribution of dislocations is present. Our formulation is based on new differential geometric concepts, which in particular relate
to Lie groups. We then consider the static case, which describes crystalline bodies in equilibrium in free space. The mathematical problem in this case is the free minimization of an energy integral, and the associated
Euler-Lagrange equations constitute a nonlinear elliptic system of partial differential equations. We solve the problem in the simplest cases of interest.
</p>projecteuclid.org/euclid.atmp/1408562452_20140820152051Wed, 20 Aug 2014 15:20 EDTQuantum Riemann surfaces in Chern-Simons theoryhttp://projecteuclid.org/euclid.atmp/1408562471<strong>Tudor Dimofte</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 3, 479--599.</p><p><strong>Abstract:</strong><br/>
We construct from first principles the operators $\hat A_M$ that annihilate the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups $SU(2)$, $SL(2,\mathbb{R})$,
or $SL(2,\mathbb{C})$ on knot complements $M$. The operator $\hat A_M$ is a quantization of a knot complement's classical $A$-polynomial $A_M(\ell,m)$. The construction proceeds by decomposing three-manifolds
into ideal tetrahedra, and invoking a new, more global understanding of gluing in topological quantum field theory to put them back together. We advocate in particular that, properly interpreted, "gluing $=$ symplectic
reduction." We also arrive at a new finite-dimensional state integral model for computing the analytically continued "holomorphic blocks" that compose any physical Chern-Simons partition function.
</p>projecteuclid.org/euclid.atmp/1408562471_20140820152112Wed, 20 Aug 2014 15:21 EDTConifold transitions in $M$-theory on Calabi-Yau fourfolds with background fluxeshttp://projecteuclid.org/euclid.atmp/1408562472<strong>Kenneth Intriligator</strong>, <strong>Hans Jockers</strong>, <strong>Peter Mayr</strong>, <strong>David R. Morrison</strong>, <strong>M. Ronen Plesser</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 3, 601--699.</p><p><strong>Abstract:</strong><br/>
We consider topology changing transitions for $M$-theory compactifications on Calabi-Yau fourfolds with background $G$-flux. The local geometry of the transition is generically a genus g curve of conifold singularities,
which engineers a 3d gauge theory with four supercharges, near the intersection of Coulomb and Higgs branches. We identify a set of canonical, minimal flux quanta which solve the local quantization condition on $G$
for a given geometry, including new solutions in which the flux is neither of horizontal nor vertical type. A local analysis of the flux superpotential shows that the potential has flat directions for a subset of these fluxes and
the topologically different phases can be dynamically connected. For special geometries and background configurations, the local transitions extend to extremal transitions between global fourfold compactifications with flux.
By a circle decompactification the $M$-theory analysis identifies consistent flux configurations in four-dimensional $F$-theory compactifications and flat directions in the deformation space of branes with bundles.
</p>projecteuclid.org/euclid.atmp/1408562472_20140820152112Wed, 20 Aug 2014 15:21 EDTThe integrable structure of nonrational conformal field theoryhttp://projecteuclid.org/euclid.atmp/1408626423<strong>A. Bytsko</strong>, <strong>J. Teschner</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 4, 701--740.</p><p><strong>Abstract:</strong><br/>
Using the example of Liouville theory, we show how the separation into left- and right-moving degrees of freedom in a nonrational conformal field theory can be made explicit in terms of its integrable structure.
The key observation is that there exist separate Baxter Q-operators for left and right-moving degrees of freedom. Combining a study of the analytic properties of the Q-operators with Sklyanin’s Separation of
Variables Method leads to a complete characterization of the spectrum. Taking the continuum limit allows us in particular to rederive the Liouville reflection amplitude using only the integrable structure.
</p>projecteuclid.org/euclid.atmp/1408626423_20140821090706Thu, 21 Aug 2014 09:07 EDTWeierstrass models of elliptic toric $K3$ hypersurfaces and symplectic cutshttp://projecteuclid.org/euclid.atmp/1408626424<strong>Antonella Grassi</strong>, <strong>Vittorio Perduca</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 4, 741--770.</p><p><strong>Abstract:</strong><br/>
We study elliptically fibered $K3$ surfaces, with sections, in toric Fano 3-folds which satisfy certain combinatorial properties relevant to F-theory/heterotic duality. We show that some of these conditions are equivalent
to the existence of an appropriate notion of a Weierstrass model adapted to the toric context. Moreover, we show that if in addition other conditions are satisfied, there exists a toric semistable degeneration of the
elliptic $K3$ surface which is compatible with the elliptic fibration and F-theory/Heterotic duality.
</p>projecteuclid.org/euclid.atmp/1408626424_20140821090706Thu, 21 Aug 2014 09:07 EDTHeterotic string plus five-brane systems with asymptotic $\mathrm{AdS}_3$http://projecteuclid.org/euclid.atmp/1408626425<strong>Karl-Philip Gemmer</strong>, <strong>Alexander S. Haupt</strong>, <strong>Olaf Lechtenfeld</strong>, <strong>Christoph Nölle</strong>, <strong>Alexander D. Popov</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 4, 771--827.</p><p><strong>Abstract:</strong><br/>
>We present $NS1+NS5$-brane solutions of heterotic supergravity on curved geometries. They interpolate between a near horizon $\mathrm{AdS}_3 \times X^k \times \mathbb{T}^{7-k}$ region and
$\mathbb R^{1,1} \times c(X^k) \times \mathbb{T}^{7-k}$, where $X^k$ (with $k =3,5,6,7$) is a $k$-dimensional geometric Killing spinor manifold, $c(X^k)$, its Ricci-flat cone and $\mathbb{T}^{7-k}$ a $(7-k)$-torus.
The solutions require first-order $\alpha'$-corrections to the field equations, and special point-like instantons play an important role, whose singular support is a calibrated submanifold wrapped by the
$\mathrm{NS5}$-brane. It is also possible to add a gauge anti-five-brane. We determine the super isometries of the near horizon geometry, which are supposed to appear as symmetries of the holographically dual
two-dimensional conformal field theory.
</p>projecteuclid.org/euclid.atmp/1408626425_20140821090706Thu, 21 Aug 2014 09:07 EDTInitial data sets with ends of cylindrical type: II. The vector constraint equationhttp://projecteuclid.org/euclid.atmp/1408626426<strong>Piotr T. Chruściel</strong>, <strong>Rafe Mazzeo</strong>, <strong>Samuel Pocchiola</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 4, 829--865.</p><p><strong>Abstract:</strong><br/>
We construct solutions of the vacuum vector constraint equations on manifolds with cylindrical ends.
