Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 46 (String Theory)

Use target signature $(-,+,\dots ,+)$ and worldsheet signature $(-,+)$. The Polyakov action has local worldsheet diffeomorphism and Weyl invariance, together with global target-space Poincaré symmetry. Under a worldsheet coordinate change $\sigma \mapsto {\sigma }^{\prime }$, the embedding coordinates are scalars and the worldsheet metric is a tensor:A Weyl transformation acts as $g_{ab}\mapsto e^{2\rho (\sigma )}{g}_{ab}$ and leaves $X$ unchanged. In two dimensions $\sqrt{-g}$ scales by $e^{2\rho }$ and $g^{ab}$ by $e^{-2\rho }$, so the action is invariant. Target translations act by $X^{\mu }\mapsto X^{\mu }+a^{\mu }$, and Lorentz transformations by $X^{\mu }\mapsto {\Lambda }^{\mu }{}_{\nu }{X}^{\nu }$ with ${\Lambda }^{\mathsf{T}}\eta \Lambda =\eta$; they leave $g$ unchanged. Infinitesimally the latter have $\delta X^{\mu }={\omega }^{\mu }{}_{\nu }{X}^{\nu }$, where ${\omega }_{\mu \nu }=-{\omega }_{\nu \mu }$.

Locally, the two diffeomorphism functions put the metric into the form $g_{ab}=e^{2\rho }{\eta }_{ab}$, and Weyl invariance removes the conformal factor. In this conformal gauge, the embedding coordinates are free scalar fields and satisfyThe metric equation must still be imposed after gauge fixing. Varying the ungauged metric gives the worldsheet stress-energy tensor constraintWith a dot and prime denoting the two coordinate derivatives, the independent Virasoro constraints areequivalently $(\dot{X}\pm X^{\prime }{)}^2=0$. Local conformal gauge leaves residual conformal transformations; on general worldsheet topologies it also leaves global moduli, rather than removing all metric information.

For target translations, differentiation of the Lagrangian with respect to ${\partial }_a{X}_{\mu }$ gives the Noether currentThe embedding equation is ${\partial }_a{j}^{a\mu }=0$. In conformal gauge, $j^{\tau \mu }={\dot{X}}^{\mu }/(2\pi {\alpha }^{\prime })$, so integration over the closed string givesPeriodic boundary conditions make its time derivative vanish. For an infinitesimal Lorentz transformation, the antisymmetric coefficient of ${\omega }_{\mu \nu }$ gives the current $j^{a,\mu \nu }=X^{\mu }{j}^{a\nu }-X^{\nu }{j}^{a\mu }$. Its divergence vanishes because the remaining product of embedding derivatives is symmetric in $\mu ,\nu$. Thus the Noether charge isThese are the target-space Noether charges of a string.

For the circular solution take $R>0$, and let ${\mathbf{e}}_r=(\cos \sigma ,\sin \sigma )$ and ${\mathbf{e}}_{\sigma }=(-\sin \sigma ,\cos \sigma )$. Its derivatives areEach spatial coordinate has equal second derivatives in $\tau$ and $\sigma$, while the time coordinate has both second derivatives zero. Furthermoreso both the field equations and the Virasoro constraints hold, including the collapse instants.

At target time $t=X^0=R\tau$, the string is a circle in the $X^1{X}^2$ plane of radius $R|\cos (t/R)|$. It contracts from radius $R$ to zero and re-expands; the negative cosine reverses material labels by a half-turn rather than creating a negative radius. The spatial configuration repeats after target time $\pi R$. This is a pulsating circular string. Its induced metric degenerates at collapse, but the conformal-gauge embedding and constraints give its continuation.

The integrals of $\sin \sigma$ and $\cos \sigma$ vanish, so the momentum isThe spatial velocity is radial, giving zero angular momentum density in the plane; all boost-charge terms integrate to zero as well. Hence $J^{\mu \nu }=0$ for every pair of indices.