</p>projecteuclid.org/euclid.atmp/1408626426_20140821090706Thu, 21 Aug 2014 09:07 EDTEnergy functionals for Calabi-Yau metricshttp://projecteuclid.org/euclid.atmp/1408626508<strong>Matthew Headrick</strong>, <strong>Ali Nassar</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 2, 867--902.</p><p><strong>Abstract:</strong><br/>
We identify a set of “energy” functionals on the space of metrics in a given Kähler class on a Calabi-Yau manifold, which are bounded below and minimized uniquely on the Ricci-flat metric in that class. Using these
functionals, we recast the problem of numerically solving the Einstein equation as an optimization problem. We apply this strategy, using the “algebraic” metrics (metrics for which the Kähler potential is given in terms
of a polynomial in the projective coordinates), to the Fermat quartic and to a one-parameter family of quintics that includes the Fermat and conifold quintics. We show that this method yields approximations to
the Ricci-flat metric that are exponentially accurate in the degree of the polynomial (except at the conifold point, where the convergence is polynomial), and therefore orders of magnitude more accurate than
the balanced metrics, previously studied as approximations to the Ricci-flat metric. The method is relatively fast and easy to implement. On the theoretical side, we also show that the functionals can be used
to give a heuristic proof of Yau’s theorem.
</p>projecteuclid.org/euclid.atmp/1408626508_20140821090829Thu, 21 Aug 2014 09:08 EDTWeaving worldsheet supermultiplets from the worldlines withinhttp://projecteuclid.org/euclid.atmp/1408626509<strong>T. Hübsch</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 2, 903--974.</p><p><strong>Abstract:</strong><br/>
Using the fact that every worldsheet is ruled by two (light-cone) copies of worldlines, the recent classification of off-shell supermultiplets of $N$-extended worldline supersymmetry is extended to construct standard off-shell
and also unidextrous (on the half-shell) supermultiplets of worldsheet $(p, q)$-supersymmetry with no central extension. In the process, a new class of error-correcting ( even-split doubly-even linear block) codes
is introduced and classified for $p + q \leqslant 8$, providing a graphical method for classification of such codes and supermultiplets. This also classifies quotients by such codes, of which many are not tensor products
of worldline factors. Also, supermultiplets that admit a complex structure are found to be depictable by graphs that have a hallmark twisted reflection symmetry.
</p>projecteuclid.org/euclid.atmp/1408626509_20140821090829Thu, 21 Aug 2014 09:08 EDT3-Manifolds and 3d indiceshttp://projecteuclid.org/euclid.atmp/1408626510<strong>Tudor Dimofte</strong>, <strong>Davide Gaiotto</strong>, <strong>Sergei Gukov</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 2, 975--1076.</p><p><strong>Abstract:</strong><br/>
We identify a large class $\mathcal{R}$ of three-dimensional $\mathcal{N} = 2$ superconformal field theories. This class includes the effective theories $T_M$ of M5-branes wrapped on 3-manifolds $\mathcal{M}$,
discussed in previous work by the authors, and more generally comprises theories that admit a UV description as abelian Chern–Simons-matter theories with (possibly non-perturbative) superpotential.
Mathematically, class $\mathcal{R}$ might be viewed as an extreme quantum generalization of the Bloch group; in particular, the equivalence relation among theories in class $\mathcal{R}$ is a quantum-field-theoretic
"2 to 3 move." We proceed to study the supersymmetric index of theories in class $\mathcal{R}$, uncovering its physical and mathematical properties, including relations to algebras of line operators and to 4d indices.
For 3-manifold theories $T_M$, the index is a new topological invariant, which turns out to be equivalent to non-holomorphic $SL(2,\mathbb{C})$ Chern-Simons theory on $\mathcal{M}$ with a previously unexplored
"integration cycle."
</p>projecteuclid.org/euclid.atmp/1408626510_20140821090829Thu, 21 Aug 2014 09:08 EDTMagic coset decompositionshttp://projecteuclid.org/euclid.atmp/1408626511<strong>Sergio L. Cacciatori</strong>, <strong>Bianca L. Cerchiai</strong>, <strong>Alessio Marrani</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 2, 1077--1128.</p><p><strong>Abstract:</strong><br/>
By exploiting a "mixed" non-symmetric Freudenthal-Rozenfeld-Tits magic square, two types of coset decompositions are analyzed for the non-compact special Kähler symmetric rank-3
coset $E_{7(-25)}/ [(E_{6(-78)} \times U(1)) / \mathbb{Z}_3]$, occurring in supergravity as the vector multiplets'scalar manifold in $\mathcal{N} = 2, \mathcal{D} = 4$ exceptional Maxwell-Einstein theory.
The first decomposition exhibits maximal manifest covariance, whereas the second ( triality-symmetric ) one is of Iwasawa type, with maximal $SO(8)$ covariance. Generalizations to conformal
non-compact, real forms of nondegenerate, simple groups "of type E7" are presented for both classes of coset parametrizations, and relations to rank-3 simple Euclidean Jordan algebras and
normed trialities over division algebras are also discussed.
</p>projecteuclid.org/euclid.atmp/1408626511_20140821090829Thu, 21 Aug 2014 09:08 EDTModuli spaces of instantons on toric noncommutative manifoldshttp://projecteuclid.org/euclid.atmp/1408626512<strong>Simon Brain</strong>, <strong>Giovanni Landi</strong>, <strong>Walter D. van Suijlekom</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 2, 1129--1193.</p><p><strong>Abstract:</strong><br/>
We study analytic aspects of $\mathrm{U}(n)$ gauge theory over a toric noncommutative manifold $M_\theta$. We analyse moduli spaces of solutions to the self-dual Yang-Mills equations on $\mathrm{U}(2)$ vector bundles
over four-manifolds $M_\theta$, showing that each such moduli space is either empty or a smooth Hausdorff manifold whose dimension we explicitly compute. In the special case of the four-sphere $S^4_\theta$ we find
that the moduli space of $\mathrm{U}(2)$ instantons with fixed second Chern number $k$ is a smooth manifold of dimension $8k - 3$.
</p>projecteuclid.org/euclid.atmp/1408626512_20140821090829Thu, 21 Aug 2014 09:08 EDTSmall resolutions of SU(5)-models in F-theoryhttp://projecteuclid.org/euclid.atmp/1408626541<strong>Mboyo Esole</strong>, <strong>Shing-Tung Yau</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 6, 1195--1253.</p><p><strong>Abstract:</strong><br/>
We provide an explicit desingularization and study the resulting fiber geometry of elliptically fibered four-folds defined by Weierstrass models admitting a split $\tilde{A}_4$ singularity over a divisor of the discriminant locus.
Such varieties are used to geometrically engineer SU(5) grand unified theories in F-theory. The desingularization is given by a small resolution of singularities. The $\tilde{A}_4$ fiber naturally appears after resolving
the singularities in codimension-one in the base. The remaining higher codimension singularities are then beautifully described by a four-dimensional affine binomial variety which leads to six different small resolutions
of the elliptically fibered four-fold. These six small resolutions define distinct four-folds connected to each other by a network of flop transitions forming a dihedral group. The location of these exotic fibers in the base
is mapped to conifold points of the three-folds that defines the type IIB orientifold limit of the F-theory. The full resolution has interesting properties, specially for fibers in codimension-three: the rank of the singular fiber
does not necessary increase and the fibers are not necessary in the list of Kodaira and some are not even (extended) Dynkin diagrams.
</p>projecteuclid.org/euclid.atmp/1408626541_20140821090902Thu, 21 Aug 2014 09:09 EDTPhysical aspects of quantum sheaf cohomology for deformations of tangent bundles of toric varietieshttp://projecteuclid.org/euclid.atmp/1408626542<strong>Ron Donagi</strong>, <strong>Josh Guffin</strong>, <strong>Sheldon Katz</strong>, <strong>Eric Sharpe</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 6, 1255--1301.</p><p><strong>Abstract:</strong><br/>
In this paper, we will outline computations of quantum sheaf cohomology for deformations of tangent bundles of toric varieties, for those deformations describable as deformations of toric Euler sequences. Quantum sheaf
cohomology is a heterotic analogue of quantum cohomology, a quantum deformation of the classical product on sheaf cohomology groups, that computes nonperturbative corrections to analogues of $\overline{27}^3$
couplings in heterotic string compactifications. Previous computations have relied on either physics-based gauged linear sigma model (GLSM) techniques or computation-intensive brute-force Cech cohomology
techniques. This paper describes methods for greatly simplifying mathematical computations, and derives more general results than previously obtainable with GLSM techniques. We will outline recent results
(rigorous proofs will appear elsewhere).
</p>projecteuclid.org/euclid.atmp/1408626542_20140821090902Thu, 21 Aug 2014 09:09 EDTModuli stacks of maps for supermanifoldshttp://projecteuclid.org/euclid.atmp/1408626543<strong>Tim Adamo</strong>, <strong>Michael Groechenig</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 6, 1303--1342.</p><p><strong>Abstract:</strong><br/>
We consider the moduli problem of stable maps from a Riemann surface into a supermanifold; in twistor-string theory, this is the instanton moduli space. By developing the algebraic geometry of supermanifolds
to include a treatment of super-stacks we prove that such moduli problems, under suitable conditions, give rise to Deligne-Mumford superstacks (where all of these objects have natural definitions in terms
of super-geometry). We make some observations about the properties of these moduli super-stacks, as well as some remarks about their application in physics and their associated Gromov-Witten theory.
</p>projecteuclid.org/euclid.atmp/1408626543_20140821090902Thu, 21 Aug 2014 09:09 EDTAn algebraic geometry method for calculating DOS for 2D tight binding modelshttp://projecteuclid.org/euclid.atmp/1408626544<strong>Koushik Ray</strong>, <strong>Siddhartha Sen</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 6, 1343--1355.</p><p><strong>Abstract:</strong><br/>
An algebraic geometry method is used to calculate the moments of the electron density of states as a function of the energy for lattices in the tight binding approximation. Interpreting the moments as the Mellin transform
of the density allows writing down a formula for the density as an inverse Mellin transform. The method is illustrated by working out the density function for the two-dimensional square and honeycomb lattices.
</p>projecteuclid.org/euclid.atmp/1408626544_20140821090902Thu, 21 Aug 2014 09:09 EDTThe quantization of gravity in globally hyperbolic spacetimeshttp://projecteuclid.org/euclid.atmp/1408626545<strong>Claus Gerhardt</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 6, 1357--1391.</p><p><strong>Abstract:</strong><br/>
We apply the Arnowitt-Deser-Misner approach to obtain a Hamiltonian description of the Einstein-Hilbert action. In doing so we add four new ingredients: (i) we eliminate the diffeomorphism constraints, (ii) we replace
the densities $\sqrt{g}$ by a function $\varphi(x, g_{ij})$ with the help of a fixed metric $\chi$ such that the Lagrangian and hence the Hamiltonian are functions, (iii) we consider the Lagrangian to be defined in a fiber
bundle with base space $\mathcal{S}_0$ and fibers $F(x)$ which can be treated as Lorentzian manifolds equipped with the Wheeler-DeWitt metric. It turns out that the fibers are globally hyperbolic, and (iv) the
Hamiltonian operator H is a normally hyperbolic operator in the bundle acting only in the fibers and the Wheeler-DeWitt equation $Hu = 0$ is a hyperbolic equation in the bundle. Since the corresponding Cauchy
problem can be solved for arbitrary smooth data with compact support, we then apply the standard techniques of Algebraic Quantum Field Theory (QFT) which can be naturally modified to work in the bundle.
</p>projecteuclid.org/euclid.atmp/1408626545_20140821090902Thu, 21 Aug 2014 09:09 EDTOn twisted large $N = 4$ conformal superalgebrashttp://projecteuclid.org/euclid.atmp/1408626546<strong>Zhihua Chang</strong>, <strong>Arturo Pianzola</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 6, 1393--1415.</p><p><strong>Abstract:</strong><br/>
We explicitly compute the automorphism group of the large $N = 4$ conformal superalgebra and classify the twisted loop conformal superalgebras based on the large $N = 4$ conformal superalgebra. By considering the
corresponding superconformal Lie algebras, we validate the existence of only, two (up to isomorphism) such algebras as described in the physics literature. Our approach is based on viewing the objects to be classified
as "étale twisted forms" of objects over the Laurent polynomial ring $\mathbb{C}[t \pm 1]$. This allows methods from non-abelian cohomology (torsors) to enter into the picture. It is worth pointing out that the group of
automorphisms of the large $N = 4$ conformal superalgebra is larger than the one described in the physics literature. Remarkably enough, both groups have the same étale cohomology over $\mathbb{C}[t \pm 1]$
which explains the agreement on the classification of the corresponding superconformal Lie algebras).
</p>projecteuclid.org/euclid.atmp/1408626546_20140821090902Thu, 21 Aug 2014 09:09 EDTOn the vector bundles associated to the irreducible representations of cocompact lattices of $\text{SL}(2,{\mathbb C})$http://projecteuclid.org/euclid.atmp/1408626547<strong>Indranil Biswas</strong>, <strong>Avijit Mukherjee</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 6, 1417--1424.</p><p><strong>Abstract:</strong><br/>
We prove the following: let $\Gamma\, \subset\, \text{SL}(2,{\mathbb C})$ be a cocompact lattice and let $\rho\,:\, \Gamma\, \longrightarrow\, \text{GL}(r,{\mathbb C})$ be an irreducible representation. Then the holomorphic
vector bundle $E_\rho\, \longrightarrow\, \text{SL}(2,{\mathbb C})/ \Gamma$ associated to $\rho$ is polystable. The compact complex manifold $\text{SL}(2,{\mathbb C})/ \Gamma$ has natural Hermitian structures;
the polystability of $E_\rho$ is with respect to these natural Hermitian structures. We show that the polystable vector bundle $E_\rho$ is not stable in general.
</p>projecteuclid.org/euclid.atmp/1408626547_20140821090902Thu, 21 Aug 2014 09:09 EDTBerglund–Hübsch–Krawitz mirrors via Shioda mapshttp://projecteuclid.org/euclid.atmp/1408626548<strong>Tyler L. Kelly</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 17, Number 6, 1425--1449.</p><p><strong>Abstract:</strong><br/>
We prove the birationality of multiple Berglund-Hübsch-Krawitz (BHK) mirrors by using Shioda maps. We do this by creating a birational picture of the BHK correspondence in general. We give an explicit quotient of a
Fermat variety to which the mirrors are birational.