At maximum expansion, all string elements are instantaneously at rest, and the length is $L_0=2\pi R$. Thus the rest mass per unit length, equal to the string tension, isAt other times the instantaneous circumference is $L=2\pi R|\cos \tau |$, and the elements have speed $|\sin \tau |$. Their Lorentz factor is $1/|\cos \tau |$, so the energy remains $TL/\sqrt{1-v^2}=2\pi TR$. The rest-energy density is $T$; the total rest-frame energy divided by the changing circumference is not constant. This distinguishes the requested intrinsic mass density from the larger energy density of moving string elements.

In a two-dimensional conformal field theory, a quasi-primary operator of conformal weights $(h,\tilde{h})$ obeysfor global conformal, or Möbius, transformations. A primary operator obeys this law for every locally invertible holomorphic and antiholomorphic conformal coordinate change. Quasi-primary is therefore weaker: the stress tensor at nonzero central charge is quasi-primary, but its Schwarzian transformation term prevents it from being primary under a general map.

A primary state is a simultaneous eigenstate of $L_0,{\tilde{L}}_0$ with eigenvalues $h,\tilde{h}$, satisfyingA quasi-primary state need only satisfy the corresponding $n=1$ conditions. Negative modes of the Virasoro algebra generate descendants.

In radial quantization, a local operator at the origin prepares a state on a surrounding circle; logarithmic radius is Euclidean time. The state–operator correspondence is $|\mathcal{O}⟩=\mathcal{O}(0)|0⟩$. Acting with a Virasoro generator means inserting its stress-tensor contour around that operator. For a primary operator, the operator product expansion isMultiplication by $z^{n+1}$ and the contour residue give $L_0|\mathcal{O}⟩=h|\mathcal{O}⟩$, $L_{-1}|\mathcal{O}⟩=|\partial \mathcal{O}⟩$, and $L_n|\mathcal{O}⟩=0$ for $n>0$. Conversely the contour modes are the coefficients of the singular expansion; vanishing positive modes removes every pole of order three and higher. The remaining double and simple poles generate precisely the primary transformation law. The antiholomorphic argument is identical. This proves the state-operator correspondence for Virasoro primaries in both directions.

An integrated string vertex operator is integrated with $d^{2}z$. The measure transforms with weights $(-1,-1)$, so conformal invariance requires its matter operator to have weights $(1,1)$. In old covariant string quantization the same requirement is $L_{n>0}={\tilde{L}}_{n>0}=0$ and $L_0={\tilde{L}}_0=1$. The ghost-dressed unintegrated vertex $c\bar{c}V$ has total weights $(0,0)$. Physical states are identified modulo null string states, or more systematically by BRST cohomology; raw counting of all primary operators includes gauge redundancy.

For a free embedding boson the holomorphic stress tensor is $T=-:\partial X\cdot \partial X:/{\alpha }^{\prime }$. Contracting it with $:e^{ik\cdot X}:$ gives weight ${\alpha }^{\prime }{k}^2/4$; every holomorphic oscillator of level $r$ adds $r$ to that weight. ThusWith Lorentzian target signature, $M^2=-k^2$, so the closed-bosonic-string mass shell from primary weights isThe ground state is tachyonic, level one is massless, and higher levels are massive. The primary conditions also constrain the polarization tensors; the two zero-mode conditions fix the mass and level matching.

For the commuting beta-gamma system, write $r=z-w$. The displayed exchange leaves the arguments attached to their respective fields. Exchanging the arguments instead givesThis radial-ordering sign in commuting beta-gamma contractions must be kept when differentiating. Expanding the total derivative in the proposed stress tensor givesThe contraction $\gamma (z)\beta (w)\sim 1/r$ and its derivative $\partial \gamma (z)\beta (w)\sim -1/r^2$ giveSimilarly $\partial \beta (z)\gamma (w)\sim 1/r^2$ and $\beta (z)\gamma (w)\sim -1/r$ giveThere are no higher poles. Both fields are holomorphic and $\bar{T}=0$, so

To compute the central charge, put $A=:\partial \beta \gamma :$ and $B=:\beta \partial \gamma :$. Only double Wick contractions can give the fourth-order pole in $TT$. Their coefficients arewhere only the double-contraction terms are shown. For example, the mixed contraction uses $\partial \beta (z)\partial \gamma (w)\sim 2/r^3$ and $\gamma (z)\beta (w)\sim 1/r$. All Wick pairings have bosonic signs. Therefore the fourth-order coefficient isThe stress-tensor operator product expansion has fourth-order coefficient $c/2$, yieldingThis is the stress tensor of a commuting beta-gamma system calculation. Single contractions supply the usual $2T/r^2+\partial T/r$ terms, with the potential third-order terms canceling.

The Euclidean dilaton term is $S_{\Phi }=(4\pi {)}^{-1}\int \sqrt{h}\Phi (X)R^{(2)}$. A constant mode ${\Phi }_0$ therefore gives $S_{\Phi }={\Phi }_0\chi$ by the Gauss-Bonnet theorem. Setting the string coupling to $g_s=e^{{\Phi }_0}$, an oriented closed worldsheet of genus $g$ has weightEach canonically normalized external closed-string vertex supplies a further factor $g_s$. Thus the connected $n$-point genus-$g$ contribution carriesThis is the string genus expansion. Four external strings on a sphere give $g_s^2$, as in the displayed tree amplitude.

Use the conventional complex-coordinate measure $d^{2}z=2d(\mathrm{Re}z)d(\mathrm{Im}z)$, with $\partial =({\partial }_x-i{\partial }_y)/2$. The action then has the standard real-coordinate normalization $(4\pi {\alpha }^{\prime }{)}^{-1}\int (\nabla X{)}^2{d}^{2}x$. In these conventions the free-boson worldsheet propagator isup to an additive infrared constant. Indeed the quadratic kernel after integration by parts is $-(\pi {\alpha }^{\prime }{)}^{-1}\partial \bar{\partial }$, and $\partial \bar{\partial }\log |z-w{|}^2=2\pi {\delta }^{(2)}(z-w)$ for this measure. Their product gives the required delta function. A simultaneous change of the coordinate-measure convention and action coefficient leaves this physical normalization unchanged.

Separate the constant mode $x_0$ of $X$. Its integral supplies $(2\pi {)}^D{\delta }^{(D)}({\sum }_i{p}_i)$, so four-momentum conservation is essential, even when that delta function is suppressed in the amplitude notation. Treat the exponential vertices as normal ordered: self-contractions are then absent. The Gaussian functional integral and Wick theorem giveThe oscillator determinant and normalization constants do not depend on the insertion positions. Substitution therefore gives the requested Koba-Nielsen factor under the four insertion integrals, with the sphere's residual conformal-group volume divided out.

For the Möbius transformation $z^{\prime }=(az+b)/(cz+d)$, the unit determinant condition givesThe distance-product factor attached to insertion $i$ is $|c{z}_i+d{|}^{-{\alpha }^{\prime }{\sum }_{j\ne i}{p}_i\cdot p_j}$. Momentum conservation changes its exponent to ${\alpha }^{\prime }{p}_i^2$. Thus the total factor at each insertion, including the measure, isOn the tachyon mass shell $p_i^2=4/{\alpha }^{\prime }$ it is one. This proves the Möbius invariance of on-shell closed-string vertex integrals. The correlator is covariant; the correlator times all four measures is invariant. This is the invariant integrand needed to quotient by the residual $SL(2,\mathbb{C})$ transformations.

For the massless Type II superstring theory amplitude, use $s=-(p_1+p_2{)}^2$ in Lorentzian target signature and $s+t+u=0$. At generic fixed $t$, the numerator factor $\Gamma (-{\alpha }^{\prime }s/4)$ has simple poles atAt these points $\Gamma (1+{\alpha }^{\prime }s/4)=\Gamma (1+n)=n!$ is finite and nonzero. The $t,u$ factors are also finite and nonzero for generic $t$, so the poles persist; isolated residue zeros at special kinematics do not remove the general mass level. Tree-level factorization interprets a pole at $s=M^2$ as exchange of a state of that mass. The Type II gamma-factor mass poles therefore exhibit a massless level and an infinite tower of massive levels, with equally spaced squared masses $4/{\alpha }^{\prime }$.