</p>projecteuclid.org/euclid.atmp/1408626548_20140821090902Thu, 21 Aug 2014 09:09 EDTIdempotents for Birman–Murakami–Wenzl algebras and reflection equationhttp://projecteuclid.org/euclid.atmp/1412953897<strong>A. P. Isaev</strong>, <strong>A. I. Molev</strong>, <strong>O. V. Ogievetsky</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 1, 1--25.</p><p><strong>Abstract:</strong><br/>
A complete system of pairwise orthogonal minimal idempotents for Birman-Murakami-Wenzl algebras is obtained by a consecutive evaluation of a rational function in several variables on sequences of quantum contents
of up-down tableaux. A by-product of the construction is a one-parameter family of fusion procedures for Hecke algebras. Classical limits to two different fusion procedures for Brauer algebras are described.
</p>projecteuclid.org/euclid.atmp/1412953897_20141010111139Fri, 10 Oct 2014 11:11 EDT$\mathcal{N}=2$ quantum field theories and their BPS quivershttp://projecteuclid.org/euclid.atmp/1412953898<strong>Murad Alim</strong>, <strong>Sergio Cecotti</strong>, <strong>Clay Córdova</strong>, <strong>Sam Espahbodi</strong>, <strong>Ashwin Rastogi</strong>, <strong>Cumrun Vafa</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 1, 27--127.</p><p><strong>Abstract:</strong><br/>
We explore the relationship between four-dimensional $\mathcal{N}=2$ quantum field theories and their associated BPS quivers. For a wide class of theories including super-Yang-Mills theories, Argyres-Douglas models,
and theories defined by M5-branes on punctured Riemann surfaces, there exists a quiver which implicitly characterizes the field theory. We study various aspects of this correspondence including the quiver interpretation
of flavor symmetries, gauging, decoupling limits, and field theory dualities. In general a given quiver describes only a patch of the moduli space of the field theory, and a key role is played by quantum mechanical dualities,
encoded by quiver mutations, which relate distinct quivers valid in different patches. Analyzing the consistency conditions imposed on the spectrum by these dualities results in a powerful and novel mutation method
for determining the BPS states. We apply our method to determine the BPS spectrum in a wide class of examples, including the strong coupling spectrum of super-Yang-Mills with an ADE gauge group and fundamental
matter, and trinion theories defined by M5-branes on spheres with three punctures.
</p>projecteuclid.org/euclid.atmp/1412953898_20141010111139Fri, 10 Oct 2014 11:11 EDTSuperconformal indices, Sasaki-Einstein manifolds, and cyclic homologieshttp://projecteuclid.org/euclid.atmp/1412953899<strong>Richard Eager</strong>, <strong>Johannes Schmude</strong>, <strong>Yuji Tachikawa</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 1, 129--175.</p><p><strong>Abstract:</strong><br/>
The superconformal index of the quiver gauge theory dual to type IIB string theory on the product of an arbitrary smooth Sasaki-Einstein manifold with five-dimensional AdS space is calculated both from the gauge theory
and gravity viewpoints. We find complete agreement. Along the way, we find that the index on the gravity side can be expressed in terms of the Kohn-Rossi cohomology of the Sasaki-Einstein manifold and that the index
of a quiver gauge theory equals the Euler characteristic of the cyclic homology of the Ginzburg dg algebra associated to the quiver.
</p>projecteuclid.org/euclid.atmp/1412953899_20141010111139Fri, 10 Oct 2014 11:11 EDTUniqueness of the AdS spacetime among static vacua with prescribed null infinityhttp://projecteuclid.org/euclid.atmp/1412953900<strong>Oussama Hijazi</strong>, <strong>Sebastián Montiel</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 1, 177--203.</p><p><strong>Abstract:</strong><br/>
We prove that an $(n+1)$-dimensional spin static vacuum with negative cosmological constant whose null infinity has a boundary admitting a non-trivial Killing spinor field is the AdS spacetime. As a consequence, we
generalize previous uniqueness results by X. Wang and by Chruściel-Herzlich and introduce, for this class of spin static vacua, some Lorentzian manifolds which are prohibited as null infinities.
</p>projecteuclid.org/euclid.atmp/1412953900_20141010111139Fri, 10 Oct 2014 11:11 EDTBosonic part of $4d$ $N=1$ supersymmetric gauge theory with general couplings: local existencehttp://projecteuclid.org/euclid.atmp/1412953901<strong>Fiki T. Akbar</strong>, <strong>Bobby E. Gunara</strong>, <strong>Triyanta </strong>, <strong>Freddy P. Zen</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 1, 205--227.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove the local existence of the bosonic part of $N=1$ supersymmetric gauge theory in four dimensions with general couplings. We start with the Lagrangian of the vector and chiral multiplets with general
couplings and the scalar potential is turned on. Then, for the sake of simplicity, we set all fermions to vanish at the level of equations of motion, so we only have the bosonic parts of the theory. We apply Segal’s general
theory to show the local existence of solutions of equations of motion by taking Kähler potential to be bounded above by $U(n)$ symmetric Kähler potential and the first derivative of gauge couplings to be at most
linear growth functions.
</p>projecteuclid.org/euclid.atmp/1412953901_20141010111139Fri, 10 Oct 2014 11:11 EDTMultiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theoryhttp://projecteuclid.org/euclid.atmp/1414414836<strong>Domenico Fiorenza</strong>, <strong>Hisham Sati</strong>, <strong>Urs Schreiber</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 2, 229--321.</p><p><strong>Abstract:</strong><br/>
The world-volume theory of coincident M5-branes is expected to contain a nonabelian 2-form/nonabelian gerbe gauge theory that is a higher analog of self-dual Yang-Mills theory. But the precise details—in particular
the global moduli/instanton/magnetic charge structure—have remained elusive. Here we deduce from anomaly cancellation a natural candidate for the holographic dual of this nonabelian 2-form field,
under $\mathrm{AdS}_7 \; / \; \mathrm{CFT}_6$ duality. We find this way a 7-dimensional nonabelian Chern-Simons theory of String 2-connection fields, which, in a certain higher gauge, are given locally
by non-abelian 2-forms with values in an affine Kac-Moody Lie algebra. We construct the corresponding action functional on the entire smooth moduli 2-stack of field configurations, thereby defining the theory
globally, at all levels and with the full instanton structure, which is nontrivial due to the twists imposed by the quantum corrections. Along the way we explain some general phenomena of higher nonabelian gauge
theory that we need.
</p>projecteuclid.org/euclid.atmp/1414414836_20141027090038Mon, 27 Oct 2014 09:00 EDTProperties of $c_2$ invariants of Feynman graphshttp://projecteuclid.org/euclid.atmp/1414414837<strong>Francis Brown</strong>, <strong>Oliver Schnetz</strong>, <strong>Karen Yeats</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 2, 323--362.</p><p><strong>Abstract:</strong><br/>
The $c_2$ invariant of a Feynman graph is an arithmetic invariant which detects many properties of the corresponding Feynman integral. In this paper, we define the $c_2$ invariant in momentum space and prove
that it equals the $c_2$ invariant in parametric space for overall log-divergent graphs. Then we show that the $c_2$ invariant of a graph vanishes whenever it contains subdivergences. Finally, we investigate how
the $c_2$ invariant relates to identities such as the four-term relation in knot theory.
</p>projecteuclid.org/euclid.atmp/1414414837_20141027090038Mon, 27 Oct 2014 09:00 EDTIntersection spaces, perverse sheaves and type IIB string theoryhttp://projecteuclid.org/euclid.atmp/1414414838<strong>Markus Banagl</strong>, <strong>Nero Budur</strong>, <strong>Laurenţu Maxim</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 2, 363--399.</p><p><strong>Abstract:</strong><br/>
The method of intersection spaces associates rational Poincaré complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes
in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA theory. For complex projective hypersurfaces with an isolated singularity, we show that the cohomology of
intersection spaces is the hypercohomology of a perverse sheaf, the intersection space complex, on the hypersurface. Moreover, the intersection space complex underlies a mixed Hodge module, so its hypercohomology
groups carry canonical mixed Hodge structures. For a large class of singularities, e.g., weighted homogeneous ones, global Poincar´e duality is induced by a more refined Verdier selfduality isomorphism for this perverse
sheaf. For such singularities, we prove furthermore that the pushforward of the constant sheaf of a nearby smooth deformation under the specialization map to the singular space splits off the intersection space complex
as a direct summand. The complementary summand is the contribution of the singularity. Thus, we obtain for such hypersurfaces a mirror statement of the Beilinson-Bernstein-Deligne decomposition of the pushforward of
the constant sheaf under an algebraic resolution map into the intersection sheaf plus contributions from the singularities.
</p>projecteuclid.org/euclid.atmp/1414414838_20141027090038Mon, 27 Oct 2014 09:00 EDTSpecial polynomial rings, quasi modular forms and duality of topological stringshttp://projecteuclid.org/euclid.atmp/1414414839<strong>Murad Alim</strong>, <strong>Emanuel Scheidegger</strong>, <strong>Shing-Tung Yau</strong>, <strong>Jie Zhou</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 2, 401--467.</p><p><strong>Abstract:</strong><br/>
We study the differential polynomial rings which are defined using the special geometry of the moduli spaces of Calabi-Yau threefolds. The higher genus topological string amplitudes are expressed as polynomials in the
generators of these rings, giving them a global description in the moduli space. At particular loci, the amplitudes yield the generating functions of Gromov-Witten invariants. We show that these rings are isomorphic
to the rings of quasi modular forms for threefolds with duality groups for which these are known. For the other cases, they provide generalizations thereof. We furthermore study an involution which acts on the quasi
modular forms. We interpret it as a duality which exchanges two distinguished expansion loci of the topological string amplitudes in the moduli space. We construct these special polynomial rings and match them
with known quasi modular forms for non-compact Calabi-Yau geometries and their mirrors including local $\mathbb{P}^2$ and local del Pezzo geometries with $E_5$, $E_6$, $E_7$ and $E_8$ type singularities.
We provide the analogous special polynomial ring for the quintic.
</p>projecteuclid.org/euclid.atmp/1414414839_20141027090038Mon, 27 Oct 2014 09:00 EDTAlgebraic renormalization and Feynman integrals in configuration spaceshttp://projecteuclid.org/euclid.atmp/1414414840<strong>Ozgór Ceyhan</strong>, <strong>Matilde Marcolli</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 2, 469--511.</p><p><strong>Abstract:</strong><br/>
This paper continues our previous study of Feynman integrals in configuration spaces and their algebro-geometric and motivic aspects. We consider here both massless and massive Feynman amplitudes, from the point
of view of potential theory. We consider a variant of the wonderful compactification of configuration spaces that works simultaneously for all graphs with a given number of vertices and that also accounts for the external
structure of Feynman graph. As in our previous work, we consider two version of the Feynman amplitude in configuration space, which we refer to as the real and complex versions. In the real version, we show that we
can extend to the massive case a method of evaluating Feynman integrals, based on expansion in Gegenbauer polynomials, that we investigated previously in the massless case. In the complex setting, we show that
we can use algebro-geometric methods to renormalize the Feynman amplitudes, so that the renormalized values of the Feynman integrals are given by periods of a mixed Tate motive. The regularization and
renormalization procedure is based on pulling back the form to the wonderful compactification and replace it with a cohomologous one with logarithmic poles. A complex of forms with logarithmic poles, endowed
with an operator of pole subtraction, determine a Rota-Baxter algebra on the wonderful compactifications. We can then apply the renormalization procedure via Birkhoff factorization, after interpreting the regularization
as an algebra homomorphism from the Connes-Kreimer Hopf algebra of Feynman graphs to the Rota-Baxter algebra. We obtain in this setting a description of the renormalization group.We also extend the period
interpretation to the case of Dirac fermions and gauge bosons.
</p>projecteuclid.org/euclid.atmp/1414414840_20141027090038Mon, 27 Oct 2014 09:00 EDTNoncommutative connections on bimodules and Drinfeld twist deformationhttp://projecteuclid.org/euclid.atmp/1414762185<strong>Paolo Aschieri</strong>, <strong>Alexander Schenkel</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 3, 513--612.</p><p><strong>Abstract:</strong><br/>
Given a Hopf algebra $H$, we study modules and bimodules over an algebra $A$ that carry an $H$-action, as well as their morphisms and connections. Bimodules naturally arise when considering noncommutative
analogues of tensor bundles. For quasitriangular Hopf algebras and bimodules with an extra quasi-commutativity property we induce connections on the tensor product over $A$ of two bimodules from connections
on the individual bimodules. This construction applies to arbitrary connections, i.e. not necessarily Hequivariant ones, and further extends to the tensor algebra generated by a bimodule and its dual. Examples of these
noncommutative structures arise in deformation quantization via Drinfeld twists of the commutative differential geometry of a smooth manifold, where the Hopf algebra $H$ is the universal enveloping algebra of vector
fields (or a finitely generated Hopf subalgebra).
We extend the Drinfeld twist deformation theory of modules and algebras to morphisms and connections that are not necessarily $H$-equivariant. The theory canonically lifts to the tensor product structure.
</p>projecteuclid.org/euclid.atmp/1414762185_20141031092950Fri, 31 Oct 2014 09:29 EDTThe Sen limithttp://projecteuclid.org/euclid.atmp/1414762186<strong>Adrian Clingher</strong>, <strong>Ron Donagi</strong>, <strong>Martijn Wijnholt</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 3, 613--658.</p><p><strong>Abstract:</strong><br/>
$F$-theory compactifications on elliptic Calabi-Yau manifolds may be related to IIb compactifications by taking a certain limit in complex structure moduli space, introduced by A. Sen. The limit has been characterized on the
basis of $SL(2, \mathrm{Z})$ monodromies of the elliptic fibration. Instead, we introduce a stable version of the Sen limit. In this picture the elliptic Calabi-Yau splits into two pieces, a $\mathbf{P}^1$-bundle and a conic
bundle, and the intersection yields the IIb space-time.We get a precise match between $F$-theory and perturbative type IIb. The correspondence is holographic, in the sense that physical quantities seemingly spread
in the bulk of the $F$-theory Calabi-Yau may be rewritten as expressions on the log boundary. Smoothing the $F$-theory Calabi-Yau corresponds to summing up the $D(-1)$-instanton corrections to the IIb theory.