There is no negative-$s$ pole in this exchanged tower. In particular, at negative integer values of ${\alpha }^{\prime }s/4$ the reciprocal s-channel denominator can instead have zeros. This is consistent with the absence of the bosonic tachyon in the Type II superstring mass spectrum. The factor alone does not determine degeneracies, spins or every possible uncoupled state; external polarizations and the rest of the amplitude control their residues.

The string nonlinear sigma model regards the target-space metric, the Kalb–Ramond field and the dilaton as couplings of the two-dimensional worldsheet theory. Schematically its Euclidean terms areQuantization produces running background couplings and a possible worldsheet Weyl anomaly. Gauge-fixed matter bosons contribute central charge $D$, while the reparameterization bc ghost system contributes $-26$ in each chirality. In the critical flat background cancellation therefore requires $D=26$. In a curved background, vanishing improved sigma-model beta functions further imposes field equations for $G,B,\Phi$. For example, the leading metric condition isThese are spacetime conditions because the couplings are functions of the embedding coordinates, not additional worldsheet coordinates.

The leading equations arise by varying the string-frame massless effective actionThe factor $e^{-2\Phi }$ is the sphere contribution in the string genus expansion. Higher sigma-model orders generate higher-derivative terms in the ${\alpha }^{\prime }$ expansion; higher genera give string-loop corrections. Thus conformal consistency of the quantum worldsheet determines the low-energy target equations and their action. This is a massless-sector truncation: the bosonic tachyon is not removed by writing an action only for $G,B,\Phi$.

For a brane embedding $X^{\mu }(\xi )$, the D-brane worldvolume pullbacks areThe dilaton is evaluated at the embedding. When $T_p$ denotes the physical tension in a background with constant mode ${\Phi }_0$, the convention is $\tilde{\Phi }=\Phi (X)-{\Phi }_0$, so $e^{-\tilde{\Phi }}=1$ in that vacuum. Equivalently one can use $\tilde{\Phi }=\Phi (X)$ and a coefficient with the constant $e^{-{\Phi }_0}$ not already included. The physics is the same; the two conventions must not count that constant factor twice.

Several features motivate the Dirac-Born-Infeld action. Without gauge or two-form fields, it is the relativistic induced-volume action for a tensionful extended object. An open-string boundary couples to its gauge potential $A$, so a bulk transformation $B\mapsto B+d\Lambda$ is canceled by $A\mapsto A-P[\Lambda ]/(2\pi {\alpha }^{\prime })$. Thus only $P[B]+2\pi {\alpha }^{\prime }F$ can enter. For slowly varying constant field strength, the open-string disk partition function resums its dependence into the determinant; expansion gives the Maxwell kinetic term. The disk has Euler characteristic one, giving $e^{-\Phi }$ rather than the closed-string sphere's $e^{-2\Phi }$. Worldvolume covariance and the induced metric then combine these ingredients into the determinant expression.

Explicitly, in flat background and static gauge $X^a={\xi }^a$ with transverse fluctuations $X^I$, the weak-field expansion starts asThe linear term in the antisymmetric matrix $F$ vanishes; the quadratic term follows from expanding $\frac{1}{2}\mathrm{Tr}\log (1+2\pi {\alpha }^{\prime }{\eta }^{-1}F)$. Thus the DBI action contains both the scalar and gauge kinetic energies, while higher powers and derivatives are string-scale corrections.

For two coincident D-branes, an oriented open string has independent labels $r,s\in \{1,2\}$ at its ends. Its Chan-Paton factors are the matrix units $E_{rs}$. The four massless vector modes span the Lie algebra of $U(2)$: diagonal strings supply the two Abelian fields, and oppositely oriented interbrane strings supply the off-diagonal modes. Joining strings multiplies these matrices, producing the non-Abelian commutators. Transverse vector polarizations become adjoint scalar fields. This is Coincident-D-brane gauge enhancement.