</p>projecteuclid.org/euclid.atmp/1414762186_20141031092950Fri, 31 Oct 2014 09:29 EDTAn exact expression for photon polarization in Kerr geometryhttp://projecteuclid.org/euclid.atmp/1414762187<strong>Anusar Farooqui</strong>, <strong>Niky Kamran</strong>, <strong>Prakash Panangaden</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 3, 659--686.</p><p><strong>Abstract:</strong><br/>
We analyze the transformation of the polarization of a photon
propagating along an arbitrary null geodesic in Kerr geometry.
The motivation comes from the problem of an observer trying to
communicate quantum information to another observer in Kerr
spacetime by transmitting polarized photons. It is essential that
the observers understand the relationship between their frames of
reference and also know how the photon’s polarization transforms
as it travels through Kerr spacetime. Existing methods to calculate
the rotation of the photon polarization (Faraday rotation) depend
on choices of coordinate systems, are algebraically complex and
yield results only in the weak-field limit.
We give a closed-form expression for a parallel propagated frame
along an arbitrary null geodesic using Killing-Yano theory, and
thereby solve the problem of parallel transport of the polarization
vector in an intrinsic, geometrically-motivated fashion. The symmetries
of Kerr geometry are utilized to obtain a remarkably compact
expression for the geometrically induced phase of the photon’s
polarization. We show that this phase vanishes on the equatorial
plane and the axis of symmetry.
</p>projecteuclid.org/euclid.atmp/1414762187_20141031092950Fri, 31 Oct 2014 09:29 EDTArea inequalities for stable marginally outer trapped surfaces in Einstein-Maxwell-dilaton theoryhttp://projecteuclid.org/euclid.atmp/1414762188<strong>David Fajman</strong>, <strong>Walter Simon</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 3, 687--707.</p><p><strong>Abstract:</strong><br/>
We prove area inequalities for stable marginally outer trapped surfaces in Einstein-Maxwell-dilaton theory. Our inspiration comes on the one hand from a corresponding upper bound for the area in terms of the charges obtained
recently by Dain, Jaramillo and Reiris in the pure Einstein-Maxwell case without symmetries, and on the other hand from Yazadjiev's inequality in the axially symmetric Einstein-Maxwell-dilaton case. The common issue in
these proofs and in the present one is a functional $\mathcal{W}$ of the matter fields for which the stability condition readily yields an upper bound. On the other hand, the step which crucially depends on whether
or not a dilaton field is present is to obtain a lower bound for $\mathcal{W}$ as well. We obtain the latter by first setting up a variational principle for $\mathcal{W}$ with respect to the dilaton field $\phi$, then by
proving existence of a minimizer $\psi$ as solution of the corresponding Euler-Lagrange equations and finally by estimating $\mathcal{W} (\psi)$. In the special case that the normal components of the electric and
magnetic fields are proportional we obtain the area bound $A \geq 8\pi PQ$ in terms of the electric and magnetic charges. In the generic case our results are less explicit but imply rigorous 'perturbation' results for
the above inequality. All our inequalities are saturated for a 2-parameter family of static, extreme solutions found by Gibbons. Via the Bekenstein-Hawking relation $A = 4S$ our results give positive lower bounds for
the entropy $S$ which are particularly interesting in the Einstein-Maxwell-dilaton case.
</p>projecteuclid.org/euclid.atmp/1414762188_20141031092950Fri, 31 Oct 2014 09:29 EDTTheory of intersecting loops on a torushttp://projecteuclid.org/euclid.atmp/1414762189<strong>J. E. Nelson</strong>, <strong>R. F. Picken</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 3, 709--740.</p><p><strong>Abstract:</strong><br/>
We continue our investigation into intersections of closed paths on a torus, to further our understanding of the commutator algebra of Wilson loop observables in $2+1$ quantum gravity, when the cosmological constant is
negative.We give a concise review of previous results, e.g. that signed area phases relate observables assigned to homotopic loops, and present new developments in this theory of intersecting loops on a torus. We
state precise rules to be applied at intersections of both straight and crooked/rerouted paths in the covering space $\mathbb{R}^2$. Two concrete examples of combinations of different rules are presented.
</p>projecteuclid.org/euclid.atmp/1414762189_20141031092950Fri, 31 Oct 2014 09:29 EDTQuantum phase transition of light in coupled optical cavity arrays: A renormalization group studyhttp://projecteuclid.org/euclid.atmp/1414762190<strong>Sujit Sarkar</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 3, 741--760.</p><p><strong>Abstract:</strong><br/>
We study the quantum phase transition of light of a system when atom trapped in microcavities and interacting through the exchange of virtual photons. We predict the quantum phase transition between the photonic Coulomb
blocked induce insulating phase and anisotropic exchange induced photonic superfluid phase in the system due to the existence of two Rabi frequency oscillations. The renormalization group equation shows explicitly
that for this system there is no self-duality. The system also shows two Berezinskii-Kosterlitz-Thouless (BKT) transitions for the different physical situation of the system. The presence of single Rabi frequency oscillation
in the system leads to the BKT transition where system shows the quantum phase transition from photonic metallic state to the Coulomb blocked induced insulating phase. For the other BKT transition when the z-component
of exchange interaction is absent, the system shows the transition from the photonic metallic state to the photonic superfluid phase. We also predict the commensurate to incommensurate transition under
the laser field detuning.
</p>projecteuclid.org/euclid.atmp/1414762190_20141031092950Fri, 31 Oct 2014 09:29 EDTA McKay-like correspondence for $(0,2)$-deformationshttp://projecteuclid.org/euclid.atmp/1415818562<strong>Paul S. Aspinwall</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 4, 761--797.</p><p><strong>Abstract:</strong><br/>
We present a local computation of deformations of the tangent bundle for a resolved orbifold singularity $\mathbb{C}^d/G$. These correspond to $(0, 2)$-deformations of $(2, 2)$-theories. A McKay-like correspondence
is found predicting the dimension of the space of first-order deformations from simple calculations involving the group. This is confirmed in two dimensions using the Kronheimer-Nakajima quiver construction. In higher
dimensions such a computation is subject to nontrivial worldsheet instanton corrections and some examples are given where this happens. However, we conjecture that the special crepant resolution given by the
$G$-Hilbert scheme is never subject to such corrections, and show this is true in an infinite number of cases. Amusingly, for three-dimensional examples where $G$ is abelian, the moduli space is associated to a
quiver given by the toric fan of the blow-up. It is shown that an orbifold of the form $\mathbb{C}^3 / \mathbb{Z}_7$ has a nontrivial superpotential and thus an obstructed moduli space.
</p>projecteuclid.org/euclid.atmp/1415818562_20141112135603Wed, 12 Nov 2014 13:56 ESTThe phase space for the Einstein-Yang-Mills equations and the first law of black hole thermodynamicshttp://projecteuclid.org/euclid.atmp/1415818563<strong>Steven McCormick</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 4, 799--825.</p><p><strong>Abstract:</strong><br/>
We use the techniques of Bartnik to show that the space of solutions to the Einstein-Yang-Mills constraint equations on an asymptotically flat manifold with one end and zero boundary components, has a Hilbert manifold
structure; the Einstein-Maxwell system can be considered as a special case. This is equivalent to the property of linearisation stability, which was studied in depth throughout the 70s [1, 2, 9, 11, 13, 18, 19].
This framework allows us to prove a conjecture of Sudarsky and Wald, namely that the validity of the first law of black hole thermodynamics is a suitable condition for stationarity. Since we work with a single end and no
boundary conditions, this is equivalent to critical points of the ADM mass subject to variations fixing the Yang-Mills charge corresponding exactly to stationary solutions. The natural extension to this work is to prove the
second conjecture from Sudarksy and Wald, which is the case where an interior boundary is present; this will be addressed in future work.
</p>projecteuclid.org/euclid.atmp/1415818563_20141112135603Wed, 12 Nov 2014 13:56 ESTTopological strings, D-model, and knot contact homologyhttp://projecteuclid.org/euclid.atmp/1415818564<strong>Mina Aganagic</strong>, <strong>Tobias Ekholm</strong>, <strong>Lenhard Ng</strong>, <strong>Cumrun Vafa</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 4, 827--956.</p><p><strong>Abstract:</strong><br/>
We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov-Witten disk amplitudes
of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also implies the equality between the $Q$-deformed $A$-polynomial and the augmentation polynomial of knot contact
homology (in the irreducible case). We also generalize this relation to the case of links and to higher rank representations for knots. The generalization involves a study of the quantum moduli space of special Lagrangian
branes with higher Betti numbers probing the Calabi-Yau. This leads to an extension of SYZ, and a new notion of mirror symmetry, involving higher dimensional mirrors. The mirror theory is a topological string,
related to D-modules, which we call the “D-model”. In the present setting, the mirror manifold is the augmentation variety of the link. Connecting further to contact geometry, we study intersection properties of branches
of the augmentation variety guided by the relation to D-modules. This study leads us to propose concrete geometric constructions of Lagrangian fillings for links. We also relate the augmentation variety with the large
$N$ limit of the colored HOMFLY, which we conjecture to be related to a $Q$-deformation of the extension of $A$-polynomials associated with the link complement.
</p>projecteuclid.org/euclid.atmp/1415818564_20141112135603Wed, 12 Nov 2014 13:56 ESTStandard modules, induction and the structure of the Temperley-Lieb algebrahttp://projecteuclid.org/euclid.atmp/1416929529<strong>David Ridout</strong>, <strong>Yvan Saint-Aubin</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 5, 957--1041.</p><p><strong>Abstract:</strong><br/>
The basic properties of the Temperley-Lieb algebra $\mathsf{TL}_n$ with parameter $\beta = q + q^{-1} , q \in \mathbb{C} \backslash \{ 0 \}$, are reviewed in a pedagogical way. The link and standard
(cell) modules that appear in numerous physical applications are defined and a natural bilinear form on the standard modules is used to characterise their maximal submodules. When this bilinear form has a
non-trivial radical, some of the standard modules are reducible and $\mathsf{TL}_n$ is non-semisimple. This happens only when $q$ is a root of unity. Use of restriction and induction allows for a finer description
of the structure of the standard modules. Finally, a particular central element $F_n \in \mathsf{TL}_n$ is studied; its action is shown to be non-diagonalisable on certain indecomposable modules and this leads to
a proof that the radicals of the standard modules are irreducible.Moreover, the space of homomorphisms between standard modules is completely determined. The principal indecomposable modules are then
computed concretely in terms of standard modules and their inductions. Examples are provided throughout and the delicate case $\beta = 0$, that plays an important role in physical models, is studied systematically.
</p>projecteuclid.org/euclid.atmp/1416929529_20141125103211Tue, 25 Nov 2014 10:32 ESTA unified quantum theory I: gravity interacting with a Yang-Mills fieldhttp://projecteuclid.org/euclid.atmp/1416929530<strong>Claus Gerhardt</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 5, 1043--1062.</p><p><strong>Abstract:</strong><br/>
Using the results and techniques of a previous paper where we proved the quantization of gravity we extend the former result by adding a Yang-Mills functional and a Higgs term to the Einstein-Hilbert action.
</p>projecteuclid.org/euclid.atmp/1416929530_20141125103211Tue, 25 Nov 2014 10:32 ESTGeometric engineering of (framed) BPS stateshttp://projecteuclid.org/euclid.atmp/1416929531<strong>Wu-yen Chuang</strong>, <strong>Duiliu-Emanuel Diaconescu</strong>, <strong>Jan Manschot</strong>, <strong>Gregory W. Moore</strong>, <strong>Yan Soibelman</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 5, 1063--1231.</p><p><strong>Abstract:</strong><br/>
BPS quivers for $\mathcal{N} = 2 \: SU(N)$ gauge theories are derived via geometric engineering from derived categories of toric Calabi-Yau threefolds. While the outcome is in agreement of previous
low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of $N$, relating the field theory BPS spectrum to large radius D-brane
bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely
many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed
BPS degeneracies in terms of motivic and cohomological Donaldson-Thomas invariants. We verify the conjectured absence of BPS states with "exotic" $SU(2)_R$ quantum numbers using motivic DT invariants.
This application is based in particular on a complete recursive algorithm which determines the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative Donaldson-Thomas invariants
for framed quiver representations.
</p>projecteuclid.org/euclid.atmp/1416929531_20141125103211Tue, 25 Nov 2014 10:32 ESTTopological field theory on a lattice, discrete theta-angles and confinementhttp://projecteuclid.org/euclid.atmp/1416929532<strong>Anton Kapustin</strong>, <strong>Ryan Thorngren</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 5, 1233--1247.</p><p><strong>Abstract:</strong><br/>
We study a topological field theory describing confining phases of gauge theories in four dimensions. It can be formulated on a lattice using a discrete 2-form field talking values in a finite abelian group
(the magnetic gauge group). We show that possible theta-angles in such a theory are quantized and labeled by quadratic functions on the magnetic gauge group. When the theta-angles vanish, the theory
is dual to an ordinary topological gauge theory, but in general it is not isomorphic to it. We also explain how to couple a lattice Yang-Mills theory to a TQFT of this kind so that the ’t Hooft flux is well-defined,
and quantized values of the theta-angles are allowed. The quantized theta-angles include the discrete theta-angles recently identified by Aharony, Seiberg and Tachikawa.