The massless gauge/scalar dynamics reduce to dimensionally reduced Yang-Mills theory in a flat background with constant dilaton and vanishing pulled-back $B$, at energies and field variations well below the string scale. More precisely, derivative corrections satisfy ${\alpha }^{\prime }{E}^2\ll 1$, and one expands in small dimensionless field strengths and scalar gradients rather than retaining the full nonlinear determinant. Weak string coupling suppresses loop corrections. After subtracting the constant brane vacuum energy and using the scalar normalization fixed below, the coefficient isA bosonic brane additionally has an open-string tachyon; the displayed action governs its massless sector and is not a stable complete truncation. For supersymmetric branes the corresponding leading action also has the massless fermions.

There is a sign qualification in the printed scalar potential. With the stated $D_a\varphi ={\partial }_a\varphi +i[A_a,\varphi ]$, take $A_a,{\varphi }^I$ Hermitian and $[{\varphi }^I,{\varphi }^J{]}^2$ to mean the literal matrix product. Then dimensional reduction hasConsequently $F_{IJ}{F}^{IJ}=-[{\varphi }^I,{\varphi }^J{]}^2$. The consistent low-energy action isThis is the Hermitian commutator potential in D-brane Yang-Mills theory. Its potential energy is $-\sum \mathrm{Tr}[{\varphi }^I,{\varphi }^J{]}^2/(4g_{\mathrm{YM}}^2)\ge 0$, because a Hermitian-matrix commutator is anti-Hermitian. For a concrete check, take ${\varphi }^1=v{\sigma }_1$, ${\varphi }^2=v{\sigma }_2$ with Pauli matrices. Then $[{\varphi }^1,{\varphi }^2{]}^2=-4v^{4}I$, so the literal printed plus sign would make the static energy negative and unbounded below. The printed plus sign is consistent only if its square denotes the positive norm $[{\varphi }^I,{\varphi }^J{]}^{†}[{\varphi }^I,{\varphi }^J]$, rather than a literal square. The gauge-boson mass calculation below uses the unambiguous kinetic terms.

In an ordinary separated-brane vacuum the scalar expectation values commute and can be simultaneously diagonalized:Their eigenvalue vectors specify the transverse positions, up to the normalization to be determined. The trace encodes the center of mass; their difference encodes separation. When the two vectors coincide, every $U(2)$ transformation commutes with the vacuum. When they are distinct, only independent diagonal phases do, so the unbroken group is $U(1)\times U(1)$. The off-diagonal vectors acquire mass by the Higgs mechanism.

Write the off-diagonal entries of $A_a$ as $W_a$ and $W_a^{\ast }$ and put $\Delta v^I=v_1^I-v_2^I$. At a constant diagonal vacuum, $D_a{\varphi }^I=i[A_a,⟨{\varphi }^I⟩]$, soThe quadratic off-diagonal gauge kinetic term is $-f_{ab}^{\ast }{f}^{ab}/(2g_{\mathrm{YM}}^2)$ with $f_{ab}={\partial }_a{W}_b-{\partial }_b{W}_a$, and the scalar kinetic term supplies $-{\sum }_I(\Delta v^I{)}^2{W}_a^{\ast }{W}^a/g_{\mathrm{YM}}^2$. Comparing these with the complex massive-vector action givesHere the term W-boson means this broken-generator vector, not specifically an electroweak particle. The normalization places the gauge coupling outside the action, so no extra $g_{\mathrm{YM}}$ multiplies these $v_r^I$; rescaling to canonically normalized scalar fields moves that coupling into the usual Higgs mass formula.

The corresponding open-string vector oscillator stretched over distance $L=|X_1-X_2|$ has classical stretching mass $TL=L/(2\pi {\alpha }^{\prime })$. Its unstretched mass contribution vanishes because it is the vector level, rather than the bosonic tachyon level. Equating it with the gauge-theory mass givesThe component relation follows by rotational covariance and a common choice of transverse origin; a uniform translation adds the same constant to both brane coordinates. This is the D-brane scalar-position normalization. Together with the string tension determined in Question 1, it fixes the exact normalization of scalar eigenvalues as brane positions.