</p>projecteuclid.org/euclid.atmp/1416929532_20141125103211Tue, 25 Nov 2014 10:32 ESTPrecanonical quantization and the Schrödinger wave functional revisitedhttp://projecteuclid.org/euclid.atmp/1417707818<strong>Igor V. Kanatchikov</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 6, 1249--1265.</p><p><strong>Abstract:</strong><br/>
We address the issue of the relation between the canonical functional Schrödinger representation in quantum field theory and the approach of precanonical field quantization proposed by the author, which requires
neither a distinguished time variable nor infinite-dimensional spaces of field configurations. We argue that the standard functional derivative Schrödinger equation can be derived from the precanonical Dirac-like
covariant generalization of the Schrödinger equation under the formal limiting transition $\gamma^0 \varkappa \to \delta(0)$, where the constant $\varkappa$ naturally appears within precanonical quantization as
the inverse of a small “elementary volume” of space. We obtain a formal explicit expression of the Schrödinger wave functional as a continuous product of the Dirac algebra valued precanonical wave functions, which
are defined on the finite-dimensional covariant configuration space of the field variables and space-time variables.
</p>projecteuclid.org/euclid.atmp/1417707818_20141204104343Thu, 04 Dec 2014 10:43 ESTT-duality for Langlands dual groupshttp://projecteuclid.org/euclid.atmp/1417707819<strong>Calder Daenzer</strong>, <strong>Erik van Erp</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 6, 1267--1285.</p><p><strong>Abstract:</strong><br/>
This article addresses the question of whether Langlands duality for complex reductive Lie groups may be implemented by T-dualization. We prove that for reductive groups whose simple factors are of Dynkin
type A, D, or E, the answer is yes.
</p>projecteuclid.org/euclid.atmp/1417707819_20141204104343Thu, 04 Dec 2014 10:43 ESTA new approach to the $N$-particle problem in QMhttp://projecteuclid.org/euclid.atmp/1417707820<strong>Joachim Schröter</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 6, 1287--1334.</p><p><strong>Abstract:</strong><br/>
In this paper the old problem of determining the discrete spectrum of a multi-particle Hamiltonian is reconsidered. The aim is to bring a fermionic Hamiltonian for arbitrary numbers $N$ of particles by analytical means
into a shape such that modern numerical methods can successfully be applied. For this purpose the Cook-Schroeck Formalism is taken as starting point. This includes the use of the occupation number representation.
It is shown that the $N$-particle Hamiltonian is determined in a canonical way by a fictional 2-particle Hamiltonian. A special approximation of this 2-particle operator delivers an approximation of the $N$-particle
Hamiltonian, which is the orthogonal sum of finite dimensional operators. A complete classification of the matrices of these operators is given. Finally the method presented here is formulated as a work program
for practical applications. The connection with other methods for solving the same problem is discussed.
</p>projecteuclid.org/euclid.atmp/1417707820_20141204104343Thu, 04 Dec 2014 10:43 ESTThe mirror symmetry of K3 surfaces with non-symplectic automorphisms of prime orderhttp://projecteuclid.org/euclid.atmp/1417707821<strong>Paola Comparin</strong>, <strong>Christopher Lyons</strong>, <strong>Nathan Priddis</strong>, <strong>Rachel Suggs</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 6, 1335--1368.</p><p><strong>Abstract:</strong><br/>
We consider K3 surfaces that possess a non-symplectic automorphism of prime order $p>2$ and we present, for these surfaces, a correspondence between the mirror symmetry of Berglund-Hübsch-Chiodo-Ruan
and that for lattice polarized K3 surfaces presented by Dolgachev.
</p>projecteuclid.org/euclid.atmp/1417707821_20141204104343Thu, 04 Dec 2014 10:43 ESTD-brane probes, branched double covers, and noncommutative resolutionshttp://projecteuclid.org/euclid.atmp/1417707822<strong>Nicolas M. Addington</strong>, <strong>Edward P. Segal</strong>, <strong>Eric R. Sharpe</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 6, 1369--1436.</p><p><strong>Abstract:</strong><br/>
This paper describes D-brane probes of theories arising in abelian gauged linear sigma models (GLSMs) describing branched double covers and noncommutative resolutions thereof, via nonperturbative effects rather
than as the critical locus of a superpotential. As these theories can be described as IR limits of Landau- Ginzburg models, technically this paper is an exercise in utilizing (sheafy) matrix factorizations. For Landau-Ginzburg
models which are believed to flow in the IR to smooth branched double covers, our D-brane probes recover the structure of the branched double cover (and flat nontrivial $B$ fields), verifying previous results.
In addition to smooth branched double covers, the same class of Landau-Ginzburg models is also believed to sometimes flow to ‘noncommutative resolutions’ of singular spaces. These noncommutative resolutions
are abstract conformal field theories without a global geometric description, but D-brane probes perceive them as a non-Kähler small resolution of a singular Calabi-Yau. We conjecture that such non-Kähler resolutions
are typical in D-brane probes of such theories.
</p>projecteuclid.org/euclid.atmp/1417707822_20141204104343Thu, 04 Dec 2014 10:43 ESTT-duality for circle bundles via noncommutative geometryhttp://projecteuclid.org/euclid.atmp/1417707823<strong>Varchese Mathai</strong>, <strong>Jonathan Rosenberg</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 6, 1437--1462.</p><p><strong>Abstract:</strong><br/>
Recently Baraglia showed how topological T-duality can be extended to apply not only to principal circle bundles, but also to non-principal circle bundles. We show that his results can also be recovered via two other methods:
the homotopy-theoretic approach of Bunke and Schick, and the noncommutative geometry approach which we previously used for principal torus bundles. This work has several interesting byproducts, including a study
of the $K$-theory of crossed products by $\tilde{O}(2) = \mathrm{Isom}(\mathbb{R})$, the universal cover of $O(2)$, and some interesting facts about equivariant $K$-theory for $\mathbb{Z}/ 2$. In the final section of this
paper, some of these results are extended to the case of bundles with singular fibers, or in other words, non-free $O(2)$-actions.
</p>projecteuclid.org/euclid.atmp/1417707823_20141204104343Thu, 04 Dec 2014 10:43 ESTBuilding blocks for generalized heterotic/F-theory dualityhttp://projecteuclid.org/euclid.atmp/1417707824<strong>Jonathan J. Heckman</strong>, <strong>Hai Lin</strong>, <strong>Shing-Tung Yau</strong>. <p><strong>Source: </strong>Advances in Theoretical and Mathematical Physics, Volume 18, Number 6, 1463--1503.</p><p><strong>Abstract:</strong><br/>
In this note we propose a generalization of heterotic/F-theory duality. We introduce a set of non-compact building blocks which we glue together to reach compact examples of generalized duality pairs. The F-theory
building blocks consist of non-compact elliptically fibered Calabi-Yau fourfolds which also admit a $K3$ fibration. The compact elliptic model obtained by gluing need not have a globally defined $K3$ fibration. By
replacing the $K3$ fiber of each F-theory building block with a $T^2$, we reach building blocks in a heterotic dual vacuum which includes a position dependent dilaton and three-form flux. These building blocks
are glued together to reach a heterotic flux background. We argue that in these vacua, the gauge fields of the heterotic string become localized, and remain dynamical even when gravity decouples. This enables
a heterotic dual for the hyperflux GUT breaking mechanism which has recently figured prominently in F-theory GUT models. We illustrate our general proposal with some explicit examples.
</p>projecteuclid.org/euclid.atmp/1417707824_20141204104343Thu, 04 Dec 2014 10:43 EST